In this paper, we compare dispersions of a scalar field in Euclidean quantum gravity with stochastic inflation. We use Einstein gravity and a minimally coupled scalar field with a quadratic potential. We restrict our attention to small mass and small field cases. In the Euclidean approach, we introduce the ground state wave function which is approximated by instantons. We used a numerical technique to find instantons that satisfy classicality. In the stochastic approach, we introduce the probability distribution of Hubble patches that can be approximated by locally homogeneous universes down to a smoothing scale. We assume that the ground state wave function should correspond to the stationary state of the probability distribution of the stochastic universe. By comparing the dispersion of both approaches, we conclude three main results. (1) For a statistical distribution with a certain value, we can find a corresponding instanton in the Euclidean side, and it should be a complex-valued instanton. (2) The size of the universe of the Euclidean approach corresponds to the smoothing scale of the stochastic side; the universe is homogeneous up to the Euclidean instanton. (3) In addition, as the mass increases up to a critical value, both approaches break at the same time. Hence, generation of classical inhomogeneity in the stochastic approach and the instability of classicality in the Euclidean approach are related.
In this paper, we investigate complex-valued Euclidean wormholes in the Starobinsky inflation. Due to the properties of the concave inflaton potential, the classicality condition at both ends of the wormhole can be satisfied, as long as the initial condition of the inflaton field is such that it is located sufficiently close to the hilltop. We compare the probabilities of classicalized wormholes with the Hartle-Hawking compact instantons and conclude that the Euclidean wormholes are probabilistically preferred than compact instantons, if the inflation lasts more than 50 e-foldings. Our result …
We investigate the initial state of the inflationary universe. In our recent publications, we showed that requesting the gauge invariance in the local observable universe to the initial state guarantees the infrared (IR) regularity of loop corrections in a general single clock inflation. Following this study, in this paper, we show that choosing the Euclidean vacuum ensures the gauge invariance in the local universe and hence the IR regularity of loop corrections. It has been suggested that loop corrections to inflationary perturbations may yield the secular growth, which can lead to the break…
Accelerating dynamics from Born-Infeld-$f(R)$ gravity are studied in a simplified conformal approach without matter. Explicit unification of inflation with late-time acceleration is realized within this singular inflation approach, which is similar to Odintsov-Oikonomou singular $f(R)$ inflation. Our model turns out to be consistent with the latest release of Planck data.
Motivated by the fact that the pre-inflationary era may evolve in an exotic way, in this work we formalize anisotropic evolution in the context of modified gravity, focusing on pre-inflationary and near the vicinity of the inflationary epochs. We specialize on specific metrics like Bianchi and Taub and we formalize the inflationary theory in vacuum $F(R)$ gravity, in $F(R)$ gravity with an extra scalar field and in Gauss-Bonnet gravity. We discuss the qualitative effects of the anisotropies on the evolution of the Universe and also we consider several specific solutions, like the de Sitter sol…
We generalize the notion of constant-roll inflation earlier introduced in General Relativity (GR) and $f(R)$ gravity to inflationary models in more general scalar-tensor gravity. A number of novel exact analytic solutions for a FLRW spatially flat cosmological background is found for this case. All forms of the scalar field potential and its coupling to gravity producing the exact de Sitter solution, while the scalar field is varying, are presented. In the particular cases of induced gravity and GR with a non-minimally coupled scalar field, all constant-roll inflationary solutions are found. I…
We investigate inflation in frames of two classes of $F(R)$ gravity and check its consistency with Planck data. It is shown that $F(R)$ inflation without self-reproduction may be constructed in close analogy with the corresponding scalar example proposed by Mukhanov for the resolution the problems of multiverse, predictability and initial conditions.
We elaborate on the inflationary model starting from multidimensional Lagrangian and gravity with second-order curvature terms. The effective scalar field is related to the Ricci scalar of extra dimensions. It is shown that the Kretschmann and the Ricci tensor square terms dominate during inflation. The observable values of the spectral index and the tensor-to-scalar ratio are obtained for specific values of the model parameters.
