Effective field theory approach to modified gravity with applications to inflation and dark energy

Shinji Tsujikawa

Hot Topics in General RelativityAnd Gravitation, Quy Nhon, 2015

Tokyo University of Science

Collaboration with A. De Felice (Kyoto)L. Gergely (Sedged)R. Kase (TUS)K. Koyama (Portsmouth)

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 theoriesHorndeski (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.

Most general scalar-tensor theories with second-order equations

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 the3+1 decomposition of space-time.

Constant time hypersurface

We choose the unitary gauge on the flat FLRW cosmological background

Then, the scalar perturbation can beabsorbed into the gravitational sector:

ADM metric

We have several geometric tensors:

Extrinsic curvature:

3-dimensional Ricci tensor:

Horndeski Lagrangian in the ADM Language 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

Several scalar quantities can be constructed:

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 ______________Higher spatial derivativesappearing in Horava gravity

____Present in GLPV theories

In the absence of higher spatial derivative, the no-ghost and no-instability conditions are

De Felice and ST (2014)

Equations of motion for tensor perturbations

where

where

We can use this result to derive the tensor power spectrumgenerated 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-orderspectrum

___________________Slow-roll corrections

__________Corrections from

are much smaller than 1.

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-orderaction 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:

Conformaltransformation

Disformal transformation

The GLPV action in the transformed frame reads

with the relations among coefficients

where

Gleyzes et al, JCAP (2014)

Bekenstein (1993)

Same form as that in the original frame

Disformal invariance of cosmological perturbations

Consider the perturbed metric

where

_Curvature perturbations

__Tensorperturbations

ST (2014), See also Minamitsuji (2014)for the case

where

Transformation to the Einstein frame

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 readsSame as the GRtensor spectrum(Stewart and Lyth, 1993)

ST, JCAP (2014)

Creminelli et al, PRL (2014),

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

__Mattersector

Three functions 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 perturbed line element in the longitudinal gauge is

The four velocity of non-relativistic matter is

Effective gravitational coupling with matter

where

The gauge-invariant density contrast

The growth rate of matter perturbations is constrained from peculiar velocities of galaxies in red-shift space distortion measurements.

obeys

___

Recent observations favor weak gravity on cosmological scales.

GR

Planck LCDM fit

RSD fit

Effective gravitational coupling in Horndeski theories

In the massless limit, the effective gravitational coupling in Horndeski theories reads

____ _______Tensor contribution

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.

ST,1505.02459(2015)

This correspond to the intrinsicmodification of the gravitational part.

A model realizing weak gravity beyond the Horndeski domain

ST (2015)

where

In the scaling matter era,

____

Negative for

It would be of interest to see the feature of weak gravity persists in future observations.

Black points are RSD data.

Summary and outlook