Tests of strong gravity:
what will astrophysics ever do for us?
Talk largely based on arXiv:1501.07274
Special thanks to:
Enrico Barausse, Vitor Cardoso, Leonardo Gualtieri, Paolo Pani,
Ulrich Sperhake, Leo Stein, Norbert Wex, Kent Yagi
Emanuele Berti, University of Mississippi/Caltech
Leiden, Feb 4 2015
1)
2)
3)
4)
5)
Weak- vs strong-field tests of GR
Strong-field tests: against what?
Black holes in modified gravity
Compact stars in modified gravity
Dynamics - smoking guns?
Not in this talk: pulsars (Wex), GWs (van den Broeck)
Weak-field tests
vs
strong-field tests
various materials toward local topography on Earth,
movable
laboratory
masses,
produce
a narrow
resonance
line whose shift could be accurately determined. Other experiments
13
galaxy [249, 19], and have reached levels of 3since
⇥ 10 1960
[2].measured
The resulting
upperof spectral lines in the Sun’s gravitational field and the change in
the shift
ummarized in Figure 1 (TEGP 14.1 [281]; for a rate
bibliography
of clocks
experiments
up to aloft on aircraft, rockets and satellites. Figure 3 summarizes the
of atomic
transported
The founda+ons of general rela+vity important redshift experiments that have been performed since 1960 (TEGP 2.4 (c) [281]).
TESTS OF
40
LOCAL POSITION INVARIANCE
TESTS OF THE
WEAK EQUIVALENCE PRINCIPLE
10-8
10-1
Eötvös
Renner
Free-fall
10-9
η
Null
Redshift
10-2
Fifth-force
searches
10-10
Pound
Snider
Boulder
Saturn
Princeton
10-11
Eöt-Wash
Moscow
10-12
α
10-3
Hmaser
Eöt-Wash
LLR
THE PARAMETER (1+γ)/2
10-4
Cli↵ord M. Will
10-13
Null
Redshift
Solar spectra
η=
10-14
[Will, 1403.7377]
Millisecond Pulsar
PoundRebka
a1 -a 2
10-5
(a1+a2)/2
1.10
Radio
Clocks in rockets
spacecraft & planes
DEFLECTION
OF LIGHT
Optical
00
20
90
19
80
19
60
70
19
19
00
20
90
19
80
19
70
19
60
19
40
19
20
19
00
19
YEAR OF EXPERIMENT
TESTS OF
ed tests of theLOCAL
weak equivalence
principle,
showing bounds on ⌘, which measures
LORENTZ
INVARIANCE
1.05
YEAR OF EXPERIMENT
Δν/ν = (1+α)ΔU/c2
2X10-4
VLBI
1.00
ence in acceleration of di↵erent materials or bodies. The free-fall and Eöt-Wash
e originally performed to search for a fifth force (green region, representing many
Figure 3: Selected
tests
of local
The blue band shows evolving bounds on ⌘ for gravitating
bodies from
lunar
laser position invariance via gravitational redshift experiments, showing
JPL
Michelson-Morley
Joos
TPA
10-6
Centrifuge
Living Reviews in Relativity
Cavities
http://www.livingreviews.org/lrr-2006-3
10-10
δ
Brillet-Hall
10-14
Hipparcos
PSR 1937+21
a Doppler shift caused by complex convective and turbulent motions in the
this e↵ect is actually
photosphere and lower chromosphere, and is expected to be minimized by observing at the solar
10-18
SHAPIRO
TIME
DELAY
Implies gravity is a metric theory: gravity is space+me curvature 1.00
Living Reviews in Relativity
Hughes Drever
NIST
Voyager
Harvard
http://www.livingreviews.org/lrr-2006-3
U. Washington
10-22
10-26
Weak Equivalence Principle+ Local Lorentz Invariance+ After almost 50 years of inconclusive or contradictory measurements, the gravitational redshift
Posi8on Invariance= of solar spectral Local lines was finally
measured reliably.
During the early years of GR, the failure
to measure this e↵ect in solar lines was siezed upon by some as reason to doubt the theory.
Unfortunately, the
measurement is not E
simple.
Solar spectral lines P
arerinciple subject to the “limb e↵ect”,
Einstein’s quivalence a variation of spectral line wavelengths between the center of the solar disk and its edge or1.05
“limb”;
bounds on ↵, which measures degree of deviation of redshift from the formula ⌫/⌫ = U/c2 . In
null redshift experiments, the bound is on the di↵erence in ↵ between di↵erent kinds of clocks.
0.95
(1+γ)/2
10-2
δ=
1/c2
-1
10
20
00
20
90
19
80
19
70
19
60
19
40
19
20
19
00
19
YEAR OF EXPERIMENT
Best test of space8me curvature: 0.95
Cassini bound Cassini
(1X10-5)
Viking
1920
1940
1960
1970
1980
1990
YEAR OF EXPERIMENT
2000
Why should we bother looking for modifica+ons of GR? (Circa 1919) Journalist: “Herr Einstein, what if the theory turned out to be wrong?” Einstein: “I would feel sorry for the dear Lord. The theory is correct.” (Circa 1970) Chandrasekhar to his postdoc Clifford Will: “Why do you spend so much Bme tesBng GR? We know the theory is right.” 1)  Theory: GR is not renormalizable It becomes renormalizable if one adds high-­‐order curvature terms to the ac+on 2)  Experiments: dark maRer, dark energy Due to modified gravity? Problem: GR is extremely well tested “in between” these two regimes “Short blanket problem” for modifica+ons of GR What is “strong” gravity? a visualization of experimental constraints,
es 2.1.
where
the under-tested
of gravGravityregimes
Parameter Space
Categories
of Systems
devoted to
a discussion of our results.
-10
ational systems
considered in this paper fall
per we
10 will work in conformal time ⌘,
tegories: laboratory, astrophysical and cosvatives with
respect to ⌘ by a dot. The
-14discussion
Most of10
our
will focus on the latter
bble
factor
is
H
⌘
ȧ/a.
Fractional energy
es.
hphysical
as ⌦M (⌘)
-18 denote time-dependent quantisystems are nearly all spherically
10
day
values
are
indicated
by a subscript
zero,
and many
can
be
approximated
by
a
test
-22 signature used is { , +,WD
The
metric
+,
+}.
