Scalar-Tensor Theories of Gravity
by
Deli Zhang (CID: 00777660)
Supervisor: Dr Ali Mozaffari
Department of Physics
Imperial College London
Submitted in partial fulfilment of the requirements for the degree of
Master of Science of Imperial College London, 2015-2016
Astract
Modified theories of gravity have receive increased attention lately due to combine motivation
coming from high-energy physics, cosmology and astrophysics. There is a class of modified gravity
called scalar-tensor theories in which the Ricci scalar R is coupled to a scalar field φ. One of
the simplest examples is the so-called Brans-Dicke theory. It is based on the introduction of a
fundamental scalar field that effectively incorporates a dynamics to the gravitational coupling,
which is determined by all matter in the universe. In spite of the diverse motivations and the rich
phenomenology that comes from its solutions, Solar System experiments impose more and more
stringent constraints on the Brans-Dicke theory, the theory thus becomes indistinguishable from
general relativity. However, Brans-Dicke and scalar-tensor gravity have been noteworthy frequently
since for several reasons: 1) Scalar fields are ubiquitous in particle physics and in cosmology, such
as Higgs boson, inflaton, string dilaton etc. 2) Brans-Dicke theory provide a graceful exit from
an inflationary epoch in the early universe; 3) The discovery that the universe is undergoing an
accelerated expansion today has spurred interest in scalar-tensor theories which account for such
features with a long-range gravitational scalar field; 4) The low-energy limit of the bosonic string
theory equivalent to a Brans-Dicke theory with parameter ω = −1 and scalar-tensor gravity exhibits
certain similarities to supergravity and string theory. Giving these development, the scalar-tensor
gravity might play a significant role in understanding of the beginning of our universe, and of the
most fundamental properties of matter. With these motivations, I review here Brans-Dicke and
scalar-tensor gravity attempt to comprehensively present their most important aspects.
2
Acknowledgements
I would like to thank my supervisor Ali Mozaffari for his help and support throughout this project.
3
Notations and Conventions
The following notations and conventions are used in this dissertation. The metric signature convention used is {− + ++} (or {− + +...+} in D > 4 spacetime dimensions). I will denote the
flat-space or Minkowski metric by ηµν , which takes the form diag(-1,1,1,1) in Cartesian coordinates
in four dimensions. Greek indices µ, ν, ... run over four dimensions of spacetime, while Latin indices
i, j, ...run over three spatial dimensions.
Partial derivatives are denoted by ∂ and covariant derivatives by ∇. Commas and semicolons
in indices will occasionally used to represent partial and covariant derivatives respectively, i.e.
φ,µ ≡ ∂µ φ and φ;µ ≡ ∇µ φ.
The covariant derivative of the stress tensor is defined as
∇µ T µν = ∂µ T µν + Γµµλ T λν + Γνµλ T µλ .
Bold font is used to indicate four-vectors, for example, v. Arrows used to indicate spatial threevectors, for example, ~v .
Nature presents us with three fundamental dimensionful constants: the speed of light c, Plank’s
constant (divided by 2π) } and Newton’s constant GN . I will work with “natural” units, de2
fined by c = } = 1. The reduced Planck mass is Mpl
= 1/8πGN in these units. A subscript 0
is used to denote a quantity evaluated at the present instant of time in the dynamics of the universe.
4
Contents
1 Introduction
7
2 General Relativity and Cosmology
11
2.1
Curvature of a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
The Gravitational Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.4
FRW Universe and Inflationary Cosmology . . . . . . . . . . . . . . . . . . . . . .
14
2.5
Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3 Modified Gravity
18
3.1
Cosmological Constant Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
Lovelock’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.3
Alternative Theories of Gravity with Extra Fields . . . . . . . . . . . . . . . . . . .
20
3.4
Higher Derivative and Non-Local Theories of Gravity . . . . . . . . . . . . . . . . .
21
3.5
Higher Dimensional Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . .
21
3.6
Screening Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4 Brans-Dicke Theory
25
4.1
Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.2
The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.3
Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.4
The weak-field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.5
The parameterised post-Newtonian approximation . . . . . . . . . . . . . . . . . .
32
4.6
Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.7
Black Holes in Brans-Dickes Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5 Cosmology in Brans-Dicke Theory
43
5.1
Duality Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5.2
Exact solutions of Brans-Dicke cosmology . . . . . . . . . . . . . . . . . . . . . . .
44
5.2.1
The O’Hanlon and Tupper solution . . . . . . . . . . . . . . . . . . . . . . .
45
5.2.2
The Brans-Dicke dust solution . . . . . . . . . . . . . . . . . . . . . . . . .
45
5.2.3
The Nariai solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5.2.4
Generalising Nariai’s solution . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.3
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5.4
Brans-Dicke Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
5.5
Accelerating universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.5.1
54
Quintessence in Brans-Dicke theory
5
. . . . . . . . . . . . . . . . . . . . . .
5.5.2
Acceleration without quintessence . . . . . . . . . . . . . . . . . . . . . . .
6 General Scalar-Tensor Theories
55
57
6.1
Constraint from BBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
6.2
Extended and Hyperextended Inflationary . . . . . . . . . . . . . . . . . . . . . . .
61
7 Equivalence of theories
63
7.1
Equivalence with f(R) theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
7.2
Relation to Kaluza-Klein theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
7.3
The dilaton from string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
8 Discussion
68
6
7
1
Introduction
The 17th century was a very auspicious time for the sciences, with major breakthroughs occurring
in the field of mathematics, physics and astronomy. Some of the greatest developments in the period include the development of the heliocentric model of the Solar System by Nicolaus Copernicus,
the pioneering work with telescopes and observational astronomy by Galileo Galilei, and the development of Kepler’s laws of planetary motion. Kepler’s laws resolved the remaining mathematical
issues raised by Copernicus’ heliocentric model, thus removing all doubt that it was the correct
model of the Universe. Working from these, Sir Isaac Newton began considering gravitation and
its effect on the orbits of planets.
It was not until 1665, when Isaac Newton introduced the now renowned “inverse-square gravitational force law”, that he was grappling with the idea of how terrestrial gravity extends to celestial
gravity. However, it would take him two more decades to fully develop his theories to the point
that he was able to offer mathematical proofs. In July of 1687, he published the Philosophiae
Naturalis Principia Mathematica, contained the well-known Newton’s three laws of motion, which
were derived from Kepler’s laws, and his own mathematical description of gravity. He deduced
that the same force that makes an object fall to the ground was responsible for the other orbital
motions. Hence, he named it “universal gravitation.” Newton used his laws of mechanics to suggest
that all masses attract each other with a force proportional to the product of their masses and
inversely proportional to their separation distance squared, which can be encapsulated by
F = GN
Mm
r2
(1.1)
where F is the force due to gravity, and two bodies separated by a distance r. The constant
GN ≈ 6.7 × 10−11 m3 /kgs2 was measured today in laboratory and solar system environments.
Newton’s gravitational theory was spectacularly successful at describing everythingthat it was applied to. From the motion of projectiles to rolling objects; from the weight of objects to the ticking
of a pendulum clock; from the buoyancy of a boat to the planets orbiting the sun, Newtons gravity
never failed. As of the mid-1800s, the observed orbits of all planets except two agreed precisely
with prediction from Newtonian gravity. The orbit of Uranus had orbital discrepancies, and the
nearest planet to the Sun, Mercury, had begun to show a stranger violation of the laws of gravity.
One of the simplest explanations for these discrepancies was to invent two extra planets. Indeed,
Neptune was predicted and found to exist in the vicinity of Uranus. Vulcan was predicted to exit
even interior to Mercury to explain the orbital discrepancy from Mercury. Yet after an exhaustive
search, this innermost planet was never found [132]. The reason that Vulcan does not exist is
8
actually because it was the expectation that Newton’s understanding of gravity breaks down when
it is applied in extreme gravitational fields, i.e. planet is so close to the Sun.
Although Newton was able to posit the existence of gravity, he was unable to explain how it
functioned. The solution came in 1915, when Einstein published his general theory of relativity
to the Prussian Academy of Sciences. He computed the spectacular figure that the contribution
of the extra curvature of space predicted an additional precession of 43 arcsecond per century of
the Sun-Mercury system (where Newtonian gravity had failed), exactly the right figure needed to
explain this observation.
Einsteins theory of general relativity that not only reproduced all of Newtons successes, but explained Mercurys orbital anomalies and made predictions about the bending of starlight by gravitational sources that could be tested. Within only a few short years, general relativity was confirmed
by experimental findings like the Lense-Thirring gravitomagnetic precession (1918) [83] and the
gravitational deflection of light by the Sun as measured in 1919 during a Solar eclipse by Arthus
Eddington.
For a century, the consensus best theory to describe the gravity is Einstein’s general theory of
gravity. This consensus is not without reason: practically all experiments and observations supports to this theory, from classical weak-field observations such as the precession of Mercury’s
perihelion and the bending of starlight around the Sun, to the loss of orbital energy to gravitational waves in binary pulsar systems and most recently discovered merging black holes from LIGO.
Despite its spectacular successful in describing our universe, there are reasons to anticipate new
gravitational physics beyond general relativity. Indeed, it predicts its own demise in two scales:
• Ultraviolet scales In the ultraviolet, i.e. at short distances and high energies, it is well
known that General Relativity is not renormalizable when one considers the graviton-matter
interactions (one-loop diagrams) and graviton-graviton interactions (two-loops diagrams),
which means the loop corrections at higher and higher order generate a never-ending set of
counter-terms, thus cannot be extended to a quantum theory. Due to the fact that General
Relativity is non-renormalizable at different orders, its validity is restricted to low-energy
domain at large scales.
• Infrared Scales From different cosmological and astrophysical observations (such as galaxy
rotation curves, the cosmic microwave background, distances to supernovae, gravitational
lensing and structure formation) together with general relativity and the Standard Model
9
of particle physics, observers and theorists conclude that our universe is consisted of about
4.9% of the energy density inferred from Plank’s results [3]. The remaining 95% of the overall
energy density must be ‘dark’ in order to be explain the observed dynamics and structure
of the universe. About 25% of this dark material is in the form of a non-relativistic, noninteracting form of matter called dark matter, and that the other 70% is in the form of a
non-clustering form of energy density with a negative pressure known as dark energy. Dark
matter was introduced to explain why rotation curves of galaxies tend to flatten out a large
radii, and why the clusters of galaxies appear to have deeper potential wells than would be
inferred from baryonic matter, whilst dark energy was introduced to as an explanation for
the present phase of accelerated expansion of the universe, as well as the features of the
spectrum of the cosmic microwave background radiation, and to solve the age of the universe
problem. However, the nature of them being most certainly amongst the deepest problems
of modern physics.
Based on these findings, the Λ-Cold Dark Matter (ΛCDM ) paradigm has thus emerged as the
standard model of cosmology. In this model, dark energy is well-presented by a cosmological
constant Λ in Einstein’s field equations, and the preferred dark matter candidate is a collection of stable, neutral, elementary particles which are known as ‘cold dark matter’(CDM)
particles. There however remain some heavy dark clouds on our global understanding, especially on galaxy scales, the reader is referred to [52, 30] for further discussion of problems
and solutions of ΛCDM model.
These shortcomings point out that General Relativity cannot to be the final theory of gravity.We
should note that a key assumption in the ΛCDM model is that validity of General Relativity on
all scales, including galactic and cosmological scales. Applying Newtonian gravitational theory to
a system it was never tested in had the consequence of requiring an entire planet to be invented.
Today, applying general relativity to cosmological scales has meant 95% of the universe must be
invented. The analogy should now be obvious: perhaps general relativity is not the gravitational
theory that should be applied on cosmological scales. It is possible that dark energy does not exist
and that the acceleration comes about because physicists’ understanding of gravity is not quite
right. Instead, it is possible that modifications in General Relativity could be a viable approach
to solve IR and UV scales.
The structured of this dissertation is as follows: in section 2, I will present the key aspects of
general relativity and cosmology. Section 3 will explore why and how can we construct alternative
to general relativity. The basic properties of Brans-Dicke theory, one of the most well-known
alternative to Einstein’s theory, are introduced in section 4. I develop the weak-field approximation
10
of Brans-Dicke theory which is useful in many applications. The parameterized post-Newtonian
approximation is explained. The concept of conformal transformation is introduced in general
terms but briefly. I also explore the properties of black hole. In section 5, I review previous results
in the literature and present the general analytical solution for the cosmological field equations.
The inflation in the Brans-Dicke gravity is studied. I also discuss the general scalar-tensor theories
in section 6. Section 7 links Brans-Dicke theory to f(R), Kaluza-Klein gravity and string theory.
Finally, in section 8, I discuss and draw conclusions.
11
2
General Relativity and Cosmology
The General Theory of Relativity is the standard theory of gravity, together with quantum field
theory, it is now widely considered to be one of the two pillars of modern physics. The general
relativity is given in the language of differential geometry and was the first such mathematical
physics theory, leading the way for other mathematical theories in physics such as the gauge
theories and string theories. I will briefly recap some of its essential features and foundations in
this section. I will also describe cosmology from the point of view of general relativity, including
Friedmann-Robertson-Walker solutions, and cosmological perturbation theory.
2.1
Curvature of a Manifold
Perhaps Einstein’s most profound insight was to realize that gravitational physics admits a geometrical interpretation. This idea led him to postulate that spacetime could be describe as a
pseudo-Riemannian manifold (a manifold endowed with a metric whose determinant is negative
definite), that is, one on which a spacetime interval ds can be written in the form
ds2 = gµν (x)dxµ dxν
(2.1)
where gµν is the metric tensor and xµ are a system of coordinates on the manifold. The metric
gµν is necessarily symmetric, so, in principle, it has ten components. In Speical Relativity, the
components of gµν are constants,
gµν = diag(−1, 1, 1, 1) = ηµν
(2.2)
This is the Minkowski metric. The line-element for Minkowski space, thus, can be written as
ds2 = −dt2 + dx2 + dy 2 + dz 2 . Now, we used the notion of distance to compute the “shortest”
distance between two points on a spacetime manifold of arbitrary geometry. By integrating the
infinitesimal path length ds (2.1) along the length of the trajectory, the total length of the trajectory
is obtained, through a spacetime having a given metric:
Z
Z
p
s = ds =
gµν dxµ dxν
(2.3)
The shortest distance can be obtained by extremizing the path length with respect to variations
in the coordinates that lie on this particular trajectory,
δS
=0
δxµ
(2.4)
To calculate the values of the coordinates along this trajectory we introduce an affine parameter
λ along the curve γ, so that xµ = xµ (λ). This allows the path length to be written as
Z
p
s=
dλ gµν ẋµ ẋν
γ
(2.5)
2.2
Einstein’s field equations
12
where an over-dots here mean differentiation with respect to affine parameter, ẋµ = dxµ /dλ.
Treating the integrand in (2.5) as a Lagrangian, the principle of least action can be used to obtain
the Euler-Lagrange equation whose solution extremizes the path length. The Euler-Lagrange
equation one obtain is
ẍµ + Γµαβ ẋα ẋβ = 0
(2.6)
where Γµαβ are the components of the Christoffel symbol and are given by
Γµαβ =
1 µν
g (gβν ,α +gαν ,β −gαβ ,ν )
2
(2.7)
Equation (2.6) is called the equation of an affinely parametrized geodesic. For flat space, Γµαβ = 0,
so that the geodesic (2.6) becomes ẍµ = 0, which means that geodesics in flat space are straight
lines.
We have seen that the information about curvature of spacetime is contained in the metric; the
question is how to extract it? You can’t get it easily from the Christoffel symbols, Γµαβ , for
instance, since they can be zero or non-zero depending on coordinate system. The information
about curvature is contained in the Ricci tensor, Rµν , which is given by
α
α
β
β
α
Rµν = ∂α Γα
µν − ∂ν Γµα + Γαβ Γµν − Γµα Γνβ
(2.8)
α
where Rµν = Rµαν
. All of the component of Rµν vanish if and only if the space is flat. The trace
of the Ricci tensor yields the Ricci scalar,
R = Rµµ = g µν Rµν
2.2
(2.9)
Einstein’s field equations
The Ricci tensor, the Ricci scalar and metric are combined to produce the Einstein’s field equations:
1
Gµν = Rµν − Rgµν = 8πGN Tµν
2
(2.10)
where Gµν is the Einstein tensor, Tµν is the energy-momentum tensor of matter fields in the
spacetime. The meaning of the different components of the energy-momentum tensor is:

 

T00 T0i
Energy Density
Energy F lux
=

Tµν = 
Ti0 Tij
M omentum Density Stress T ensor
(2.11)
The symmetric tensor Tµν encompasses all we need to know about the energy and momentum of
matter fields, which act as a source for gravity. We will consider a perfect-fluid souce of energymomentum, for which
T µν = (ρ + P )uµ uν + g µν P
(2.12)
2.3
The Gravitational Action
13
where uµ = (−1, 0, 0, 0) is the fluid four-velocity and ρ and P are the rest-frame energy and pressure of the fluid respectively.
Also, we will often be interested in the Einstein’s equation in vacuum, where Tµν = 0. This means
that the vacuum Einstein equation is simply
Rµν = 0
(2.13)
Another feature of the Einstein field equation is that the Einstein tensor always obeys a conservation equation
∇µ Gµν = 0
(2.14)
This implies that the divergence of the energy-momentum tensor must vanish:
∇µ T µν = 0
(2.15)
The covariant derivative of a scalar is just the partial derivative, so (2.15) tells that T is constant
throughout the spacetime. The ν = 0 component of this expression gives an evolution equation
for the stress-energy content of a spacetime.
2.3
The Gravitational Action
The formulation of the gravitational field equations we have presented has been geometrical: the
Einstein tensor was constructed from various combinations of derivatives of the metric, and was
equated to the energy-momentum tensor. The field equations can be derived in an alternative
route, using the principle of least action.
