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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 red-shifts 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.

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.

We apply an Internal Robustness (iR) analysis to the recently released Dark Energy Spectroscopic Instrument (DESI) baryon acoustic oscillations dataset. This approach examines combinations of data subsets through a fully Bayesian model comparison, aiming to identify potential outliers, subsets possibly influenced by systematic errors, or hints of new physics. Using this approach, we identify three data points at \\(z= 0.295,\\,0.51,\\,0.71\\) as potential outliers. Excluding these points improves the internal robustness of the dataset by minimizing statistical anomalies and enables the recovery of \\(\\Lambda\\)CDM predictions with a best-fit value of \\(w_0 = -1.050 \\pm 0.128\\) and \\(w_a = 0.208 \\pm 0.546\\). These results raise the intriguing question of whether the identified outliers signal the presence of systematics or point towards new physics.

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\\).

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.

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.

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.

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.

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.

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.