10
rbit around
a central mass, e.g. a planet
ed
calculations
sequestered
MS in the
Satellite
ar, a star orbitingare
close
to a supermassive
Let us
first consider
example
of a test pa
rectly
observable
(and the
therefore
coordinate-ind
ated
at a radial
distance
r from a central
obje
5 the gravitational
quantity,
namely
redshift
o
M
.
The
deviation
of
the
metric
from
Minkows
lines from a star or similar object. Hence eq
characterized
by the for
magnitude
theapproxim
Newton
assessingof
the
Gravita+onal fi
eld NSis a valid parameter
tational
nitude ofpotential,
the components of the metric outsid
BH in vacuum.
object
GM
2 .
We will measure the✏ ⌘
approximate
magnitu
rc
Riemann curvature tensor through the Kre
R
The
accessible
scalarstrongest
(R↵ Curvature Rgravitational
)1/2(Kretschmann . The fields
Kretschmann
sca
↵
s
calar) server
correspond
to
the
limit
✏
!
O(1),
whe
Schwarzschild metric is
-26
PSRs
tral object is a black hole and thep test part
10
BBN of the test particle’s motc. Observations
GM
1/2
↵ horizon. Although
close
to
the
event
equatio
⇠
=
R
R
=
48
.
sidered as-30a probe of the gravitational fieldMW SMBH
↵
3 c2
r
10
coordinate-dependent statement, it can be link
body.
ANTIFYING
GRAVITATIONAL FIELDS
coordinate-ind
rectly
observable
therefore
-34 e.g. the CMB, must instead be
The first
equality(and
above
is coordinate-indepen
cal systems,
S stars
2.1. 10
CategoriesThese
of Systems
of ⇠; the
namely
thedefinition
gravitational
redshift
serves as our
formal
secondo
ower
spectra.
require more carefulM87 quantity,
-38
lines
from
a star Strongest or similar
merely
an illustratory
example
for
choice
tobject.
est of the
GHence
R: eqo
tional10
systems
considered
this paper
fall
gravitational
field
mustSS
be in
assigned
to each
isSchwarzchild
a valid parameter
for assessing
the approxim
M ` astrophysical
coordinates.
The corresponding
k or angular
mode
in the power spectrum.
egories:
laboratory,
and cosPSR J
0348+0432, P
=2.46hr, -42
nitude
of theobjects
components
of complicated
the metric outsid
for rotating
is more
(Hen
define
quantifiers
analogous
to those
ost of our
discussion
will focus
on theapplied
latter
10
-­‐3 v/c=2x10
object
in
vacuum.
However,
the
additional
prefactors
will make li
ical systems,
so
that
comparisons
between
s.
-46
Last
scattering
We on
willthemeasure
the approximate
magnitu
ence
axis
ranges
used 1in
this paper
(se
[Antoniadis+, 304.6875]
gories
are
possible.
hysical10
systems
are nearly all spherically
Riemann
tensor through the Kre
and so willcurvature
be neglected.
peaks
set
the
system
will use to assess
nd out
many
be we
approximated
byCMB
a gravtest
-50can
↵
1/2
10
The(R
parameters
above
define a two-di
scalar
R↵ ✏ )and. ⇠The
Kretschmann
sca
dbit
strengths
the astrophysical
around for
a central
mass,
a cosmoplanet
Galaxies e.g.and
Clusters
space on which
we can
Schwarzschild
metric
is place the gravitatio
ories.
In -54
§4.5
we will
explain
equivalent
ar,
a star
orbiting
close
to ahow
supermassive
10
Lambda
probed by di↵erent objects,1/2observations
are
to two
c. assigned
Observations
of specific
the testlaboratory
particle’s tests
mop GM an
↵
-58
ments.
For
⇠ =simplicity
R
Rwe
= 48 3 refe
.
Accn.
dered 10
as a probe of the gravitational field
↵ will informally
2
r
c
P(k)| z=0
scale
parameters
as the ‘curvature’ and the ‘potent
body.
-62
Gravitational
spacetime,
thoughabove
this isisnot
strictly accurate
The
first equality
coordinate-indepen
al2.2.
systems,
e.g. theQuantifiers
CMB, must instead be
10
contexts
we formal
consider.
We stress
our par
-12These
-10require
-8 more
-6 careful
-4
-2
0 our
serves
as
definition
of ⇠;that
the second
ower
spectra.
ee tensors
make
up
10
10the description
10
10 of space10
10 merely
10
and ⇠ depart
from theexample
physicalfor
potential
ando
an illustratory
the choice
ravitational
field equations:
must be assigned
to each
ters the Einstein
the metric,
the
Figure: [Baker+, 1412.3455] Potential,
7 ε
Schwarzchild coordinates.
The corresponding
Curvature,
ξ
-2
(cm )
Strong-­‐field probes: black holes and neutron stars “Curvature desert” Tests of strong gravity: what will astrophysics ever do for us? “My summary of the workshop so far: We don't know what drives jets. We don't know how jets are related to the BHs or NSs responsible. We don't know how jets are related to the associated accre8on disk. We don't know how jets are related to their host galaxies. We don't know how BHs co-­‐evolve with their host galaxies. We don't know how AGN feedback works. We don't know anything.” (Anonymous) Tests of strong gravity: what will astrophysics ever do for us? Tests of
strong gravity
A guiding principle to modify GR: Lovelock’s theorem “In four spaceBme dimensions the only divergence-­‐free symmetric rank-­‐2 tensor Testing General Relativity
14
constructed solely from the metric and its derivaBves up to second differenBal order, and preserving diffeomorphism invariance, is the Einstein tensor plus a cosmological term.” WEP
WEPviolations
violations
Higher
Higherdimensions
dimensions
Lovelock
Lovelock
theorem
theorem
Diff-invar.
