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Current and future large-scale structure surveys aim to constrain cosmological parameters with unprecedented precision by analyzing vast amounts of data. This imposes a pressing need to develop fast and accurate methods for computing the matter power spectrum $P(k)$, or equivalently, the matter transfer function $T(k)$. In previous works, we introduced precise fitting formulas for these quantities within the standard cosmological model, including extensions such as the presence of massive neutrinos and modifications of gravity. However, these formulations overlooked a key characteristic imprinted in $P(k)$: the baryon acoustic oscillation signal. Here, we leverage our understanding of the well-known physics behind this oscillatory pattern to impose constraints on our genetic algorithm, a machine learning technique. By employing this ``physics-informed'' approach, we introduce an expression that accurately describes the matter transfer function with sub-percent mean accuracy. The high interpretability of the output allows for straightforward extensions of this formulation to other scenarios involving massive neutrinos and modifications of gravity. We anticipate that this formula will serve as a competitive fitting function for $P(k)$, meeting the accuracy requirements essential for cosmological analyses., Comment: 15 pages, 6 figures. Comments are very welcome!

We propose a novel approach using neural networks (NNs) to differentiate between cosmological models, especially in the case where they are nested and the additional model parameters are close to zero, making it difficult to discriminate them with traditional approaches. Our method complements Bayesian analyses for cosmological model selection, which heavily depend on the chosen priors and average the unnormalized posterior over potentially large prior volumes. By analyzing simulated realistic data sets of the growth rate of the large scale structure (LSS) of the Universe, based on current galaxy-clustering survey specifications, for the cosmological constant and cold dark matter ($\Lambda$CDM) model and the Hu-Sawicki $f(R)$ model, we demonstrate the potential of NNs to enhance the extraction of meaningful information from cosmological LSS data. We find that the NN can successfully distinguish between $\Lambda$CDM and the $f(R)$ models, by predicting the correct model with approximately $97\%$ overall accuracy, thus demonstrating that NNs can maximise the potential of current and next generation surveys to probe for deviations from general relativity., Comment: 8 pages, 7 figures, comments are welcome

The Universe's assumed homogeneity and isotropy is known as the cosmological principle. It is one of the assumptions that lead to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) metric and it is a cornerstone of modern cosmology, because the metric plays a crucial role into the determination of the cosmological observables. Thus, it is of paramount importance to question this principle and perform observational tests that may falsify this hypothesis. Here we explore the use of galaxy cluster counts as a probe of a large-scale inhomogeneity, which is a novel approach for the study of inhomogeneous models, and to determine the precision with which future galaxy cluster surveys will be able to test the cosmological principle. We present forecast constraints on the inhomogeneous Lema\^{\i}tre-Tolman-Bondi (LTB) model with a cosmological constant and cold dark matter, from a combination of simulated data according to a compilation of `Stage-IV' galaxy surveys following a methodology that involves the use of a mass function correction from numerical $N$-body simulations of an LTB cosmology. When considering the \lcdm fiducial model as a baseline for constructing our mock catalogs, we find that our combination of the forthcoming cluster surveys, will improve the constraints on the cosmological principle parameters as well on the FLRW parameters by about $50\%$ with respect to previous similar forecasts performed using geometrical and linear growth of structure probes, with $\pm20\%$ variations depending on the level of knowledge of systematic effects.These results indicate that galaxy cluster abundances are sensitive probes of inhomogeneity, and that next-generation galaxy cluster surveys, will thoroughly test homogeneity at cosmological scales, tightening the constraints on possible violations of the cosmological principle in the framework of $\Lambda$LTB scenarios. (Abridged), Comment: 9 pages, 5 figure, accepted as is and published in A&A journal after adding Fig. 1 for illustrative purposes

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 table. Changes match published version

