Some aspects of the cosmological conformal equivalence between the `Jordan frame' and the `Einstein frame'

We review the conformal equivalence in describing the background expansion of the universe by [Formula: see text] gravity both in the Jordan frame and the Einstein frame. In the Jordan frame, we present the general analytic expression for [Formula: see text] models that have the same expansion history as the [Formula: see text]CDM model. This analytic form can provide further insights on how cosmology can be used to test the [Formula: see text] gravity at the largest scales. Moreover we present a systematic and self-consistent way to construct the viable [Formula: see text] model in Jordan frame using the mass dilation rate function from the Einstein frame through the conformal transformation. In addition, we extend our study to the linear perturbation theories and we further exhibit the equivalence of the [Formula: see text] gravity presented in the Jordan frame and Einstein frame in the perturbed space–time. We argue that this equivalence has solid physics root.

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Big bounce with finite-time singularity: The F(R) gravity description

International Journal of Modern Physics D ◽

10.1142/s0218271817500857 ◽

2017 ◽

Vol 26 (08) ◽

pp. 1750085 ◽

Cited By ~ 29

Author(s):

S. D. Odintsov ◽

V. K. Oikonomou

Keyword(s):

Big Bang ◽

Einstein Frame ◽

Jordan Frame ◽

Cosmological Perturbations ◽

Scale Invariant ◽

Type Iv ◽

Big Rip ◽

Cosmological Scenario ◽

The Big Bang ◽

Big Bounce

An alternative to the Big Bang cosmologies is obtained by the Big Bounce cosmologies. In this paper, we study a bounce cosmology with a Type IV singularity occurring at the bouncing point in the context of [Formula: see text] modified gravity. We investigate the evolution of the Hubble radius and we examine the issue of primordial cosmological perturbations in detail. As we demonstrate, for the singular bounce, the primordial perturbations originating from the cosmological era near the bounce do not produce a scale-invariant spectrum and also the short wavelength modes after these exit the horizon, do not freeze, but grow linearly with time. After presenting the cosmological perturbations study, we discuss the viability of the singular bounce model, and our results indicate that the singular bounce must be combined with another cosmological scenario, or should be modified appropriately, in order that it leads to a viable cosmology. The study of the slow-roll parameters leads to the same result indicating that the singular bounce theory is unstable at the singularity point for certain values of the parameters. We also conformally transform the Jordan frame singular bounce, and as we demonstrate, the Einstein frame metric leads to a Big Rip singularity. Therefore, the Type IV singularity in the Jordan frame becomes a Big Rip singularity in the Einstein frame. Finally, we briefly study a generalized singular cosmological model, which contains two Type IV singularities, with quite appealing features.

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FRAME TRANSFORMATIONS OF GRAVITATIONAL THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x13500425 ◽

2013 ◽

Vol 28 (12) ◽

pp. 1350042 ◽

Cited By ~ 19

Author(s):

XAVIER CALMET ◽

TING-CHENG YANG

Keyword(s):

Gravitational Theory ◽

Space Time ◽

Einstein Frame ◽

Jordan Frame ◽

Quantum Field ◽

Quantum Fields ◽

Theoretical Level ◽

Curved Space ◽

Inflation Model ◽

Gravitational Theories

We show how to map gravitational theories formulated in the Jordan frame to the Einstein frame at the quantum field theoretical level considering quantum fields in curved space–time. As an example, we consider gravitational theories in the Jordan frame of the type F(ϕ, R) = f(ϕ)R-V(ϕ) and perform the map to the Einstein frame. Our results can easily be extended to any gravitational theory. We consider the Higgs inflation model as an application of our results.

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Is Spacetime Non-metric?