Diff-invar.violations
violations
Extra
Extrafields
fields
Nondynamical
Nondynamicalfields
fields
Dynamical
Dynamicalfields
fields
(SEP
violations)
(SEP violations)
Palatini f(R)
Eddington-Born-Infeld
Scalars
Vectors
Scalar-tensor, Metric f(R) Einstein-Aether
Horndeski, galileons
Horava-Lifshitz
Quadratic gravity, n-DBI
Massive
Massivegravity
gravity
Lorentz-violations
Lorentz-violations
dRGT theory
Massive bimetric
gravity
Einstein-Aether
Horava-Lifshitz
n-DBI
Tensors
TeVeS
Bimetric gravity
[EB+, arXiv:1501.07274] Figure: Paolo Pani …and about what?
Lovelock theorem as a map for the modified gravity zoo WEP
WEPviolations
violations
Higher
Higherdimensions
dimensions
Lovelock
Lovelock
theorem
theorem
Diff-invar.
Diff-invar.violations
violations
Extra
Extrafields
fields
Nondynamical
Nondynamicalfields
fields
Dynamical
Dynamicalfields
fields
(SEP
(SEPviolations)
violations)
Palatini f(R)
Eddington-Born-Infeld
Scalars
Vectors
Scalar-tensor, Metric f(R) Einstein-Aether
Horndeski, galileons
Horava-Lifshitz
Quadratic gravity, n-DBI
Tensors
TeVeS
Bimetric gravity
Massive
Massivegravity
gravity
Lorentz-violations
Lorentz-violations
dRGT theory
Massive bimetric
gravity
Einstein-Aether
Horava-Lifshitz
n-DBI
(| |)@a
@
V (| |) + Lmat
, A (| |)gab
Tests o2f general rela+vity –⇤ against what? abcd
|)RGB + f2 (| |)Rabcd R
1 (| principle §  +f
Ac8on §  Well-­‐posed §  Testable predic8ons §  Cosmologically viable, BHs, neutron stars L
= f0 ( )R
!( )@a @ a
M ( ) + Lmat
⇥
2
, A ( )gab
+f1 ( )(R2 4Rab Rab + Rabcd Rabcd )
+f2 ( )Rabcd ⇤ Rabcd + Lorentz-­‐viola8ng terms… ⇤
Alterna8ve theories usually: Introduce more corresponding
fields (scalars, vectors) r higher-­‐curvature Riemann
tensor
to othe
metric gab ,terms Rab
eed strong-­‐field tests! Challenge pillars of general rela8vity: and Ndenotes
matter
fields.
The
functions
fi (| |) (i =
§  Equivalence principle nciple § arbitrary,
but
they
are
not
all
independent
(in
Lorentz invariance (Einstein-­‐aether, TeVeS…) m can § beParity set equal
to one via field redefinitions without
conserva8on… [Gair+,1212.5575; Cligon+, 1106.2476] Theories with strong gravity predic+ons 1)  Scalar fields a) Scalar-­‐tensor theories i) Scalar-­‐tensor (Bergmann-­‐Wagoner) theories ii) Tensor-­‐mul8scalar theories iii) Horndeski theories b) Metric f(R) theories c) Quadra+c gravity i) Einstein-­‐dilaton-­‐Gauss-­‐Bonnet ii) Dynamical Chern-­‐Simons 2)  Lorentz-­‐viola+ng theories Einstein-­‐aether, Khronometric gravity, Horava gravity, n-­‐DBI 3)  Massive gravity 4)  Auxiliary fields i) Pala8ni f(R) theories ii) Eddington-­‐inspired Born-­‐Infeld gravity Proper+es of (some) modified gravity theories Theory
Extra scalar field
Scalar-tensor
Multiscalar
Metric f (R)
Quadratic gravity
Gauss-Bonnet
Chern-Simons
Generic
Horndeski
Lorentz-violating
Æ-gravity
Khronometric/
Hořava-Lifshitz
n-DBI
Massive gravity
dRGT/Bimetric
Galileon
Nondynamical fields
Palatini f (R)
Eddington-Born-Infeld
Others, not covered here
TeVeS
f (R)Lm
f (T )
Field
content
Strong EP
Massless
graviton
Lorentz
symmetry
Linear Tµ⌫
Weak EP
Wellposed?
Weak-field
constraints
S
S
S
7
7
7
X
X
X
X
X
X
X
X
X
X
X
X
X [30]
X?
X [35, 36]
[31–33]
[34]
[37]
S
P
S/P
S
7
7
7
7
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X?
7X? [39]
?
X?
[38]
[40]
SV
7
X
7
X
X
X?
[41–44]
S
S
7
7
X
X
7
7
X
X
X
X
X?
?
[43–46]
none ( [47])
SVT
S
7
7
7
X
X
X
X
X
X
X
?
X?
[16]
[16, 48]
–
–
X
X
X
X
X
X
7
7
X
X
X
?
none
none
SVT
?
?
7
?
7
X
X
X
X
X
7
X
X
X
X
7
X
?
?
?
[33]
[49]
Table 1. Catalog of several theories of gravity and their relation with the assumptions of Lovelock’s theorem. Each theory violates at least one assumpt
(see also Figure 2.1), and can be seen as a proxy for testing a specific principle underlying GR. See text for details of the entries. Key to abbreviations:
scalar; P: pseudoscalar; V: vector; T: tensor; ?: unknown; X?: not explored in detail or not rigorously proven, but there exist arguments to expect X. T
occurrence of 7X? means that there exist arguments in favor of well-posedness within the EFT formulation, and against well-posedness for the full theo
Weak-field constraints (as opposed to strong-field constraints, which are the main topic of this review) refer to Solar System and binary pulsar tests. Entr
[EB+, arXiv:1501.07274] What can astrophysics ever do for strong-­‐gravity tests? a)  Scalar fields:
Spontaneous scalarization/dynamical scalarization
Strong constraints [talk by Wex]…but what about tensor-multiscalar?
Quadratic gravity
Well posed? Astrophysical corrections Planck-scale suppressed?
…not a problem? effective field theory
b)  Lorentz violation
Strongly constrained…but no exploration of strong-field dynamics
Does Lorentz violation imply renormalizability?
