Einstein Frame Research Articles

Fifth forces and broken scale symmetries in the Jordan frame

We study the origin of fifth forces in scalar-tensor theories of gravity in the so-called Jordan frame, where the modifications to the gravitational sector are manifest. We focus on theories of Brans-Dicke type in which an additional scalar field is coupled directly to the Ricci scalar of General Relativity. We describe how the necessary diffeomorphism invariance of the modified gravitational sector leads to a modification of the usual gauge fixing term (for the harmonic gauge), as compared to Einstein gravity. This allows us to perform a consistent linearization of the gravitational sector in the weak-field limit, which gives rise to a kinetic mixing between the non-minimally coupled scalar field and the graviton. It is through this mixing that a fifth force can arise between matter fields. We are then able to compute the matrix elements for fifth-force exchanges directly in the Jordan frame, without the need to perform a conformal transformation to the so-called Einstein frame, wherein the gravitational sector is of Einstein-Hilbert form. We obtain results that are in agreement with the equivalent Einstein-frame calculations and illustrate, still in the Jordan frame, the pivotal role that sources of explicit scale symmetry breaking in the matter sector play in admitting fifth-force couplings.

On the Conformal Frames in f(R) Gravity

We discuss gravitational physics in the Jordan and Einstein frames of f(R) gravity coupled to the Standard Model. We elucidate the way in which the observed gravitational coupling arises in the Einstein frame for generic f(R). We point out that the effect of “running units” in the Einstein frame is related to the fact that the explicit and implicit quantum parameters of the Standard Model, such as the Higgs vacuum expectation value and the parameter ΛQCD, are modified by the conformal transformation of the metric and matter fields and become scalaron-dependent. Considering the scalaron of f(R) gravity describing dark matter, we show that the effect of running units in this case is extremely weak, making two frames practically equivalent.

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

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.

Subcritical hybrid inflation in a generalized superconformal model

We study a generalized superconformal model that gives rise to a subcritical regime of $D$-term hybrid inflation. Exhibiting the model both in a Jordan frame and in the Einstein frame, the effective potential of the subcritical regime is derived in the Einstein frame. It turns out the inflaton-waterfall field dynamics leads to various types of inflaton potential. Consequently the tensor-to-scalar ratio is found to range from $10^{-4}$ ($10^{-3}$) to $0.1$ for getting 60 (50) $e$-folds before the end of inflation.

Thermodynamic structure of a generic null surface and the zeroth law in scalar-tensor theory

We show that the equation of motion of scalar-tensor theory acquires thermodynamic identity when projected on a generic null surface. The relevant projection is given by $E_{ab}l^ak^b$, where $E_{ab} =8\pi T_{ab}^{(m)}$ represents the equation motion for gravitational field in presence of external matter, $l^a$ is the generator of the null surface and $k^a$ is the corresponding auxiliary null vector. Our analysis is done completely in a covariant way. Therefore all the thermodynamic quantities are in covariant form and hence can be used for any specific form of metric adapted to a null surface. We show this both in Einstein and Jordan frames and find that these two frames provide equivalent thermodynamic quantities. This is consistent with the previous findings for a Killing horizon. Also, a concrete proof of the zeroth law in scalar-tensor theory is provided when the null surface is defined by a Killing vector.

Weak-field regime of the generalized hybrid metric-Palatini gravity

In this work we explore the dynamics of the generalized hybrid metric-Palatini theory of gravity in the weak-field, slow-motion regime. We start by introducing the equivalent scalar-tensor representation of the theory, which contains two scalar degrees of freedom, and perform a conformal transformation to the Einstein frame. Linear perturbations of the metric in a Minkowskian background are then studied for the metric and both scalar fields. The effective Newton constant and the PPN parameter $\ensuremath{\gamma}$ of the theory are extracted after transforming back to the (original) Jordan frame. Two particular cases where the general method ceases to be applicable are approached separately. A comparison of these results with observational constraints is then used to impose bounds on the masses and coupling constants of the scalar fields.

Scalarons mimicking dark matter in the Hu–Sawicki model of f(R ) gravity

In this paper, we conduct a study on the scalar field obtained from [Formula: see text] gravity via Weyl transformation of the spacetime metric [Formula: see text] from the Jordan frame to the Einstein frame. The scalar field is obtained as a result of the modification in the geometrical part of Einstein’s field equation of General Relativity. For the Hu–Sawicki model of [Formula: see text] gravity, we find the effective potential of the scalar field and calculate its mass. Our study shows that the scalar field (also named as scalaron) obtained from this model has the chameleonic property, i.e. the scalaron becomes light in the low-density region, while it becomes heavy in the high-density region of matter. Then it is found that the scalaron can be regarded as a dark matter (DM) candidate since the scalaron mass is found to be quite close to the mass of ultralight axions, a prime DM candidate. Thus, the scalaron in the Hu–Sawicki model of [Formula: see text] gravity behaves as DM. Further, a study on the evolution of the scalaron mass with the redshift is also carried out, which depicts that scalaron becomes light with expansion of the Universe and with different rates at different stages of the Universe.