We investigate the inflationary realization in the context of unimodular $F(T)$ gravity, which is based on the $F(T)$ modification of teleparallel gravity, in which one imposes the unimodular condition through the use of Lagrange multipliers. We develop the general reconstruction procedure of the $F(T)$ form that can give rise to a given scale-factor evolution, and then we apply it in the inflationary regime. We extract the Hubble slow-roll parameters that allow us to calculate various inflation-related observables, such as the scalar spectral index and its running, the tensor-to-scalar ratio,…
In this work we study certain classes of $F(R,{\cal G})$ gravity which have appealing phenomenological features, with respect to the successful realization of the dark energy and of the inflationary era. Particularly, we discuss the general formalism and we demonstrate how several inflationary and dark energy evolutions can be described in the context of $F(R,{\cal G})$ gravity. Also we propose a unified model, in the context of which the early and late-time dynamics are controlled by the $F(R,{\cal G})$ gravity, thus producing inflation and the dark energy era, while the intermediate era is a…
We discuss the cosmological inflation in the context of $f(Q)$ non-metric gravity, where $Q$ is the non-metric scalar. After introducing conformal transformations for $f(Q)$ gravity, we first focus on the potential-slow-roll inflation by studying the corresponding potentials for different forms of the function $f(Q)$ in the Einstein frame. Secondly, we investigate the Hubble-slow-roll inflation for three classes of inflationary potentials considered for the specific form $f(Q)\propto Q^{2}$, in the Jordan frame. We compare results in both approaches with observations coming from Planck and BIC…
We consider a class of exponential models of $f(R)$ gravity, in the presence of a canonical scalar field, for which at early times the effective Lagrangian of the theory becomes that of a rescaled canonical scalar field with the Einstein-Hilbert term becoming $\sim αR$, with $α$ a dimensionless constant. This rescaled Einstein-Hilbert scalar field theory at early times alters the inflationary phenomenology of well-known scalar field models of inflation, but more importantly, in the context of this rescaled theory, the Swampland criteria are easily satisfied, assuming that the scalar field is s…
We consider the effects of the quantum stress tensor fluctuations of a conformal field in generating gravity waves in inflationary models. We find a non-scale invariant, non-Gaussian contribution which depends upon the total expansion factor between an initial time and the end of inflation. This spectrum of gravity wave perturbations is an illustration of a negative power spectrum, which is possible in quantum field theory. We discuss possible choices for the initial conditions. If the initial time is taken to be sufficiently early, the fluctuating gravity waves are potentially observable both…
Nucleating an Inflationary Universe: Euclidean Wormholes and their No-Boundary Limit
Department of Theoretical Physics, A.Razmadze Mathematical Institute at I.Javakhishvili Tbilisi State University（伊万·贾瓦希什维利第比利斯国立大学A.拉兹马泽数学研究所理论物理系）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））
Since the standard inflationary paradigm is based on quantum field theory on classical space-times, it excludes the Planck era. Using techniques from loop quantum gravity, the paradigm is extended to a self-consistent theory from the Planck scale to the onset of slow roll inflation, covering some 11 orders of magnitude in energy density and curvature. This pre-inflationary dynamics also opens a small window for novel effects, e.g. a source for non-Gaussianities, which could extend the reach of cosmological observations to the deep Planck regime of the early universe.