Do Lorentz violations in gravity leak into Standard Model?
c)  Massive gravity:
no FRW solution in “standard” dRGT theory…but ways out?
no Schwarzschild/Kerr solutions, instabilities…maybe testable?
d)  Nondynamical (auxiliary) fields:
well posed? curvature singularities at stellar surface?
indistinguishable from a modified EOS…but unique in
probing matter dynamics, solve some singularity problems…
Black holes in
modified gravity
Black holes in (some) modified gravity theories Theory
Extra scalar field
Scalar-tensor
Multiscalar/Complex scalar
Metric f (R)
Quadratic gravity
Gauss-Bonnet
Chern-Simons
Generic
Horndeski
Lorentz-violating
Æ-gravity
Khronometric/
Hořava-Lifshitz
n-DBI
Massive gravity
dRGT/Bimetric
Galileon
Nondynamical fields
Palatini f (R)
Eddington-Born-Infeld
Solutions
Stability
Geodesics
Quadrupole
[56–62]
?
[65, 66]
–
?
?
–
[63, 64]
?
NR [67–69]; SR [70, 71]; FR [72]
SR [78–80]; FR [81]
SR [75]
[87–89]
[73, 74]
NR [82–85]; SR [74]
?
? [90, 91]
SR [70, 75, 76]; FR [72]
[69, 86]
[75]
?
[71, 77]
[80]
Eq. (3.12)
?
NR [92–94]
?
[93, 94]
?
NR, SR [93–96]
NR [98, 99]
? [97]
?
[93, 94]
?
?
?
[104–107]
?
?
?
?
?
–
–
–
–
–
–
⌘GR [50–55]
GR [51, 63, 64]
GR [53, 54]
GR, NR [100–103]
[108]
⌘GR
⌘GR
Table 2. Catalogue of BH properties in several theories of gravity. The column “Solutions” refers to asymptotically-flat, regular solutions. Legend: ST=“S
Tensor,” ⌘GR=“Same solutions as in GR,” GR=“GR solutions are also solutions of the theory,” NR=“Non rotating,” SR=“Slowly rotating,” FR=
rotating/Generic rotation,” ?=unknown or uncertain.
[EB+, arXiv:1501.07274] Black hole solu+ons Most interesting targets for strong-gravity modifications? Maybe not!
Observations probe at most the horizon scale – curvatures not so large!
Scalar fields:
Tensor-(multi)scalar and f(R) theories:
black holes same as in GR…unless boundary conditions are nontrivial
Quadratic gravity:
Hairy solutions known numerically (EdGB), in a slow-rotation
approximation (dCS) or in the nonrotating limit (Horndeski)
Corrections suppressed by small coupling…but effective field theory?
Lorentz-violating theories:
“Universal horizons” with unclear stability properties…testable?
Massive gravity:
Schwarzschild (and Kerr) solutions unstable for all (astrophysical) masses
End-state of instability unclear…testable?
Auxiliary fields:
Black holes same as in GR
Compact stars in
modified gravity
2.1. Mass-Radius diagram for neutron stars
“Internal” tests: neutron stars of the TOV Eq. (1) yields mass-radius curves (Fig. 4).
Strong-fieldIntegration
signatures:
Dramaticin
differences
among normal
EOS’s and also between
high curvatures
interior, exist
spontaneous
scalarization…
normal and strange quark matter EOS’s. In contrast to Newtonian
3)1/2
Observables?
themass
Hartle-Thorne
expansion
in Ω/(M/R
gravity,Consider
a maximum
for each EOS
exists in GR.
In addition,
a minimum mass exists, but is only about 0.09 M( (not shown
Zero order in rotation: M(R) - mass-radius relation
since Rmin ) 200 km). It is interesting that many normal nucleonic
Radii hard to measure, both in binaries and in isolated systems
Rhoades
equation of
maximum
state
for
qs = 2.7 , 10
finds for th
the most co
this represe
et al. (1990
ning (1992
to find the
limit is sho
2.4. Maxima
The abso
ding” limit
Prigid
min ¼ 0:55
for a rigid s
rotation in
model, as d
the rotating
“Internal” tests: neutron stars Strong-field signatures:
high curvatures in interior, spontaneous scalarization…
Observables? Consider the Hartle-Thorne expansion in Ω/(M/R3)1/2
Zero order in rotation: M(R) - mass-radius relation
Radii hard to measure, both in binaries and in isolated systems
Corrections:
Moment of inertia I may be measurable in binary pulsars
[Lattimer-Schutz, Kramer, Wex…]
Tidal “Love number” may be measurable in binary inspirals
[Mora-Will, Berti-Iyer-Will, Read, Hinderer, Lang, Binnington, Poisson,
Vines, Damour, Nagar, Bernuzzi, Villain, Favata, Yagi, Yunes…]
Quadrupole Q or higher-order moments: light curves or QPOs
[Laarakkers-Poisson, Berti-Stergioulas, BWMB, Baubock+, Pappas…]
Stellar oscillations
Main problem: degeneracy between EOS and modifications of GR!
Neutron stars in (some) modified gravity theories Theory
Extra scalar field
Scalar-Tensor
Multiscalar
Metric f (R)
Quadratic gravity
Gauss-Bonnet
Chern-Simons
Horndeski
Lorentz-violating
Æ-gravity
Khronometric/
Hořava-Lifshitz
n-DBI
Massive gravity
dRGT/Bimetric
Galileon
Nondynamical fields
Palatini f (R)
Eddington-Born-Infeld
NR
Structure
SR
Collapse
Sensitivities
Stability
Geodesics
FR
[109–114]
?
[141–153]
[112, 115, 116]
?
[154]
[117–119]
?
[155]
[120–127]
?
[156, 157]
[128]
?
?
[129–139]
?
[158, 159]
[118, 140]
?
?
[160]
⌘ GR
?
[160]
[25, 40, 161–163]
?
[77]
?
?
?
?
?
?
[162]
?
?
?
?
?
?
?
[164, 165]
?
?
[166]
[43, 44]
[158]
?
[167]
?
?
?
?
?
?
?
[43, 44]
?
?
?
?
?
[168, 169]
[170]
?
[170]
?
?
?
[171, 172]
?
?
?
?
?
?
[173–177]
[178–184]
?
[178, 179]
?
?
?
[179]
–
–
?
[185, 186]
?
?
Table 3. Catalog of NS properties in several theories of gravity. Symbols and abbreviations are the same as in Table 2.