Constraints on the cosmological concordance model parameters from observables at different redshifts are usually obtained using the locally measured value of the gravitational constant $G_N$. Here we relax this assumption, by considering $G$ as a free parameter, either constant over the redshift range or dynamical but limited to differ from fiducial value only above a certain redshift. Using CMB data and distance measurements from galaxy clustering BAO feature, we constrain the cosmological parameters, along with $G$, through a MCMC bayesian inference method. Furthermore, we investigate whether the tensions on the matter fluctuation $\sigma_8$ and Hubble $H_0$ parameter could be alleviated by this new variable. We used different parameterisations spanning from a constant $G$ to a dynamical $G$. In all the cases investigated in this work we found no mechanism that alleviates the tensions when both CMB and BAO data are used with $\xi_{\mathrm{g}} = G / G_N$ constrained to 1.0$\pm0.04$ (resp. $\pm0.01$) in the constant (resp. dynamical) case. Finally, we studied the cosmological consequences of allowing a running of the spectral index, since the later is sensitive to a change in $G$. For the two parameterisations adopted, we found no significant changes to the previous conclusions., Comment: Swiflty accepted and published in JCAP. Minor changes and references added in this version after proof reading. 18 pages, 9 figures

We study the variation of the gravitational Newton's constant on cosmological scales in scalar-tensor theories of gravity. We focus on the simplest models of scalar-tensor theories with a coupling to the Ricci scalar of the form $F(\sigma) = N_{pl}^2 + \xi\sigma^2$, such as extended Jordan-Brans-Dicke ($N_{pl}=0$), or a non-minimally coupled scalar field with $N_{pl}=M_{pl}$, which permits the gravitational constant to vary self-consistently in time and space. In addition, we allow the gravitational constant to differ from the Newton's constant $G$, i.e. $G_{\rm eff}(z=0) = G(1+\Delta)^2$. Combining the information from {\em Planck} 2018 CMB temperature, polarization and lensing, together with a compilation of BAO measurements from BOSS, we constrain the imbalance to $\Delta = -0.022 \pm 0.023$ (68% CL) and the coupling to $10^3\, \xi < 0.82$ (95% CL) for JBD and for a non-minimally coupled scalar field we constrain the imbalance to $\Delta > -0.018$ ($< 0.021$) and the coupling parameter to $\xi < 0.089$ ($\xi > - 0.041$) both at 95% CL. These constraints correspond to a variation of the gravitational constant now respect to the one in the radiation era to be smaller than 3% (95% CL) and to the ratio of the gravitational Newton's constant measured from cosmological scales and the one measured in a Cavendish-like experiment to be smaller than 4-15% (95% CL). With current data, we observe that the degeneracy between $\Delta$, the coupling $\xi$, and $H_0$ allows for a larger value of the Hubble constant increasing the agreement between the measurement of the Hubble constant by the SH0ES team and its value inferred by CMB data. Future data such as the combination of CMB anisotropies from LiteBIRD and CMB-S4, and large-scale structures galaxy clustering from DESI and galaxy shear from LSST will reduce the uncertainty to $\sigma(\Delta) = 0.004$., Comment: 33 pages, 18 figures, 6 tables

We test the cosmological implications of a varying absolute magnitude of Type Ia supernovae using the Pantheon compilation, by reconstructing different phenomenological approaches that could justify a varying absolute magnitude, but also approaches based on cosmic voids, modified gravity models and a binning scheme. In all the cases considered in this work, we find good agreement with the expected values of the standard $\Lambda$CDM model and no evidence of new physics., Comment: 10 pages, 5 figures, 6 tables

The jackknife method gives an internal covariance estimate for large-scale structure surveys and allows model-independent errors on cosmological parameters. Using the SDSS-III BOSS CMASS sample, we study how the jackknife size and number of resamplings impact the precision of the covariance estimate on the correlation function multipoles and the error on the inferred baryon acoustic scale. We compare the measurement with the MultiDark Patchy mock galaxy catalogues, and we also validate it against a set of log-normal mocks with the same survey geometry. We build several jackknife configurations that vary in size and number of resamplings. We introduce the Hartlap factor in the covariance estimate that depends on the number of jackknife resamplings. We also find that it is useful to apply the tapering scheme to estimate the precision matrix from a limited number of resamplings. The results from CMASS and mock catalogues show that the error estimate of the baryon acoustic scale does not depend on the jackknife scale. For the shift parameter $\alpha$, we find an average error of 1.6%, 2.2% and 1.2%, respectively from CMASS, Patchy and log-normal jackknife covariances. Despite these uncertainties fluctuate significantly due to some structural limitations of the jackknife method, our $\alpha$ estimates are in reasonable agreement with published pre-reconstruction analyses. Jackknife methods will provide valuable and complementary covariance estimates for future large-scale structure surveys., Comment: 13 pages, 8 figures, 3 tables