Expert Opinion on Astronomy and Astrophysics ◽

10.18063/eoaa.v2i1.384 ◽

2018 ◽

Vol 2 (1) ◽

Cited By ~ 2

Author(s):

Mark D. Roberts

Keyword(s):

General Relativity ◽

Higher Dimensions ◽

Christoffel Symbol ◽

Einstein Frame ◽

Jordan Frame ◽

Scalar Tensor Theory ◽

Field Equations ◽

Extra Information ◽

Tensor Theory ◽

Palatini Variation

If one assumes higher dimensions and that dimensional reduction from higher dimensions produces scalar-tensor theory and also that Palatini variation is the correct method of varying scalar-tensor theory then spacetime is nonmetric. Palatini variation of Jordan frame lagrangians gives an equation relating the dilaton to the object of non-metricity and hence the existence of the dilaton implies that the spacetime connection is more general than that given soley by the Christoffel symbol of general relativity. Transferring from Jordan to Einstein frame, which connection, lagrangian, field equations and stress conservation equations occur are discussed: it is found that the Jordan frame has more information, this can be expressed in several ways, the simplest is that the extra information corresponds to the function multiplying the Ricci scalar in the action. The Einstein frame has the advantages that stress conservation implies no currents and that the field equations are easier to work with. This is illustrated by application to Robertson-Walker spacetime.

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A NOTE ON THE UNIFIED FIRST LAW IN f(R) GRAVITY THEORY

International Journal of Modern Physics D ◽

10.1142/s0218271812500344 ◽

2012 ◽

Vol 21 (04) ◽

pp. 1250034 ◽

Cited By ~ 4

Author(s):

YI ZHANG ◽

YUNGUI GONG ◽

ZONG-HONG ZHU

Keyword(s):

Field Theory ◽

Scalar Field ◽

Conformal Transformation ◽

Dynamical Equation ◽

Einstein Frame ◽

Jordan Frame ◽

View Point ◽

Minimal Coupling ◽

Friedmann Equations ◽

Dynamical Equivalence

Because of the dynamical equivalence between the f(R) gravity and the Brans-Dicke theory, the dynamical equation in the f(R) gravity is suggested to be derived from a view point of thermodynamics here. By a conformal transformation, the Brans–Dicke theory in the Jordan frame could be expressed as a minimal coupling scalar field theory in Einstein frame. Using the entropy-area relation [Formula: see text], the correct Friedmann equations could be gotten in both frames. Furthermore, we also discuss the corresponding generalized Misner–Sharp energies for theoretical consistence.

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Scalar–tensor theories of gravity: validity of the cosmic no-hair conjecture

Canadian Journal of Physics ◽

10.1139/p11-074 ◽

2011 ◽

Vol 89 (9) ◽

pp. 937-940

Author(s):

Sudeshna Mukerji ◽

Nairwita Mazumder ◽

Ritabrata Biswas ◽

Subenoy Chakraborty

Keyword(s):

Coupling Parameter ◽

Matter Field ◽

Einstein Frame ◽

Energy Conditions ◽

Jordan Frame ◽

Scalar Tensor Theory ◽

Bianchi Models ◽

Tensor Theory ◽

Theories Of Gravity ◽

Theory Of Gravity

This paper deals with the cosmic no-hair conjecture for anisotropic Bianchi models in the scalar–tensor theory of gravity. Herein, we have considered both the Jordan frame and the Einstein frame to describe the scalar–tensor theory of gravity and examine the conjecture. In the Jordan frame, one should restrict both the coupling function of the scalar field and the coupling parameter, in addition to the usual energy conditions for the matter field, to maintain the validity of the cosmic no-hair conjecture, while in the Einstein frame, the restrictions are purely on the energy conditions.

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f(R) dual theories of quintessence: expansion-collapse duality

Journal of Cosmology and Astroparticle Physics ◽

10.1088/1475-7516/2021/12/016 ◽

2021 ◽

Vol 2021 (12) ◽

pp. 016

Author(s):

Dipayan Mukherjee ◽

H.K. Jassal ◽

Kinjalk Lochan

Keyword(s):