We start from the Einstein-Hilbert action:
Z
√
1
d4 x −gR + Smatter (gµν , ψ),
(2.16)
S=
16πGN
√
We introduce the invariant volume element −gd4 x to preserve the action under coordinate transformations, and Smatter is the action for the matter fields, ψ, coupled to gravity. Now, varying the
action with respect to metric g µν gives,
Z
√
√
1
δS =
d4 x(δ −gR + −gδR) + δSmatter (gµν , ψ)
16πGN
(2.17)
The variation of the metric determinant is obtained using ln(detM ) = T r(logM ), it follows that
√
1√
1√
δ −g =
−gg µν δgµν = −
−ggµν δg µν
2
2
(2.18)
and the variation of the Ricci scalar is given by
µα
δR = Rµν δg µν + ∇α (g µν δΓα
δΓββµ )
µν − g
(2.19)
2.4
FRW Universe and Inflationary Cosmology
The second term of equation (2.19) can be neglected. Plugging all of this in we find that
Z
√
1
1
δS =
d4 x −g(Rµν − gµν R)δg µν + δSmatter (gµν , ψ)
16πGN
2
14
(2.20)
We define the energy momentum tensor to be
2 δSmatter
Tµν = − √
−g δg µν
(2.21)
This allows us to recover the complete Einstein’s field equation,
1
Rµν − Rgµν = 8πGN Tµν
2
(2.22)
or equivalently Gµν = 8πGN Tµν .
Moreover, there are a variety of other formalisms that one can use to derive Einstein’s equations.
The Palatini procedure [115], for instance, is a different way of deriving the Einstein equation from
the Einstein-Hilbert action. Instead of using the standard metric variation of the Einstein-Hilbert
action, it uses an independent variation with respect to the metric and an independent connection
(Palatini variation). The action is formally the same but now the Riemann tensor and the Ricci
tensor are constructed with the independent connection. Further alternative formulation of general
relativity includes metric-affine gravity [63], vierbein formalism [101], Plebanski formalism [118]
and Eddington formalism [49].
2.4
FRW Universe and Inflationary Cosmology
The standard model of cosmological model, usually known ΛCDM ‘concordance model’. Its key
features are that, to leading order, the expansion rate of our universe can be describe through the
evolution of the scale factor a(t) that appears in the Friedmann-Robertson-Walker (FRW) metric
in spherical symmetric coordinate system as:
dr2
2
2
2
2
2
2
2
ds = −dt + a (t)
+ r (dθ + sin θdφ )
(1 − kr2 )
(2.23)
where parameter k specifies the curvature of spatial hypersurfaces. If k = 0, the hyper-surfaces of
constant t are flat, if k > 0 they are positively curved, and if k < 0 they are negatively curved.
The dynamics of the scale factor is given by the Friedmann equation, which is obtained as the
‘time-time’ part of (2.22) when using the metric of (2.23):
2
ȧ
8πGN
k
=
ρ− 2
a
3
a
ä
4πGN
=−
(ρ + 3P )
a
3
(2.24)
(2.25)
Here, ρ and P should be understood as the sum of all contributions to the energy density and
pressure in the universe. We write ρr for the contribution from radiation, ρm for the contribution
2.4
FRW Universe and Inflationary Cosmology
15
by matter, neutrinos ρν and ρκ for spatial curvature. The first Friedmann equation is often written
in terms of the Hubble parameter, H ≡ ȧ/a,
H2 =
8πGN
k
ρ− 2
3
a
(2.26)
with H0 = H(a = 1) being the Hubble constant, i.e. the value of the Hubble parameter evaluated
at present. We may also define the dimensionless density parameters
Ωi ≡
where ρc is the total energy density ρc =
ρi
ρc
(2.27)
3H 2
8πGN
, and one can express the energy density ρi of
P
the i component of a multi-component cosmic fluid in a FRW university. Since ρ = i ρi , the
P
Friedmann equation then becomes the constraint i Ωi = 1 in the flat universe.
th
The evolution of the energy density of the ith component is evaluated with a perfect fluid stressenergy tensor and an FRW metric:
ρ̇ + 3Hγρ = 0
(2.28)
where P = (γ − 1)ρ is the equation of state parameter of the matter component. For the known
forms of matter, γm = 1, γr = 34 , γκ = 23 , and γν is in the range [1, 43 ].
In standard inflationary theories, the universe is dominated by a scalar field φ, the inflaton, with
potential V (φ) and minimally coupled to Ricci curvature; standard inflation corresponds to acceleration of the universe ä > 0 and is described by the action
Z
1
1 µν
4 √
S = d x −g
R + g ∂µ φ∂ν φ − V (φ) ,
16πGN
2
(2.29)
where no matter action is included other than φ, since the inflaton dominates the dynamics during
inflation, it sources the evolution of the FRW background. Consistency with the symmetries of
the FRW spacetime requires that the background value of the inflaton only depends on time, φ(t).
The stress-energy tensor of φ assumes the canonical form
1
φ
Tµν
= ∂µ ∂ν − gµν ( ∂ σ ∂σ − V (φ))
2
(2.30)
When the stress-energy tensor is evaluated in this homogeneous configuration it takes the form of
a perfect fluid, with density and pressure
1 2
φ̇ + V (φ)
2
1
Pφ = φ̇2 − V (φ)
2
ρφ =
(2.31)
(2.32)
The scalar potential is usually taken to be positive because during inflation, in which φ̇ ' 0,
ρ ' V (φ) is dominated by the potential of the scalar and is therefore required to be non-negative.
2.5
Perturbation theory
16
Assume the metric is that of a flat FRW spacetime. Substituting ρφ into the Friedmann equations
(2.24) and (2.25), we get
1 2
φ̇ + V ,
2
i
8πGN h 2
(φ̇) − V ,
Ḣ = −H 2 −
3
8πGN
H =
3
2
(2.33)
(2.34)
while the scalar φ satisfies the Klein-Gordon equation
φ̈ + 3H φ̇ = −V 0
(2.35)
where V 0 ≡ dV /dφ. The potential acts like a force, while the expansion of the universe adds friction.
The equation of inflation (2.24) - (2.35) and those satisfied by cosmological perturbations are solved
in the slow-roll approximation
(φ̇)2
V,
2
φ̈ H φ̇,
(2.36)
in which equations (2.24) - (2.35) reduce to
8πGN
V,
3
3H φ̇ ' −V 0 ,
H2 '
(2.37)
(2.38)
The slow-roll approximation assumes that the solution of φ(t) of the field equations rolls slowly
over a shallow section of the scalar field potential V (φ), which then mimics the effect of a cosmological constant Λ ' V (φ). Inflation stops when the potential V (φ) ends its plateau and quickly
decreases to a zero minimum. Then φ quickly accelerates toward this minimum, overshoots it, and
oscillates around the minimum of the potential. These oscillations are damped by particle creation
due to explicit coupling of φ to other fields or to the Ricci curvature, dissipating the kinetic energy
of φ and raising the temperature of the universe after the inflationary expansion. This process is
called reheating characterized by the generation of relativistic particles. The superluminal expansion solves the flatness, horizon and monopole problems of the classical cosmology and provides a
mechanism for the generation of density perturbations. These are generated by quantum fluctuations of the scalar field and seed the formation of large scale, which was formed after the end of
inflationary and radiation eras.
2.5
Perturbation theory
The universe we live in is not isotropic and homogeneous. To understand the formation and
evolution of large-scale structures, we have to move beyond the simplified FRW treatment. We
will introduce linear energy density perturbations about the homogeneous background, which turns
out is an indispensable tool for making predictions for a variety of cosmological observations. Here
2.5
Perturbation theory
17
we shall only consider scalar fluctuations, because only scalar perturbations exhibit the growing
modes needed to cause the formation of large-scale structure in the universe. Thus the metric can
be written as:
→
−
1
ds2 = a2 −(1 − 2Ξ)dτ 2 − 2( ∇ i β)dτ dxi + 1 + χ qij + Dij ν dxi dxj ,
3
(2.39)
−
→−
→
where Dij ≡ ∇i ∇j − 31 qij ∆ is traceless spatial derivative operator that projects out the longitudinal, traceless, spatial part of the metric perturbation, and qij is the flat spatial metric. Because
the metric is perturbed, all geometrical quantities computed from the metric will also become
perturbed from their value in an FRW background. In general relativity, not all of the components
of the metric (2.39) satisfy dynamical equations of motion and some of the components of the
metric are merely constraints. These correspond to gauge freedom in the theory, and it is usual
to make coordinate system choice to remove this freedom; there are four components which can
be removed. Two of the popular choices are the synchronous gauge Ξ = β = 0 and conformal
Newtonian gauge ν = β = 0. In the conformal Newtonian gauge only scalar perturbations to the
metric can be studied.
Bardeen introduced two popular gauge-invariant combinations of the metric perturbations [13]:
1
1
Φ̂ ≡ − (χ − ∆ν) + H(ν 0 + 2β),
6
2
1 00
1
Ψ̂ ≡ −Ξ − (ν + 2β 0 ) − H(ν 0 + 2β).
2
2
(2.40)
(2.41)
This reduce to the familiar potential Φ̂ = Φ = − 61 χ and Ψ̂ = Ψ = −Ξ of the conformal Newtonian
gauge.
One of the main sources of information in cosmology is through the observation of perturbations
about a Friedmann background. Such perturbations can be probed through their effects on the
dynamics of particles and light, including density fluctuations, peculiar velocities, anisotropies in
the Cosmic Microwave Background (CMB), and weak lensing.
18
3
3.1
Modified Gravity
Cosmological Constant Problem
Einstein’s original field equations are given in equation (2.10). He was interested in finding static
(ȧ = 0) solutions of equation (2.24), both due to his hope that general relativity would embody
Mach’s principle that matter determines inertia, and simply to account for the astronomical data
as they were understood at the time. A static universe with a positive energy density is compatible
with equation (2.24) if the spatial curvature is positive (k = +1) and the density is appropriately
tuned; however equation (2.25) implies that ä will never vanish in such a spacetime if the pressure
is also non-negative. Einstein therefore suggested a modification of his equation, to
1
Rµν − Rgµν + gµν Λ = 8πGN Tµν
2
(3.1)
which equivalent to Gµν + gµν Λ = 8πGN Tµν , where Λ is the cosmological constant, which is a
dimensionful parameter with units of (length)−2 . The cosmological constant term can enter the
Einstein field equations in two ways: as a geometric term, and on the right-hand side as an energy
density belonging to the vacuum,
Λ
(3.2)
8πGN
This equivalence is the origin of the identification of cosmological constant with the energy of the
ρvac = ρΛ ≡
vacuum. The cosmological constant is naturally one of the simplest explanation for the observed
acceleration of the universe [117, 123]. In order to reproduce the observed expansion rate of the
universe, the effective cosmological constant would need to have a value of Λ = H02 (where H0 is
the Hubble factor at redshift zero). When expressed as an vacuum energy density this is equivalent
to:
(obs)
ρΛ
2
= ΛMpl
∼ 2 × 10−10 erg/cm3
(3.3)
Quantum Field theory tells us to consider the energy of the vacuum as arising from a simple
harmonic oscillator of mass m at every point in spacetime, with ground-state energy E = w/2 =
√
k 2 + m2 /2. If we are confident that we can use ordinary quantum field theory all the way up
to the Planck scale, we can integrate over all possible wavenumbers up to a certain ultraviolet
momentum cut-off scale kmax , this gives energy density of :
(pl)
ρΛ
≈
4
kmax
∼ 2 × 10110 erg/cm3
16π 2
(3.4)
The ratio of (3.3) to (3.4) is the origin of the famous discrepancy of 120 orders of magnitude
between the theoretical and observational values of the cosmological constant. We know of no
special symmetry which could enforce a vanishing vacuum energy while remaining consistent with
the known laws of physics; this conundrum is the“cosmological constant problem”. For a comprehensive discussion of cosmological constant problem, the reader is referred to [99].
3.2
Lovelock’s Theorem
19
The situation is made worse still by the coincidence problem: the cosmological constant has an
energy density of the same order of magnitude as the average matter density in the universe today,
i.e. ρΛ ∼ ρmatter . In other words, the ρΛ and ρmatter will different in the past, and will be different
in future.
There are some efforts to understand these problems based on the idea of the anthropic principle [139]. It is also possible that there exists some new field in our universe that does not belong to
the Standard Model of particle physics. Such theories is known as dark energy. Another possible
explanation for the observations is that general relativity is not complete description of gravity
in all regimes of our universe, e.g. general relativity is not valid on cosmological scales. There
are collectively theories different from General Relativity known as modified gravities, which may
provide an alternative explanation to dark energy for the observed acceleration of the universe.
3.2
Lovelock’s Theorem
Lovelock’s theorem [96, 97] states that the only local, second-order (or less) gravitational field
equations that can be derived from the action that consists solely of the four-dimensional metric
tensor of spacetime are the Einstein field equations and/or with a cosmological constant term.
This does not, however, imply that the Einstein-Hilbert action is the only action constructed from
gµν that results in the Einstein equations. In fact, in four dimensions or less one finds that the
most general possibility is
L=
√
µν ρλ
αβ
−g[αR − 2Λ + βµνρλ Rµν
Rαβρλ + γ(R2 − 4Rνµ Rµν + Rρλ
Rµν )]
(3.5)
where µνρλ is the four-dimensional Levi-Civita symbol and α, β and γ are constants. However,
the third and fourth term in the equation above does not contribute to the Euler-Lagrange equations [39]. The fourth term, also known as the Gauss-Bonnet term [79, 108], reduces to a surface
term in four dimensions or fewer. Therefore (3.5) can be considered as an equivalent action for
general relativity.
The consequence of Lovelock’s theorem is that to construct metric theories of gravity with field
equations that is different from General Relativity one must implement one (or more) of the
following:
• Introducing other fields into the gravitational action, beyond the metric tensor;
• Consider a spacetime with dimensionality different from four;
• Construct a theory contain higher than second derivatives of the metric in the field equations;
• Employ a non-local Lagrangian;
3.3
Alternative Theories of Gravity with Extra Fields
20
• Give up on either rank (2, 0) tensor field equations, or divergence-free field equations.
• Relinquish the principle of least action
The first four of these can be used to initiate a classification scheme for theories of modified gravity.
Fig. 1 shows classification of modified theories of gravity in which they violate Lovelock’s theorem.
I will briefly discuss each case in the following.
Figure 1: Tree diagram of modified theories of gravity (produced by Tessa Baker, http://users.
ox.ac.uk/~lady2729/Research.html).
3.3
Alternative Theories of Gravity with Extra Fields
Einstein’s general theory of relativity is a geometrical theory of spacetime. The fundamental building block is a metric field. For this reason the theory may be called a “tensor theory”. There is
no reason that why we cannot introduce other fields in the field equations of gravity. The simplest
scenario that one could consider is the addition of an extra scalar field, but one might also choose
to consider extra vector field, tensor field, or any combination of them (such as TeVeS). Examples
involving each type of new field can be refer to Fig. 1. Additional fields need to be suppressed at
scales where general relativity has been well tested, such as lab or Solar System. This is usually
achieved by making the new fields couple weakly to ordinary matter fields.
3.4
Higher Derivative and Non-Local Theories of Gravity
21
In some sense, the additional fields are an easy way to replace the cosmological constant. This
faces, however, two difficulties: i) a modified gravity is consistent not only with observations
of the background expansion rate, but also with observables that depend on the linear perturbations of a gravity theory; ii) finding a physical explanation for the forms/values of the new
functions/constants of the theory. Otherwise, it will be just fine-tuning, i.e. constructing a model
specifically fit the observational data without any physical basis.
3.4
Higher Derivative and Non-Local Theories of Gravity
Recall form Lovelock’s theorem that general relativity represents the most general theory describing
a single metric that in four dimensions has field equations with at most second-order derivatives.
An alternative method to extend general relativity is to construct a theory to higher than second
order. Gravity theory constructed in such way has advantage to improve the renormalizability
properties, because it will cause the graviton propagator to fall more quickly in the UV. The
caveat is that modifying gravity in this way can introduce instabilities into the theory, such as
the Ostrogradski instability [112] and ghost (refer to a scalar field whose kinetic terms have the
opposite sign to the canonical case).
One of the well-known and have been intensively studied example of a higher-derivative theory
is f(R) gravity, in which the Einstein-Hilbert action is extended to be a more general function of
the Ricci scalar then the simple linear one that leads to Einstein’s equations. This results in field
equations that contain fourth-order derivatives. f(R) gravity can map onto a scalar-tensor theory
which will be discussed in section 7.1. In Hǒrava-Lifschitz gravity, as another example, one allows
for higher-order spatial derivatives but remain second-order time derivatives. However, one must
give up the principle of Lorentz invariance in this case. Both examples can deviate considerably
from general relativity, while still maintaining some basic stability properties.
3.5
Higher Dimensional Theories of Gravity
Although general relativity is formulated in 3 + 1 dimensional manifold, Riemannian geometry is
not restricted to this. It is therefore possible to study gravitational theories in higher dimensions.
Indeed, this is more than just a theoretical curiosity. Shortly after the advent of General Relativity,
Kaluza and Klein attempted to unify the forces of gravity and electromagnetism by postulating
a fifth dimension that is curled up into an observably small circle leaving four-dimensional spacetime extended infinitely. The treatment of compact dimensions in this way experienced a revival
with the advent of superstring theory, which can only be formulated consistently in 10 dimensions.
Instead of the simple circle of Kaluza-Klein theory, the additional six dimensions are compactified
on geometrically complex spaces known as Calabi-Yau manifolds.
3.6
Screening Mechanism
22
Another classes of higher-dimensional theories are models in which the additional dimension(s) are
large or infinite in extent. Our four-dimensional universe is then referred to as a brane existing in
the higher-dimensional bulk spacetime. A key feature of such theories is that all Standard Model
fields are confined our four-dimensional brane, with only gravity able to propagate into the bulk
dimensions. The most well-known examples of branworld theories are Dvali-Gabadadze-Porrati
(DGP) gravity [48] and Randall-Sundrum braneworld models [120, 121].
The third general classes of higher-dimensional theories are models based on the Gauss-Bonnet
term and higher-dimensional generalizations of the Einstein-Hilbert action. Gauss-Bonnet theories
are primarily of interest as a low-energy limit of a heterotic string action [91].
3.6
Screening Mechanism
Despite the overwhelming evidence for the existence of dark energy and dark matter, their underlying fundamental physics remains unknown. It is possible that the dark sector includes new
light degrees of freedom that couple to both dark and baryonic matter with gravitational strength,
thereby affecting the nature of gravity and the growth of structure on the largest scales. Naively
the existence of light, gravitationally-coupled scalars are ruled out by solar system tests of gravity.