[EB+, arXiv:1501.07274] Lagrangian. ð!; "Þ parameter space.
regions
in
the
two-dimensional
des
(written
in
MATHEMATICA
and
C++
) are The
in bounds on the centra
and
a
redefinition
of
the
scalar
field;
i.e.,
by
reformulating
g
PHYSICAL REVIEW
D
84,
104035
(2011)
A “theory of theories” the theory in the EinsteinSome
frame [24].
However,
depending
results
are
shown
in
Figs. 1–3 for
different
EOS
IV.
PERFECT-FLUID
COMPACT
STARS IN
greement
with
Refs.
[5,78,79]
in
general
son the explicit form of f and #, these transformations can2 theEXTENDED
constraints
on
the
maximum
SCALAR-TENSOR
THEORIES
5 not impossible)
%
a
2
rrespond
to
the
same
!"
¼
20M
,
and
indeed
they
models
and
different
values
of
!
and
".
In
each
figure
be
hard
(if
to
write
in
closed
analytic
form.
(@ ! $ Vðj!jÞ þ f1 ðj!jÞR
L
¼
f
ðj!jÞR
$
"ðj!jÞ@
!
0
a
)limit.
A. StaticAs
solutions
For this reason we find it convenient to start from the
quantity.
shown
in Fig.
4
ost
exactly
on
top
of
each
other.
the
top
two
panels
show
the
mass-density
relation
and
Lagrangian (2), which reduces to standard
We begin byabcd
looking for static, spherically symmetric
ab scalarsgeneral
mass
is
a field
monotonically
de
þ
f
ðj!jÞR
R
þ
f
ðj!jÞR
R
tensor
theories
in
the
Jordan
frame
if
Aðj"jÞ
!
1
and
in
the
2 mass-radius
ab
3
abcd
equilibrium
solutions
of the
equationsstars.
with metric
milar
degeneracy
will
occur
for
any
other
functional
the
relation
for
static
(nonrotating)
The
Einstein frame if f ðj"jÞ ! 1=ð16$Þ and #ðj"jÞ ! 1.
The
requirement
that oth
V.advantage
COMPACT
STARS
IN
%
2
of this approach
ispanel
that,
atabcd
least
in
dr
fAnother
f1 ð!Þ,
provided
that
the
scalar
field
remains
small
bottom-left
shows
the
binding
energy
as
a
function
þ f encompass
ðj!jÞRgeneric
R theories,þ Ldsmat¼½!;
A ðj!jÞg
"BðrÞdt
þ
þ r sin %d’
abcd fðRÞ
ab (;þ r d%(2)
-principle,
it should also 4
1
"
2mðrÞ=r
ported
by
the
theory
shou
GAUSS-BONNET
GRAVITY
here,
thatthe
ato Taylor
expansion
similar
to Eq.
(29)panel we display
central
density.
In
the
bottom
right
the
which areso
equivalent
particular
scalar-tensor
theories
s(but see [45] for possible issues with this point of view). and a charged,
spherically
fieldplace a
observed
[Yunes &symmetric
Svalue,
tein, scalar
1can
101.2921] Inapplication
this sense,moment
most
of
our
results
remain
valid
for
a
of
inertia
as
a
function
of
(gravitational)
mass.
of the formalism discussed above,
[Pani+, 1109.0928] Table
III,
we consider
Mmax
TABLE
I.and
Specific
models
obtained
from
the Lagrangian
(2). Here
& ! ð16$GÞ
.
function
f
ð!Þ,
not
only
for
EDGB
gravity.
In
our
numerical
calculations,
we
observed
that
the
1% paper
abcd
ainder
of
this
we
study
neutron
stars
in
e
where R
isf thef dualf off thef Riemann
tensor,
which
M
*
1:93M
is
max
!
V
#
A # (which
L
scalar
field
in
the
interior
of
the
star
is
always
small:
in
hermore,
since
only
the
second
term
on
right
vity.
We
defer
a
more
general
study
of
the
full
cGeneral relativity
introduces possible
parity-violating
From
&
0
0
0
0 corrections
0 observation
0
1 [57].
1 [12])
perfect
fluid
in
and
$2
special
cases
it
can
be
as
large
as
!
#
10
,
but
more
ide
of
Eq.
(29)
contributes
to
the
dynamics,
it
Scalar-tensor
(Jordan
[24]
Fð"Þ
0(7) to
0 future
0
0 work.
0
Vð"Þ
#ð"Þ
1
perfect fluid
ved
from
theframe)
Lagrangian
the(Einstein
Lagrangian
above,
the
equations
of
motion
read:
eScalar-tensor
bounds
on
"#
using
the
frame) [23]
&
0
0
0
0
0
Vð"Þ
2&
Að"Þ
perfect fluid
$4
typically
!
&
10
in
most
of
the
parameter
space.
In
the
that
the
field
equations
are
(approximately)
sym2&
f
¼f
perfect fluid
fðRÞ [36]
&
0 the
0 Lagrangian
0
0
0 & (7)
gravity
is
obtained
from
rQuadratic gravity [47]
EOS
models
predic
"%
' "1, 'the
" coupling
0 Causal
0
1 in Eq.
1 (28)
perfect
fluid
& limit,
' " '
small-field
!
f
can
be
under
the
transformation
1
a real scalar field
¼ #f (or0 !0 ¼
0),
0
1
1
perfect fluid
EDGB [48]
&
e $ "4f
-ring
mum
mass,
so 1all neutron
s
Dynamical$1
Chern-Simons
[59]
&
0
0
0
("
0
0
1
perfect fluid
Taylor-expanded:
, V & 0 and 1& 0 0 0 0 ! place
j"j very
1
0
Boson stars [71]
-16&Þ
mild1 constraints
! !G$!;
!
!
$!:
(30)
þ ½H ab þ I ab þ J abomit
þ Kthis
ab '
ab ( EOS from Table
A
fe0(# ;
f
&
(28)
16#f
ð!Þ
#
!
þ
!"!:
(29
EDGB: of1 this symmetry, we
1
sadvantage
For
the
‘‘standard’’
va
present
results
only
16& 1
104035-4
p
ffiffiffi
ð!Þ
2 mat
½A
T
þ
T
¼
piffiffiffis (#
Set !
>
0.