The Baryon Acoustic Oscillation (BAO) scale in the clustering of galaxies is a powerful standard ruler to measure cosmological distances and determine the geometry of the Universe. Past surveys have detected the BAO feature in the clustering of different galaxy samples, most of them composed of redder, quiescent galaxies and bluer, star-forming ones out to redshift $z\sim1$. Besides these targets, new upcoming surveys will observe high-redshift galaxies with bright nebular emission lines out to $z\sim2$, quasars and Lyman-$\alpha$ quasars at $z\gt 2$. All these different galaxy targets will be used as multi-tracers of the same underlying dark matter field. By combining them over wide cosmological volumes, we will be able to beat cosmic variance and measure the growth of structure with unprecedented accuracy. In this work, we measure the BAO scale in the two-point auto- and cross-correlation functions of three independent populations of multi-tracers extracted from the SDSS DR7 Main galaxy sample at redshift $0.02\lt z\lt 0.22$. Combining their covariances, we find accurate constraints on the shift parameter $\alpha=1.00\pm 0.04$ and $D_{\rm{V}}(z=0.1)/r_{\rm{s}}=2.92\pm0.12$., Comment: 7 pages, 3 figures, 1 table

If light enough primordial black holes (PBH) account for dark matter, then its density decreases with time as they lose mass via Hawking radiation. We show that this time-dependence of the matter density can be formulated as an equivalent $w(z)$ dark energy model and we study its implications on the expansion history. Using our approach and comparing with the latest cosmological data, including the supernovae type Ia, Baryon Acoustic Oscillations, Cosmic Microwave Background and the Hubble expansion H(z) data, we place observational constraints on the PBH model. We find that it is statistically consistent with $\Lambda$CDM according to the AIC statistical tool. Furthermore, we entertain the idea of having a population of ultra-light PBHs, decaying around neutrino decoupling, on top of the dark matter fluid and show how this offers a natural dark matter-radiation coupling altering the expansion history of the Universe and alleviating the $H_0$ tension., Comment: 12 pages, 3 figures, 3 tables, expanded Section III. Comments welcome

We study the imprints of an effective dark energy fluid in the large scale structure of the universe through the observed angular power spectrum of galaxies in the relativistic regime. We adopt the phenomenological approach that introduces two parameters $\{Q,\eta\}$ at the level of linear perturbations and allow to take into account the modified clustering (or effective gravitational constant) and anisotropic stress appearing in models beyond $\Lambda$CDM. We characterize the effective dark energy fluid by an equation of state parameter $w=-0.95$ and various sound speed cases in the range $10^{-6}\leq c^2_s\leq 1$, thus covering K-essence and quintessence cosmologies. We calculate the angular power spectra of standard and relativistic effects for these scenarios under the $\{Q,\eta\}$ parametrization, and we compare these relative to a fiducial $\Lambda$CDM cosmology. We find that, overall, deviations relative to $\Lambda$CDM are stronger at low redshift since the behavior of the dark energy fluid can mimic the cosmological constant during matter domination era but departs during dark energy domination. In particular, at $z=0.1$ the matter density fluctuations are suppressed by up to $\sim3\%$ for the quintessence-like case, while redshift-space distortions and Doppler effect can be enhanced by $\sim15\%$ at large scales for the lowest sound speed scenario. On the other hand, at $z=2$ we find deviations of up to $\sim5\%$ in gravitational lensing, whereas the Integrated Sachs-Wolfe effect can deviate up to $\sim17\%$. Furthermore, when considering an imperfect dark energy fluid scenario, we find that all effects are insensitive to the presence of anisotropic stress at low redshift, and only the Integrated Sachs-Wolfe effect can detect this feature at $z=2$ and very large scales., Comment: 21 pages, 3 figures, comments on dark energy anisotropic stress added, results unchanged. Matches published version

The extended Jordan-Brans-Dicke (eJBD) theory of gravity is constrained by a host of astrophysical and cosmological observations spanning a wide range of scales. The current cosmological constraints on the first post-Newtonian parameter in these simplest eJBD models in which the recent acceleration of the Universe is connected with the variation of the effective gravitational strength are consistent, but approximately two order of magnitude larger than the time-delay test within the Solar System. We forecast the capabilities of future galaxy surveys in combination with current and future CMB anisotropies measurements to further constrain the simplest dark energy models within eJBD theory of gravity. By considering two cases of a monomial potential (a quartic potential or a cosmological constant), we show how Euclid-like galaxy clustering and weak lensing data in combination with BOSS and future CMB observations have the potential to reach constraints on the first post-Newtonian parameter $\gamma_{\rm PN}$ comparable to those from the Solar System., Comment: 14 pages, 4 figures, 1 table. Version accepted by JCAP

