Hot Topics in General Relativity
And Gravitation, Quy Nhon, 2015
Effective field theory approach to
modified gravity with applications
to inflation and dark energy
Shinji Tsujikawa
Collaboration with
A. De Felice (Kyoto)
L. Gergely (Sedged)
R. Kase (TUS)
K. Koyama (Portsmouth)
Tokyo University of Science
Motivation of going beyond General Relativity

Origin of inflation
--it comes from some geometric effect or from a scalar field
beyond the standard model of particle physics?

Origin of dark energy (and dark matter)
--the present cosmic acceleration may come from a
large-distance modification of gravity?
 Construction of renormalizable theory of gravity
--short-distance modification of gravity
Horndeski theories
Most general scalar-tensor theories
with second-order equations
Horndeski (1974)
Deffayet et al (2011)
Charmousis et al (2011)
Kobayashi et al (2011)
The Lagrangian of Horndeski theories is constructed to keep the equations of motion
up to second order, such that the theories are free from the Ostrogradski instability.
ADM decompositon of space-time
We can start from a general action involving all the possible
geometric scalar quantities appearing in the ADM formalism.
The ADM formalism is based on the
3+1 decomposition of space-time.
ADM metric
We choose the unitary gauge on the
flat FLRW cosmological background
Then, the scalar perturbation can be
absorbed into the gravitational sector:
We have several geometric tensors:
Extrinsic curvature:
3-dimensional Ricci tensor:
Constant time
hypersurface
Horndeski Lagrangian in the ADM Language
Several scalar quantities can be constructed:
Gleyzes et al (2013)
Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories
do NOT impose these conditions.
The action in Horndeski and GLPV theories has the dependence
Inclusion of higher spatial derivatives
In Horava-Lifshitz gravity, there are spatial derivatives like
to realize an anisotropic scaling between time and spatial derivatives.
In the healthy extension of Horava-Lifshitz grvaity, there are scalar
quantities coming from the acceleration vector
like
Then, the general action implementing GLPV theories and Horava-Lifshitz
gravity is
Tensor perturbations in the EFT approach
where
De Felice and
ST (2014)
____ ______________
Present in Higher spatial derivatives
GLPV
appearing in Horava gravity
theories
In the absence of higher spatial derivative, the no-ghost and
no-instability conditions are
Equations of motion for tensor perturbations
where
where
We can use this result to derive the tensor power spectrum
generated during inflation.
Inflationary tensor power spectrum
Then, the spatial derivatives can be treated as corrections, in which case
the tensor power spectrum under the slow-roll approximation reads
_______
___________________
Leading-order
spectrum
Slow-roll corrections
are much smaller than 1.
__________
Corrections from
Einstein frame
Is there a convenient frame in which the leading-order tensor power
spectrum is of the simpler form?
We define the Einstein frame as the one in which the second-order
action for tensor perturbation is of the same form as in GR, i.e.,
In the Einstein frame, the leading-order inflationary tensor spectrum
should be of the form
In GLPV theories it is possible to transform to the Einstein frame under
the so-called disformal transformation.
Disformal transformations
The structure of the GLPV action is preserved under the disformal transformation:
Bekenstein (1993)
Conformal
transformation
Disformal
transformation
The GLPV action in the transformed frame reads
Same form as that
in the original frame
with the relations among coefficients
Gleyzes et al, JCAP (2014)
where
Disformal invariance of cosmological perturbations
Consider the perturbed metric
ST (2014),
See also
Minamitsuji (2014)
for the case
where
_
Curvature
perturbations
where
__
Tensor
perturbations
Transformation to the Einstein frame
Creminelli et al, PRL (2014),
ST, JCAP (2014)
In GLPV theories, the next-to-leading order tensor power spectrum in the
transformed frame is given by
We can transform to the Einstein frame for the choice
Then the tensor power spectrum reads
Same as the GR
tensor spectrum
(Stewart and Lyth, 1993)
where
Application to dark energy
The EFT formalism was also applied to dark energy (Vernizzi’s talk).
Usually, the quadratic-order EFT action is written of the form (Creminelli et al):
__
__________________ _____________________
Background
Perturbations
Three functions
Matter
sector
Functions
If we specify the theories (e.g. Horndeski), there are explicit relations between
the above EFT functions and the free parameters of theories.
See Gleyzes et al (2013), ST(2014)
The EFT formalism is also implemented in the CAMB code (Silvestri et al).
Cosmological perturbations in the presence of matter
The scalar degree of freedom can give rise to
 the late-time cosmic acceleration at the background level
 interactions with the matter sector (CDM, baryons)
We take into account non-relativistic matter with the energy density
____
_______
Background Perturbations
The four velocity of non-relativistic matter is
The perturbed line element in the longitudinal gauge is
Effective gravitational coupling with matter
The growth rate of matter perturbations is constrained from peculiar velocities
of galaxies in red-shift space distortion measurements.
The gauge-invariant density contrast
___
obeys
where
Recent observations favor weak gravity
on cosmological scales.
Planck LCDM fit
DE-related
Planck
Planck+ BSH
Planck+ WL
Planck+ BAO/ RSD
µ0 − 1
0.5
0.55
8
Planck+ WL+ BAO/ RSD
0.0
− 0.5
Growth Rate, f(z) s (z)
1.0
GR
0.5
0.45
0.4
Planck LCDM
RSD fit
6dFGS
LRG
BOSS
WiggleZ
VIPERS
0.35
0.3
− 1.0
−1
0
1
η0 − 1
2
3
0
0.2
0.4
0.6
Redshift, z
0.8
RSD fit
1
Effective gravitational coupling in Horndeski theories
ST,
1505.02459
(2015)
In the massless limit, the effective gravitational coupling in Horndeski theories reads
____
Tensor
contribution
This correspond to the intrinsic
modification of the gravitational part.
_______
Scalar
contribution
Always positive under the no-ghost
and no-instability conditions:
The necessary condition to realize weaker gravity than that in GR is
The scalar-matter interaction always enhances the effective gravitational coupling,
so the realization of weak gravity is quite limited in Horndeski theories.
A model realizing weak gravity beyond the Horndeski domain
ST (2015)
where
0.60
Black points are RSD data.
0.55
In the scaling matter era,
Negative for
fs
____
8
0.50
0.45
0.40
0.35
It would be of interest to see
the feature of weak gravity
persists in future observations.
0.30
0
0.2
0.4
0.6
z
0.8
1
Summary and outlook