;The
natural string tab
heory be¼(ap-2), if we(3)consider
case
solutions
for
! c<hoice ab0(;can
2f0
( are
coupling
constants.
When
( the
¼
, this
]dately)
first
term is
constant
GB term is a
obtained
Since
inverting
sign
[2e.g. Kand
an8+, hep-­‐th/9511071] by the
simply
of
the the
0
0
2
0
2
2
2
2
2
2
2
"1
0
1
2
3
4
mat
Rf;R "f
" 2
16$Gf
;R
1
("
2
3
1
4
1
m2
2
2
"1=2
0
"1=2
;R
Testing General Relativity
Scalar-­‐tensor theory: spontaneous scalariza+on 3.5
0.4
2.5
0.10
3.0
2.0
0.3
2.5
MS1
1.5
~
~
0.2
1.5
δq / q
APR MS1
~
FPS
q /M
2.0
δM/M
FPS APR
M/MO.
78
0.05
1.0
1.0
0.1
APR
0.5
FPS
0.0
10
12
14 16
R/km
0.5
[Pani+EB, 1405.4547] MS1
18
1
2
M/MO.
3
0.0
FPS
0.0
12
1.0
2.0
M/MO.
3.0
0
1
MS1
APR
2
M/MO.
3
15
0.0
50
3
10
APR
2 2
APR
APR
2
10
MS1
FPS
FPS
1
2
M/MO.
FPS
3
0
1
2
M/MO.
3
0
g cm s ]
36
20
5
[10
2 2
g cm s ]
30
~ rot
1
MS1
36
~
MS1
10
λ
MS1
4
λ [10
2
~
~
Q [10
45
6
I [10
45
2
0
g cm ]
8
2
g cm ]
40
APR
FPS
0
1
2
M/MO.
3
0
1
2
M/MO.
3
0
Neutron star solu+ons Main advantage: they probe how matter couples to gravity
Curvatures can be very large inside neutron stars
Scalar fields:
Scalarization tightly constrained by binary pulsars
Torsional oscillations (flares) would not yield new constraints
…but anisotropic matter or multiscalarization
f(R) theories: do compact stars even exist? Dynamical studies?
Quadratic gravity:
In EdGB, NS constraints already tighter than BH constraints!
Lorentz-violating theories:
Only nonrotating stars
Massive gravity:
Maybe no solutions? [Damour+, hep-th/0212155]
Auxiliary fields:
Surface singularities may rule out theory! degeneracy with EOS
onding to ! ¼ 20M( , " ¼ 1 and ! ¼ 10M( , " ¼ 4
symmetry (30), which is exact in this c
Stellar structure in Einstein-­‐dilaton-­‐Gauss-­‐Bonnet theory 2.5
2.5
2.0
2.0
M M
2
2
1.5
2
2
2
1.0
EOS APR 0.5
dMmax/d(αβ)<0 1
No stars for ρ>ρc 2
2
2
2
2
c
1,
1,
1,
2,
2,
2,
4,
4,
4,
3
15
10 g cm
10 M 2
20 M 2
50 M 2
10 M 2
20 M 2
50 M 2
10 M 2
20 M 2
50 M 2
M M
GR
1.0
0.5
4
5
4
5
3
m M 1
0.20
0.15
0.10
0.05
1
2
c
3
15
10 g cm
3
1.5
9
10
11
12
R km
13
14
square roots, whose argument m
ars
max
M2
values
of M we
consider,
constraints using
the Cau
and from the
FPS
EOS,
respectively.
COMPACT
STARS
INthe
ALTERNATIVE
THEORIES
O
existence cof
physical
(real
Constraints on Einstein-­‐dilaton-­‐Gauss-­‐Bonnet ouplings 2
allowdensity
values
of "#belarger
than
100M
The bounds on would
the central
translated
into # .this ‘‘real
100 can!
%
1,
by
imposing
c
FPS constraints on the maximum mass, which is an observable
! M
MNo
*
1:7M
EOS
Mmax * 1:4M#
APR
max
#
max *
1
pact
stars104035
80
ORIES OF
...
PHYSICAL
D
84,
2REVIEW
2thecom
3 (20
quantity.
As
shown
in
Fig.
4,
for
a
given
EOS
maximum
128#
"
<
!
Causal
c !2
42
2
7776$P
#c
FPSIII. decreasing
"# & 30:1M
13:9M
no m
c#
mass is a monotonically
function
of&
"#.
# the"#
TABLE
Constraints
on
EDGB
parameters
from
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
60
"#
&
41:9M
APR
"#
&
50:3M
FPS"# &
supThe requirement
that
the
maximum
mass
M
#
#
observation of a nonrotating neutron star
M. For
! max
2 with
ð3Pcmass
þ #APR
c Þð3Pc !
ported by the
theory
should
be larger
than someusing
trusted
values
of
M
we
consider,
constraints
the Causal
Causal E
40
RI
RI
2 In
observed
value,
can
place
a
direct
upper
bound
on
"#.
.
would
allow
values
of
"#
larger
than
100M
#
.
.
.
PHYSICAL
REVIEW
D
84,
104035
(2011)
.5 3.0Table
3.5 III, we consider M
20 # , Mmax * 1:7M# and
max * 1:4M
S
3
candidate’’
EOS
forbound
aparameters
(nonexotic)
neutron
star
TABLE
III.EOS
Constraints
on
the
EDGB
from
an
For
a*
given
value
of
!",
theRI
con
*
1:4M
M
1:7M
M
*
1:93
M
m
PR
max
#
max
#
max
Mmax
* 1:93M
(which
is
the
lower
of
the
recent
RI
RI
#
max
observation of a nonrotating
neutronrelativity,
star with
mass
M.
For the
maximum
central
density,
#
within
general
the
results
in
Table
0
ausal
c II,
2 0.5 1.0
2 upper
observation
in
[12])
and
we
translate
them
into
1.5
2.0
2.5
3.0
3.5
FPSwe consider,
"# & 30:1M
"# &
13:9M
no good
models
2
values of M
constraints
using
the
Causal
EOS
#
#(31)
Equation
is
in
a
"
&
23:8M
.
r the parameters
"#
in"# using the data
# 2 plotted 2 in max
215
3
bounds
on
Fig.
