# Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories

## Abstract

It is known that Horndeski theories can be transformed to a sub-class of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories under the disformal transformation of the metric g{sub μν}→Ω{sup 2}(ϕ)g{sub μν}+Γ(ϕ,X)∇{sub μ}ϕ∇{sub ν}ϕ, where Ω is a function of a scalar field ϕ and Γ is another function depending on both ϕ and X=g{sup μν}∇{sub μ}ϕ∇{sub ν}ϕ. We show that, with the choice of unitary gauge, both curvature and tensor perturbations on the flat isotropic cosmological background are generally invariant under the disformal transformation. By means of the effective field theories encompassing Horndeski and GLPV theories, we obtain the second-order actions of scalar/tensor perturbations and present the relations for physical quantities between the two frames. The invariance of the inflationary power spectra under the disformal transformation is explicitly proved up to next-to-leading order in slow-roll. In particular, we identify the existence of the Einstein frame in which the tensor power spectrum is of the same form as that in General Relativity and derive the condition under which the spectrum of gravitational waves in GLPV theories is red-tilted.

- Authors:

- Department of Physics, Faculty of Science, Tokyo University of Science,1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan)

- Publication Date:

- Sponsoring Org.:
- SCOAP3, CERN, Geneva (Switzerland)

- OSTI Identifier:
- 22454541

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2015; Journal Issue: 04; Other Information: PUBLISHER-ID: JCAP04(2015)043; OAI: oai:repo.scoap3.org:10093; Article funded by SCOAP3. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; COSMOLOGICAL INFLATION; COSMOLOGY; GENERAL RELATIVITY THEORY; GRAVITATIONAL WAVES; METRICS; PERTURBATION THEORY; SCALAR FIELDS; SCALARS; SPECTRA; TENSORS

### Citation Formats

```
Tsujikawa, Shinji.
```*Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories*. United States: N. p., 2015.
Web. doi:10.1088/1475-7516/2015/04/043.

```
Tsujikawa, Shinji.
```*Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories*. United States. doi:10.1088/1475-7516/2015/04/043.

```
Tsujikawa, Shinji. Mon .
"Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories". United States.
doi:10.1088/1475-7516/2015/04/043.
```

```
@article{osti_22454541,
```

title = {Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories},

author = {Tsujikawa, Shinji},

abstractNote = {It is known that Horndeski theories can be transformed to a sub-class of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories under the disformal transformation of the metric g{sub μν}→Ω{sup 2}(ϕ)g{sub μν}+Γ(ϕ,X)∇{sub μ}ϕ∇{sub ν}ϕ, where Ω is a function of a scalar field ϕ and Γ is another function depending on both ϕ and X=g{sup μν}∇{sub μ}ϕ∇{sub ν}ϕ. We show that, with the choice of unitary gauge, both curvature and tensor perturbations on the flat isotropic cosmological background are generally invariant under the disformal transformation. By means of the effective field theories encompassing Horndeski and GLPV theories, we obtain the second-order actions of scalar/tensor perturbations and present the relations for physical quantities between the two frames. The invariance of the inflationary power spectra under the disformal transformation is explicitly proved up to next-to-leading order in slow-roll. In particular, we identify the existence of the Einstein frame in which the tensor power spectrum is of the same form as that in General Relativity and derive the condition under which the spectrum of gravitational waves in GLPV theories is red-tilted.},

doi = {10.1088/1475-7516/2015/04/043},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 04,

volume = 2015,

place = {United States},

year = {Mon Apr 27 00:00:00 EDT 2015},

month = {Mon Apr 27 00:00:00 EDT 2015}

}