In this work we perform an analysis on the recently proposed conjoined cosmic growth and cosmic expansion diagram [1] to compare several dark energy models using the Figure of Merit showed in [2], which consists in the inverse of the $1\sigma$ confidence region in the $f\sigma_8(z)-H(z)$ plot. Our analysis also consists of comparing the models by performing different statistical criteria: Bayes factor [3], the Bayesian Information Criteria [4] and the Akaike Information Criterion [5]. We also developed a 3-dimensional Figure of Merit to account simultaneously for the errors on the growth rate and the Hubble parameter. The main idea is to consider several cosmological models and compare them with the different statistical criteria in order to highlight the differences and the accuracies of each single criterion., Comment: 13 pages, 4 figures, 9 tables

We perform an Internal Robustness analysis (iR) to a compilation of the most recent $f\sigma_8(z)$ data, using the framework of 1209.1897. The method analyzes combinations of subsets in the data set in a Bayesian model comparison way, potentially finding outliers, subsets of data affected by systematics or new physics. In order to validate our analysis and assess its sensitivity we performed several cross-checks, for example by removing some of the data or by adding artificially contaminated points, while we also generated mock data sets in order to estimate confidence regions of the iR. Applying this methodology, we found no anomalous behavior in the $f\sigma_8(z)$ data set, thus validating its internal robustness., Comment: 11 pages, 1 table, 5 figures. Changes match published version

We test both the FLRW geometry and $\Lambda$CDM cosmology in a model independent way by reconstructing the Hubble function $H(z)$, the comoving distance $D(z)$ and the growth of structure $f\sigma_8(z)$ using the most recent data available. We use the linear model formalism in order to optimally reconstruct the latter cosmological functions, together with their derivatives and integrals. We then evaluate four of the null tests available in literature: $Om_{1}$ by Sahni et al., $Om_{2}$ by Zunckel \& Clarkson, $Ok$ by Clarkson et al., and $ns$ by Nesseris \& Sapone. For all the four tests we find agreement, within the errors, with the standard cosmological model., Comment: 8 pages, 7 figures and 1 table

We analyse the efficiency of future large scale structure surveys to unveil the presence of scale dependent features in the primordial spectrum --resulting from cosmic inflation-- imprinted in the distribution of galaxies. Features may appear as a consequence of non-trivial dynamics during cosmic inflation, in which one or more background quantities experienced small but rapid deviations from their characteristic slow-roll evolution. We consider two families of features: localized features and oscillatory extended features. To characterise them we employ various possible templates parametrising their scale dependence and provide forecasts on the constraints on these parametrisations for LSST like surveys. We perform a Fisher matrix analysis for three observables: cosmic microwave background (CMB), galaxy clustering and weak lensing. We find that the combined data set of these observables will be able to limit the presence of features down to levels that are more restrictive than current constraints coming from CMB observations only. In particular, we address the possibility of gaining information on currently known deviations from scale invariance inferred from CMB data, such as the feature appearing at the $\ell \sim 20$ multipole (which is the main contribution to the low-$\ell$ deficit) and a potential feature appearing at $\ell \sim 800$., Comment: 37 pp., 5 Tabs., 10 Figs, v3: changed discussion around templates II, III, added clarifications, comments and references. Matches JCAP version

Euclid is a European Space Agency medium class mission selected for launch in 2020 within the Cosmic Vision 2015 2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and redshifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid's Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission., Comment: This article provides an update of arXiv:1206.1225, with different authors. Forecasts are not updated in this version

Cosmological fluids are commonly assumed to be distributed in a spatially homogeneous way, while their internal properties are described by a perfect fluid. As such, they influence the Hubble-expansion through their respective densities and equation of state parameters. The subject of this paper is an investigation of the fluid-mechanical properties of a dark energy fluid, which is characterised by its sound speed and its viscosity apart from its equation of state. In particular, we compute the predicted spectra for the integrated Sachs-Wolfe effect for our generalised fluid, and compare them with the corresponding predictions for weak gravitational lensing and galaxy clustering, which had been computed in previous work. We perform statistical forecasts and show that the integrated Sachs-Wolfe signal obtained by cross correlating Euclid galaxies with Planck temperatures, when joined to galaxy clustering and weak lensing observations, yields a percent sensitivity on the dark energy sound speed and viscosity. We prove that the iSW effect provides strong degeneracy breaking for low sound speeds and large differences between the sound speed and viscosity parameters., Comment: 10 pages, 6 figures