Nonclassical cosmological dynamics in the low-energy limit of loop quantum scalar-tensor theory

In previous work, we showed that in loop quantum cosmology of scalar-tensor theory (STT) with the holonomy correction the background equations of motion in the Jordan frame have two branches, i.e., the ${b}_{+}$ branch and the ${b}_{\ensuremath{-}}$ branch. In the low-energy limit, the ${b}_{+}$ branch of the equations of motion reproduce the equations of motion of classical STT, while the ${b}_{\ensuremath{-}}$ branch of equations of motion do not reproduce the classical equations. In this paper, we investigate cosmological dynamics in an expanding universe whose background is described by the ${b}_{\ensuremath{-}}$ branch of equations of motion of STT, and we especially focus on the dynamics of the perturbations in the low-energy limit because it is most relevant to the current observational range. First, we show that the low-energy limit of the ${b}_{\ensuremath{-}}$ branch of equations of motion can be a stable attractor in the expansion phase of a universe. Then, we find a low-energy effective Hamiltonian on the spatially flat Friedmann-Robertson-Walker background. The background part of this Hamiltonian can yield the low-energy limit of the ${b}_{\ensuremath{-}}$ branch of equations, and this Hamiltonian consists of constraints whose constraint algebra is different from the classical case but also closed up to arbitrary order of perturbations. Remarkably, we find that this Hamiltonian can be transformed into the Hamiltonian of the Einstein frame by field redefinitions different from the classical case. Moreover, we also develop the linear cosmological perturbation theory and apply it to study the slow-roll inflation in this context. Finally, we study a specific model of STT. In this model, a contracting universe described by classical STT in the remote past can pass through the bounce and evolve into an expanding universe whose background dynamics is described by the ${b}_{\ensuremath{-}}$ branch of equations of motion. It is also shown that the slow-roll inflation can take place in this case, and the spectral indices of the slow-roll inflation agree well with the observations. The results in this paper indicate that there exists an alternative consistent theory which is different from the classical theory in the low-energy limit of loop quantum STT.

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

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.

Inflation story: slow-roll and beyond

We present constraints on inflationary dynamics and features in the primordial power spectrum of scalar perturbations using the Cosmic Microwave Background temperature, polarization data from Planck 2018 data release and updated likelihoods. We constrain the slow-roll dynamics using Hilltop Quartic Potential and Starobinsky R + R 2 model in the Einstein frame using the Planck 2018 binned Plik likelihood. Using the Hilltop as base potential, we construct Whipped Inflation potential to introduce suppression in the scalar power spectrum at large angular scales. We notice marginal (68% C.L.) preference of suppression from the large scale temperature angular power spectrum. However, large-scale E-mode likelihood based on high frequency instrument cross spectrum, does not support this suppression and in the combined data the preference towards the suppression becomes negligible. Based on the Hilltop and Starobinsky model, we construct the Wiggly Whipped Inflation potentials to introduce oscillatory features along with the suppression. We use unbinned data from the recently released CamSpec v12.5 likelihood which updates Planck 2018 results. We compare the Bayesian evidences of the feature models with their baseline slow-roll potentials. We find that the complete slow-roll baseline potential is moderately preferred against potentials which generate features. Compared to Planck 2015 PlikHM bin1 likelihood, we find that the significance of sharp features has decreased owing to the updates in the data analysis pipeline. We also compute the bispectra for the best fit candidates obtained from our analysis.

Late-time acceleration due to a generic modification of gravity and the Hubble tension

We consider a scenario of modified gravity, which is generic to late-time acceleration, namely, acceleration in the Jordan frame and no acceleration in the Einstein frame. The possibility is realized by assuming an interaction between dark matter and the baryonic component in the Einstein frame which is removed by going to the Jordan frame using a disformal transformation giving rise to an exotic effective fluid responsible for causing phantom crossing at late times. In this scenario, past evolution is not distinguished from $\Lambda$CDM but late time dynamics is generically different due to the presence of phantom crossing that causes a monotonous increase in the expansion rate giving rise to distinctive late-time cosmic feature. The latter can play a crucial role in addressing the tension between the observed value of Hubble parameter by CMB (Cosmic Microwave Background) measurements and the local observations. We demonstrate that the Hubble tension significantly reduces in the scenario under consideration for the chosen scale factor parametrizations. The estimated age of the universe in the model is well within the observational bounds in the low and high red-shift regimes.