Over the last decade, however, it has been realized that scalar fields can in fact be clever and evade
detection from local experiments through screening mechanisms [73].
There are three widely recognized screening mechanisms, all based on scalar fields:
• The Chameleon mechanism.
The Chameleon Mechanism [75, 74] operates whenever a scalar field couples to matter in
such a way that its mass depends on the local matter density of the gravitational system.
On cosmological distance scales the mean mass density is low, the scalar is light and able
to propagate, mediating a ‘fifth force’ of gravitational strength, but in relatively dense environments such as near the Earth the local density is high, the scalar acquires a large mass,
making its effects too short range to be detectable.
We explore this by constructing Chameleon field theories as:
!
Z
2
Mpl
1
4 √
2
Scham = d x −g
R − (∂φ) − V (φ) + Smatter [gµν A2 (φ)].
2
2
(3.6)
where we include a scalar potential V (φ) and a general coupling A(φ) to matter fields. The
3.6
Screening Mechanism
23
equation of motion for φ that derives from this action is
φ = V,φ − A3 (φ)A,φ Te,
(3.7)
where Te = geµν Teµν is the trace of the energy-momentum tensor defined with respect to the
Jordan-frame metric geµν = A2 (φ)gµν . We can approximate the metric as flat space, ignore
time derivatives, and focus on the case of a non-relativistic, pressureless source, i.e. Te00 = −e
ρ,
in order to study the field on solar system and galactic scales. In terms of an energy density
ρ = A3 (φ)e
ρ conserved in Einstein frame, we obtain
∇2 φ = V,φ + A,φ ρ.
(3.8)
Hence, the scalar field is affected by ambient matter density. The field profile is governed by
an effective potential
Vef f (φ) = V (φ) + A(φ)ρ.
(3.9)
Figure 2: The overall chameleon effective potential Vef f (solid curve) is the sum of the actual
potential V (φ) (dashed curve), and a density-dependent term from its coupling to matter (dotted
curve).
For suitable chosen V (φ) and A(φ), this effective potential can develop a minimum at some
finite field value φmin , such that Vef f,φ (φmin ) = 0 (as illustrated in Fig.2 ). This tells us the
amount of mass of the chameleon field m2min = V,φφ (φm in) + A,φφ (φm in)ρ. However, the
3.6
Screening Mechanism
24
depth and position of this minimum changes over with the surrounding density of matter,
hence providing of a mechanism to have light fields on cosmological scales but heavy near
the Earth.
As can be seen from Fig.2, the effective mass can be a steeply growing function of the ambient
density. The chameleon mechanism therefore exploits the large difference between local and
cosmic densities to generate a wide range in mass scales.
• The Vainshtein Mechanism.
The Vainshtein Mechanism [137, 11] relies on derivative couplings of a scalar field present
in the Lagrangian and field equations. These terms become large in the vicinity of massive
sources. When the action is canonically normalized, this has the effect of weakening their
interactions with matter, thereby recovering general relativity. This mechanism is essential
to the viability of DGP gravity [48], Galileon theories [43] and massive gravity [9, 10].
• The Symmetron Mechanism.
The Symmetron Mechanism [66, 67] relies on a scalar field with a density-dependent potential.
In high mass density regions the vacuum expectation value (VEV) of the potential is small,
and becoming large in regions of low mass density. The coupling of the scalar to matter is
proportional to the VEV, so that the scalar couples with gravitational strength in regions of
low density, but is decoupled and screened in regions of high density.
25
4
4.1
Brans-Dicke Theory
Brief History
Scalars are the simplest fields. It has been the protagonist of physical theories for a long time.
The first gravity theory of scientific interest was developed by Newton, using a scalar potential
field. Before publication of general relativity, G. Nordström attempted to construct a scalar theory
of gravity by promoting the Newtonian potential function to a Lorentz scalar. A few years later,
after general relativity was published, Kaluza and Klein unified gravity and electromagnetism in
five dimensional spacetime [70, 78].
The essential feature of scalar-tensor theories is that the gravitational constant is time-dependent:
this idea dates back to the work of P.A.M. Dirac’s in 1937 related to the large number hypothesis(LNH) [47], which obviously beyond what can be understood with the scope of general relativity.
Dirac was inspired by the coincidence between some representative numbers in physics, like the
ratio between the gravitational and electric force between the electron and the proton in the hydrogen atom, and characteristic time of the sub-atomic phenomena and the age of the universe,
both giving numbers of about 1040 . There should be physical explanation for the “coincidence”.
Among these large number coincidences is GN M/R ∼ 1, for cosmological mass M and radius R.
Since the age of the universe is an increasing quantity, the gravitational “constant” is proposed
may change with time, which is determined by the totality of matter in the universe. It expresses
the ability of mass-energy to interact gravitationally.
P. Jordan further developed this idea by taking the form of a complete gravitational theory in
which GN was promoted to the role of gravitational scalar field. Jordan started to embed a
four-dimensional curved manifold in five-dimensional flat space-time [69]. He also discussed the
possible connection of his theory with Kaluza’s five dimensional unified field theory. Jordan and
his colleagues went beyond the five dimensional origins of this scalar and proposed purely four
dimensional field equations involving a scalar field related to Newton’s constant.
Finally, the idea of a scalar-tensor theory reached full maturity with the work of C. Brans and R.
H. Dicke [25]. They published a new theory that was to become the prototype of the alternative
theory to general relativity. In Brans-Dicke theory, Machs ideas on induction, that the total mass
distribution in the universe should determine local inertial properties, were of main concern. In
fact, Sciama [131] had proposed a model theory of inertial induction. His idea was to relate the
“inertial force” experienced during acceleration of a reference frame relative to the average mass
of the universe to gravitational forces between the local test mass and the fixed stars. Base on this
4.1
Brief History
26
assumption, Dicke proposed that the gravitational constant should be a function of mass distribution in the universe.
Combining the Dirac’s LNH and the Mach’s principle, Dicke suggested to consider a field theory
by introducing a new field φ, that
φ ∼ 1/GN ∼ M/R
(4.1)
What does it mean that 1/GN is a scalar field? Paradoxically, it would mean that GN is varying
gravitational constant, it would likely change over time. In principle, it could also vary from location to location and time to time. An object at a point in space where constant gravitational
constant is small will be less massiveand warp the local spacetime lessthan an identical object at a
point where the constant is large. In the framework of general relativity, the fact that the new field
is a scalar is significant. In Einstein’s theory, the gravitational field can be “transformed away”, at
least locally, gravity can be made to vanish. However, a scalar field cannot be transformed away in
a similar manner. Thus it possesses a type of permanence which would harmonize with its being
dependent on an overall quantity such as the mass of the observable universe.
In fact, scalar fields have had a long and controversial life in gravity theories, with a history of
deaths and resurrections. The interest in Brans-Dicke gravity had considerably decreased in the
1970s, probably due to the more stringent constraints imposed on the theory by the Solar System
experiments, rendering it indistinguishable from general relativity. The astrophysical observation
constraint the value of the Brans-Dicke free parameter ω to be large, which is expect to be of
order unity. The lower bound on ω became larger as more precise experiments were carried out,
which corresponds to fine-tuning ω to satisfy observational limits. However, cosmology implies less
restrictive bounds on ω.
Later on the resurge of interest is owing to the following reasons: 1) The new importance of scalar
field in modern unified theories, in particular the dilaton and string theories. The low-energy
limit of the string theory yields a Brans-Dicke theory and the scalar-tensor gravity exhibits certain
similarities to supergravity and string theory. Brans-Dicke theory can be derived from classical
Kaluza-Klein theory, after compactification of the extra spatial dimensions. 2) Theorists discovered
some plausible mechanisms that allows the parameter to be order unity in the early universe but
diverge later. 3) After the idea of Inflation proposed by Guth and Linde in the early 1980s,
the General Relativity fail to provide a graceful exit from an inflationary epoch, however, scalar
field allows the inflationary epoch to end via bubble nucleation without the need for fine-tuning
cosmological parameters. 4) Brans-Dicke theory account for dark energy, which is in a form of
negative pressure, with a long-range gravitational scalar field.
4.2
4.2
The Lagrangian
27
The Lagrangian
Brans and Dicke proposed the basic Lagrangian [25]:
1 √
ω
LBD =
−g φR − g µν ∂µ φ∂ν φ + Lmatter (gµν , Ψ).
16π
φ
(4.2)
where ω is a dimensionless parameter. The factor φ in the denominator of second term is introduced in the Lagrangian to make ω dimensionless. Matter is not directly coupled to φ, in the sense
that the Lagrangian density Lmatter does not depend on φ, but φ is directly coupled to the Ricci
scalar. Since gravity is universally coupled to all physics, the direct coupling of φ to geometry,
φR, means that φ is universally coupled in some sense. The gravitational field is described by the
metric tensor gµν and by the Brans-Dicke scalar field φ, which together with the matter variables
describe the dynamics. The Brans-Dicke theory becomes indistinguishable from general relativity
in the limit ω → ∞ of the Brans-Dicke parameter. This statement is true in most situations,
however, several exact solutions of Brans-Dicke fail to yield the corresponding general relativistic
solution in this limit [6, 114, 125, 126, 128].
The original motivation for the introduction of Brans-Dicke theory was the search for a theory
containing Mach’s principle, which is not completely or explicitly embodied in general relativity.
A local problem such as study the Brans-Dicke equivalent of the Schwarzschild solution of general
relativity takes into account the cosmological matter distributed in the universe, and that the
gravitational coupling is generated by this matter through the cosmological field φ. By contrast,
in general relativity the gravitational coupling is constant and one can consider the Schwarzschild
solution by neglecting the rest of the universe.
The first term of equation (4.2) is called a “non-minimal” coupling, which marked the birth of scalar
tensor theory. It might be useful to explain what does “non-minimal” mean. In curved spacetime,
physical laws should exhibit general covariance, which means they should be independent of any
choice of basis or coordinate chart. In special relativity, we restrict attention to coordinate systems
corresponding to inertial frames. The laws of physics should exhibit special covariance: take the
same form in any inertial frame (this is the principle of relativity). By doing the following, we can
convert such laws of physics into general covariant laws:
ηµν → gµν
and
∂µ → ∇µ
(4.3)
We replace the Minkowski metric by a curved spacetime metric, and replace partial derivatives
with covariant derivatives. The differentiation for a scalar field equals to its covariant derivative,
f;µ = f,µ . According to this standard procedure, the second term on the right-hand side of (4.2) is
obtained from − ωφ η µν ∂µ φ∂ν φ. In this context, the field φ comes to couple to gravity only through
4.2
√
The Lagrangian
28
−gg µν . The gravitational coupling obtained by applying this “minimum” rule is called a minimal
coupling in analogy with a similar rule for charged fields in electrodynamics. The first term on the
right-hand side of (4.2) cannot be obtained by this rule; in flat Minkowski space-time this term
simply goes away. This is the origin of the name “nonminimal”.
Another important point in the Brans-Dicke model is that they demanded that the scalar field
must be decoupled from matter Lagrangian in order to preserve the weak equivalent principle
(WEP), which states that the trajectory of a freely falling test body is independent of its internal
structure and composition. I will brief argue why matter Lagrangian coupled to scalar field will
violate the WEP [56].
In general relativity, the equation of motion of a point mass is determined by geodesic, which is
given from an action
Z
Sp = −mI
dτ
(4.4)
where τ is a proper time. We can see that the inertial mass m appears only as an overall factor, so
it does not affect the trajectory in spacetime. This implies universal free-fall, a major expression
of the WEP. If the Lagrangian contains scalar field, it will enter (4.4) via the presence of “source”
term on the right hand side, hence it will violate the WEP.
Einstein’s theory is based on the WEP, but it is of course not unique to general relativity. Any
theory whose matter field couple to a unique metric tensor, e.g. the Brans-Dicke theory, satisfies the WEP. The WEP has been supported ever more strongly by experimental measurements.
Eötvös-type experiments, for instance, test the WEP by measuring the fractional difference in the
acceleration of freely falling bodies of different composition, e.g. η parameter. The current best
limit comes from the torsion balance experiment measured by Schlamminger [130], which gives
η=2
|a1 − a2 |
= (0.3 ± 1.8) × 10−13
|a1 + a2 |
(4.5)
where, a1 and a2 are the relative difference in accelerations of the two bodies. A number of projects
are in the development or launched to push the bounds on η even lower. The project MICROSCOPE is launched on 25 April 2016 to probe these limits further and test WEP with a precision
on the order of 10−15 . Another test mission, called the Satellite Test of the Equivalence Principle
(STEP), is being developed by Stanford University and an international team of collaborators,
with the goal of a 10−18 test.
Although the Brans-Dicke theory obeys the weak principle, it violates the strong equivalence principle (SEP). The SEP, to quote Brans-Dicke, states that all “the laws of physics, including numerical
4.3
Field equations
29
content (i.e. dimensionless physical constants), as observed locally in a freely falling laboratory,
are independent of the location in time or space of the laboratory” [25]. In other words the proportionality between inertial and gravitational masses holds true up to the higher order effects of
gravity.
When the WEP is satisfied, scalar-tensor theories are also constrained by tests of the gravitational
inverse-square law. Fischbach et al. [54] suggested the existence of a fifth force of nature, with a
strength of about a percent of gravity, but with a range of a few hundred meters. A scalar field
with mass m mediating a force of range λ and coupling strength α, corresponding to the Yukawa
potential
U = −α
GN M −r/λ
e
r
(4.6)
Experimental tests can be charaterized as providing limits on λ and α. The inverse-square law
holds (α ∼ O(1)) down to a length scale λ = 56µm [72].
4.3
Field equations
By varying the action derived from integrating (4.2) over all space with respect to g µν and using
the properties
√
1√
−ggµν δg µν ,
δ( −g) = −
2
√
√
√
1
δ( −gR) = −g(Rµν − gµν R)δg µν ≡ −gGµν δg µν ,
2
(4.7)
(4.8)
one obtains the field equation
Gµν = 8πφ−1 Tµν +
ω
1
(∂µ φ∂ν φ − gµν (∂φ)2 ) + φ−1 (∇µ ∇ν φ − gµν φ).
φ2
2
√
δ
√−2
( −gLmatter ) is the
−g δg µν
√
and φ = √1−g ∂µ ( −gg µν ∂ν φ),
(4.9)
where Tµν ≡
stress-energy tensor of ordinary matter, (∂φ)2 ≡
g µν ∂µ φ∂ν φ,
which is a covariant d’Alembertian for a scalar
field. The first term on the right hand side of (4.9) is the source term of general relativity, but
with the variable gravitational coupling parameter φ−1 . The second term is the energy-momentum
tensor of the scalar field, also coupled with the gravitational coupling φ−1 . The third term is unfamiliar and results from the presence of second derivatives of the metric tensor in R. These second
derivatives are eliminated by integration by parts to give a divergence and the extra terms.
We take the trace of (4.9) by using g µν Gµν = −R, we obtain
−R =
8πT
ω
3φ
− 2 (∂φ)2 −
φ
φ
φ
where T is the trace of Tµν , which T = g µν Tµν .
(4.10)
4.4
The weak-field approximation
30
The model not only contains metric tensor, but also contain the dynamical scalar field φ, and we
must vary the action derived from Eq. (4.2) with respect to φ gives the field equation
φ −
1
1φ
(∂φ)2 +
R=0
2φ
2ω
(4.11)
The field equation for φ can be rewritten by substituting the Equation (4.10) into Equation (4.11),
we obtained a field equation for the scalar field φ:
φ =
8π
T
3 + 2ω
(4.12)
Equation (4.12) shows that the scalar field has its source given by the trace of the matter energymomentum tensor. It is also clear that the coupling to matter of the scalar field vanishes in the
limit ω → ∞. In other words, the theory approaches Einstein’s theory in this limit.
In addition to the field equations, we have the conservation law in the Brans-Dicke theory
∇ν T µν = 0
(4.13)
as in general relativity. This is a property related to the preservation of the general diffeomorphism
invariance in this theory.
The form of the Lagrangian (4.2) or of the field equation (4.9) suggests that the varying gravitational coupling is described through a long range scalar field φ
Gef f (φ) =
1
φ
(4.14)
which becomes a function of the spacetime point. For this reason, the range of values φ > 0
corresponding to attractive gravity is usually chosen. We may expect that φ is spatially uniform,
but varies slowly with cosmic time, as suggested by Dirac.
4.4
The weak-field approximation
We try to see what physical results can be derived from these field equations. We first study the
limit in which the fields are weak. This would correspond to the Newtonian approximation of
Einstein’s equation. The scalar field can be written as φ = φ0 + ξ, where φ0 is a constant with
dimension of mass, |δφ| 1, corresponding to a “vacuum expectation value” in the quantum field
theory. Also it will play the role of the “background field” in cosmology.
We have φ = φ0 to the zeroth approximation for which the first on the right-hand side of (4.2) is
1 √
simply 16π
−gφ0 R. The weak-field solution to Eq. (4.12) is
φ = ξ = ∇2 ξ −
∂2ξ
8πT
=
∂t2
(3 + 2ω)
(4.15)
4.4
The weak-field approximation
31
The retarded-time solution to the equation is given by [25],
Z
2
T 3
ξ=−
d x
3 + 2ω
r
(4.16)
In the Newtonian limit, we consider matter objects at rest. In terms of the mass density ρ, we
write T = −ρ. Since the solution ξ of Eq. (4.15) is time-independent, it will be reduced to
∇ξ 2 (r) = −
8πρ
(3 + 2ω)
(4.17)
For a point mass with total mass M, the Poisson equation has solution
ξ(r) =
2M
(3 + 2ω)r
(4.18)
Linearization with respect to the metric field can also applied in the same way. As in general
relativity the metric tensor is written as
gµν = ηµν + hµν
(4.19)
By computing the Christoffel symbols, Ricci tensor and Ricci scalar, we obtain the Einstein tensor:
Gµν =
1
[∂µ ∂λ hλν + ∂ν ∂λ hλµ − ∂µ ∂ν h − hµν − ηµν (∂ρ ∂σ hρσ − h)]
2
(4.20)
where we defined the trace of the perturbation as h = η µν hµν = hµµ , and the d’Alembertian is
simply the one from flat space, = −∂t2 + ∂x2 + ∂y2 + ∂z2 . Equation (4.9) can be written to first
order in hµν and ξ as
hµν − ∂µ ∂λ hλν − ∂ν ∂λ hλµ + ∂µ ∂ν h + ηµν (∂ρ ∂σ hρσ − h) − 2φ−1
0 (ηµν − ∂µ ∂ν )σ = −
16π
Tµν (4.21)
φ0
The last two terms on the left-hand side come from the nonminimal coupling term and represent
mixing between two fields, hµν and ξ. We may remove the mixing terms to further simplified the
the Eq. (4.21) by introducing another field χµν defined by
1
χµν = hµν − ηµν h − ηµν ξφ−1
0 ,
2
(4.22)
Equation (4.21) then becomes
χµν = −
16π
Tµν
φ0
(4.23)
with the retarded-time solution
χµν
4
=
φ0
Z
Tµν 3
d x
r
(4.24)
For a stationary point of mass M, the solution of Eq.(4.23) is given by
χ(r) =
other components are zero.