We present the analytical solutions for the evolution of matter density perturbations, for a model with a constant dark energy equation of state $w$ but when the effects of the dark energy perturbations are properly taken into account. We consider two cases, the first when the sound speed of the perturbations is zero $c_s^2=0$ and the general case $010 H_0$ or equivalently $k/h>0.0033 \textrm{Mpc}^{-1}$. We also estimate the corrections to the growth index $\gamma(z)$, commonly used to parametrize the growth-rate. We find that these corrections due to the DE perturbations affect the growth index $\gamma$ at the $3\%$ level. We also compare our new expressions for the growth index with other expressions already present in the literature and we find that the latter are less accurate than the ones we propose here. Therefore, our analytical calculations are necessary as the theoretical predictions for the fundamental parameters to be constrained by the upcoming surveys need to be as accurate as possible, especially since we are entering in the precise cosmology era where parameters will be measured to the percent level., Comment: 8 pages, 4 figures

We systematically study the null-test for the growth rate data first presented in [S. Nesseris and D. Sapone, arXiv:1409.3697] and we reconstruct it using various combinations of data sets, such as the $f\sigma_8$ and $H(z)$ or Type Ia supernovae (SnIa) data. We perform the reconstruction in two different ways, either by directly binning the data or by fitting various dark energy models. We also examine how well the null-test can be reconstructed by future data by creating mock catalogs based on the cosmological constant model, a model with strong dark energy perturbations, the $f(R)$ and $f(G)$ models, and the large void LTB model that exhibit different evolution of the matter perturbations. We find that with future data similar to an LSST-like survey, the null-test will be able to successfully discriminate between these different cases at the $5\sigma$ level., Comment: 15 pages; 10 figures; 5 tables

Current and upcoming surveys will measure the cosmological parameters with an extremely high accuracy. The primary goal of these observations is to eliminate some of the currently viable cosmological models created to explain the late time accelerated expansion (either real or only inferred). However, most of the statistical tests used in cosmology have a strong requirement: the use of a model to fit the data. Recently there has been an increased interest on finding tests that are model independent, i.e. to have a function that depends entirely on observed quantities and not on the model, see for instance [1]. In this letter we present an alternative consistency check at the perturbative level for a homogeneous and isotropic Universe filled with a dark energy component. This test makes use of the growth of matter perturbations data and it is able to not only test the homogeneous and isotropic Universe but also, within the framework of a Friedmann-Lema\^itre-Robertson-Walker Universe, if the dark energy component is able to cluster, if there is a tension in the data or if we are dealing with a modification of gravity., Comment: 4 pages, 1 figure

We compare four different methods that can be used to analyze the type Ia supernovae (SnIa) data, ie to use piecewise-constant functions in terms of: the dark energy equation of state $w(z)$, the deceleration parameter $q(z)$, the Hubble parameter $H(z)$ and finally the luminosity distance $d_L$. These four quantities cover all aspects of the accelerating Universe, ie the phenomenological properties of dark energy, the expansion rate (first and second derivatives) of the Universe and the observations themselves. For the first two cases we also perform principal component analysis (PCA) so as to decorrelate the parameters, while for the last two cases we use novel analytic expressions to find the best-fit parameters. In order to test the methods we create mock SnIa data (2000 points, uniform in redshift $z\in[0,1.5]$) for three fiducial cosmologies: the cosmological constant model ($\Lambda$CDM), a linear expansion of the dark energy equation of state parameter $w(a)=w_0+w_a(1-a)$ and the Hu-Sawicki $f(R)$ model. We find that if we focus on the two mainstream approaches for the PCA, i.e. $w(z)$ and $q(z)$, then the best piecewise-constant scheme is always $w(z)$. Finally, to our knowledge the piecewise-constant method for $H(z)$ is new in the literature, while for the rest three methods we present several new analytic expressions., Comment: 23 pages, 3 figures, 8 tables. Changes match published version