Emergence of $R^2$ inflation which is the best fit framework for CMB observations till date comes from the attempts to attack the problem of quantization of gravity which in turn have resulted in the trace anomaly discovery. Further developments in trace anomaly and different frameworks aiming to construct quantum gravity indicate an inevitability of non-locality in fundamental physics at small time and length scales. A natural question would be to employ the $R^2$ inflation as a probe for signatures of non-locality in the early Universe physics. Recent advances of embedding $R^2$ inflation in…
We investigate a (super-)renormalizable and ghost-free theory of gravity, showing that under a natural (exponential) ansatz of the form factor and a suitable truncation it can give rise to the Starobinsky inflationary theory in cosmological frameworks, and thus offering a theoretical justification of its origin. We study the corresponding inflationary evolution and we examine the generation of curvature perturbations, adapting the $f(R)$-like equations in a symmetry-reduced FLRW metric. Furthermore, we analyze how the ultraviolet regime of a simply renormalizable and unitary theory of gravity …
Cosmological models in 1+1 dimensions are an ideal setting for investigating the quantum structure of inflationary dynamics -- gravity is renormalizable, while there is room for spatial structure not present in the minisuperspace approximation. We use this fortuitous convergence to investigate the mechanism of slow-roll eternal inflation. A variant of 1+1 Liouville gravity coupled to matter is shown to model precisely the scalar sector of cosmological perturbations in 3+1 dimensions. A particular example of quintessence in 1+1d is argued on the one hand to exhibit slow-roll eternal inflation a…
Inflation is studied in the context of induced gravity (IG) $γσ^2 R$, where $R$ is the Ricci scalar, $σ$ a scalar field and $γ$ a dimensionless constant. We study in detail cosmological perturbations in IG and examine both a Landau-Ginzburg (LG) and a Coleman-Weinberg (CW) potential toy models for small field and large field (chaotic) inflation and find that small field inflationary models in IG are constrained to $γ\lesssim 3 \times 10^{-3}$ by WMAP 5 yrs data. Finally we describe the regime of coherent oscillations in induced gravity by an analytic approximation, showing how the homogeneous …
A general equation, describing the lowest order corrections coming from quantum gravitational effects to the spectrum of cosmological scalar fluctuations is obtained. These corrections are explicitly estimated for the case of a de Sitter evolution.
An intermediate inflationary universe model within the context of non-minimally coupled to the scalar curvature is analyzed. We will conduct our analysis under the slow roll approximation of the inflationary dynamics and the cosmological perturbations considering a coupling of the form $F(φ)=κ+ξ_nφ^n$. Considering the trajectories in the $r-n_s$ plane from Planck data, we find the constraints on the parameter-space in our model.
We explore trace-anomaly driven inflation in modified gravity. It is explicitly shown that in $T^2$ teleparallel gravity, the de Sitter inflation can occur, although quasi de Sitter inflation happens in $R^2$ gravity. Furthermore, we investigate the influence of the trace anomaly on inflation. It is found that in $f(T)$ gravity, the de Sitter inflation can end because it becomes unstable due to the trace anomaly, whereas also in higher derivative gravity, the de Sitter inflation can be realized and it will be over thanks to the trace anomaly for smaller parameter regions in comparison with tho…
We review inflationary cosmology in modified gravity such as $R^2$ gravity with its extensions in order to generalize the Starobinsky inflation model. In particular, we explore inflation realized by three kinds of effects: modification of gravity, the quantum anomaly, and the $R^2$ term in loop quantum cosmology. It is explicitly demonstrated that in these inflationary models, the spectral index of scalar modes of the density perturbations and the tensor-to-scalar ratio can be consistent with the Planck results. Bounce cosmology in $F(R)$ gravity is also explained.
We study slow-roll inflation and the generation of primordial fluctuations in $F(T)$ gravity's rainbow. We obtain the second order action for scalar and tensor perturbations and then calculate the primordial power spectrum for them. Thus, after calculating the inflationary observables up to first order in slow-roll approximation, namely the scalar spectral index $n_{s}$ and the tensor-to scalar ratio $r$, we confront the predictions of the model with the current PLANCK and BICEP/Keck data.
We use the Wilsonian functional Renormalisation Group (RG) to study quantum corrections for the Higgs inflationary action including the effect of gravitons, and analyse the leading-order quantum gravitational corrections to the Higgs' quartic coupling, as well as its non-minimal coupling to gravity and Newton's constant, at the inflationary regime and beyond. We explain how within this framework the effect of Higgs and graviton loops can be sufficiently suppressed during inflation, and we also place a bound on the corresponding value of the infrared RG cut-off scale during inflation. Finally, …
We study cosmological inflation on a warped DGP braneworld where inflaton field is non-minimally coupled to induced gravity on the brane. We present a detailed calculation of the perturbations and inflation parameters both in Jordan and Einstein frame. We analyze the parameters space of the model fully to justify about the viability of the model in confrontation with recent observational data. We compare the results obtained in these two frames also in order to judge which frame gives more acceptable results in comparison with observational data.