4M
φ0 r
(4.25)
4.5
The parameterised post-Newtonian approximation
32
The weak-field approximation is useful in many applications. We can compare the prediction of
bending light or planetary orbits in the Brans-Dicke theory to general relativity in this approximation. Consider the planet, for example, the difference of gravitational potential energy can be
computed as
VBD − VGR
1
=
VGR
2ω + 3
(4.26)
It is obvious that the Brans-Dicke theory deviates considerably from general relativity unless the
parameter ω is very large. The deviation from general relativity is greatest in the limit ω → −3/2.
This is the strong coupling limit of Brans-Dicke gravity when matter couples to the scalar with
infinite strength.
4.5
The parameterised post-Newtonian approximation
The early experimental successes of general relativity were the consistency with the perihelion
precession of Mercury and the deflection of light by the Sun. Later on, a number of experimental
tests of general relativity were carried out, including the measurement of gravitational redshift,
the Shapiro time delay, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational
wave damping has been detected in an amount that agrees with general relativity to better than
half a percent using the Hulse-Taylor binary pulsar , which further established the validity of GR
in the weak field regime.
Solar system tests of alternate theories of gravity are commonly phrased in the language of the
Parameterised Post-Newtonian (PPN) formalism, which is widely used by both theoretical and
observational gravitational physicists. The PPN formalism applies to metric theories of gravity in
the weak field, e.g. slow motion limit. In this limit, the metric can then be written as an expansion
about the Minkowski metric in terms of dimensionless gravitational potentials of varying degrees
of smallness. The “order of smallness” is defined by
U (x, t) ∼ v 2 ∼ p/ρ ∼ Π ∼ (4.27)
where U is the Newtonian gravitational potential and is nowhere larger than 10−5 , v is the 3velocity of a fluid element, P is the pressure of fluid, ρ is its rest-mass desnsity (p/ρ is ≈ 10−5
in the Sun), Π is the ratio of energy density to rest-mass density (Π is ≈ 10−5 in the Sun), and
≈ GM/R is the principal figure that distinguishes strong from weak gravity. In the regime of
strong gravity: ≈ 0.5 near the event horizon of a non-rotating black hole; for neutron stars,
≈ 0.2. For the solar system < 10−5 , which is in the regime of weak gravity.
The post-Newtonian limit for time-like particles requires a knowledge of g00 to O(4), g0i to O(3),
and gij to O(2). On the other hand, the post-Newtonian limit of null particles requires a knowledge
4.5
The parameterised post-Newtonian approximation
33
of g00 and gij both to O(2).
In spite of the differences in structure between different metric theories of gravity, the calculation
of the post-Newtonian limit is summarised as follows. I will focus on scalar-tensor theories. First
one identifies the variables, including dynamical gravitational variables such as the metric gµν and
scalar field φ , and prior-geometrical variables such as a flat background metric ηµν . All dynamical
fields should then be perturbed from their expected background values, and the perturbations
assigned an appropriate order of smallness each. For the scalar-tensor theory the expansion is
usually
(0)
gµν = gµν
+ hµν ,
φ = φ0 + ϕ
(4.28)
Generally, the post-Newtonian orders of these perturbations are given by
h00 ≈ O(2) + O(4),
h0j ≈ O(3),
hij ≈ O(2),
ϕ ≈ O(2) + O(4)
Substitute these forms into the field equations. Then solving for h00 to O(2). Only the lowest postNewtonian order equations are needed. With this solution in hand, one then proceeds to solve for
hij to O(2) and h0j to O(3), and finally h00 to O(4) can be solved for. The energy-momentum
tensor must also expanded to post-Newtonian order. We obtain
T00 = ρ(1 + Π + v 2 − h00 ) + O(6)
T0i = −ρv i + O(5)
Tij = ρvi vj + P δij + ρO(6).
(4.29)
We have sofar outlined the procedure that one needs to follow in order to gain the appropriate
form of the metric that couples to matter fields in the weak-field limit. Once done, we can compare
the result of gµν to the ‘PNN test metric’ below:
g00
=
−1 + 2GN U − 2βG2N U 2 − 2ξG2N ΦW + (2γ + 2 + α3 + β1 − 2ξ)GN Φ1
+2(1 + 3γ − 2β + β2 + ξ)G2N Φ2 + 2(1 + β3 )GN Φ3 − (β1 − 2ξ)GN A
goi
gij
+2(3γ + 3β4 − 2ξ)GN Φ4
1
1
= − (3 + 4γ + α1 − α2 + β1 − 2ξ)GN Vi − (1 + α2 − β1 + 2ξ)GN Wi
2
2
= (1 + 2γGN U )δij
(4.30)
Here β, γ, ξ, α1 , α2 , α3 , β1 , β2 , β3 , and β4 are PPN parameters, U is the Newtonian gravitational
potential, and ΦW , Φ1 , Φ2 , Φ3 , Φ4 , A, Vi and Wi are the metric potentials (the precise formula of
these potential is given in [140]).
4.5
The parameterised post-Newtonian approximation
34
Following the method described above, we obtain for the post-Newtonian metric of Brans-Dicke
theory
g00
goi
gij
3 + 2ω
= −1 + 2GN U −
+4
GN Φ1
4 + 2ω
1 + 2ω
1+ω
2
+4
GN Φ2 + 2GN Φ3 + 6
GN Φ4
4 + 2ω
2+ω
1 10 + 7ω
1
= −
GN Vi − GN Wi
2
2+ω
2
1+ω
GN U δij
=
1+2
2+ω
2G2N U 2
(4.31)
The PPN parameters can be read off:
1+ω
,
β = 1,
ξ = 0,
2+ω
α1 = α2 = α3 = β1 = β2 = β3 = β4 = 0
γ=
(4.32)
The value of Newton’s constant can also be shown to be given by
Gef f =
4 + 2ω −1
φ
3 + 2ω
(4.33)
Notice that if φ changes as a result of the evolution of the universe, then Gef f may change from
its present value. According to equation (4.33), we find Gef f → φ−1 in the limit ω → ∞. This is
because the scalar coupling to matter vanishes in this limit.
In general relativity we have that γ = β = 1, and otherwise = 0. Solar system experiments have
been analysed in terms of these parameters. As mentioned, observations that involve only null
(2)
geodesics are sensitive to the Newtonian part of the metric, g00 to O(2) and the term gij to O(2)
only. These two terms involve the PPN parameter γ only. We know about that
• the predicted deflection of light around the Sun that one should observe is [140]:
∆θ =
1+γ
∆θGR
2
(4.34)
∆t =
1+γ
∆tGR
2
(4.35)
• Shapiro time delay effect [140]:
where the subscript GR represents the value of quantity as predicted by general relativity. The
latest observed value of ∆t yielded [21]:
γ − 1 = (2.1 ± 2.3) × 10−5
which agrees with the general relativistic value of γ = 1.
(4.36)
4.5
The parameterised post-Newtonian approximation
35
Table 1: The experimental tests of Brans-Dicke theory
Experimental Tests
ω
Remarks
reference
Nordtvedt Effect
>79
Lunar laser ranging
[142]
Deflection of light
>10,000
Radio interferometry
[82]
Shapiro delay
≥ 500
VLBI
[122]
Spinning gyroscope
≥ 118
Viking spacecraft
[50]
Now, we can consider observations involve time-like geodesics. the perihelion shift of Mercury is
given by [140]
6πM
∆ω =
p
2 − β + 2γ
1
µ
+ (2α1 − α2 + α3 + 2β2 )
+ J2
3
6
M
r2
2M p
,
(4.37)
where M is the total mass of the two bodies involved, µ is their reduced mass; p ≡ a(1 − d2 ) is the
semi-latus rectum of the orbit, with the semi-major axis a and the eccentricity e; and J2 is dimensionless measure of its quadrupole moment, which is introduced due to non-spherical symmetry of
the mass distribution. Owing to the solar spin, there is centrifugal force, generating oblateness,
and hence J2 can be estimated to be ∼ 10−7 .
The most stringent constraint on the Brans-Dicke theory was reported in 2003 using the Doppler
tracking data of the Cassini spacecraft while it was on its way to Saturn, with Eq.(4.36) and Eq.
(4.32), this gives the 2σ constraint on the coupling parameter [21]
ω > 50000.
(4.38)
This was obtained in the solar system test for spherically symmetric solutions. Some other constraints are listed in table 1. However, the limitation of such experiments is that they are “weak
field” experiments and probe only a very limited range of space and time. They could no reveal
spatial or time variation of gravitational on larger scales. As we have discussed, the larger the
value of ω, the smaller the effects of the scalar field, and in the limit ω → ∞, the theory becomes
indistinguishable from general relativity in all its predictions. This unusually large value is hardly
appealing since one expects dimensionless coupling parameters to be of order unity. Thus, BransDicke theory was once considered as a prototype of more general class of theories including a scalar
field.
The reader who is interested in delving further into the many experimental tests of the Brans-Dicke
theory can refer to [140, 141] for more information.
4.6
4.6
Conformal transformations
36
Conformal transformations
It is well known that Brans-Dicke theory is related to general relativity through a conformal
transformation. By this we mean that under a transformation of the metric that changes scales,
but not angles, one can find a new metric that obeys the Einstein equation, with the scalar
contributing as an ordinary matter field. A conformal transformation transforms a metric gµν into
ḡµν according to the rule
gµν = φ−1 ḡµν
(4.39)
where φ is the Brans-Dicke scalar field. Symbols with bars over metric refer to quantities in the
Einstein (general relativistic) conformal frame and symbols without bars represent quantities in
the Jordan (Brans-Dicke) conformal frame. The frame in which matter universally couples to the
gravitating metric is called the Jordan frame and the frame in which matter couples to a different
metric is called the Einstein frame. General speaking, except in the weak field limit, physics looks
different in two different conformal frames.
From equation (4.39) we find
g µν = φḡ µν ,
√
√
−g = φ−2 −ḡ
(4.40)
We do the same for the Christoffel symbol and obtain:
Γµνλ = Γ̄µνλ − (Ξ,ν δλµ + Ξ,λ δνµ − Ξ̄,µ ḡνλ )
(4.41)
where Γ̄µνλ is a Christoffel symbol in the new conformal frame, and Ξ = lnφ−1/2 . Once we have
the result on Γµνλ , we then compute Ricci tensor Rµν , and Ricci scalar R,
¯
Rµν = R̄µν − 2Ξ,µν + 2Ξ,µ Ξ,ν − (2Ξ,µ Ξ,ν + Ξ)ḡ
µν
(4.42)
¯ − 6ḡ µν Ξ,µ Ξ,ν )
R = φ(R̄ − 6Ξ
(4.43)
where the d’ALambertian operator transforms as Ξ =
√
√1 ∂µ ( −ḡḡ µν ∂ν Ξ).
−ḡ
Under these trans-
formation, I will now show how the Brans-Dicke Lagrangian in the Jordan frame can be transformed
into the Einstein frame.
From Eq. (4.2), by applying a conformal transformation, we obtain
1 √
¯ − 6ḡ µν Ξ,µ Ξ,ν ) − ω g µν φ,µ φ,ν ] + L̄matter (φ−1 ḡµν , Ψ)
−g[φ2 (R̄ − 6Ξ
(4.44)
16π
φ
q
8π
where L̄matter is the conformally transformed matter Lagrangian. We set φ = exp ψ ω+3/2
,
L=
and also note that g µν Ξ,µ Ξ,ν = 14 g µν φ,µ φ,ν . The term Ξ disappears on integration by parts so
we can discard it. Applying these substitution we can write the transformed Lagrangian (4.44) as
√
1
1 µν
R̄ − ḡ ∂µ ψ∂ν ψ + L̄matter (φ−1 ḡµν , Ψ).
(4.45)
L = −ḡ
16π
2
4.6
Conformal transformations
37
This is the Lagrangian in the new conformal frame. We can see that the non-minimal coupling
to the Ricci scalar has been removed. In the Einstein frame, we explicitly see that matter now
couples to the scalar φ and the new metric ḡµν . This means we have a fifth force mediated by φ
acting on standard model fields, and results in the violation of the SEP. The strength of the force
is controlled by α given in (4.6).
In last section, we derived field equation (4.12) in the Jordan frame representing coupling to matter
of the scalar field, which I will now re-express in the Einstein frame. First the left-hand side of
equation (4.12) can be written as
√
1
φ = √ ∂µ ( −gg µν ∂ν φ)
−g
(4.46)
We use
s
φ = exp ψ
√
√
−g =
8π
ω + 3/2
g
= ḡ
µν
,
s
−ḡexp −2ψ
s
µν
!
exp ψ
8π
ω + 3/2
!
!
,
8π
ω + 3/2
(4.47)
finding
s
φ =
8π
exp 2ψ
ω + 3/2
s
8π
ω + 3/2
!
¯
ψ
(4.48)
Now we will find the transformation of energy-momentum tensor. We recall that
1 δLmatter
Tµν = −2 √
−g δg µν
(4.49)
δḡ ρσ δLmatter
δLmatter
δLmatter
=
= φ−1
δg µν
δg µν δḡ ρσ
δḡ µν
(4.50)
From chain rule we obtain,
Hence we can write
Tµν = φT̄µν
q
8π
with φ = exp ψ ω+3/2
. In this way we have
T = g µν Tµν = φ2 T̄
(4.51)
(4.52)
We finally arrived
¯ =
ψ
r
8π
T̄
3 + 2ω
(4.53)
which implies that ψ has its source given also by the trace of matter energy-momentum tensor.
Combining this with the fact that ψ is no longer mixed with the metric tensor in the Einstein
frame, we can conclude that ψ has direct coupling to matter in Lagrangian [56].
4.7
Black Holes in Brans-Dickes Theory
38
The conformal transformations presented above establish a mathematical equivalence between
Brans-Dicke theory and Einstein’s general relativity. One might ask whether the mathematical
equivalence need to be physically equivalent. There are three incompatible points of view occurring
in the literature: i) the Jordan and the Einstein frame are physically equivalent; ii) the Jordan
frame is physical one, and the Einstein frame is unphysical; iii) the Jordan frame is unphysical,
and the Einstein frame is physical. So, which one should we use? There is no satisfactory answer
at present: the Einstein frame version with fixed units is always well-behaved in terms of energies,
while there are situations in which the Jordan frame version is unviable. However, the former may
completely lack the physical motivation of the latter [53].
4.7
Black Holes in Brans-Dickes Theory
One of most profound prediction from general relativity is the existence of Black Hole. The recent
direct detection of gravitational waves by the LIGO experiment [1, 2] is considered as a clear evidence for the existence of these objects since the measured gravitational wave carries the signature
of the merging of two massive black holes. The objective of this section is to explore whether black
holes in Brans-Dicke theory are the same as in general relativity and study the thermodynamics
of black holes.
There are four most well-known black hole solutions in General Relativity: the uncharged static
black hole, given by the Schwarzschild solution; the charged static black hole, the ReissnerNordström solution; the uncharged rotating black hole, the Kerr solution; and the charged, rotating black hole, known as the Kerr-Newman solution. The properties of these solutions led to
the so-called no-hair conjecture: the final state of gravitational collapse is characterised by its
mass M, angular momentum J, and charge Q. There is strong evidence that this conjecture may
be violated in some situations [64], in particular, when new sources like scalar fields are consider.
The Schwarzschild metric is a solution to the vacuum Einstein equations Rµν = 0. The line element
is given by:
ds2 = (1 −
2GN M 2
2GN M −1 2
)dt − (1 −
) dr − r2 (dθ2 − sin2 θdϕ2 )
r
r
(4.54)
where 0 < r < ∞ is a radial coordinate. The line-element is the unique spherically symmetric solution to the vacuum Einstein equations. This result is known as Birkhoff ’s theorem. The
Schwarzschild metric breaks down at r = 2M and r = 0. But it turns out that at r = 2M is just
the coordinate singularity. We call the surface r = 2M the event horizon and the region r < 2M
the black hole. Nothing will escape when cross the event horizon.
4.7
Black Holes in Brans-Dickes Theory
39
One of the important results in the study of black hole physics is the discovery of the thermodynamics behaviour of those objects. The temperature and entropy of the Schwarzschild black are
given by
TH =
1
,
8πM
SBH =
A
4lP2
where A is the area of the event horizon surface, and lP2 is the planck length defined as lP =
(4.55)
√
GN }.
Since the gravitational collapse and the subsequent black hole formation is of great importance in
classical gravity, many authors have investigated these aspects in Brans-Dicke theory [55, 92, 46,
71, 129, 128]. Hawking proved in four dimensions, the stationary and vacuum Brans-Dicke solution
is just the Schwarzschild solution with a constant scalar field everywhere [62]. Cai and Myung have
proved that in four dimensions, the charged black hole solution in the Brans-Dicke-Maxwell theory
is just the Reissner-Nordstrom̈ solution with a constant scalar field [31]. The Kerr-Newman type
black solutions, which are different from the solutions in general relativity, have been constructed
for −5/2 < ω < −3/2 in [77]. We now explore the black holes in Brans-Dicke theory.