We test the FLRW cosmology by reconstructing in a model-independent way both the Hubble parameter $H(z)$ and the comoving distance $D(z)$ via the most recent Hubble and Supernovae Ia data. In particular we use: data binning with direct error propagation, the principal component analysis, the genetic algorithms and the Pad\'e approximation. Using our reconstructions we evaluate the Clarkson {\it et al} test known as $\Omega_K(z)$, whose value is constant in redshift for the standard cosmological model, but deviates elsewise. We find good agreement with the expected values of the standard cosmological model within the experimental errors. Finally, we provide forecasts, exploiting the Baryon Acoustic Oscillations measurements from the Euclid survey., Comment: 22 pages, 17 figures; changes match published version

Recent work has demonstrated that it is important to constrain the dynamics of cosmological perturbations, in addition to the evolution of the background, if we want to distinguish among different models of the dark sector. Especially the anisotropic stress of the (possibly effective) dark energy fluid has been shown to be an important discriminator between modified gravity and dark energy models. In this paper we use approximate analytical solutions of the perturbation equations in the presence of viscosity to study how the anisotropic stress affects the weak lensing and galaxy power spectrum. We then forecast how sensitive the photometric and spectroscopic Euclid surveys will be to both the speed of sound and the viscosity of our effective dark energy fluid when using weak lensing tomography and the galaxy power spectrum. We find that Euclid alone can only constrain models with very small speed of sound and viscosity, while it will need the help of other observables in order to give interesting constraints on models with a sound speed close to one. This conclusion is also supported by the expected Bayes factor between models., Comment: 16 pages, 8 figures; matches published version

Euclid is a European Space Agency medium class mission selected for launch in 2019 within the Cosmic Vision 2015-2025 programme. The main goal of Euclid is to understand the origin of the accelerated expansion of the Universe. Euclid will explore the expansion history of the Universe and the evolution of cosmic structures by measuring shapes and redshifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid's Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission., Comment: 236 pages, minor edits to match the journal version 2013

The characterisation of dark energy is one of the primary goals in cosmology especially now that many new experiments are being planned with the aim of reaching a high sensitivity on cosmological parameters. It is known that if we move away from the simple cosmological constant model then we need to consider perturbations in the dark energy fluid. This means that dark energy has two extra degrees of freedom: the sound speed $\cs$ and the anisotropic stress $\sigma$. If dark energy is inhomogenous at the scales of interest then the gravitational potentials are modified and the evolution of the dark matter perturbations is also directly affected. In this paper we add an anisotropic component to the dark energy perturbations. Following the idea introduced in \cite{Sapone:2009mb}, we solve analytically the equations of perturbations in the dark sector, finding simple and accurate approximated solutions. We also find that the evolution of the density perturbations is governed by an effective sound speed which depends on both the sound speed and the anisotropic stress parameter. We then use these solutions to look at the impact of the dark energy perturbations on the matter power spectrum and on the Integrated Sachs-Wolfe effect in the Cosmic Microwave Background., Comment: 14 pages, 8 figures, to be submitted to PRD

We provide exact solutions to the cosmological matter perturbation equation in a homogeneous FLRW universe with a vacuum energy that can be parametrized by a constant equation of state parameter $w$ and a very accurate approximation for the Ansatz $w(a)=w_0+w_a(1-a)$. We compute the growth index $\gamma=\log f(a)/\log\Om_m(a)$, and its redshift dependence, using the exact and approximate solutions in terms of Legendre polynomials and show that it can be parametrized as $\gamma(a)=\gamma_0+\gamma_a(1-a)$ in most cases. We then compare four different types of dark energy (DE) models: $w\Lambda$CDM, DGP, $f(R)$ and a LTB-large-void model, which have very different behaviors at $z\gsim1$. This allows us to study the possibility to differentiate between different DE alternatives using wide and deep surveys like Euclid, which will measure both photometric and spectroscopic redshifts for several hundreds of millions of galaxies up to redshift $z\simeq 2$. We do a Fisher matrix analysis for the prospects of differentiating among the different DE models in terms of the growth index, taken as a given function of redshift or with a principal component analysis, with a value for each redshift bin for a Euclid-like survey. We use as observables the complete and marginalized power spectrum of galaxies $P(k)$ and the Weak Lensing (WL) power spectrum. We find that, using $P(k)$, one can reach (2%, 5%) errors in $(w_0, w_a)$, and (4%, 12%) errors in $(\gamma_0, \gamma_a)$, while using WL we get errors at least twice as large. These estimates allow us to differentiate easily between DGP, $f(R)$ models and $\Lambda$CDM, while it would be more difficult to distinguish the latter from a variable equation of state parameter or LTB models using only the growth index.}, Comment: 29 pages, 7 figures, 6 tables