Ghost-free [formula omitted] gravity

In this work we shall address the ghost issue of F(R,G) gravity, which is known to be plagued with ghost degrees of freedom. These ghosts occur due to the presence of higher than two derivatives in the field equations, and can arise even when considering cosmological perturbations, where superluminal modes may arise in the theory. If we consider the quantum theory, the ghosts generate the negative norm states, which give the negative probabilities, and therefore the ghosts are physically inconsistent. Motivated by the importance of F(R,G) gravity for providing viable inflationary and dark energy phenomenologies, in this work we shall provide a technique that can render F(R,G) gravity theories free from ghost degrees of freedom. This will be done by introducing two auxiliary scalar fields, and by employing the Lagrange multiplier technique, the theory is ghost free in the Einstein frame. Also the framework can be viewed as a reconstruction technique and can be used as a method in order to realize several cosmological evolutions of interest. We demonstrate how we can realize several cosmologically interesting phenomenologies by using the reconstruction technique.

Critical analysis of modulus stabilization in a higher dimensional F(R) gravity

An exact solution for the bulk 5-dimensional geometry is derived for F(R) gravity with non-flat de-Sitter 3-branes located at the $M_4 \times Z_2$ orbifold boundaries. The corresponding form of F(R) that leads to such an exact solution of the bulk metric is derived which turns out to have all positive integer powers of R.In such a scenario the stability issue of the modulus (radion field) is analyzed critically for different curvature epochs in both Einstein and Jordan frames. The radion in the effective 4-d theory exhibits a phantom epoch making this model viable for a non-singular bounce. Simultaneous resolution of the gauge hierarchy problem is exhibited through the resulting stable vaue of the radion field in the effective $3+1$ dimensional theory.

On the Trans-Planckian Censorship Conjecture and the generalized non-minimal coupling

We investigate the Trans-Planckian Censorship Conjecture (TCC) and the arising bounds on the inflationary cosmology caused by that conjecture. In that investigation, we analyze TCC bounds for both Jordan and Einstein frames in the presence of a generic non-minimal coupling (to gravity) term. That term allows us to use the functional freedom it brings to the inflationary Lagrangian as an effective Planck mass. In this sense, we argue one should consider the initial field value of the effective Planck mass for the TCC. We show that as a result, one can remove the TCC upper bounds without the need to produce a new process or go beyond the standard inflation mechanism, with the generalized non-minimal coupling, and for Higgs-like symmetry-breaking potentials.

Scalar fields near compact objects: resummation versus UV completion

Low-energy effective field theories containing a light scalar field are used extensively in cosmology, but often there is a tension between embedding such theories in a healthy UV completion and achieving a phenomenologically viable screening mechanism in the IR.Here, we identify the range of interaction couplings which allow for a smooth resummation of classical non-linearities (necessary for kinetic/Vainshtein-type screening), and compare this with the range allowed by unitarity, causality and locality in the underlying UV theory.The latter region is identified using positivity bounds on the 2→2 scattering amplitude, and in particular by considering scattering about a non-trivial background for the scalar we are able to place constraints on interactions at all orders in the field (beyond quartic order).We identify two classes of theories can both exhibit screening and satisfy existing positivity bounds, namely scalar-tensor theories of P(X) or quartic Horndeski type in which the leading interaction contains an odd power of X.Finally, for the quartic DBI Galileon (equivalent to a disformally coupled scalar in the Einstein frame),the analogous resummation can be performed near two-body systems and imposing positivity constraints introduces a non-perturbative ambiguity in the screened scalar profile.These results will guide future searches for UV complete models which exhibit screening of fifth forces in the IR.

Integrable cosmological models with an additional scalar field

We consider modified gravity cosmological models that can be transformed into two-field chiral cosmological models by the conformal metric transformation. For the R^2 gravity model with an additional scalar field and the corresponding two-field model with the cosmological constant and nonstandard kinetic part of the action, the general solutions have been obtained in the spatially flat FLRW metric. We analyze the correspondence of the cosmic time solutions obtained and different possible evolutions of the Hubble parameters in the Einstein and Jordan frames.