Brans [26] actually has provided exact static and spherical symmetric solution to the vacuum BransDicke field equations in four possible forms depending on the values of the arbitrary constants
appearing in the solution. Here, we consider only the class I Brans solution since it is the only
form that is permitted for all values of ω, the other three forms are only valid for ω < −3/2 which
implies non-positive contribution of matter to effective gravitational constant. It has been shown
that Brans class III and class IV solutions are not different in [22], and class I and class II solutions
are equivalent in [23]. The Brans-Dicke vacuum field equations can be written as:
Rµν = ωφ−2 ∂µ φ∂ν φ + φ−1 ∇µ ∇ν φ,
(4.56)
φ = 0
(4.57)
In terms of Schwarzschild coordinates, the Brans class I solution takes the form [33]:
ds2 = −A(r)m+1 dt2 + A(r)n−1 dr2 + r2 A(r)n (dθ2 + sin2 θϕ2 )
(4.58)
where A(r) = 1 − 2 rr0 , and m, n, φ0 and r0 are arbitrary constants. This metric is asymptotically
flat for every value of m and n. The metric reduce to Schwarzschild metric when m = n = 0.
The scalar field is given by [33]:
φ(r) = φ0 A(r)−(m+n)/2 ;
(4.59)
The coupling constant is found from:
ω = −2
(m2 + n2 + nm + m − n)
(m + n)2
(4.60)
4.7
Black Holes in Brans-Dickes Theory
40
Either from (4.58) or (4.59) we can compute the Ricci tensor and eventually get curvature R:
R = −2(m2 + n2 + mn + m − n)
r02
A(r)−n−1 .
r4
(4.61)
The Ricci scalar vanishes like r−4 as r → ∞.
In order to study the occurrence of true singularities of the metric (4.58), we will examine the
scalar invariant:
I
=
Rµνρσ Rµνρσ =
=
4r02 (r − r0 )−2(n+1) r−4+2n
r 2
0
r
I1 (m, n) + 4
r 0
r
I2 (m, n) + 6I3 (m, n),
(4.62)
where:
I1 (m, n)
=
48 + 56m + 41m2 + 10m3 + m4 − 56n − 34mn
−20m2 n − 2m3 n + 29n2 + 6mn2 + 3m3 n2 − 8n3 + n4 ,
(4.63)
I2 (m, n)
= −12 − 13m − 8m2 − m3 + 13n + 4mn + 2m2 n − 6n2 + n3 ,
(4.64)
I3 (m, n)
=
(m + 1)2 + (n − 1)2
(4.65)
The scalar invariant goes to zero as r → ∞, unless m = −1 and n = 1. In this case, I3 (−1, 1) =
I2 (−1, 1) = 0. Hence, we obtain
I(−1, 1) = 48
r04
r8
(4.66)
This can be compared to the Schwarzschild metric:
IS = 48
r02
r6
(4.67)
So, what is the behaviour of the scalar invariant as r → 2r0 ? From equation (4.62), we see that
I → O[(r − 2r0 )−2−2n ] as r → 2r0 . Thus, if n 6 −1, there is no singularity occur at r = 2r0 .
In case of r = 0, we see that I → 0, except when n = −m = 2, where I1 = I2 = 0. Thus as r → 0
r2
we obtain I(−2, 2) = 48 r06 , which is the same as in the Schwarzschild case.
In order for this metric solution to represent a black hole spacetime ,the parameter (m,n) appearing in the solution should satisfy certain conditions. One straightforward way of obtaining such
conditions is to consider under what circumstances an event horizon forms. It turned out that
under condition m − n + 1 > 0, by studying outgoing null geodesics, the surface at r = 2r0 could
behave as an event horizon [33].
Putting them altogether, the condition for the metric solution in eq.(4.58) to represent a black
hole spacetime with regular event horizon tuns out to be
m−n−1>0
and
n 6 −1
(4.68)
4.7
Black Holes in Brans-Dickes Theory
41
Therefore it appears that for these values of the parameters appearing in the solution, the metric in
eq.(4.58) could represent a non-trivial black hole spacetime different from the usual Schwarzschild
solution in general relativity. One of the outstanding results on black holes in Brans-Dicke theory
is Hawking’s theorem, which states that the Schwarzschild metric is the only spherically symmetric solution of vacuum Brans-Dicke theory. To prove this theorem, Hawking assumed that the
Brans-Dicke scalar field φ is constant outside the black hole and satisfies the weak energy condition. The problem is that the non-trivial Brans-Dicke black hole solution is different from the
Schwarzschild black hole, which is apparently contradict to the Hawking’s theorem. This is in fact
not the case because the non-trivial Brans-Dicke black hole solutions violate the WEC, which is
satisfied if Tab ta tb ≥ 0 for stress-energy tensor Tab and all timelike vector ta , on the scalar field.
√
In order for type I solution to be physically acceptable, it demands −2 < ω < −(2 + 1/ 2).
Also, the Brans class I solution can be used to describe a wormhole provided ω lies in the interval
√
−2 −
3
3
< ω < −2 [134].
We now turn to the investigation of the thermodynamics of non-trivial BD black hole solutions
discussed thus far. From section 4.5, we note that for finite ω the only parameter that deviates from
its general relativistic value is PPN parameter γ. It is interesting to note that it is not the value
of that one should expect to measure outside of a black hole that has formed from gravitational
collapse in this theory. Such an object can be shown to have an external gravitational field with
γ = 1, as predicted by Hawking [62]. This does not, however, mean that gravitational collapse to
a black hole proceeds in the same way in Brans-Dicke theory as it does in general relativity. In the
Brans-Dicke case, the violation of the WEC in the Jordan frame is responsible for the violation of
the second law of black hole thermodynamics, which states that the surface area of a black hole
never decrease. Moreover, the violation of the WEC is also responsible for the breakdown of the
apparent theorem, that is, an apparent horizons are allowed to pass outside of the event horizon.
The scalar gravitational waves are emitted during the collapse, and the horizon area can decrease
with time, which does not occur in general relativity. This problem is addressed by Kang [71].
Kang realised that the problem is not in the area law itself but rather in the expression of the
black hole entropy, which is not simply one quarter of the area in these theories. The expression
for the entropy is rather
SBH
1
=
4
Z
Σ
d2 x
p
g (2) φ =
φA
,
4
(4.69)
(2)
where φ is the Brans-Dicke scalar and g (2) is the determinant of the restriction gµν ≡ gµν |Σ of the
metric gµν to the horizon surface Σ. This expression can be understood by the simple replacement
of the Newton constant GN with the effective gravitational coupling Gef f = φ−1 in Brans Dicke
theory so that SBH = A/4Gef f . The quantity SBH is always non-decreasing, even if the area
decreases.
4.7
Black Holes in Brans-Dickes Theory
42
Following this, one can consider the Einstein frame representation of Brans-Dicke theory given by
√
the conformal transformation gµν → geµν ≡ GN φgµν , accompanied by the scalar field redefinition
q
2ω+3 dφ
dφe = 16πG
. In the Einstein frame the gravitational coupling is a true constant but matter
N φ
couples explicitly to the scalar field and what were massive test particles following timelike geodesics
of the Jordan frame metric gµν do not follow geodesics of the rescaled metric geµν . Null geodesics
are left unchanged by the conformal rescaling, as well as null vectors and all forms of conformally
invariant matter. A black hole event horizon, being a null surface, is also unchanged. The area of
an event horizon is not, and the change in the entropy formula SBH = A/4GN → A/4Gef f = φA/4
can be understood as the change in the area due to the conformal rescaling of gµν . We can now
obtain the Einstein frame area is
Z
Z
p
p
e=
d2 x ge(2) =
d2 xGN φ g (2) = GN φA
A
Σ
(4.70)
Σ
assuming that the scalar field is constant on the horizon. Therefore, the entropy-area relation
e
SeBH = A/4G
N still holds in the Einstein frame. Therefore, the Jordan frame and Einstein frame
entropies are equivalent. One of consequence for using the Einstein frame representation of BransDicke gravity is that if the scalar field vanishes on the horizon of a Brans-Dicke black hole the
latter is attributed zero temperature and zero entropy . These black holes with vanishing φ has
been called “cold black holes” [27, 28, 143]. These cold black holes have zero gravity and an infinite
area, which imply infinite tidal forces at the horizon [29].
43
5
Cosmology in Brans-Dicke Theory
Let us now proceed to derive the Brans-Dicke equations for a FRW universe described by the metric
(2.23), and assuming a perfect fluid matter content. The Brans-Dicke scalar φ only depends on
the cosmic time in this metric one has
(∂φ)2 = −(φ̇)2 ,
φ = −(φ̈ + 3H φ̇) = −
(5.1)
1 d 3
(a φ̇).
a3 dt
(5.2)
the time-time component of the Brans-Dicke field equation (4.9) yields the constraint equation
H2 =
φ̇ ω φ̇2
8π
k
ρ− 2 −H +
3φ
a
φ
6 φ2
(5.3)
which provides a first integral. By Using R = 6(Ḣ + 2H 2 + k/a2 ), the trace equation (4.10) yields,
k
4πT
ω
Ḣ + 2H 2 + 2 = −
−
a
3φ
6
φ̇
φ
!2
+
1 φ
2 φ
(5.4)
From equations (5.3), (5.4) and the expression of the trace T = 3P −ρ, one obtains the acceleration
equatioin
−8π
ω
Ḣ =
[(ω + 2)ρ + ωP ] −
(2ω + 3)φ
2
φ̇
φ
!2
+ 2H
φ̇
k
+ 2
φ a
(5.5)
In addition, the dynamical equation (4.12) for the scalar field reduces to
φ̈ =
8π
(ρ − 3P ) − 3H φ̇.
2ω + 3
(5.6)
where H = ȧ/a is the Hubble rate, over-dots denote differentiation with respect to the proper
time of a comoving observer. In the presence of a radiative fluid with equation of state P = ρ/3,
equation (5.6) can be integrated, yielding the solution φ = constant or the equation
φ̇ = φ0 a−3
(5.7)
where φ0 is an integration constant. Equations (5.3) and (5.6) can be re regarded as the two
independent equations regulating the dynamics of spatially homogeneous and isotropic Brans-Dicke
cosmology. Each non-interacting fluid source P (ρ) separately satisfies a conservation equation:
ρ̇ + 3H(ρ + P ) = 0
(5.8)
Equation (5.6) is of second order in φ(t), while equations (5.3) and (5.8) are of first order in a,φ
and ρ. To solve them, we define the equation of state P = (γ − 1)ρ, with γ = constant is assigned,
and the equations are solved. One can integrate (5.8) to obtain
ρ = ρ0 a−3γ
where ρ0 is an integration constant.
(5.9)
5.1
5.1
Duality Symmetry
44
Duality Symmetry
One of interests to study the Brans-Dicke cosmology is that the equations for spatially flat FRW
models in vacuum have duality symmetry [89, 90]. By redefining the scale factor and the BransDicke field as follows,
α ≡ lna,
Φ ≡ −ln(Gφ)
the duality transformation gives [89, 90]
3ω + 2
ω+1
α→
α−2
Φ
3ω + 4
3ω + 4
6
3ω + 2
Φ→
α−
Φ
3ω + 4
3ω + 4
(5.10)
(5.11)
This transformation generalises the scale factor duality present in the effective action of string
theories [138, 136, 57, 58]
α → −α,
Φ → Φ − 6α
(5.12)
For ω = −1, this duality is reproduced by equations (5.11), which means the Brans-Dicke parameter
that yields low-energy string theory.
5.2
Exact solutions of Brans-Dicke cosmology
Let us now begin to study the solutions corresponding to a spatially flat FRW universe. Many of
solutions presented in the following are of the power-law type
a(t) ∝ tx ,
φ(t) ∝ ty ,
(5.13)
with 3x + y ≥ 1. In Brans-Dicke theory, a power-law solution plays a role analogous to that of the
inflationary de Sitter attractor in general relativity.
The scalar field can be re-expressed in term of γ
φ̈ + 3H φ̇ =
8π
(4 − 3γ)ρ,
2ω + 3
(5.14)
while the cosmic fluid obeys equation (5.9). If the scale factor described by a power-law a = a0 tx ,
the equation for the Brans-Dicke scalar becomes
φ̈ +
3x
8π(4 − 3γ)ρ0 −3γx
φ̇ =
t
.
t
(2ω + 3)a3γ
0
(5.15)
The general integral of this equation is then
φ = φ0 t y +
with y = 0 or 3x + y = 1.
(2ω +
8πρ0 (4 − 3γ)
t2−3γx
− 3γx)[1 + 3x(1 − γ)]
3)a3γ
0 (2
(5.16)
5.2
Exact solutions of Brans-Dicke cosmology
5.2.1
45
The O’Hanlon and Tupper solution
At early time, O’Hanlon and Tupper [109] approach the vacuum solutions with the range of BransDicke parameter ω > −3/2, ω 6= 0, −4/3, and a vanishing cosmological constant:
x±
t
a(t) = a0
,
t0
y±
t
,
φ(t) = φ0
t0
where
"
#
r
1
2ω + 3
x± =
ω+1±
,
3ω + 4
3
y± =
1∓
(5.17)
(5.18)
p
3(2ω + 3)
,
3ω + 4
(5.19)
and satisfies 3x + y = 1. The solution corresponding to exponent (x− , y− ) is called the slow solution, while (x+ , y+ ) is fast solution. The terminology is due to the fact that at early time, the
slow solution describes increasing scalar field and decreasing gravitational coupling GN , while fast
solution associates with decreasing φ and increasing gravitational coupling. Both solutions are
interchanged under the duality transformation (5.11).
In the limit ω → ∞, the O’Hanlon and Tupper vacuum solution is
a(t) ∝ t1/3 ,
φ = constant
(5.20)
This limit does not reproduce the corresponding general relativistic solution which is Minkowski
spacetime.
In the limit ω = −4/3, the solution approaches the de Sitter space
a(t) = a0 exp(Ht),
φ(t) = φ0 exp(−3Ht)
(5.21)
with H =constant.
5.2.2
The Brans-Dicke dust solution
The Brans-Dicke dust solution corresponds to a pressureless dust fluid (P = 1, γ = 1) and to a
matter-dominated universe with ω 6= −4/3,
with x =
2(ω+1)
3ω+4 ,
y=
2
3ω+4 ,
−6(ω+1)
3ω+4 ,
(5.22)
φ(t) = φ0 ty
(5.23)
and satisfies 3x + y = 2; while
ρ=
with z = −3q =
a(t) = a0 tx
and ρ =
C
.
a30
C
= ρ0 tz ,
a3γ
(5.24)
5.2
Exact solutions of Brans-Dicke cosmology
5.2.3
46
The Nariai solution
At late time Nariai [105] approached a particular power-law exact solutions with ω 6= −4[3γ(2 −
γ)]−1 < 0 and a vanishing cosmological constant:
a(t) = a0 tx
(5.25)
φ(t) = φ0 ty ,
(5.26)
where
2[1 + 2ω(2 − γ)]
,
[4 + 3ωγ(2 − γ)]
[2(4 − 3γ)]
y=
[4 + 3ωγ(2 − γ)]
x=
(5.27)
(5.28)
which satisfy the relation y + 3γx = 2. The energy density of ordinary matter scales as
ρ(t) =
C
= ρ0 t z ,
a3γ
z = −3γx.
(5.29)
We will now consider some particular cases of the Nariai solution in the following:
A. Solution for matter dominated universe
For solution of a dust fluid with γ = 1, P = 0, ω 6= −4/3), one have
a(t) = a0 t
2(ω+1)
3ω+4
,
2
φ(t) = φ0 t 3ω+4 ,
ρ(t) = ρ0 t
−6(ω+1)
3ω+4
(5.30)
(5.31)
(5.32)
This reproduces the Brans-Dicke dust solution (5.24). If ω = −1, one has a Minkowski space;
if ω < −4/3, or ω > −1, the solution is expanding; if −4/3 < ω < −1, the solution is instead
pole-like. In this case, there are two disconnected branches, one expanding for t < −t0 and the
other contracting for t > −t0 .
B. Solution for radiative universe
Another particular case of the Nariai solution is that of a radiative fluid (γ = 4/3, P = ρ/3)), with
Tαα = 0:
1
a(t) = a0 t 2 ,
(5.33)
φ(t) = φ0 = const,
C
ρ0
ρ(t) = 3P (t) = 4 =
.
a
(1 + δt)2
(5.34)
(5.35)
The solution implies φ̇ = 0. This gives the usual radiative solution of general relativity theory, with
1
a ∝ t 2 . This is quite attractive, since the fact the radiative phase of the standard cosmological
5.2
Exact solutions of Brans-Dicke cosmology
47
model can be reproduced, implies that a Brans-Dicke cosmological scenario can preserve the results
from nucleosynthesis and those connected with the relic radiation that permeates the universe
today. Note that, this solution is independent of ω and corresponds to a vanishing curvature
R = 6(Ḣ + 2H 2 ).
C. Universe dominated by the vacuum energy density
Another interesting Nariai’s general solution is that of a universe with cosmological constant [98, 81]
corresponding to γ = 0, P = −ρ,
1
a(t) = a0 tω+ 2 ,
(5.36)
φ(t) = φ0 t2 .
(5.37)
This solution is important for the extended inflationary scenario. It is not the only solution
describing a universe dominated by a cosmological constant, but it is an in phase space. Other
spatially flat FRW solution with −3/2 < ω < −4/3 and a cosmological constant are discussed in
ref [124].
