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Register a new userGeneral slow-roll inflation in $f(R)$ gravity under the Palatini approach
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Diego S\\'aez-Chill\\'on G\\'omez
Publication date
2020
fields
Physics
and research's language is
English
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Slow-roll inflation is analyzed in the context of modified gravity within the Palatini formalism. As shown in the literature, inflation in this framework requires the presence of non-traceless matter, otherwise it does not occur just as a consequence of the non-linear gravitational terms of the action. Nevertheless, by including a single scalar field that plays the role of the inflaton, slow-roll inflation can be performed in these theories, where the equations lead to an effective potential that modifies the dynamics. We obtain the general slow-roll parameters and analyze a simple model to illustrate the differences introduced by the gravitational terms under the Palatini approach, and the modifications on the spectral index and the tensor to scalar ratio predicted by the model.
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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 $R + eta T$, where $R$ denotes the Ricci scalar, $T$ the trace of the energy-momentum tensor and $eta$ 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 the scalar to tensor ratio is a bit higher. We then proceeded to analyze the behavior of an inflation driven by $f(R,T)$ gravity coupled with a real scalar field. By taking the slow-roll approximation, we generated interesting scenarios where a Klein Gordon potential leads to observationally consistent inflation observables. Our results makes it clear-cut that in addition to the Ricci scalar and scalar fields, the trace of energy momentum tensor also play a major role in driving inflationary scenarios.
After giving a pedagogical review we clarify that the stochastic approach to inflation is generically reliable only at zeroth order in the (geometrical) slow-roll parameter $epsilon_1$ if and only if $epsilon_2^2ll 6/epsilon_1$, with the notable exception of slow-roll. This is due to the failure of the stochastic $Delta N$ formalism in its standard formulation. However, by keeping the formalism in its regime of validity, we showed that, in ultra-slow-roll, the stochastic approach to inflation reproduces the power spectrum calculated from the linear theory approach.
We investigate cosmological dynamics based on $f(R)$ gravity in the Palatini formulation. In this study we use the dynamical system methods. We show that the evolution of the Friedmann equation reduces to the form of the piece-wise smooth dynamical system. This system is is reduced to a 2D dynamical system of the Newtonian type. We demonstrate how the trajectories can be sewn to guarantee $C^0$ extendibility of the metric similarly as `Milne-like FLRW spacetimes are $C^0$-extendible. We point out that importance of dynamical system of Newtonian type with non-smooth right-hand sides in the context of Palatini cosmology. In this framework we can investigate singularities which appear in the past and future of the cosmic evolution. We consider cosmological systems in both Einstein and Jordan frames. We show that at each frame the topological structures of phase space are different.
We focus on a series of $f(R)$ gravity theories in Palatini formalism to investigate the probabilities of producing the late-time acceleration for the flat Friedmann-Robertson-Walker (FRW) universe. We apply statefinder diagnostic to these cosmological models for chosen series of parameters to see if they distinguish from one another. The diagnostic involves the statefinder pair ${r,s}$, where $r$ is derived from the scale factor $a$ and its higher derivatives with respect to the cosmic time $t$, and $s$ is expressed by $r$ and the deceleration parameter $q$. In conclusion, we find that although two types of $f(R)$ theories: (i) $f(R) = R + alpha R^m - beta R^{-n}$ and (ii) $f(R) = R + alpha ln R - beta$ can lead to late-time acceleration, their evolutionary trajectories in the $r-s$ and $r-q$ planes reveal different evolutionary properties, which certainly justify the merits of statefinder diagnostic. Additionally, we utilize the observational Hubble parameter data (OHD) to constrain these models of $f(R)$ gravity. As a result, except for $m=n=1/2$ of (i) case, $alpha=0$ of (i) case and (ii) case allow $Lambda$CDM model to exist in 1$sigma$ confidence region. After adopting statefinder diagnostic to the best-fit models, we find that all the best-fit models are capable of going through deceleration/acceleration transition stage with late-time acceleration epoch, and all these models turn to de-Sitter point (${r,s}={1,0}$) in the future. Also, the evolutionary differences between these models are distinct, especially in $r-s$ plane, which makes the statefinder diagnostic more reliable in discriminating cosmological models.
The $f(R,T)$ gravity is a theory whose gravitational action depends arbitrarily on the Ricci scalar, $R$, and the trace of the stress-energy tensor, $T$; its field equations also depend on matter Lagrangian, $mathcal{L}_{m}$. In the modified theories of gravity where field equations depend on Lagrangian, there is no uniqueness on the Lagrangian definition and the dynamics of the gravitational and matter fields can be different depending on the choice performed. In this work, we have eliminated the $mathcal{L}_{m}$ dependence from $f(R,T)$ gravity field equations by generalizing the approach of Moraes [Eur. Phys. J. C 79(8), 674 (2019)]. We also propose a general approach where we argue that the trace of the energy-momentum tensor must be considered an unknown variable of the field equations. The trace can only depend on fundamental constants and few inputs from the standard model. Our proposal resolves two limitations: first the energy-momentum tensor of the $f(R,T)$ gravity is not the perfect fluid one; second, the Lagrangian is not well-defined. As a test of our approach we applied it to the study of the matter era in cosmology, and the theory can successfully describe a transition between a decelerated Universe to an accelerated one without the need for dark energy.
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