4.
The
"#
&
& 33:6M
"#
&larger
50:3Mthan
g cm "#
. 41:9M
would allowAPR
values of
"#
100M
results,
shown
in
Fig.
5. T
#
#
c as10
#
This
bound
should
be
compared
to
the
boun
odels. Only
the
combiCausal EOS models predict an unrealistically
large
exclusion
plot:maxiit shows the ma
oximation
!
'
1
(cf.
that
comes
from
requiring
the
existence
of bla
* neutron
1:4M
*online).
1:7M
*plot
1:93M
EOS
Mso
RI online).
color
Maximum
mass
as FIG.
a# function
of
the
maxall
maxwith
# !"
max
#
5M
(color
Exclusion
for
the
parameter
&M
would
mum
mass,
stars
Mmax
p
ffiffiffi
as2:8M
a function
of
the
maximu
#
FIG.
6
lines
no
compact
star
!"
of the
EDGB
coupling
parameters,
for
different
EOS
limit
and
for
nonrotating
models.
Only
the
the
small
!
solutions
in
the
theory
[48,53].
For
#
¼
2
,
this
2
2
c 13:9M different
mild&constraints
on alternative
theories,
andmodels
we and
non
.0 3.5place
FPSvery "#
30:1M#
"#
&
no EOS
models
#
value
nd the
in the
nonrotating
main
text for
In
region
abovecase
the (cf. the
nation
"#details).
is bounded,
due toof the
approximation
!
'
2 implies
2 values
2 on
ment
!"
(i.e.,
the
rig
candidate’’
EOS
for
a
(nonexotic)
neutron
star
inter
omit
this
EOS
from
Table
III.
"#
&
41:9M
"#
&
33:6M
APR
"#
&
50:3M
#
#
#
gr
ft of the filled
circle,
this maximum mass
corresponds
ability),
static
configuEq. (29)).
In thetoregion
above theindotted
lines nothe
comp
maximum
the
mass-density
r
For
the
‘‘standard’’
value
of
the
EDGB
coupling
within
general
relativity,
the
results
in
Table
III
im
#
"
al
stability
criterion;
to
the
right,
it
corresponds
to
the
Table
solution can be constructed
(cf. Eq.
(31)).2Indensity
the region
a
pffiffiffi text
max
ations (cf. the main
" maximum
M
central
#
2
BH
c
m central(#
density
can
construct
static
& we
23:8M
¼density
), which
if" we
consider
the(marked
APR as
EOS
asRadial
our Instability),
‘‘best
from
parameters
"#
in2for
& 70
;Þ, i.e. to
thick
‘‘RI,’’
static
# . lines
mum
central
of
2
maximum
of
Mð#
the
M
c
um models. The recent measurement
[12]
of
a
neutron
10M
#
#
rations
are
unstable
against
radial
perturbations
(cf.
the
m
This
bound
should
be
compared
to
the
bound
on
Only the combimass-radius
relation.
If
our
stab
forforadetails).
(nonexotic)
star
interior central de
is marked by EOS
a horizontal
line.
Only
the neutron
h M ' 2M( candidate’’
Markers
indicate
the
maximum
o
n α
a
lready c
omes f
rom N
Ss, n
ot BHs!
tion ! ' 1Best (cf. bound that comes from requiringnothe
existence
of black h
stable
static
configurations
ating configurations and the radius for nonrotating configurations.
Are we tes+ng the EOS or gravity? Universal rela+ons M* (APR) [Mo.]
10
1.2
0.78
1
-2
|I-I
(fit)
|/I
(fit)
10
1.7
APR
SLy
LS220
Shen
PS
PCL2
Polytrope (n=1)
SQM1
SQM2
SQM3
fit
I
10
2.2
2
10
-3
10
0
10
1
10
2
λ
10
3
10
4
(tid)
I-­‐Love-­‐Q and three-­‐hair rela+ons could help tell theories apart ¯
Model-independent tests can also be obtained by measuring the first four multipol
moments,
i.e.,
angular
momentum,
quadrupole
and spin
octupole [26].
Are wmass,
e tes+ng the EOS or gmass
ravity? Universal rela+ons Issues: igure 4.11. Left panel: The I-Love relation in dCS gravity for various EOSs. The shaded regio
most plane
theories epresents a hypothetical error box in theIn I-Love
resulting
from independent measurements o
he moment
of inertia
and ofChern-­‐Simons the tidal Love(e.g. number
(the black Easterisk
marks the hypotheticall
other than dynamical scalar-­‐tensor, ddington-­‐inspired gravity) measured values). The black solid
line shows
the I-Love
relation
inR GR. The top axis shows the N
universal rela+ons same as in G
mass M⇤ for the Shen EOS. An alternative theory
is consistent with the measurement only if th
modified I-Love relation
passes throughtheories: the errorubox.
Right
panel: n
Love-compactness
relation i
R2, Lorentz-­‐viola8ng niversal rela8ons ot studied GR for various EOSs. The shaded region represents a hypothetical measurement error box in th
Massive gravity, general Horndeski: no studies of serror
tellar box,
structure
ove-compactness plane. Different NS EOSs are consistent
with
the
and
therefore
it ma
e possibleAll totheories carry outin tests
GR (this
conclusionpdoes
not apply to QSs).
[From
[25].]
one ofsweep? post-­‐TOV, ost-­‐Hartle-­‐Thorne [Silva’s poster] Gravitational-wave
smoking guns?
s 0
2
ffi s
ions
general
relativity.
occurs
in all
theories
m
−
µesxample ) ,of gravity
(12)with Kerr BHs as
=
2 Ė∞ from
pa
p Θ(Ω
, so that the angular velocity
of the
horizon
Expect t
he u
nexpected: n 4
12π
r
ground solutions and a scalar field of mass @!s coup
(here and below we set G ¼ c0¼ 1). A
þ
scalars and superradiant instabili+es uency ! < m!Massive H incident on the BH (where
here
Θ(x) isnumber)
the w
Heaviside
function.
For generic modes,
uthal
quantum
is amplified
in
a Superradiance hen ω
< m
Ω
H
responsible
for
many
interesting
effects
[9–16].
