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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.

The notion of self-acceleration has been introduced as a convenient way to theoretically distinguish cosmological models in which acceleration is due to modified gravity from those in which it is due to the properties of matter or fields. In this paper we review the concept of self-acceleration as given, for example, by [1], and highlight two problems. First, that it applies only to universal couplings, and second, that it is too narrow, i.e. it excludes models in which the acceleration can be shown to be induced by a genuine modification of gravity, for instance coupled dark energy with a universal coupling, the Hu-Sawicki f(R) model or, in the context of inflation, the Starobinski model. We then propose two new, more general, concepts in its place: force-acceleration and field-acceleration, which are also applicable in presence of non universal cosmologies. We illustrate their concrete application with two examples, among the modified gravity classes which are still in agreement with current data, i.e. f(R) models and coupled dark energy.

We present a first application to photometric galaxy clustering and weak lensing of wavelet based multi-scale higher order summary statistics: starlet peak counts and starlet \\(\\ell_1\\)-norm. Peak counts are the local maxima in the map and the \\(\\ell_1\\)-norm is computed via the sum of the absolute values of the starlet (wavelet) decomposition coefficients of a map, providing a fast multi-scale calculation of the pixel distribution, encoding the information of all pixels in the map. We employ the cosmo-SLICS simulations sources and lenses catalogues and we compute wavelet based higher order statistics in the context of combined probes and their potential when applied to the weak lensing convergence maps and galaxy maps. We get forecasts on the matter density parameter \\(\\Omega_{\\rm m}\\), the reduced Hubble constant \\(h\\), the matter fluctuation amplitude \\(\\sigma_8\\), and the dark energy equation of state parameter \\(w_0\\). We find that, in our setting for this first application, considering the two probes as independent, starlet peaks and the \\(\\ell_1\\)-norm represent interesting summary statistics that can improve the constraints with respect to the power spectrum also in the case of photometric galaxy clustering and when the two probes are combined.

We present a new summary statistic for weak lensing observables, higher than second order, suitable for extracting non-Gaussian cosmological information and inferring cosmological parameters. We name this statistic the 'starlet \\(\\ell_1\\)-norm' as it is computed via the sum of the absolute values of the starlet (wavelet) decomposition coefficients of a weak lensing map. In comparison to the state-of-the-art higher-order statistics -- weak lensing peak counts and minimum counts, or the combination of the two -- the \\(\\ell_1\\)-norm provides a fast multi-scale calculation of the full void and peak distribution, avoiding the problem of defining what a peak is and what a void is: The \\(\\ell_1\\)-norm carries the information encoded in all pixels of the map, not just the ones in local maxima and minima. We show its potential by applying it to the weak lensing convergence maps provided by the MassiveNus simulations to get constraints on the sum of neutrino masses, the matter density parameter, and the amplitude of the primordial power spectrum. We find that, in an ideal setting without further systematics, the starlet \\(\\ell_1\\)-norm remarkably outperforms commonly used summary statistics, such as the power spectrum or the combination of peak and void counts, in terms of constraining power, representing a promising new unified framework to simultaneously account for the information encoded in peak counts and voids. We find that the starlet \\(\\ell_1\\)-norm outperforms the power spectrum by \\(72\\%\\) on M\\(_{\\nu}\\), \\(60\\%\\) on \\(\\Omega_{\\rm m}\\), and \\(75\\%\\) on \\(A_{\\rm s}\\) for the Euclid-like setting considered; it also improves upon the state-of-the-art combination of peaks and voids for a single smoothing scale by \\(24\\%\\) on M\\(_{\\nu}\\), \\(50\\%\\) on \\(\\Omega_{\\rm m}\\), and \\(24\\%\\) on \\(A_{\\rm s}\\).

The Planck collaboration has recently published maps of the Cosmic Microwave Background (CMB) radiation, in good agreement with a LCDM model, a fit especially valid for multipoles l > 40. We explore here the possibility that dark energy is dynamical and gravitational attraction between dark matter particles is effectively different from the standard one in General Relativity: this is the case of coupled dark energy models, where dark matter particles feel the presence of a fifth force, larger than gravity by a factor beta^2. We investigate constraints on the strength of the coupling beta in view of Planck data. Interestingly, we show that a non-zero coupling is compatible with data and find a likelihood peak at beta = 0.036 \\pm 0.016 (Planck + WP + BAO) (compatible with zero at 2sigma). The significance of the peak increases to beta = 0.066 \\pm 0.018 (Planck + WP + HST) (around 3.6sigma) when Planck is combined to Hubble Space Telescope data. This peak comes mostly from the small difference between the Hubble parameter determined with CMB measurements and the one coming from astrophysics measurements. In this sense, future observations and further tests of current observations are needed to determine whether the discrepancy is due to systematics in any of the datasets. Our aim here is not to claim new physics but rather to show how Planck data can be used to provide information on dynamical dark energy and modified gravity, allowing us to test the strength of an effective fifth force between dark matter particles with precision smaller than 2%.