5.2.4
Generalising Nariai’s solution
Nariai’s solution is not the most general power-law solution for a spatially flat FRW model and
it can be generalised [60] by employing the new time coordinate τ defined by dτ =
dt
,
a3(γ−1)
which
only coincides with t for a dust fluid with γ = 1. Such solutions can be found for any equations of
state. The generalised Nariai solution for ω > −3/2 is
√
a(τ ) = a0 (τ − τ+ )
ω
3 1+ω(2−γ)∓
√
1±
1+ 2 ω
3
(τ − τ− )
1+ 2 ω
3
ω
3 1+ω(2−γ)±
√
1∓
√
1+ 2 ω
3
(5.38)
1+ 2 ω
3
√ 2
√ 2
φ(τ ) = φ0 (τ − τ+ ) 1+ω(2−γ)∓ 1+ 3 ω (τ − τ+ ) 1+ω(2−γ)± 1+ 3 ω
(5.39)
where a0 , φ0 and τ± are integration constants, and we assumed that t+ > t− . Nariai’s solution is
recovered for τ+ = τ− . If τ+ 6= τ− , the solution approaches the O’Hanlon-Tupper vacuum solution
as τ → τ± . For ω < −3/2 we find
 q
2
a(τ ) = a0 [(τ + τ− )2 + τ+
]
1+ω(2−γ)
2A
exp ±

2
φ(τ ) = φ0 [(τ + τ− )2 + τ+
]
where A =
3
2ω
1 + ω(2 − γ) −
q
1 + 23 ω
4−3ω
2A
2
3 |ω|
−1
A
q
exp ∓3(2 − γ)

τ
+
τ
−
tan−1
,
τ+
2
3 |ω|
A
−1
(5.40)

tan−1
τ + τ− 
τ+
(5.41)
q
1 + ω(2 − γ) + 1 + 32 ω . If we consider only the ra-
diation dominated solutions with γ = 4/3, for ω > −3/2(ω < −3/2) we will see that the scale factor
undergoes an initial period of rapid (slow) expansion and at late time a(τ ) ∝ τ . Similarly, φ can be
5.3
Perturbations
48
seen to be changing rapidly at early times and slowly at late time. The different behaviour at early
and late times can be attributed to periods of free scalar-field domination and radiation domination
respectively. In the case of ω < −3/2 the field φ and matter are decoupled, which means they never
exchange energy, and the initial singularity can be avoided which result in a bouncing universe [17].
On the other hand we can consider solutions for the pressureless matter dominated era. In the
case ω > −3/2, since t+ > t− , the initial singularity occurs at t = t+ . Hence, when t → t+ , the
solution reads:
√
1+ω±
a ∝ (t − t+ )
1+ 2 ω
3
4+3ω
(5.42)
When t → ∞, we can neglect t± and the solution takes the form given by (5.30). In fact, equation
(5.30) implies several possibilities: i) if −3/2 < ω < −4/3, the solution exhibits an accelerated
expansion, but with a negative gravitational coupling; ii) if −4/3 < ω < −1, the exponent is
negative, and it is also an accelerated universe, but with positive gravitational coupling; iii) if
−1 < ω, the solution represents a decelerating expanding universe. In the other asymptote, iv) if
−4/3 < ω < 0, the exponent is negative, and the second derivative of the scale factor is positive;
v) if ω > 0 the expansion is decelerating. If ω < −3/2, the main characteristic is the absence of
singularities. When ω < −3/2 there is a phantom scalar field in the minimally coupled version of
the Brans-Dicke theory.
Unlike in general relativity, in Brans-Dicke theory it is also possible to have spatially positive
(k = 1) or negative (k = −1) curved exact vacuum solutions, which can be referred to [104, 94, 84,
14, 102, 103].
5.3
Perturbations
The standard big bang model of the universe is a very successful one supported by three major
pieces of evidence i) the expansion of the universe, which can be interpreted from the measurement
of redshift of galaxies; ii) the existence of a cosmic microwave background (CMB), the predicted
spectrum of the CMB is that of a blackbody with temperature of 2.7K, in remarkable agreement
with the observations, and iii) primordial nucleosynthesis, the predicted relative abundances of
Hydrogen, Helium and other light elements are in agreement with observations.
If one replaces general relativity with Brans-Dicke theory one needs to test whether the theory will
be in agreement with the observed recession of galaxies, spectrum and temperature of the CMB,
and primordial nucleosynthesis. To study this, we first consider perturbed FRW spacetime. The
perturbation of a FRW universe are extremely important because they leave an imprint as temperature fluctuations in the CMB, which have been detected by Wilkinson Microwave Anisotropy
5.3
Perturbations
49
Probe (WMAP) [19, 65] and Planck [3]. For the Brans-Dicke theory the perturbed spacetime have
been studied many times before. Here we will present these equations in the synchronous gauge
which Ξ = β = 0. Now, we can write the perturbed metric as
gµν = ḡµν + a2 (τ )hµν
(5.43)
where ḡµν is the unperturbed FRW metric and hµν is the perturbation that satisfies h00 = h0ν =
hν0 = 0 in the synchronous gauge. The remaining non-zero hij perturbations can be broken up into
scalar, vector and tensor parts. Let us denote a cosmic fluid component (e.g., baryons, neutrinos,
photons, cold dark matter, etc.) by index f . The perturbations to the energy density and pressure
are written as δρf and δPf , with peculiar velocity potentials and anisotropic stress written as θf
e f is the tensor contribution to the anisotropic stress of the fluid f .
and Σf , respectively. The Σ
If we consider only scalar and tensor perturbations, then the perturbed Einstein and Brans-Dicke
equations are
e
h00T + 2He
h0T
1 φ0
−2k 2 η + H +
h0
2φ
2η 0
8πa2 X
ef ,
(ρf + Pf )Σ
φ
f
0
0
2 X
8πa
φ
δφ
=
ρf δf + ω − 3H
φ
φ
φ
f
ω φ02 δφ
− k 2 + 3H2 +
2 φ2 φ
2 X
1 0 1
8πa
φ0
(ρf + Pf )θf + δφ −
=
H−ω
δφ
φ
φ
φ
φ
=
(5.44)
(5.45)
(5.46)
f
1 00
φ0
ν + H+
+η
2
2φ
1
δφ00 + 2Hδφ + φ0 h0
2
=
=
8πa2
δφ
(ρ + P )Σf +
φ
φ
8πa2 X
(δρf − 3δPf ).
2ω + 3
(5.47)
(5.48)
f
where primes denote differentiation with respect to conformal time, τ .
The background cosmological evolution and perturbations can be used to place constraints on
Brans-Dicke theory from CMB and primordial nucleosynthesis. The CMB can be used to place
constraints on the Brans-Dicke parameter ω. This has been done a number of times in the literature [85, 86], with the latest results based on constraints given by Planck, with Baryon Acoustic Oscillation (BAO) data, and observations of type Ia supernovae (SNeIa), giving 1249.5 > ω > 302.53
S
at the 68% confidence level (CL) and ω < −9999.5 ω > 232.06 at the 95% CL [86].
The limits on the variation of the gravitational constant was derived from Planck + BAO + SN
data [86], the upper bounds on Ġ/G is given as:
Ġ
= 0.2649 × 10−12 year−1
G
(5.49)
5.4
Brans-Dicke Inflation
50
This limit is below 10−10 years−1 by an order of magnitude, but may still appear to be roughly
consistent with Dirac’s prediction. However, Fujii and Maeda [56] argue that the upper bounds is
outside the range for which the LNH works, therefore, the LNH is completely ruled out.
Another cosmological probe that has been extensively applied to Brans-Dicke theory is that of
the primordial nucleosynthesis of light elements [38, 34, 35]. During the radiation era, when the
temperature drops to about 109 K due to the cosmic expansion, nucleosynthesis begins with the
production of approximately 25% by mass of 4 He and small amounts of 2 H, 3 He and 7 Li.
The abundance of 4 He and other light elements is sensitive to the expansion rate of the universe.
Hence one can constrain the Brans-Dicke theory by using observational limits on the relative
abundance of light elements. At the time of nucleosynthesis, vast majority of neutrons will end up
in the helium nuclei, hence the mass fraction X(4 He) of helium depends on the ratio of neutron
and proton number densities nn and np ,
X(4 He) = 2
x
|nucleosynthesis
1+x
(5.50)
where x = nn /np . Before freeze-out tf o the weak interaction maintains neutrons and protons in
chemical equilibrium; after freeze-out, the value of x is frozen at the time tf o
mn − mp
x|tf o = exp
kB Tf o
(5.51)
where mn and mp are the neutron and proton masses, respectively, kB is the Boltzmann constant,
and Tf o is the temperature at freeze-out. The neutrons decay freely between the time freeze-out
and nucleosynthesis. The final abundance of neutrons at nucleosynthesis and the mass fraction of
4
He produced is determined by the duration of this period. If the expansion of universe is faster,
the ratio x is closer to unity. If the cosmic expansion is slower, 4 He is underproduction. I will
further discuss the deviation from general relativity in scalar-tensor theories in section 6.
In the Brans-Dicke theory the process of nucleosynthesis therefore is similar as in a general relativity
cosmology, but with a different value of G during this process, which result in a different expansion
rate. The typical bounds that can be achieved on the coupling parameter from observations of
element abundances are then given by ω > 300 or ω 6 −30, assuming the power-law solutions
(5.25) and (5.26).
5.4
Brans-Dicke Inflation
Recent cosmological data indicate that the universe has undergone two acceleration phases: an
early acceleration phase called ‘inflation’ has occurred before the radiation-dominated era, and a
5.4
Brans-Dicke Inflation
51
more recent era of accelerate expansion which appears to continue today. The gravitationally repulsive stress that is responsible for the current acceleration of the universe is called ‘dark energy’
and must possess sufficient negative pressure to exert gravitational repulsion. Whilst a range of
“exotic” fluids and modifications of the gravitational action can provide cosmological acceleration,
scalar fields are the simplest candidates to explain the acceleration phases of the universe. Let us
first consider inflation in the Brans-Dicke theory and will discuss present accelerated expansion
in 5.5.
The form of the action in the Jordan frame becomes
Z
√
1
1 ω µν
S = d4 x −g φR −
g ∂µ φ∂ν φ − V (φ) .
2
2φ
(5.52)
It should be note that the original Brans-Dicke theory does not have the field potential (V (φ) = 0).
The evolution equations for a spatially flat FRW universe is derived:
φ̇
3 H+
2φ
φ̈ + 3H φ̇ +
!2
(2ω + 3)
−
4
φ̇
φ
!2
−
V
= 0,
φ
2
(φV,φ − 2V ) = 0
(2ω + 3)
(5.53)
(5.54)
Considering the slow-roll conditions |φ̇| |Hφ| and |φ̈| |3H φ̇|, the equations reduce to
3H 2 φ − V ' 0,
2
3H φ̇ +
(φV,φ − 2V ) ' 0.
(2ω + 3)
(5.55)
(5.56)
From (5.55) and (5.56), one can get H and φ̇ in terms of the potential V (φ) in the slow-roll approximation.
The amount of inflation is measured by the number of e-folds of accelerated expansion
Z a
Z t
N=
dlna =
H(t)dt
ae
(5.57)
te
where ae is the scale factor at the end of inflation. In the slow regime we can use
H(t)dt =
H
(2ω + 3)
V
dφ =
dφ
2
φ(φV,φ − 2V )
φ̇
(5.58)
The CMB temperature anisotropies correspond to the perturbations whose wavelengths crossed
the Hubble radius around N ≈ 50 − 60 before the end of inflation [87]. Substituting H and φ̇ from
equations (5.55), (5.56) and (5.58) into (5.57), we obtain
N'
(2ω + 3)
2
Z
φ
φe
V
φ(φV,φ − 2V )
dφ,
(5.59)
5.4
Brans-Dicke Inflation
52
where φe is the scalar field at the end of inflation. To determine analytically the scalar field at the
end of inflation, we set 1 = 1 (see (5.60)), because the slow-roll conditions are violated at the end
of inflation.
To obtain the power spectrum of the curvature perturbation, we first introduce the slow-roll parameters in Brans-Dicke theory [44, 68]
1 = −
Ḣ
,
H2
2 =
φ̈
,
H φ̇
3 =
φ̇
,
2Hφ
4 =
Ė
.
2HE
(5.60)
where the parameter E = ω + 3/2 is constant in Brans-Dicke theory and therefore the fourth
slow-roll parameter vanishes, i.e. 4 = 0. We then obtain the scalar spectral index for this model
ns ' 1 − 41 − 22 + 23 .
(5.61)
We thus can calculate the running of the scalar spectral index for Brans-Dicke theory as
dns
' −821 + 222 − 423 − 21 2 + 41 3 ,
dlnk
(5.62)
we have used the relation k = aH which is valid at the horizon crossing. From Planck 2015 TT, TE,
EE+lowP data [4], the measurement of this parameter gives dns /dlnk = 0.0057 ± 0.0071 at 68%
confidence level. In the slow-roll approximation, the power spectrum of the curvature perturbation
evaluated at the horizon crossing takes the form [44]
2
1
H
Ps '
k=aH
Qs 2π
(5.63)
where
Qs =
φ̇2 (2ω + 3)
2
φ̇
2H 2 φ 1 + 2Hφ
(5.64)
From Planck 2015 TT,TE,EE+lowP data [4], the of the scalar perturbation amplitude has been
estimated as Ps = (2.139 ± 0.063) × 10−9 . The power spectrum of the tensor perturbations is given
by [44]
Pt '
2 H 2 .
π 2 F k=aH
(5.65)
An important inflationary observable is the tensor-to-scalar ratio which is defined as
r≡
Pt
.
Ps
(5.66)
We then obtain the tensor-to-scalar for Brans-Dicke theory in the slow-roll approximation
r ' 4(2ω + 3)
φ̇2
.
H 2 φ2
(5.67)
The recent constraint on this observable has been estimated r < 0.1 (95% CL, Planck 2015
TT,TE,EE+lowP data) [4]. If we consider the inflationary observables in terms of the potential in the slow-roll approximation, using the above equasions, the scalar spectral index is obtained
5.5
Accelerating universe
53
as
ns ' 1 +
2
[φ(6V V,φ + 2φV V,φφ − 3φVφ2 − 4V 2 ].
(2ω + 3)V 2
(5.68)
The tensor-to-scalar ratio can be derived as
r'
16(φV,φ − 2U )2
.
(2ω + 3)U 2
(5.69)
Another inflationary observable which can be used to discriminate between inflationary models,
is the non-Gaussianity parameter fN L . Different inflationary models predict maximal signal for
different shapes of non-Gaussianity. Therefore, the shape of non-Gaussianity is potentially a powerful probe of the mechanism that generate the primordial perturbations [18]. For the single field
inflationary models with non-canonical kinetic terms, the non-Gaussianity parameter has peak in
the equilateral shape. The issue of primordial non-Gaussianities in the Brans-Dicke theory was
widely analysed in the Ref. [45].
The equilateral non-Gaussianity parameter (related to non-standard kinetic terms) is in the BransDicke theory [7] equal to
5
5
equil
fN
L = − 2 + 3 .
4
6
(5.70)
The equilateral non-Gaussianity is of order of the slow-roll parameters which are very smaller
than unity in the slow-roll regime in the Brans-Dicke theory. On the other hand, the slow-roll
conditions can be perfectly satisfied in the Brans-Dicke gravity.Therefore, the equilateral nonGaussianity parameter in the Brans-Dicke gravity can be in agreement with the Planck 2015
equil
prediction, fN
L = −16 ± 70 (68% CL) [135].
It has been showed that in the Brans-Dicke theory and in the slow-roll approximation, the relations
for the scalar index and tensor-to-scalar ratio are identical in the Einstein and Jordan frames [135].
The result above can be applied for various inflationary potentials, which can be checked in light
of the Planck 2015 observational results [4]. The studied from [135] shows that in the Brans-Dicke
theory, the power-law, inverse power-law and exponential potentials are ruled out by the Planck
2015 data. The D-brane, SB SUSY and displaced quadratic potentials can be in well agreement
with the observational data since their results can lie inside the 68% CL region of Planck 2015
data. Moreover, for the hilltop, Higgs, Coleman-Weinberg and natural potentials, the predictions
of the model are in well agreement with the Planck 2015 data at 95% CL.
5.5
Accelerating universe
In 1998 it was discovered that the expansion of the universe at the present cosmological epoch is
accelerated, i.e. ä > 0. This conclusion is based on observations of distance type Ia supernovae [117,
5.5
Accelerating universe
54
123], which can be used as standard candles to reliably determine the cosmological parameters.
Indeed, low redshift supernovae can be used to obtain the Hubble constant, H0 , while supernovae at
greater distances allow probing the deceleration parameter, q0 . The observation revealed that these
supernovae are fainter than expected, which strongly suggest that we are living in an accelerating,
low-matter density universe. The consistency relationship between these cosmological parameters
and the luminosity distance, DL , of a supernovae is given, for moderate redshifts z, by
H0 DL = z + (1 − q0 )z 2 /2 + ...
(5.71)
where q ≡ −äa/(ȧ)2 . For an accelerating universe (q < 0), DL is larger than for a decelerating one
with q > 0. If the universe expanded more slowly in the past than today, it is actually older than
previously thought and apparent faintness of the supernovae is explained.
The observational evidence is now rather compelling for a spatially flat universe with total energy
density Ω = ρ/ρc , in units of critical density ρc =
X
3H02
8πGN
, given by
Ωi = Ωq + Ωm = 1
(5.72)
i
where Ωm ∼ 0.3 is the matter density, of small percent is baryonic, and most of the matter density
should come from dark matter, and Ωq ∼ 0.7 describes an unknown form of energy called dark
energy or quintessence. The Friedmann equation (2.25) allows an accelerated expansion if and
only if the dominant form of energy in the universe is distributed homogeneously and is described
by an effective equation of state satisfying P < −ρ/3.
An obvious candidate for dark energy is the cosmological constant Λ which has been discussed in
section 3.1. The cosmological constant explanation is currently rejected by most authors in favour
of an alternative calls for a scalar field endowed with a potential which can give rise to a dynamical
vacuum energy, the so-called “quintessence”. Various of quintessence models are proposed, most
of them invoking a scalar field with a very shallow potential, which until recently was overdamped
by the expansion of the universe, so that its energy density was smaller than the radiation energy
density at early times. The reader is referred to [53] for more examples of quintessence models,
including Brane-world models, a rolling tachyon field in string theories, a Chaplygin gas and so on.
5.5.1
Quintessence in Brans-Dicke theory
Let us consider simple Brans-Dicke theory.
One should be noted that a massless, non-self-
interacting Brans-Dicke scalar cannot be a form of quintessence. Recall that in section 5.2, I
discussed that the power-law solution is accelerating only if −3/2 < ω < −4/3, a range of BransDicke parameter that is in contradict with the observational limit. Similarly, a pole-like solution
5.5
Accelerating universe
55
is obtained with a non self-interacting Brans-Dicke scalar only in the range −4/3 < ω < −1.