Dark Energy ◽

Cosmological Constant ◽

Einstein Frame ◽

Time Limit ◽

Gravity Models ◽

Accelerated Expansion ◽

Jordan Frame ◽

Late Time ◽

Perturbative Solution ◽

The Universe

Abstract The accelerated expansion of the universe demands presence of an exotic matter, namely the dark energy. Though the cosmological constant fits this role very well, a scalar field minimally coupled to gravity, or quintessence, can also be considered as a viable alternative for the cosmological constant. We study f(R) gravity models which can lead to an effective description of dark energy implemented by quintessence fields in Einstein gravity, using the Einstein frame-Jordan frame duality. For a family of viable quintessence models, the reconstruction of the f(R) function in the Jordan frame consists of two parts. We first obtain a perturbative solution of f(R) in the Jordan frame, applicable near the present epoch. Second, we obtain an asymptotic solution for f(R), consistent with the late time limit of the Einstein frame if the quintessence field drives the universe. We show that for certain class of viable quintessence models, the Jordan frame universe grows to a maximum finite size, after which it begins to collapse back. Thus, there is a possibility that in the late time limit where the Einstein frame universe continues to expand, the Jordan frame universe collapses. The condition for this expansion-collapse duality is then generalized to time varying equations of state models, taking into account the presence of non-relativistic matter or any other component in the Einstein frame universe. This mapping between an expanding geometry and a collapsing geometry at the field equation level may have interesting potential implications on the growth of perturbations therein at late times.

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Question of quantum equivalence between Jordan frame and Einstein frame

Physical Review D ◽

10.1103/physrevd.91.084033 ◽

2015 ◽

Vol 91 (8) ◽

Cited By ~ 77

Author(s):

Alexander Yu. Kamenshchik ◽

Christian F. Steinwachs

Keyword(s):

Einstein Frame ◽

Jordan Frame

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Jordan and Einstein Frames Hamiltonian Analysis for FLRW Brans-Dicke Theory

Universe ◽

10.3390/universe8010014 ◽

2021 ◽

Vol 8 (1) ◽

pp. 14

Author(s):

Matteo Galaverni ◽

Gabriele Gionti S. J.

Keyword(s):

Equations Of Motion ◽

Einstein Frame ◽

Jordan Frame ◽

Inverse Transformation ◽

Conformal Transformations ◽

Hamiltonian Equations ◽

Flrw Universe ◽

Jordan And Einstein Frames ◽

Hamiltonian Analysis

We analyze the Hamiltonian equivalence between Jordan and Einstein frames considering a mini-superspace model of the flat Friedmann–Lemaître–Robertson–Walker (FLRW) Universe in the Brans–Dicke theory. Hamiltonian equations of motion are derived in the Jordan, Einstein, and anti-gravity (or anti-Newtonian) frames. We show that, when applying the Weyl (conformal) transformations to the equations of motion in the Einstein frame, we did not obtain the equations of motion in the Jordan frame. Vice-versa, we re-obtain the equations of motion in the Jordan frame by applying the anti-gravity inverse transformation to the equations of motion in the anti-gravity frame.

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COMPARISON OF FRAMES: JORDAN VERSUS EINSTEIN FRAME FOR A NON-MINIMAL DARK ENERGY MODEL

Modern Physics Letters A ◽

10.1142/s0217732309031053 ◽

2009 ◽

Vol 24 (38) ◽

pp. 3143-3155 ◽

Cited By ~ 12

Author(s):

KOUROSH NOZARI ◽

S. DAVOOD SADATIAN

Keyword(s):

Dark Energy ◽

Scalar Field ◽

Conformal Transformation ◽

Einstein Frame ◽

Jordan Frame ◽

Energy Component ◽

Phantom Divide ◽

Dark Energy Component ◽

Phantom Divide Line ◽

Line Crossing

We construct a dark energy model where a scalar field non-minimally coupled to gravity plays the role of the dark component. We compare cosmological consequences of this non-minimal coupling of the scalar field and gravity in the spirit of the dark energy paradigm in Jordan and Einstein frames. Some important issues such as phantom divide line crossing, existence of the bouncing solutions and the stability of the solutions are compared in these two frames. We show that while a non-minimally coupled scalar field in the Jordan frame is a suitable dark energy component with capability to realize phantom divide line crossing, its conformal transformation in the Einstein frame does not have this capability. The conformal transformation from Jordan frame to Einstein frame transforms the equation of state parameter of the dark energy component to its minimal form with a redefined scalar field and in this case it is impossible to realize a phantom phase with possible crossing of the phantom divide line.

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