Spherically Symmetric Space-Times in Generalized Hybrid Metric-Palatini Gravity

We discuss vacuum static, spherically symmetric asymptotically flat solutions of the generalized hybrid metric-Palatini theory of gravity (generalized HMPG) suggested by B\"ohmer and Tamanini, involving both a metric $g_{\mu\nu}$ and an independent connection $\hat \Gamma_{\mu\nu}{}^\alpha$; the gravitational field Lagrangian is an arbitrary function $f(R,P)$ of two Ricci scalars, $R$ obtained from $g_{\mu\nu}$ and $P$ obtained from $\hat \Gamma_{\mu\nu}{}^\alpha$. The theory admits a scalar-tensor representation with two scalars $\phi$ and $\xi$ and a potential $V(\phi,\xi)$ whose form depends on $f(R,P)$. Solutions are obtained in the Einstein frame and transferred back to the original Jordan frame for a proper interpretation. In the completely studied case $V \equiv 0$, generic solutions contain naked singularities or describe traversable wormholes, and only some special cases represent black holes with extremal horizons. For $V(\phi,\xi) \ne 0$, some examples of analytical solutions are obtained and shown to possess naked singularities. Even in the cases where the Einstein-frame metric $g^E_{\mu\nu}$ is found analytically, the scalar field equations need a numerical study, and if $g^E_{\mu\nu}$ contains a horizon, in the Jordan frame it turns to a singularity due to the corresponding conformal factor.

Conformal transformation with multiple scalar fields and geometric property of field space with Einstein-like solutions

Multiple scalar fields appear in vast modern particle physics and gravity models. When they couple to gravity non-minimally, conformal transformation is utilized to bring the theory into Einstein frame. However, the kinetic terms of scalar fields are usually not canonical, which makes analytic treatment difficult. Here we investigate under what conditions the theories can be transformed to the quasi-canonical form, in which case the effective metric tensor in field space is conformally flat. We solve the relevant nonlinear partial differential equations for arbitrary number of scalar fields and present several solutions that may be useful for future phenomenological model building, including the $\sigma$-model with a particular non-minimal coupling. We also find conformal flatness can always be achieved in some modified gravity theories, for example, Starobinsky model.

Scale-invariant supergravity theory in component formulation

We formulate N=1 supergravity theory in four-dimensions with local scale invariance in semi-on-shell component formulation in four dimensions. The algebra we adopt has the generators (Pm,Mmn,Qα,S) consisting of the generators in super-Poincaré algebra: (Pm,Mmn,Qα) with 4+6+4 components, and that of local scale-transformation S with one additional component. Our field content consists of the on-shell multiplet of supergravity (eμm,ψμα), the tensor multiplet (Bμν,χα,φ) and the ‘Weylon multiplet’ (Sμ,ρα,Cμνρ), where Sμ is the Weylon that gauges local scale symmetry, ψμ,χ and ρ are Majorana spinors, while Cμνρ is a tensor auxiliary field. Similar to the non-supersymmetric case that has no minimal coupling between the Sμ-field and spin 1/2 field, the Weylon does not minimally couple to spin 3/2 gravitino field in our locally supersymmetric generalization. By going to the Einstein frame from the original Jordan frame, we show that our system is equivalent to a system with the gauged dilaton-shift symmetry. This implies that our system bypasses persistent dilaton-related problems, avoiding a long-distance force for a massless dilaton, a minimum value of its potential, or a run-away vacuum.

Conformal gravity does not predict flat galaxy rotation curves

We reconsider the widely held view that the Mannheim--Kazanas (MK) vacuum solution for a static, spherically-symmetric system in conformal gravity (CG) predicts flat rotation curves, such as those observed in galaxies, without the need for dark matter. This prediction assumes that test particles have fixed rest mass and follow timelike geodesics in the MK metric in the vacuum region exterior to a spherically-symmetric representation of the galactic mass distribution. Such geodesics are not conformally invariant, however, which leads to an apparent discrepancy with the analogous calculation performed in the conformally-equivalent Schwarzschild-de-Sitter (SdS) metric, where the latter does not predict flat rotation curves. This difference arises since the mass of particles in CG must instead be generated dynamically through interaction with a scalar field. The energy-momentum of this required scalar field means that, in a general conformal frame from the equivalence class of CG solutions outside a static, spherically-symmetric matter distribution, the spacetime is not given by the MK vacuum solution. A unique frame does exist, however, for which the metric retains the MK form, since the scalar field energy-momentum vanishes despite the field being non-zero and radially dependent. Nonetheless, we show that in both this MK frame and the Einstein frame, in which the scalar field is constant, massive particles follow timelike geodesics of the SdS metric, thereby resolving the apparent frame dependence of physical predictions and unambiguously yielding rotation curves with no flat region. We also comment on how our analysis resolves the long-standing uncertainty regarding gravitational lensing in the MK metric. (Abridged)