For a massive scalar, potential function V = V (φ) = m2 φ2 /2 and for ω < 0, accelerating power-law
solutions exist [20] with
4/3
t
,
t0
−2
t
φ(t) = φ0
,
t0
a(t) = a0
(5.73)
(5.74)
with the deceleration parameter q0 = −1/4 and the time variation of the gravitational coupling is
Ġ/G = 3H0 /2, which is consistent with the observations. However, the age of the universe turns
out to be
s
t0 =
4|ω + 2| −1
H0 ≥ 471H0−1
3Ωm
(5.75)
if the limit |ω| > 50000 is used. This value is much older than is currently observed. In order to
obtain a meaningful age of universe it implies that the ω must be negative. Interestingly, negative
values for ω are found in the Brans-Dicke effective low-energy models arising from Kaluza-Klein
(see section 7.2) and string theory (see section 7.3). A perturbation analysis [20] shows that energy
density perturbation in a accelerated expanding universe can grow for negative values of ω.
5.5.2
Acceleration without quintessence
The above presented an accelerating model for the spatially flat universe in a modified Brans-Dicke
theory using a potential which is a function of the Brans-Dicke scalar field itself. The solution also
can be obtained in Brans-Dicke theory without the potential in the following.
Recall the scalar field equation given in (5.16), for the pressureless dust fluid (γ = 1), the scalar
field takes the form
φ=
(3ω + 4)ρ0 2/(3ω+4)
t
.
2a30 (2ω + 3)
(5.76)
where x = 2(ω + 1)/(3ω + 4). It is easily seen that one can generate different accelerating solutions
for different negative values of ω. For ω = −5/3, we get φ ∝ t−2 , which reproduced the result
obtained in quintessence case. In this case, the deceleration parameter q0 = −1/4. For ω = −8/5,
the scalar field φ ∝ t−5/2 , giving q0 = −1/3, which means the acceleration rate is higher than in
the quintessence case. This model yields a set of accelerating solutions fo different values of ω in
the range −2 ≤ ω ≤ −3/2. In fact, a sufficiently negative ω may effectively lead to a negative
pressure and thus drive a positive acceleration or uniform expansion without the violation of the
energy condition by normal matter.
5.5
Accelerating universe
56
It is important to note that there are problems in bridging this result with the radiation-dominated
decelerated universe for the same values of the BransDicke parameter, in this case the value of
ω would be in the range −3/2 ≤ ω ≤ 0. The indicated range of values of ω, which drives an
accelerated expansion for a matter-dominated model, does not produce a consistent radiation
model which explains the primordial nucleosynthesis. The solution to this problem lies in the
natural generalisation of the theory by allowing ω to be a function of the Brans-Dicke scalar field
φ rather than a constant. With the equation of state for radiation (P = 13 ρ), equation (6.9) gives
a first integral
φ̇(2ω + 3)1/2 =
A
,
a3
(5.77)
where A is an integration constant. A simple choice like [12]
(2ω + 3) = (φ − 1)2
(5.78)
when used in equation (5.77) along with a ∝ t1/2 yields
B
(φ − 1)2 = √
t
(5.79)
B is a constant of integration. It clearly shows that ω decreases with time towards a constant
value -3/2, which indeed produces an accelerated expansion for a late dust-dominated universe as
we have seen. In this case, the primordial nucleosynthesis can be successfully explained.
57
6
General Scalar-Tensor Theories
Brans-Dicke and scalar-tensor theories have repeatedly generated interest for several reasons. First,
a gravitational scalar field appears in scalar-tensor theories together with the metric tensor, and a
fundamental scalar coupled to gravity is an unavoidable feature of superstring, supergravity, and
M-theories. Scalar fields are ubiquitous in particle physics and in cosmology, such as the Higgs
boson of the Standard Model, the superpartner of spin 1/2 particles in supergravity, and the string
dilaton appearing in higher-dimensional graviton. Second, as we have discussed, the duality symmetry of string theory (5.12) has an analogue in the symmetry (5.11) of Brans-Dickes theory. The
potent motivation for the study of scalar tensor theories comes from the reality that the low energy
limit of the bosonic string theory corresponds to a Brans-Dicke theory with coupling parameter
ω = −1 (see section 7.3), also ω = −3 is obtained from a less conventional string theory [32, 95].
A further interest in Brans-Dicke and scalar-tensor gravity emanates from the extended and hyperextended inflationary scenarios of the early universe discussed in section 6.2. In addition,
Brans-Dicke theory is recovered in the 4-dimensional brane moving in a higher-dimensional space
in the Randall-Sundrum brane-world [120, 121]. Let us now consider more general scalar-tensor
theories.
A general form of the scalar-tensor theory can be derived from the action in the Jordan frame
Z
ω(φ)
1
4 √
µ
d x −g φR −
∇µ φ∇ φ − 2Λ(φ) + Smatter (gµν , Ψ)
(6.1)
SST =
16π
φ
where ω(φ) is coupling parameter, Λ is φ dependent generalisation of the cosmological constant,
R
√
and Smatter = d4 x −gLmatter does not explicitly depend on the scalar field φ. This theory reduces to the Brans-Dicke theory in the limit ω → constant and Λ → 0, and approaches to general
relativity in the limit ω → ∞, ω 0 /ω 2 → 0 and Λ → constant. The distinguishing feature of the
action (6.1) is that the Brans-Dicke parameter ω(φ) is now a function of the Brans-Dicke like scalar
and therefore varies with the spacetime point. In the spatially homogeneous and isotropic universe
relevant for cosmology, ω only varies with the cosmic time. Also, the scalar-tensor theories with
multiple scalar-fields are systematically investigated [40, 119].
The variation of the action derived from integrating over all space, gives the field equations
ω(φ)
1
Gµν = 8πφ−1 Tµν + 2 ∇µ φ∇ν φ − (∇φ)2 + φ−1 (∇µ ∇ν φ − gµν φ) − φ−1 Λgµν(6.2)
φ
2
1 φ =
8πT − ω 0 (∇φ)2 − 4Λ − 2φΛ0
(6.3)
2ω + 3
where primes denote differentiation with respect to φ. These are the field equations of the scalartensor theories of gravity. The field equations can be expressed in Einstein frame by conformal
58
transformation
gµν = e2Γ(x) ḡµν ,
(6.4)
where Γ(x) is an arbitrary function of the spacetime coordinates xµ . Under this transformation
we can derive the Ricci tensor and Ricci scalar, and obtain the Einstein frame field equations
1¯
α
¯
¯
¯
∇α ψ ∇ ψ + V ḡµν
Ḡµν = 8π T̄µν + ∇µ ψ ∇ν ψ −
(6.5)
2
√
4π
¯ − dV = −
T̄
(6.6)
ψ
dψ
(3 + 2ω)2
where we define 8πV (ψ) ≡ e4Γ Λ and the energy-momentum tensor T̄µν with respect to ḡµν so that
T̄ µν = e6Γ T µν . It can now be explicitly seen that while the Einstein frame energy-momentum
¯ µ T̄ µν =
tensor is not covariantly conserved, ∇
√
4π
¯ν
(3+2ω)2 T̄ ∇ ψ.
Let us now consider the case without Λ(φ), i.e. Λ(φ) = 0. This is often used to model the
possibility of having a coupling parameter ω in the early universe that is small enough to have
some interesting effects, while being large enough in the late universe to be compatible with the
stringent bounds imposed upon such couplings by observations of gravitational phenomena in the
solar system, and other nearby astrophysical systems. This interest is bolstered by the presence of
an attractor mechanism that ensures general relativity is recovered as a stable asymptote at late
times in FLRW cosmology [41].
Following the procedure from section 4.5, we can read off the PPN parameters:
γ=
1+ω
2+ω
and
β =1+
ω0
(4 + 2ω)(3 + 2ω)2
(6.7)
with all other parameters equalling zero. The value of γ is the same as in the Brans-Dicke theory,
while the value of β reduces to the Brans-Dicke theory when ω is constant. The variation of ω
with φ can be constraint by observations of post-Newtonian phenomena, early universe, or near
black hole and so on.
Assuming a perfect fluid matter content and taking the FRW line-element (2.23), the field equations
in scalar-tensor theories of gravity reduce to
!2
ω φ̇
κ
+
− 2,
6 φ
a
8π
ω̇
φ̈ =
(ρ − 3P ) − 3H +
φ̇.
2ω + 3
2ω + 3
8π
H =
ρ−H
3φ
2
φ̇
φ
!
(6.8)
(6.9)
One can obtain
ω
−8π
Ḣ =
[(ω + 2)ρ + ωP ] −
(2ω + 3)φ
2
φ̇
φ
!
+
φ̇
(φ̇)2
dω
κ
+
2H
+
2
a
φ 2(2ω + 3)φ dφ
(6.10)
59
These equations differ from the equations of Brans-Dicke cosmology by the term in ω̇ = φ̇dω/dφ,
and by the fact that now ω is a function of φ. The exact solutions with κ = 0 have been studied
in many literature [14, 15, 107]. We will not reproduce these solutions here. Now we return to
attractor mechanism, which is most easily seen in the Einstein conformal frame [41, 39]. For a
spatially flat FRW geometry the evolution equation for the scalar field can be written as
√
8π
ψ 00 + 4π(2 − γ)ψ 0 + 4π(4 − 3γ)α = 0.
02
3 − 4πψ
(6.11)
where primes denote differentiation with respect to lnā, and γ is the equation of state P = (γ −1)ρ.
The α = 1/(3 + 2ω)2 denotes the strength of coupling between the scalar and tensor degrees of
freedom. This equation is clear the equation for a simple harmonic oscillator with a driving force
√ R
given by gradient potential (1−3ω)Γ, where Γ = 4π αdψ. The interpretation of Γ as an effective
potential is often expanded as
Γ = α0 (ψ − ψ0 ) +
β0
(ψ − ψ0 )2 + O((ψ − ψ)3 ),
2
(6.12)
where ψ0 is an assumed local minimum of Γ(ψ), and α0 and β0 are constants. In terms of this
parameterisation the PPN parameters become
β =1+
α02 β0
,
2(1 + α02 )2
γ =1−
2α02
.
1 + α02
(6.13)
Let us now turn to perturbations around a general FRW background in scalar-tensor theories.
We will work on conformal Newtonian gauge, see section 2.5. The scalar part of the perturbed
line-element takes the form
ds2 = a2 [−(1 + 2ψ)dτ 2 + (1 − 2Φ)qij dxi dxj ],
(6.14)
where we have used conformal time, τ , and qij is the metric of a static 3-space with constant
curvature. The perturbed scalar field equation is given as
a0
8πa2 (δρ − 3δP )
a0
δφ00 + 2 δφ0 + k 2 δφ − 2φ00 Ψ − φ0 Ψ0 + 4 Ψ + 3Φ0 −
a
a
(2ω + 3)
2
2
0
d ω φ δφ
2 dω
a0 0
a
2 dω 0 0
02
00
(φ
δφ
−
φ
Ψ)
+
φ
+
2
φ
δφ ,
=−
+
(2ω + 3) dφ2 a2
a2 dφ
a2 dφ
a
(6.15)
as well as the condition
Φ−Ψ=
8πa2
δφ
(ρ + P )Σ +
φ
φ
(6.16)
where primes denote differentiation with respect to the conformal time, τ , and k is the wavenumber of the perturbation.
By analysing the angular power spectrum data of the CMB obtained from the Planck 2015 results,
the parameters α0 and β0 of the attractor model are constrained [111]. The authors find that
2
the present-day deviation from the Einstein gravity (α02 ) is constrained as α02 < 2.5 × 10−4−4.5β0
6.1
Constraint from BBN
60
2
(95.45% C.L) and α02 < 6.3 × 10−4−4.5β0 (99.99% C.L) for 0 < β0 < 0.4. Note that as β0 → 0
Brans-Dicke theory is recovered, and as α0 → 0 general relativity is recovered. The coupling
parameter ω is constrained as ω > 2000 (95.25%) and ω > 790 (99.99%). Also we find that the
variation of the gravitational constant in the recombination epoch, G(φrec )/G0 , in the nonflat
universe is constrained as G(φrec )/G0 < 1.0062 (95.45%) and G(φrec )/G0 < 1.0125 (99.99%).
6.1
Constraint from BBN
Big bang nucleosynthesis has also been explored in the context of general scalar theories. It is
found that, for compatibility with BBN, the present value of the scalar coupling, i.e. the present
level of deviation from Einstein’s theory, must must satisfy the constraint [42]
α02
.
10−6.5 β0−1
Ωm h2
0.15
−1.5
,
(6.17)
when β0 & 0.5. These bounds are weakened for β0 . 0.5.
The deviation from general relativity is quantified by the expansion rate in the scalar-tensor HST ,
measured in units of the expansion rate H that one would have in general relativity with the same
form and amount of matter:
ξn ≡
p
HST
= HST 8πGN 3ρ
HGR
(6.18)
This ratio is called speed-up factor and assumes the value unity in general relativity. A calculation
yields
ξn =
1
r
Ω(φn ) 1 +
where Ω(φ) =
√
h
1
dΩ
GN Ω dφ
i2 ,
(6.19)
GN φ is the scale factor of the conformal transformation mapping the scalar-
tensor theory into the Einstein frame, and φn and φ0 are the values of the scalar field at the
time of nucleosynthesis now, respectively. If ξn deviates from unity there is underproduction and
overproduction of 4 He during nucleosynthesis. The value of ξn is estimated within the range
0.8 ≤ ξn ≤ 1.2. The primordial 4 He mass fraction, X(4 He), is given by [34, 110]:
τν − 889.8s
4
10 nb
X( He) = 0.228 + 0.010ln 10
+ 0.012(Nν − 3) + 0.185
+ 0.327lnξn (6.20)
nγ
889.8s
where nb and nγ are the number densities of baryons and photons, respectively, and Nν is the
number of light (m . 1M eV ) neutrino species, τν is the neutrino lifetime. Based on the available
experimental limits, a reasonable choice (Nν = 3, τν < 885s yields X(4 He) ≤ 0.250, which implies
lnξn ≤ 0.0797 [127]. The limits set on ξn can be used to constrain on the coupling function ω(φ).
Note that for which detailed BBN constraints are available, one can calculate the dark matter relic
abundances in the scalar-tensor gravity using a generic form A(φ) = exp(βφ2 /2) for the coupling
6.2
Extended and Hyperextended Inflationary
61
between the scalar field φ and the metric [100, 36].
6.2
Extended and Hyperextended Inflationary
The extended inflationary scenario [80], as coined by La and Steinhardt for the case of BransDicke theory, brings a new lease on life to one of the peculiar features of Guth’s old inflationary
theory [61], namely the idea that the universe undergoes a spontaneous first order phase transition
from a metastable vacuum. The motivation behind this is the possibility of producing a successful
inflationary phase transition from a false vacuum state, thus avoiding the fine tuning problems
associated with ‘new inflation’. Inflation is supposed to end due to tunnelling of ψ from the false
to the true vacuum, with spontaneous nucleation of bubbles of true vacuum. The nucleation rate
of true vacuum bubbles per Hubble time and per Hubble volume can be defined as [53]
ηV =
Γ
,
H4
(6.21)
where Γ is the nucleation rate per unit time and H −1 is the Hubble radius. In order to complete
the phase transition from false to true vacuum ending old inflation, there must be enough bubbles
nucleating per Hubble time and volume, i.e., it requires ηV ∼ 1. However, to solve the problems of
the standard big bang model, there need to be enough amount of inflation, which means we need
to constraint ηV 1, otherwise the inflation will be stopped too early.
To solve the incompatibility of these constraints, the old inflationary scenario was abandoned and
can be replaced by Brans-Dicke theory. In this scenario, the Hubble parameter is not constant and
the nucleation rate of true vacuum bubbles is not constant either,
ηV =
Γ
∝ t4 .
H4
(6.22)
This is exactly what is needed to cure the problem of old inflationary scenario: ηV is small at
early times during which only a few bubbles of true vacuum nucleate and inflation proceeds for
a sufficient number of e-folds. The nucleation of bubbles of true vacuum becomes more efficient
later on, the cosmic expansion slows down, the phase transition is completed, and the false vacuum
energy disappears as tunnelling proceeds.
Unfortunately, it was soon found that the extended inflation has a serious flaw. Although the
nucleation rate is small, it is not exactly zero at early times and a few true vacuum bubbles do
nucleate in this early epoch. The bubble collisions at the end of inflation will leave a significant
imprint in the CMB, which should be detected today. Since this is not observed, a constraint is
placed on extended inflation. A calculation of the bubble spectrum constrain [88] the limit on the
6.2
Extended and Hyperextended Inflationary
62
Brans-Dicke parameter ω ≤ 20, which is obviously conflicts with the constraint imposed by (4.38).
The problem of extended inflation can be improved by considering more general scalar-tensor
theories of gravity instead of Brans-Dicke theory, which is called hyperextended inflation. In this
way, the coupling function ω(φ) can vary to large values after the end of inflation. In the version of
hyperextended inflation proposed by Steinhardt and Accetta [133], the value of ω is initially large,
producing an acceptable spectrum of density perturbations. Later on, ω assumes low values and
only a few true vacuum bubbles are nucleated at this time, leaving no significant imprint in the
CMB. Inflation stops due to the dynamics of the scalar, not due to a first order phase transition
and nucleation of bubbles.
63
7
7.1
Equivalence of theories
Equivalence with f(R) theories
In general relativity, given the distribution of mass and energy, spacetime bends to minimise its
curvature, denoted by Ricci scalar R. An alternative way proposed that the spacetime can contort
to minimise the curvature plus some extra function of the curvature, which are known as the
f (R) theories. That changes produces an extra gravity-like force that can either attract or repel
under different conditions. The f (R) theories generalise the Einstein-Hilbert action by making the
action a more general function of the Ricci scalar then the simple linear one that leads to Einstein’s
equations, which describe by the following action:
Z
√
1
S=
d4 x −gf (R) + Smatter (gµν , ψ)
16π
(7.1)
Variation of f (R) action gives
1
δS =
16π
Z
√
d4 x −gδg µν (fR Rµν + gµν fR − ∇µ ∇ν fR ),
(7.2)
where fR = df /dR. From this we can derive the field equations:
1
1
fR Rµν − f (R) + gµν fR − ∇µ ∇ν =
Tµν
2
8π
(7.3)
This reduce to the GR field equations if we take f (R) = R.
One can introduce a new field χ and write the dynamically equivalent action
Z
√
1
d4 x −g[f (χ) + f 0 (χ)(R − χ)] + Smatter (gµν , ψ).