In this letter we shall provide a formal derivation of the inflationary slow-roll indices for $F(R)$ gravity from the power spectrum without the condition $\dotε_1=0$, frequently used in the literature, where $ε_1=-\frac{\dot{H}}{H^2}$ is the first slow-roll index. We shall employ Karamata's theorem for regularly varying functions, and as we show, we shall derive the same expressions for the scalar and tensor spectral index of $F(R)$ gravity, also appearing in the current literature, without the misleading and rather strong condition $\dotε_1=0$. The only conditions that are needed are the slo…
In this work, we studied the slow-roll approximation of cosmic inflation within the context of $f(R,T)$ gravity, where $R$ is the scalar curvature, and $T$ is the trace of the energy-momentum tensor. By choosing a minimal coupling between matter and gravity, we obtained the modified slow-roll parameters, the scalar spectral index ($n_s$), the tensor spectral index ($n_{\textrm{T}}$), and the tensor-to-scalar ratio ($r$). We computed these quantities for a general power-law potential, Natural & Quartic Hilltop inflation, and the Starobinsky model, plotting the trajectories on the $(n_s,r)$ plan…
It is sometimes argued that observation of tensor modes from inflation would provide the first evidence for quantum gravity. However, in the usual inflationary formalism, also the scalar modes involve quantised metric perturbations. We consider the issue in a semiclassical setup in which only matter is quantised, and spacetime is classical. We assume that the state collapses on a spacelike hypersurface, and find that the spectrum of scalar perturbations depends on the hypersurface. For reasonable choices, we can recover the usual inflationary predictions for scalar perturbations in minimally c…
We systematically review some standard issues and also the latest developments of modified gravity in cosmology, emphasizing on inflation, bouncing cosmology and late-time acceleration era. Particularly, we present the formalism of standard modified gravity theory representatives, like $F(R)$, $F(\mathcal{G})$ and $F(T)$ gravity theories, but also several alternative theoretical proposals which appeared in the literature during the last decade. We emphasize on the formalism developed for these theories and we explain how these theories can be considered as viable descriptions for our Universe.…
The article presents modeling of inflationary scenarios for the first time in the $f(R,T)$ theory of gravity. We assume the $f(R,T)$ functional from to be $R + ηT$, where $R$ denotes the Ricci scalar, $T$ the trace of the energy-momentum tensor and $η$ the model parameter (constant). We first investigated an inflationary scenario where the inflation is driven purely due to geometric effects outside of GR. We found the inflation observables to be independent of the number of e-foldings in this setup. The computed value of the spectral index is consistent with latest Planck 2018 dataset while th…
Quantum gravity, as a fundamental theory of space-time, is expected to reveal how the universe may have started, perhaps during or before an inflationary epoch. It may then leave a potentially observable (but probably minuscule) trace in cosmic large-scale structures that seem to match well with predictions of inflation models. A systematic quest to derive such tiny effects using one approach, loop quantum gravity, has, however, led to unexpected obstacles. Such models remain incomplete, and it is not clear whether loop quantum gravity can be consistent as a full theory. But some surprising ef…
Once quantum corrections are taken into account, the strong coupling limit of the $ξ$-attractor models (in metric gravity) might depart from the usual Starobinsky solution and move into linear inflation. Furthermore, it is well known that the metric and Palatini formulations of gravity lead to different inflationary predictions in presence of non-minimally couplings between gravity and the inflaton. In this letter we show that for a certain class of non-minimally coupled models, loop corrections will lead to a linear inflation attractor regardless of the adopted gravity formulation.
