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Register a new userGlobal dynamics and inflationary center manifold and slow-roll approximants
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We consider the familiar problem of a minimally coupled scalar field with quadratic potential in flat Friedmann-Lema^itre-Robertson-Walker cosmology to illustrate a number of techniques and tools, which can be applied to a wide range of scalar field potentials and problems in e.g. modified gravity. We present a global and regular dynamical systems description that yields a global understanding of the solution space, including asymptotic features. We introduce dynamical systems techniques such as center manifold expansions and use Pade approximants to obtain improved approximations for the `attractor solution at early times. We also show that future asymptotic behavior is associated with a limit cycle, which shows that manifest self-similarity is asymptotically broken toward the future, and give approximate expressions for this behavior. We then combine these results to obtain global approximations for the attractor solution, which, e.g., might be used in the context of global measures. In addition we elucidate the connection between slow-roll based approximations and the attractor solution, and compare these approximations with the center manifold based approximants.
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We analyse field fluctuations during an Ultra Slow-Roll phase in the stochastic picture of inflation and the resulting non-Gaussian curvature perturbation, fully including the gravitational backreaction of the fields velocity. By working to leading order in a gradient expansion, we first demonstrate that consistency with the momentum constraint of General Relativity prevents the field velocity from having a stochastic source, reflecting the existence of a single scalar dynamical degree of freedom on long wavelengths. We then focus on a completely level potential surface, $V=V_0$, extending from a specified exit point $phi_{rm e}$, where slow roll resumes or inflation ends, to $phirightarrow +infty$. We compute the probability distribution in the number of e-folds $mathcal{N}$ required to reach $phi_{rm e}$ which allows for the computation of the curvature perturbation. We find that, if the fields initial velocity is high enough, all points eventually exit through $phi_{rm e}$ and a finite curvature perturbation is generated. On the contrary, if the initial velocity is low, some points enter an eternally inflating regime despite the existence of $phi_{rm e}$. In that case the probability distribution for $mathcal{N}$, although normalizable, does not possess finite moments, leading to a divergent curvature perturbation.
There have been thousands of cosmological models for our early universe proposed in the literature, and many of them claimed to be able to give rise to scale-invariant power spectrum as was favored by the observational data. It is thus interesting to think about whether there are some relations among them, e.g., the duality relation. In this paper, we investigate duality relations between cosmological models in framework of general relativity (GR) , not only with varying slow-roll parameter $epsilon$, but also with sound speed $c_s$, which can then be understood as adiabatic duality. Several duality relationships are formulated analytically and verified numerically. We show that models with varying $epsilon$ and constant $c_s$ can be dual in scalar spectral index, but not tensor one. On the other hand, allowing both $epsilon$ and $c_s$ to vary can make models dual in both scalar and tensor spectral indices.
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.
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.
We study flat FLRW $alpha$-attractor $mathrm{E}$- and $mathrm{T}$-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of $e$-folds, is associated with a solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this `inflationary attractor solution. A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.
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