S=
16π
(7.4)
where primes denote differentiation with respect to the scalar χ. Variation of this action gives the
field equation for χ as:
f 00 (χ)(R − χ) = 0
(7.5)
Therefore, χ = R if f 00 (χ) 6= 0. Substituting R = χ into eq. (7.1) recovers the standard f(R) action. Therefore eq.(7.1) is an equivalent action for f (R) gravity, with the special case of f 00 (χ) = 0
corresponding to the Einstein-Hilbert action.
To recast the equivalent action as a scalar-tensor theory we make the definitions:
φ=
df (R)
= fR ,
dR
V (φ) = R(φ)φ − f (R(φ))
(7.6)
Then eq. (7.1) becomes:
S=
1
16π
Z
√
d4 x −g[φR − V (φ)] + Smatter (ψ, gµν )
(7.7)
7.1
Equivalence with f(R) theories
64
This is the Jordan frame representation of the action of a Brans-Dicke theory with Brans-Dicke
parameter ω = 0 in this case. It should be stressed that the scalar degree of freedom φ = fR is
quite different from a matter field, like all nonminimally coupled scalars, it can violate all of the
energy conditions [53]. Therefore, as has been observed long ago, f (R) theories are dynamically
equivalent to Brans-Dicke theory.
Noted that Palatini f (R) gravity can also be cast in the form of a Brans-Dicke theory with BransDicke parameter ω = −3/2. The classification of f (R) theories of gravity and equivalent BransDicke theories is summarised in Fig. 3.
Figure 3: Schematic diagram relating various versions of f(R) gravity and Brans-Dicke theory.
Now, let us investigate a theory of gravity whose action is composed of the Ricci scalar and its
derivatives [106]:
Z
S=
√
d4 x −gf (R, (∇R)2 , R),
(7.8)
e
where (∇R)2 = g µν ∇µ R∇ν R. We now introduce a set of Lagrange multipliers (f 0 (e
χ), f 0 (e
σ ), f 0 (ψ))
and associated auxiliary fields (χ, Σ, Ψ) to reduce the order of derivatives:
Z
√
e
χ)(χ − R) + f 0 (e
σ )(Σ + (∇R)2 ) − f 0 (ψ)(Ψ
− R)],
S = d4 x −g[f (χ, Σ, Ψ) + f 0 (e
(7.9)
Equivalently, one can rewrite this action as
Z
√
S = d4 x −g[f (χ, Σ, Ψ) + f 0 (χ)(χ − R) + f 0 (σ)(Σ + (∇R)2 ) − f 0 (ψ)(Ψ − R)],
(7.10)
e f 0 (σ) = f 0 (e
e
where f 0 (e
χ) = f 0 (e
χ) + ∇µ [f 0 (e
σ )∇µ (χ + R)] + f 0 (ψ)],
σ ), f 0 (ψ) = f 0 (ψ).
This
e to (f 0 (χ), f 0 (σ), f 0 (ψ))
replacement is justified since the transformation from (f 0 (e
χ), f 0 (e
σ ), f 0 (ψ))
e
is regular and invertible. Since the action does not include any derivative terms of f 0 (e
σ ) and f 0 (ψ),
which can be plugged back into the action:
Z
√ S = d4 x −g f χ, (∇χ)2 , χ + f 0 (e
χ)(χ − R) .
(7.11)
7.2
Relation to Kaluza-Klein theory
65
This verifies that we can indeed replace all the derivatives of R with those of χ under the replacement of R by χ and the introduction of the Lagrange multiplier.
The f (R) gravity and dark energy energy models in general relativity both are able to explain
the observed late-time accelerating expansion of the universe, they lead to different formation and
evolution of cosmic structure. However, a recent study [93] report the observational constraints
on the f (R) theory derived from weak lensing peak abundances, which are closely related to the
mass function of massive halos, agree with the predictions of dark energy and weaken the case for
f (R) theories.
7.2
Relation to Kaluza-Klein theory
Shortly after the general relativity was published, Kaluza and Klein [70, 78] attempted to unify
gravity and electrodynamics, known as KK theory. The basic idea was to consider general relativity
on a 4 + 1 dimensional manifold where one of the spatial dimensions was assume to be “compactified” to a small circle leaving four-dimensional spacetime extended infinity as we see. The KK
theory played a decisive role in making clear the importance of high-dimensional spacetime in 10
dimensional string theory and 11 dimensional supergravity/M-theory.
One of interest to study KK theory here is due to the fact that Brans-Dicke theory can be derived
from KK theory with d extra spatial dimensions, obtaining the coupling parameter [53]
ω=−
(d − 1)
d
(7.12)
In this derivation, the Brans-Dicke scalar field has a geometrical origin in the determinant of
the metric defined on the submanifold of the extra dimensions. The simplest version of classical
Kaluza-Klein theory [113] with a single dilaton field, one begins with spacetime (M ⊗ ∂M, γAB ),
where M is a 4-dimensional manifold with one timelike dimension and ∂M is a submanifold with
d spatial dimensions (d ≥ 1). Let us assume the “Ansatz” for the D = (4 + d)-dimensional metric
with d-dimensional compactified space:
γAB

gµν
=
0
0
φαβ


(7.13)
where A, B, .... = 0, 1, ..., (3 + d), µ, ν = 0, 1, 2, 3, and α, β, ... = 4, 5, ..., (3 + d). Notice that we
omitted the off-diagonal components for the gauge fields, focusing on the scalar field. We first
define general relativity in D dimensions, described by the action
Z
√
1
S[γ] =
dD X −γR
16πGD
(7.14)
7.3
The dilaton from string theory
66
where GD is D-dimensional gravitational coupling, γ = det(γAB ), γAB is the D dimensional metric
with corresponding Ricci tensor, RAB , and Ricci scalar, R = γ AB RAB . Note that we are neglecting
the matter Lagrangian for brevity. Now, introducing the determinant of the metric of the extra
dimensions [53],
ϕ ≡ |det(φαβ )|
(7.15)
ραβ ≡ ϕ−1/d φαβ ,
(7.16)
and the symmetric tensor
it is, by definition, |det(ραβ )| = 1. It is assumed that the extra spatial dimensions are compactified
on circles of size l. Then, the integral over the (4 + d) dimensions in the action (7.14) is split into
R
the product of an integral over the four spacetime dimensions d4 x and of an integral over the
R
remaining d dimensions dd x. Thus the action can be reduced to
S=
V (l)
16πGD
Z
√ √
(d − 1) (∇ϕ)2
d4 x −g ϕ (R + R∂M ) +
,
4d
ϕ2
(7.17)
where V (l) is the volume of the compact manifold ∂M of the extra dimensions and R∂M is the
√
Ricci curvature of ∂M . In terms of the scalar field φ ≡ ϕ and setting GN = GD /V (l) , the action
is
S=
1
16πGN
Z
√
(d − 1) (∇φ)2
d4 x −g φ(R + R∂M ) +
.
d
φ
(7.18)
This action describes a Brans-Dicke theory with parameter ω given by equation (7.12) in which the
scalar φ does not carry dimensions. From a practical point of view, the derivation of Brans-Dicke
theory from KK gravity provides a technique for generating exact solutions in one theory starting
from familiar solutions in the other theory.
7.3
The dilaton from string theory
One of the reasons for renewed interest in scalar-tensor theories is the relationship that exists
between the gravitational part of Lagrangian for Brans-Dicke theory and the low-energy effective
action for bosonic string theory, which has been extensive studies for the past few decades. String
theory is one of the most promising candidates for a theory of quantum gravity. It postulates
that all the elementary particles known to physics are different aspects of one and the same type
of fundamental one-dimensional object, called a string. It was shown that a closed string has a
zero-mode described by a symmetric second-rank tensor that behaves in the low-energy limit like
the space-time metric. the field equations of these zero-mode fields were derived explicitly as [59]:
1
Rµν = −2∇µ ∇ν χ + Hµρσ Hνρσ ,
4
∇λ H λµν − 2(∂λ χ)H λµν = 0,
1
R = 4 χ + (∂χ)2 + (H 2 ),
12
(7.19)
(7.20)
(7.21)
7.3
The dilaton from string theory
67
where H 2 ≡ Hµνρ H µνρ , and Hµνρ is the totally antisymmetric 3-form field. The dimension will be
either 10 or 26, depending on whether supersymmetry is included or not, respectively. The totally
antisymmetric field strength is defined by
Hµνλ = ∂µ Bνλ + cyclic permutation,
(7.22)
where Bµν is an antisymmetric second-rank tensor field. The equations (7.19) - (7.21) can be
shown to be derived from the Lagrangian:
Lst
1 √
1 2
µν
=
−gexp(−2χ) R + 4g ∂µ χ∂ν χ − H
16π
12
(7.23)
If we identify φ = exp(−2χ), and one can easily show that dφ = −2exp(−2χ)dχ. We can re-express
(7.23) as
Lst =
1 √
1
1
−g(φR − g µν ∂µ φ∂ν φ − H 2 )
16π
φ
12
(7.24)
The Lst is identical to the Brans-Dicke Lagrangian (4.2) when ω = −1 [16].
String theory provides a way to avoid the divergences that have plagued traditional field theory.
This is mainly because a string is an extended object, which is contrary to the concept of point
particles upon which field theory is based. From a more technical point of view, however, finiteness
is due to an invariance under conformal transformation in two-dimensional space-time in which
propagating strings reside. The two-dimensional conformal invariance has its descendant in Ddimensional field theory, namely dilatation invariance, which is implemented with the help of the
scalar field in precisely the same way as in the Brans-Dicke theory. The relevant transformations
are
ḡµν = Ω2 gµν ,
(7.25)
φ̄ = Ω1−D/2 φ,
(7.26)
B̄µν = Ω2 Bµν .
(7.27)
where Ω is constant. We find
√
√
−gR = Ω2−D −ḡR,
(7.28)
which implies that the Einstein-Hilbert term is not invariant unless D = 2. Also, comparing
(7.26) and (7.27) reveals that the present transformation is different from the scale transformation
involving the mass dimension of fields. The field φ and χ is naturally called a dilaton.
68
8
Discussion
It is remarkable that general relativity, born a century ago out of almost pure thought, has passed
every experimental tests. However, all attempts to quantize gravity and to unify it with the other
forces suggest that the general relativity may not be the last word. There are ongoing experiments
search for new physics beyond Einstein at different scales: the short distance scales or high energy,
and the large distance scales of the astrophysical, galactic, and cosmological realms.
In the meantime, many alternative to the Einstein’s general relativity has been proposed, many of
which have been driven by the discovery of the dark side of the Universe in last few decades. From
a theoretical perspective, model building is an important part of understanding and explaining existing data, as well as making new predictions. However, tight constraints imposed by local gravity
test mean that theorists are presented with a choice: They can either study minimal deviations
away from general relativity, or must otherwise look for mechanisms that hide modifications to
gravity on the scales probed by experiment. The latter provides an opportunity to solve some of
the cosmological puzzles that have arisen with the discovery of dark matter and dark energy.
Modified gravity necessarily involves additional fields, extra dimensions, or broken symmetries,
since we know that GR is the unique diffeomorphism invariant theory of a single rank-2 tensor
that can be constructed from the metric variation of an action in four dimensions. If we wish to
account for dark energy, or solve the cosmological constant problem using modified gravity, these
deviations must be manifest in the solutions of the Friedmann equations. We must also require,
however, that they can replicate the successful predictions of the standard cosmology, such as the
abundance of light elements, the peak positions of the CMB acoustic spectrum, or baryon acoustic
oscillations.
In this dissertation, I focus on discussing one of most well-known modified gravity: Brans-Dicke
and scalar-tensor gravity, which has appeared in the sixties as an important theory of classical
gravity to general relativity. It is known that Mach’s principle and Dirac’s suggestion for varying
gravitational constant GN were behind the advent of the Brans-Dicke theory. Mach’s principle
implying that the inertial mass of an object depends on the matter distribution in the universe so
that the gravitational constant should have time-dependence and is usually described by a scalar
field.
Ever since the Brans-Dicke first appeared, it has remained as a viable theory of classical gravity
in that it has managed to survive all the available observational/experimental tests. Sections 4
provides an overview of Brans-Dicke theory. I develop the weak-field approximation which can
69
be used to predict the planetary orbits and light bending in the solar system. The parameterized
post-Newtonian approximation, which has provided the most accurate determination of the parameter ω, is also explained. The issue of conformal transformation is discussed. By applying a
conformal transformation, we can transform nonminimal coupling term into another form, due to
the fact that general relativity is not invariant under this transformation. The question whether
Jordan frame or Einstein frame is physical remains unanswered.
In 4.7, we explore the structure and thermodynamics of black hole in Brans-Dicke theory. Brans
himself have provided static, spherically-symmetric metric and scalar field solutions to the vacuum Brans-Dicke field equations. The Brans-Dicke theory admits black holes different from the
Schwarzschild one when the WEC is not respected, otherwise only naked singularities appear. As
it it well known now that in many physical situation the WEC could be violated, the existence of
Brans-Dicke black holes in nature is an interesting possibility.
However, not all classically allowed black hole solutions may be physically realistic. They should
be further censored by quantum nature of black holes, namely their Hawking evaporation mechanism. It turned out that the non-trivial solutions that violate the WEC on the Brans-Dicke scalar
field and thus have been discounted in Hawkings theorem fail to survive the quantum censorship
and hence would not really arise in nature. After all, stationary black holes in Brans-Dicke theory
appear to be identical to those in general relativity once they settle down.
A striking feature of Brans-Dicke black holes is that they can have zero surface gravity and an
infinite area of the event horizon. Contrary to the usual black holes, these cold black holes imply
infinite tidal forces at the horizon. However, this is only true for massless fields.
In section 5, I investigated the cosmological solutions and inflation of Brans-Dickes theory. The
equations governing the background cosmology was shown in the Brans-Dicke gravity. Then, the
background equations was simplified by considering the slow-roll approximation. The relations of
the inflationary observables for the Brans-Dicke gravity was also obtained in the slow-roll approximation. A quintessence scalar field in Brans-Dicke theory is shown to give rise to an accelerated
expansion for the present universe. Moreover, Brans-Dicke theory may also explain the present
accelerated expansion of the universe without resorting to a cosmological constant or quintessence
matter. While the inflation in the early universe and the accelerated expansion at present might
be explained by the same scalar field, no convincing cosmological model has been found with this
as a natural feature.
70
Brans-Dicke theory proved to be extremely useful in solving some of the outstanding problems in
the inflationary universe scenario with the possibility of an ‘extended inflation’. Unfortunately, it
was soon found that bubble collisions at the ended of inflation produce unacceptable fluctuations in
the CMB. A solution for this problem was called ‘hyper-extended inflation’ by considering general
scalar-tensor theories instead of Brans-Dicke theory, which was discussed in 6.2.
It should not be overlooked, however, that Brans-Dicke parameter ω has to pick up a low negative
value in order to solve the acceleration and coincidence problems by its own right without having
to invoke dissipative processes or exotic fields. This squarely conflicts with the lower limit imposed
on ω by the local tests represented, for example, by the PPN analysis and this implies that the
Brans-Dicke theory is very close to the general relativity.
The problems with local tests appeared which could be cope with through some mechanisms. In
some models [24], the scalar field has a chameleon behaviour in the sense that it acquires a densitydependent effective mass, which was discussed 3.6. It has a sufficiently heavy mass in Solar System
(large-density environment) and then the local gravity constraints suppressed. Meanwhile, it can
take a small effective mass in cosmological (low-density environment) scale that can be considered
as a candidate for dark energy. However, the chameleon mechanism has its own problem. It is
therefore important for theorists to identify new ways to screen the deviations from general relativity at short distances, and this is expected to be an important avenue of future research.
Another mechanism that may turn the Brans-Dicke theory competitive even for small values of
ω [37, 51]. This mechanism is based on the concept of external field, which is a field not subject
to the variational principle. It has been showed that it is possible to join a part of a general
solution to a part of a scalar-tensor one such that the complete solution neither belongs to general
relativity nor to scalar-tensor theory, but fully satisfies the external scalar tensor field equations.
The external fields may effectively work as a type of screening mechanism for scalar-tensor theories.
In the last section 7, I explored the connections of the Brans-Dicke theory to f(R) theories, KaluzaKlein theory, and string theories. These relations have kept the Brans-Dicke theory a strongly
investigated proposal. String theory incorporates gravity, but it is necessarily accompanied dilaton, possibly a re-incarnation of the scalar field of scalar tensor theory in an entirely different and
unexpected context.
In this dissertation, I stayed mainly within the confines of the classical theory. It is also important
to incorporate the quantum corrections due to the interaction among matter fields. The reader is
71
referred to [56, 116, 51] for discussion on quantum effects and quantum solutions in Brans-Dicke
cosmology.
Furthermore, in 1979, Anthony Zee and Lee Smolin suggested that the Brans-Dicke scalar field and
Higgs field might be one and the same field. They glued the two key pieces of scalar field together
by combining the Brans-Dicke gravitational equations with a Goldstone-Higgs symmetry-breaking
potential. In this model, the local strength of gravity initially varied over space and time, with
Gef f ∝ φ−2 , but its present-day constant value emerged after the φ field settled into a minimum
of its symmetry-breaking potential, which presumably occurred in the first moments of the big
bang. In this way, they offered an explanation of why the gravitational force is so weak compared
with other forces: when the field settled into its final state, φ = ±V , it anchored φ to some large,
nonzero value, pushing Gef f to a small value.
A recent study [5] investigates one possibility follows from identifying the Higgs field with the
Brans-Dicke field or any variation thereof in a more general scalar-tensor gravity theory, so that
the Higgs vacuum expectation value determines the strength of the gravitational interaction, as
fixed by Gef f . By choosing a suitable coupling function between the field and the Ricci scalar, it
can ensure that Newton’s ”constant” in the unbroken phase, when φ = 0, but retain Gef f 6= 0 in
the broken phase. This effectively decouples matter and gravity in the unbroken phase, so that
there are still free gravitons at high energies, but there is no way to produce them other than by
vacuum quantum fluctuations. In this scenario, the symmetry restoration could “turn off” gravity
at high energy might have great implications for cosmology and quantum gravity.
The Brans-Dicke and scalar-tensor gravity, already exits more the half century old, seems to contain
yet many unexplored features that could lead to a solid alternative to the general relativity.
References
72
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