We investigate in detail a family of quintessential inflation models in the context of $R+R^2$ Palatini modified gravity. We find that successful inflation and quintessence are obtained with an inflaton scalar potential that is approximately quadratic in inflation and inverse quartic in quintessence. We show that corrections for the kination period due to Palatini modified gravity are subdominant, while the setup does not challenge constraints on modified gravity from solar system observations and microscopic experiments, in contrast to the metric case. We obtain concrete predictions regarding…
We investigate how eternal inflation is affected by quantum gravity effects. We consider general features of quantum gravity, such as renormalizability, complementarity, minimal length, definition of observables, and weak gravity conjecture. We also consider phenomenological models such as ghost inflation, non-commutative inflation, brane inflation, k-inflation and resonant tunneling. We find that all these features and models do not support eternal inflation. These evidences show hints that eternal inflation is prohibited by quantum gravity.
Starobinsky inflation and Higgs inflation in gravity, and their equivalence based on the common inflationary potential are extended to supergravity in the proper framework, where the Starobinsky and Higgs descriptions of inflation arise in two different gauges of the same supergravity model.
We study logamediate inflation in the context of $f(T)$ teleparallel gravity. $f(T)$-gravity is a generalization of the teleparallel gravity which is formulated on the Weitzenböck spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor. We consider an $f(T)$-gravity model which is sourced by a canonical scalar field. Assuming a power-law $f(T)$ function in the action, we investigate an inflationary universe with a logamediate scale factor. Our results show that, although logamediate inflation is completely ruled out by observationa…
A field kinetic coupling with the Einstein tensor leads to a gravitationally enhanced friction during inflation, by which even steep potentials with theoretically natural model parameters can drive cosmic acceleration. In the presence of this non-minimal derivative coupling we place observational constraints on a number of representative inflationary models such as chaotic inflation, inflation with exponential potentials, natural inflation, and hybrid inflation. We show that most of the models can be made compatible with the current observational data mainly due to the suppressed tensor-to-sca…
We construct a warm inflation model with inflaton field non-minimally coupled to induced gravity on a warped DGP brane. We incorporate possible modification of the induced gravity on the brane in the spirit of $f(R)$-gravity. We study cosmological perturbations in this setup. In the case of two field inflation such as warm inflation, usually entropy perturbations are generated. While it is expected that in the case of one field inflation these perturbations to be removed, we show that even in the absence of the radiation field, entropy perturbations are generated in our setup due to non-minima…
We have derived an effective potential for inflationary scenario from torsion and quantum gravity correction in terms of the scalar field hidden in torsion. A strict bound on the CP violating $θ$ parameter, ${\cal O}(10^{-10})<θ<{\cal O}(10^{-9})$ has been obtained, using {\tt Planck+WMAP9} best fit cosmological parameters.
We reconsider the toy-model of topological inflation, based on the R*4-modified gravity. By using its equivalence to the certain scalar-tensor gravity model in four space-time dimensions, we compute the inflaton scalar potential and investigate a possibility of inflation. We confirm the existence of the slow-roll inflation with an exit. However, the model suffers from the eta-problem that gives rise to the unacceptable value of the spectral index n_s of scalar perturbations.
We consider 4D quantum-dilaton gravity with the most general coupling in a homogeneous and isotropic universe, especially an inflationary one, which is essentially characterized by an exponentially expanding scale factor with time. We show that on the inflationary background this theory can be miraculously renormalized, at least at the one-loop level, which must be an effective theory during the inflation of the un-constructed complete quantum theory of gravity.
A D-dimensional induced gravity theory is studied carefully in a $4 + (D-4)$ dimensional Friedmann-Robertson-Walker space-time. We try to extract information of the symmetry breaking potential in search of an inflationary solution with non-expanding internal-space. We find that the induced gravity model imposes strong constraints on the form of symmetry breaking potential in order to generate an acceptable inflationary universe. These constraints are analyzed carefully in this paper.
