Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories
Abstract
It is known that Horndeski theories can be transformed to a subclass of GleyzesLangloisPiazzaVernizzi (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 secondorder 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 nexttoleading order in slowroll. 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 redtilted.
 Authors:
 Department of Physics, Faculty of Science, Tokyo University of Science,13, Kagurazaka, Shinjukuku, Tokyo 1628601 (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: PUBLISHERID: 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:10.1088/14757516/2015/04/043.
Tsujikawa, Shinji. Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories. United States. doi:10.1088/14757516/2015/04/043.
Tsujikawa, Shinji. 2015.
"Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories". United States.
doi:10.1088/14757516/2015/04/043.
@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 subclass of GleyzesLangloisPiazzaVernizzi (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 secondorder 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 nexttoleading order in slowroll. 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 redtilted.},
doi = {10.1088/14757516/2015/04/043},
journal = {Journal of Cosmology and Astroparticle Physics},
number = 04,
volume = 2015,
place = {United States},
year = 2015,
month = 4
}

It is known that Horndeski theories can be transformed to a subclass of GleyzesLangloisPiazzaVernizzi (GLPV) theories under the disformal transformation of the metric g{sub μ ν} → Ω{sup 2}(φ)g{sub μ ν}+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 μ ν}∇{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 secondorder actions of scalar/tensor perturbationsmore »

Spatially covariant theories of gravity: disformal transformation, cosmological perturbations and the Einstein frame
We investigate the cosmological background evolution and perturbations in a general class of spatially covariant theories of gravity, which propagates two tensor modes and one scalar mode. We show that the structure of the theory is preserved under the disformal transformation. We also evaluate the primordial spectra for both the gravitational waves and the curvature perturbation, which are invariant under the disformal transformation. Due to the existence of higher spatial derivatives, the quadratic Lagrangian for the tensor modes itself cannot be transformed to the form in the Einstein frame. Nevertheless, there exists a oneparameter family of frames in which themore » 
Cosmological disformal invariance
The invariance of physical observables under disformal transformations is considered. It is known that conformal transformations leave physical observables invariant. However, whether it is true for disformal transformations is still an open question. In this paper, it is shown that a pure disformal transformation without any conformal factor is equivalent to rescaling the time coordinate. Since this rescaling applies equally to all the physical quantities, physics must be invariant under a disformal transformation, that is, neither causal structure, propagation speed nor any other property of the fields are affected by a disformal transformation itself. This fact is presented at themore » 
Cosmological perturbations in mimetic Horndeski gravity
We study linear scalar perturbations around a flat FLRW background in mimetic Horndeski gravity. In the absence of matter, we show that the Newtonian potential satisfies a secondorder differential equation with no spatial derivatives. This implies that the sound speed for scalar perturbations is exactly zero on this background. We also show that in mimetic G {sup 3} theories the sound speed is equally zero. We obtain the equation of motion for the comoving curvature perturbation (first order differential equation) and solve it to find that the comoving curvature perturbation is constant on all scales in mimetic Horndeski gravity. Wemore » 
The two faces of mimetic Horndeski gravity: disformal transformations and Lagrange multiplier
We show that very general scalartensor theories of gravity (including, e.g., Horndeski models) are generically invariant under disformal transformations. However there is a special subset, when the transformation is not invertible, that yields new equations of motion which are a generalization of the socalled 'mimetic' dark matter theory recently introduced by Chamsedinne and Mukhanov. These conclusions hold true irrespective of whether the scalar field in the action of the assumed scalartensor theory of gravity is the same or different than the scalar field involved in the transformation. The new equations of motion for our general mimetic theory can also bemore »