Here
we
ess,
the
excess
energy
coming
from
the
large distances and for ω = mΩp > µs , scalar ral energy.
Superradiance
isinteresting
responsible
for possibility that an object in orexplore
the
Strongest i
nstability: µ
M∼1 s
ation
over
gravitational
radiation: compare
ng
effectsdominates
[9–16]. Here we
explore
the ing
bit
around
astandard
rotating
BH may excite
superradiant
modes
0705.2880] that
anwith
object
in
orbit
around
a rotating
q.ility[Dolan, (12)
the
quadrupole
formula
Ė
=
∞
superradiant
modes
ampli-10 −5 to2 appreciable
to
appreciable
As
the objectto orbits
around
2 amplitudes.
/5
(r
/M
)
m
/M
.
This
result
is
oblivious
the
For µ
=1eV, M
=M
: µ
M∼10
0
p itsun bject orbitss around the BH
loses energy
in
s
thelof
BH
itrotating
loses
energy
in bgravitational
aves,
slowly
spiraling
in,
as(or shown
experi-In
esence
the
BH.
fact,
ω > waves,
µs , theslowly spiNeed ight scalars primordial lack hfor
oles!) e Hulse-Taylor
binary
This follows
ralling
in, pulsar.
as shown
experimentally
by the
Hulse-Taylor
uxes
at
the
horizon
are
negligible.
However,
for
frelance:
if the orbital
energy
of the
particle
is
Nega+ve s
calar fl
ux a
t t
he h
orizon from
binary
pulsar.
This
follows
encies
close
to
µ
,
a
resonance
occurs
at energy
[24]: balance: if the
al (gravitational plus scalar)
s energy flux is
close to superradiant resonances at is E , and the total (gravithen
orbital
energy
of
the
particle
p
)
*
2
E_ þ E_ g þ E_ s ¼ 0:
g
s
µs M energy
(1)
2
2 plus
2 scalar)
ptational
flux
is
Ė
=
Ė
+
Ė
, then
ωres = µs − µµMs r /M (resonance) (αm /M ) Ė,FIG. n
= 0,online).
1, T... 1010Pictorial
(13)
1 α(color
description of floating
l+
1 +with
n −0.1828An orbiting
1.1 · 10 body excites
10
Es > 0, and therefore10the 4.33400288873563
orbit
shrinks
superradiant scalar modes close
10
10
21.4020987080510
−0.4881
1.6 · 10
-(dE /dt)is massive, the flux at i
it is possible that, 10
due 99.9339974413005
to superradiance,−1.1588BH horizon.
Since the scalar
field
10
2.3 · 10 s
g
10
(dE /dt)
n this
case E_ p o
¼rbits” 0, and w
the
particle
can
consists
solely
of
gravitational
radiation.
s
“Floa+ng hen Ė
+
Ė
+
Ė
=
0
.
(1)
10
p
om Eq. (10) TABLE
we see
that
Ė
<
0
in
the
superradiant
I. Orbital radius at resonance
r+ and peak scalar flux for
10
þ
s
p
0
−2
-2
s, peak
r+
crit
-3
−1
-4
−2
−3
-5
−3
−5
-2
(mp/M) (dElm/dt)
−1
-6
11
-7
22
-8
s
r+
g
T
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
n = 0, l = m = 1, a = 0.99M and several values of µ M ≪ 1.
(k < 0).
For comparison,
extreme mass ratio inspiral
gime
Closea typical
to resonance
webecomes
get (cf. also
[24])
H
=107(24)=241101(5)
241101-1
!
2011
American Physical S
detectable by space-based interferometers
at radii r /M ∼
Compa+ble w
ith c
urrent e
xperiments!
s Hz)] , where f is the lower
50 [(10gM /M )(f /10
s
0
-9
-10
10
-11
10
-12
10
-13
10
-14
10
-16
-2×10
0
-16
2×10
Δr0/rres
Usually ∞
Ė + Ėl+1 >−µ0,
and therefore the orbit shrinks
2
Mr/(l+1+n)
s
X
∼
r
e
.that, due (14)
lmω
with time.
it isCardoso++
possible1109.6021]
to superradi[Press+Teukolsky
1972,However
Detweiler 1980,
6
⊙
cut
−4
−2/3
cut
cutoff for the sensitivity threshold of the interferometer. A
floating orbit occurs for α > αcrit . Notice that αcrit is well
below current observational bounds [21] for any µs .
5
10
15
20
25
r0/M
FIG. 2. Dominant fluxes of scalar and gravitational e
Ask not what astrophysics can do for strong gravity. Ask what strong gravity can do for astrophysics! [Gerosa’s poster] Inverse problem: binary evolu+on from GW observa+ons [Gerosa+, 1302.4442; 1411.0674; poster!]
Effic
ient
Resonant plane locking
L · S1 ⇥ S2 ' 0
sin
' 0 (equilibrium)
tides
Merging BH binary
No t
i d es
sin
Stan
dard
mass
Resonant precession
' ±180
✓12 6= 0 (tail)
ratio
Free precession
' ±1 (pile-up)
0.02
0.015
Weakly precessing binaries
' 0
✓12 ' 0
tio
ss r a
a
m
r sed
Reve
0.08
FIG. 1. Predictions for the spin orientation of BH binary
as they enter the LIGO/Virgo band.
0.06
0.01
0.04
0.005
0.02
0.02
0.08
0.6
0.4
0.06
0.015
0.01
0.04
0.005
0.02
0.02
0.08
0.2
0
0.06
0.4
0.015
0.01
0.04
0.005
0.02
0.2
0
-180
0
-90
0
90
180
0
45
90
135
180
Some concluding thoughts -­‐ hope in dynamics! 1) Small couplings
small deviations from GR in the dynamical regime?
Scalar fields suggest the answer is no!
Spontaneous/dynamical scalarization
Floating orbits
2) Is nature hiding deviations from us?
Scalarization: not a cosmological attractor [Damour-Nordtvedt 1993]
Floating orbits: fine tuning…
3) When do we expect to do precision gravitational-wave astronomy
at the level necessary to test strong-field gravity?
[Buonanno’s talk]
4) LISA Pathfinder will launch soon –
could we please have eLISA before 2034?
(I would have a much easier time giving my next talk)