In this paper we investigate the inflationary dynamics of an $f(R)$ gravity in the presence of a canonical scalar field. We specifically choose the cosmological evolution to be a quasi-de Sitter evolution and also the $f(R)$ gravity is assumed to be a modified version of the $R^2$ gravity. We investigate which scalar field potential can produce the quasi-de Sitter evolution for the choice of the $f(R)$ gravity we made, and also we study in detail the inflationary dynamics of the resulting theory. As we demonstrate, the spectral index is identical to the one corresponding to the ordinary $R^2$ …
We study a model of quintessential inflation constructed in $R^2$ modified gravity with a non-minimally coupled scalar field, in the Palatini formalism. Our non-minimal inflaton field is characterised by a simple exponential potential. We find that successful quintessential inflation can be achieved with no fine-tuning on the model parameters. Predictions on the characteristics of dark energy will be tested by observations in the near future, while contrast with existing observations provides insights on the modified gravity background, such as the value of the non-minimal coupling and its run…
We discuss the introduction of a non minimal coupling between the inflaton and gravity in terms of the recently proposed $β$-function formalism for inflation\cite{Binetruy:2014zya}. Via a field redefinition we reduce to the case of minimally coupled theories. The universal attractor at strong coupling has a simple explanation in terms of the new field. Generalizations are discussed and the possibility of evading the universal attractor is shown.
We investigate the tensor perturbation in the inflation model driven by a massive-scalar field in Eddington-inspired Born-Infeld gravity. For short wave-length modes, the perturbation feature is very similar to that of the usual chaotic inflation. For long wave-length modes, the perturbation exhibits a peculiar rise in the power spectrum which may leave a signature in the cosmic microwave background radiation.
In this paper, we investigate the potential-driven inflation models with a disformal coupling to Einstein Gravity, to find out the effects of such a coupling on these models. We consider a simple coupling form which introduces only one parameter, and three inflation models, namely the chaotic inflation, the Higgs inflation, and the monodromy inflation. We find that the disformal coupling can have some modifications to the observational variables of these models such as the power spectrum, the spectral index as well as the tensor/scalar ratio, although not too large due to the constraints on th…
We consider a modified gravity framework for inflation by adding to the Einstein-Hilbert action a direct $f(ϕ)T$ term, where $ϕ$ is identified as the inflaton and $T$ is the trace of the energy-momentum tensor. The framework goes to Einstein gravity naturally when the inflaton decays out. We investigate inflation dynamics in this $f(ϕ)T$ gravity (not to be confused with torsion-scalar coupled theories) on a general basis and then apply it to three well-motivated inflationary models. We find that the predictions for the spectral tilt and the tensor-to-scalar ratio are sensitive to this new $f(ϕ…
We consider $f(R)$ gravity theories which unify $R^n$ inflation and dark energy models. First, from the final Planck data of the cosmic microwave background, we obtain a condition, $1.977 < n < 2.003$. Next, under this constraint, we investigate local-gravity tests for three models. We find that the $R^n$ term can dominate over the dark energy term even at the Earth's curvature scale, contrary to intuition; however, the $R^n$ term does not relax or tighten the constraints on the three models.
In the recent years, a host of modified gravity models have been proposed as alternatives to the dark energy. A quantum theory of gravity also requires to modify `General Theory of Relativity'. In the present article, we consider five different modified theories of gravity, and compare inflationary parameters with recent data sets released by two Planck collaboration teams. Our analysis reveals that the scalar-tensor theory of gravity is the best alternative.
The discovery of B-modes, and their effect on the fit to inflationary parameters, opens a window to explore quantum gravity. In this paper we adopt an effective theory approach to study quantum gravity effects in inflation. We apply this approach to chaotic and $ϕ^4$ inflation, and find that BICEP2 constrains these new operators to values which are consistent with the effective theory approach. This result opens the possibility to study quantum gravity in a systematic fashion, including its effect on Higgs inflation and other Starobisnky-like models.
We study the inflationary dynamics in Brane World gravity and look for observational signatures of any deviation from the standard General Relativity based results of Cosmological Perturbation Theory. We first review the standard paradigm of General Relativity based inflationary dynamics and cosmological perturbation theory and then go on to review Brane World gravity. Finally we look at the high energy corrections for some chosen models and compare the results with the Planck and WMAP (9 year) data. Then we make a summary of our results and point out certain interesting features of Brane Worl…