The characterization of dark energy is a central task of cosmology. To go beyond a cosmological constant, we need to introduce at least an equation of state and a sound speed and consider observational tests that involve perturbations. If dark energy is not completely homogeneous on observable scales then the Poisson equation is modified and dark matter clustering is directly affected. One can then search for observational effects of dark energy clustering using dark matter as a probe. In this paper we exploit an analytical approximate solution of the perturbation equations in a general dark energy cosmology to analyze the performance of next-decade large scale surveys in constraining equation of state and sound speed. We find that tomographic weak lensing and galaxy redshift surveys can constrain the sound speed of the dark energy only if the latter is small, of the order of $c_{s}\lesssim0.01$ (in units of $c$). For larger sound speeds the error grows to 100% and more. We conclude that large scale structure observations contain very little information about the perturbations in canonical scalar field models with a sound speed of unity. Nevertheless, they are able to detect the presence of "cold" dark energy, i.e. a dark energy with non-relativistic speed of sound.

In this paper we review a part of the approaches that have been considered to explain the extraordinary discovery of the late time acceleration of the Universe. We discuss the arguments that have led physicists and astronomers to accept dark energy as the current preferable candidate to explain the acceleration. We highlight the problems and the attempts to overcome the difficulties related to such a component. We also consider alternative theories capable of explaining the acceleration of the Universe, such as modification of gravity. We compare the two approaches and point out the observational consequences, reaching the sad but foresightful conclusion that we will not be able to distinguish between a Universe filled by dark energy or a Universe where gravity is different from General Relativity. We review the present observations and discuss the future experiments that will help us to learn more about our Universe. This is not intended to be a complete list of all the dark energy models but this paper should be seen as a review on the phenomena responsible for the acceleration. Moreover, in a landscape of hardly compelling theories, it is an important task to build simple measurable parameters useful for future experiments that will help us to understand more about the evolution of the Universe., Comment: Minor revisions; to appear in IJMPA

Dark energy perturbations are normally either neglected or else included in a purely numerical way, obscuring their dependence on underlying parameters like the equation of state or the sound speed. However, while many different explanations for the dark energy can have the same equation of state, they usually differ in their perturbations so that these provide a fingerprint for distinguishing between different models with the same equation of state. In this paper we derive simple yet accurate approximations that are able to characterize a specific class of models (encompassing most scalar-field models) which is often generically called "dark energy". We then use the approximate solutions to look at the impact of the dark energy perturbations on the dark matter power spectrum and on the integrated Sachs-Wolfe effect in the cosmic microwave background radiation., Comment: 11 pages, 5 figures, minor changes to match published version

Dark energy can be studied by its influence on the expansion of the Universe as well as on the growth history of the large-scale structure. In this paper, we follow the growth of the cosmic density field in early dark energy cosmologies by combining observations of the primary CMB temperature and polarisation power spectra at high redshift, of the CMB lensing deflection field at intermediate redshift and of weak cosmic shear at low redshifts for constraining the allowed amount of early dark energy. We present these forecasts using the Fisher-matrix formalism and consider the combination of Planck-data with the weak lensing survey of Euclid. We find that combining these data sets gives powerful constraints on early dark energy and is able to break degeneracies in the parameter set inherent to the various observational channels. The derived statistical 1-sigma-bound on the early dark energy density parameter is sigma(Omega_d^e)=0.0022 which suggests that early dark energy models can be well examined in our approach. In addition, we derive the dark energy figure of merit for the considered dark energy parameterisation and comment on the applicability of the growth index to early dark energy cosmologies., Comment: 25 pages, 14 figures, 3 tables; v2: very minor additions, updated to match version to be published in JCAP

We discuss the phenomenology of the dark energy in first order perturbation theory, demonstrating that the dark energy cannot be fully constrained unless the dark matter is found, and that there are two functions that characterise the observational properties of the dark sector for cosmological probes. We argue that measuring these two functions should be an important goal for observational cosmology in the next decades., Comment: 4 pages, 2 figures, uses moriond.sty. To be published in the Proceedings of the XLIII Rencontres de Moriond 'Cosmology 2008'

