Tracking the validity of the quasi-static and sub-horizon approximations in modified gravity

Within the framework of modified gravity (MG), the quasi-static (QS) and sub-horizon (SH) approximations are widely used in analyses aiming to identify departures from the concordance model at late-times. In general, it is assumed that time derivatives are subdominant with respect to spatial derivatives given that the relevant physical modes are those well inside the Hubble radius. In practice, the perturbation equations under these approximations are reduced to a tractable algebraic system in terms of the gravitational potentials and the perturbations of involved matter fields. Here, in the framework of $f(R)$ theories, we revisit standard results when these approximations are invoked using a new parameterization scheme that allows us to track the relevance of each time-derivative term in the perturbation equations. This new approach unveils terms which are neglected in the standard procedure. We assess the relevance of these differences by comparing results from both approaches against full numerical solutions for two well-known toy-models: the designer $f(R)$ model and the Hu-Sawicki model. We find that: i) the SH approximation can be safely applied to linear perturbation equations for scales $0.06 h/$Mpc $\lesssim k \lesssim 0.2 h/$Mpc, ii) in this "safety region", the QS approximation provides a very accurate description of the late-time cosmological dynamics even when dark energy significantly contribute to the cosmic budget, and iii) our new methodology performs better than the standard procedure, even for several orders of magnitude in some cases. Although, the impact of this major improvement on the linear observables is minimal for the studied cases, this does not represent an invalidation for our approach. Instead, our findings indicate that the perturbation expressions derived under these approximations in more general MG theories, such as Horndeski, should be also revisited.Comment: 28 pages, 18 figures. Changes match published versio

Using machine learning to compress the matter transfer function T(k)T(k)

The linear matter power spectrum $P(k,z)$ connects theory with large scale structure observations in cosmology. Its scale dependence is entirely encoded in the matter transfer function $T(k)$, which can be computed numerically by Boltzmann solvers, and can also be computed semi-analytically by using fitting functions such as the well-known Bardeen-Bond-Kaiser-Szalay (BBKS) and Eisenstein-Hu (EH) formulae. However, both the BBKS and EH formulae have some significant drawbacks. On the one hand, although BBKS is a simple expression, it is only accurate up to $10\%$, which is well above the $1\%$ precision goal of forthcoming surveys. On the other hand, while EH is as accurate as required by upcoming experiments, it is a rather long and complicated expression. Here, we use the Genetic Algorithms (GAs), a particular machine learning technique, to derive simple and accurate fitting formulae for the transfer function $T(k)$. When the effects of massive neutrinos are also considered, our expression slightly improves over the EH formula, while being notably shorter in comparison.Comment: 12 pages, 7 figures, 2 tables. Changes match published versio

Anisotropic Dark Energy from String Compactifications

We explore the cosmological dynamics of a minimalistic yet generic string-inspired model for multifield dark energy. Adopting a supergravity four-dimensional viewpoint, we motivate the model's structure arising from superstring compactifications involving a chiral superfield and a pure $U(1)$ gauge sector. The chiral sector gives rise to a pair of scalar fields, such as the axio-dilaton, which are kinetically coupled. However, the scalar potential depends on only one of them, further entwined with the vector field through the gauge kinetic function. The model has two anisotropic attractor solutions that, despite a steep potential and thanks to multifield dynamics, could explain the current accelerated expansion of the Universe while satisfying observational constraints on the late-times cosmological anisotropy. Nevertheless, justifying the parameter space allowing for slow roll dynamics together with the correct cosmological parameters, would be challenging within the landscape of string theory. Intriguingly, we find that the vector field, particularly at one of the studied fixed points, plays a crucial role in enabling geodesic trajectories in the scalar field space while realizing slow-roll dynamics with a steep potential. This observation opens a new avenue for exploring multifield dark energy models within the superstring landscape.Comment: Submitted to JHEP. Comments are welcome

Reconstructing the parameter space of non-analytical cosmological fixed points

Dynamical system theory is a widely used technique in the analysis of cosmological models. Within this framework, the equations describing the dynamics of a model are recast in terms of dimensionless variables, which evolve according to a set of autonomous first-order differential equations. The fixed points of this autonomous set encode the asymptotic evolution of the model. Usually, these points can be written as analytical expressions for the variables in terms of the parameters of the model, which allows a complete characterization of the corresponding parameter space. However, a thoroughly analytical treatment is impossible in some cases. In this work, we give an example of a dark energy model, a scalar field coupled to a vector field in an anisotropic background, where not all the fixed points can be analytically found. Then, we put forward a general scheme that provides a numerical description of the parameter space. This allows us to find interesting accelerated attractors of the system with no analytical representation. This work may serve as a template for the numerical analysis of highly complicated dynamical systems.Comment: 13 pages, 13 figures, 1 table. Changes match the published versio

Machine learning unveils the linear matter power spectrum of modified gravity

The matter power spectrum $P(k)$ is one of the main quantities connecting observational and theoretical cosmology. Although for a fixed redshift this can be numerically computed very efficiently by Boltzmann solvers, an analytical description is always desirable. However, accurate fitting functions for $P(k)$ are only available for the concordance model. Taking into account that forthcoming surveys will further constrain the parameter space of cosmological models, it is also of interest to have analytical formulations for $P(k)$ when alternative models are considered. Here, we use the genetic algorithms, a machine learning technique, to find a parametric function for $P(k)$ considering several possible effects imprinted by modifications of gravity. Our expression for the $P(k)$ of modified gravity shows a mean accuracy of around 1-2% when compared with numerical data obtained via modified versions of the Boltzmann solver CLASS, and thus it represents a competitive formulation given the target accuracy of forthcoming surveys.Comment: 11 pages, 2 figures, 1 tabl

Anisotropic Dark Energy from a Coupled 2-form Field

In this paper, we study the cosmological dynamics of a 2-form field coupled to a cold dark matter component in a Bianchi I background. Using a dynamical system analysis, we show that our model has two attractor solutions corresponding to late-time accelerated universes. These solutions have an important distinction between each other: one is isotropic and the other one is not, i.e. there is a non-zero shear contribution induced by the 2-form field. Due to the coupling in the dark sector, we also obtain a scaling solution where the 2-form field behaves as a pressure-less fluid. For a particular set of parameters and initial conditions, we numerically study the expansion history of the model. We find that the value of the shear at redshift zero is within the current observational bounds, which could be a potential alleviation to the quadrupole anomaly, and that the equation of state of dark energy is exactly -1.Comment: 12 pages, 7 figure