The growth factor of linear fluctuations is probably one of the least known quantity in observational cosmology. Here we discuss the constraints that baryon oscillations in galaxy power spectra from future surveys can put on a conveniently parametrized growth factor. We find that spectroscopic surveys of $5000 deg^2$ extending to $z \approx 3$ could estimate the growth index $\gamma$ within 0.06; a similar photometric survey would give $\Delta\gamma\approx 0.15$. This test provides an important consistency check for standard cosmological model and could constrain modified gravity models. We discuss the errors and the figure of merit for various combinations of redshift errors and survey size., Comment: 7 pages, 8 figures, 5 tables

We introduce a convenient parametrization of dark energy models that is general enough to include several modified gravity models and generalized forms of dark energy. In particular we take into account the linear perturbation growth factor, the anisotropic stress and the modified Poisson equation. We discuss the sensitivity of large scale weak lensing surveys like the proposed DUNE satellite to these parameters. We find that a large-scale weak-lensing tomographic survey is able to easily distinguish the Dvali-Gabadadze-Porrati model from LCDM and to determine the perturbation growth index to an absolute error of 0.02-0.03., Comment: 19 pages, 11 figures

There is now strong observational evidence that the expansion of the universe is accelerating. The standard explanation invokes an unknown "dark energy" component. But such scenarios are faced with serious theoretical problems, which has led to increased interest in models where instead General Relativity is modified in a way that leads to the observed accelerated expansion. The question then arises whether the two scenarios can be distinguished. Here we show that this may not be so easy, demonstrating explicitely that a generalised dark energy model can match the growth rate of the DGP model and reproduce the 3+1 dimensional metric perturbations. Cosmological observations are then unable to distinguish the two cases., Comment: 4 pages, 2 figures

We consider fluid perturbations close to the "phantom divide" characterised by p = -rho and discuss the conditions under which divergencies in the perturbations can be avoided. We find that the behaviour of the perturbations depends crucially on the prescription for the pressure perturbation delta-p. The pressure perturbation is usually defined using the dark energy rest-frame, but we show that this frame becomes unphysical at the divide. If the pressure perturbation is kept finite in any other frame, then the phantom divide can be crossed. Our findings are important for generalised fluid dark energy used in data analysis (since current cosmological data sets indicate that the dark energy is characterised by p ~ -rho so that p < -rho cannot be excluded) as well as for any models crossing the phantom divide, like some modified gravity, coupled dark energy and braneworld models. We also illustrate the results by an explicit calculation for the "Quintom" case with two scalar fields., Comment: 14 pages, 10 figures, v2: updated to agree with published version: more readable figures, slightly expanded discussion on modified gravity models and the interpolation across w=-1, results and conclusions unchanged

We estimate the constraints that the recent high-redshift sample of supernovae type Ia put on a phenomenological interaction between dark energy and dark matter. The interaction can be interpreted as arising from the time variation of the mass of dark matter particles. We find that the coupling correlates with the equation of state: roughly speaking, a negative coupling (in our sign convention) implies phantom energy ($w_{\phi}<-1$) while a positive coupling implies ``ordinary'' dark energy. The constraints from the current supernovae Ia Hubble diagram favour a negative coupling and an equation of state $w_{\phi}<-1$. A zero or positive coupling is in fact unlikely at 99% c.l. (assuming constant equation of state); at the same time non-phantom values ($w_{\phi}>-1$) are unlikely at 95%. We show also that the usual bounds on the energy density weaken considerably when the coupling is introduced: values as large as $\Omega_{m0}=0.7$ become acceptable for as concerns SNIa. We find that the rate of change of the mass $\dot{m}/m$ of the dark matter particles is constrained to be $\delta_{0}$ in a Hubble time, with $-10<\delta_{0}<-1$ to 95% c.l.. We show that a large positive coupling might in principle avoid the future singularity known as ``big rip'' (occurring for $w_{\phi}<-1$) but the parameter region for this to occur is almost excluded by the data. We also forecast the constraints that can be obtained from future experiments, focusing on supernovae and baryon oscillations in the power spectra of deep redshift surveys. We show that the method of baryon oscillations holds the best potential to contrain the coupling., Comment: 10 pages, 14 figures, to be submitted to JCAP; v2: minor changes, added a plot and reference updates