We consider coupled dark energy (CDE) cosmologies, where dark matter particles feel a force stronger than gravity, due to the fifth force mediated by a scalar field which plays the role of dark energy. We perform for the first time a tomographic analysis of coupled dark energy, where the coupling strength is parametrized and constrained in different redshift bins. This allows us to verify which data can better constrain the strength of the coupling and how large the coupling can be at different epochs. First, we employ cosmic microwave background data from Planck, the Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT), showing the impact of different choices that can be done in combining these datasets. Then, we use a range of low redshift probes to test CDE cosmologies, both for a constant and for a tomographic coupling. In particular, we use for the first time data from weak lensing (the KiDS-1000 survey), galaxy clustering (BOSS survey), and their combination, including 3x2pt galaxy-galaxy lensing cross-correlation data. We do not find evidence for nonzero coupling, either for a constant or tomographic case. A nonzero coupling is however still in agreement with current data. For CMB and background datasets, a tomographic coupling allows for \\(\\beta\\) values up to one order of magnitude larger than in previous works, in particular at \\(z<1\\). The use of 3x2pt analysis then becomes important to constrain \\(\\beta\\) at low redshifts, even when coupling is allowed to vary: for 3x2pt we find, at \\(0.5 < z < 1\\), \\(\\beta=0.0180_{-0.011}^{+0.007}\\), comparable to what CMB and background datasets would give for a constant coupling. This makes upcoming galaxy surveys potentially powerful probes to test CDE models at low redshifts. (abridged)

We investigate an Early Coupled Quintessence model where a light scalar mediates a fifth force stronger than gravity among dark matter particles and leads to the growth of perturbations prior to matter-radiation equality. Using cosmological data from the \\(\\textit{Planck}\\) Cosmic Microwave Background power spectra, the Pantheon+ Type 1a Supernovae, Baryon Acoustic Oscillations, and Big Bang Nucleosynthesis, we constrain the coupling strength \\(\\beta\\) and the redshift \\(z_{\\rm OFF}\\) at which the interaction becomes effectively inactive, finding a firm degeneracy between these two parameters which holds true regardless of when the scaling regime begins.

Some cosmological models with non-negligible dark energy fractions in particular windows of the pre-recombination epoch are capable of alleviating the Hubble tension quite efficiently, while keeping the good description of the data that are used to build the cosmic inverse distance ladder. There has been an intensive discussion in the community on whether these models enhance the power of matter fluctuations, leading {\\it de facto} to a worsening of the tension with the large-scale structure measurements. We address this pivotal question in the context of several early dark energy (EDE) models, considering also in some cases a coupling between dark energy and dark matter, and the effect of massive neutrinos. We fit them using the Planck 2018 likelihoods, the supernovae of Type Ia from the Pantheon compilation and data on baryon acoustic oscillations. We find that ultra-light axion-like (ULA) EDE can actually alleviate the \\(H_0\\) tension without increasing the values of \\(\\sigma_{12}\\) with respect to those found in the \\(\\Lambda\\)CDM, whereas EDE with an exponential potential does not have any impact on the tensions. A coupling in the dark sector tends to enhance the clustering of matter, and the data limit a lot the influence of massive neutrinos, since the upper bounds on the sum of their masses are too close to those obtained in the standard model. We find that in the best case, namely ULA, the Hubble tension is reduced to \\(\\sim 2\\sigma\\).

UNIONS is an ongoing deep photometric multi-band survey of the Northern sky. As part of UNIONS, CFIS provides r-band data which we use to study weak-lensing peak counts for cosmological inference. We assess systematic effects for weak-lensing peak counts and their impact on cosmological parameters for the UNIONS survey. In particular, we present results on local calibration, metacalibration shear bias, baryonic feedback, the source galaxy redshift estimate, intrinsic alignment, and the cluster member dilution. For each uncertainty and systematic effect, we describe our mitigation scheme and the impact on cosmological parameter constraints. We obtain constraints on cosmological parameters from MCMC using CFIS data and MassiveNuS N-body simulations as a model for peak counts statistics. Depending on the calibration (local versus global, and the inclusion of the residual multiplicative shear bias), the mean matter density parameter \\(\\Omega_m\\) can shift up to \\(-0.024\\) (\\(-0.5\\sigma\\)). We also see that including baryonic corrections can shift \\(\\Omega_m\\) by \\(+0.027\\) (\\(+0.5 \\sigma\\)) with respect to the DM-only simulations. Reducing the impact of the intrinsic alignment and cluster member dilution through signal-to-noise cuts can lead to a shift in \\(\\Omega_m\\) of \\(+0.027\\) (\\(+0.5 \\sigma\\)). Finally, with a mean redshift uncertainty of \\(\\Delta \\bar{z} = 0.03\\), we see that the shift of \\(\\Omega_m\\) (\\(+0.001\\) which corresponds to \\(+0.02 \\sigma\\)) is not significant. This paper investigates for the first time with UNIONS weak-lensing data and peak counts the impact of systematic effects. The value of \\(\\Omega_m\\) is the most impacted and can shift up to \\(\\sim 0.03\\) which corresponds to \\(0.5\\sigma\\) depending on the choices for each systematics. We expect constraints to become more reliable with future (larger) data catalogues, for which the current pipeline will provide a starting point.

Modified gravity theories predict in general a non standard equation for the propagation of gravitational waves. Here we discuss the impact of modified friction and speed of tensor modes on cosmic microwave polarization B modes. We show that the non standard friction term, parametrized by \\(\\alpha_{M}\\), is degenerate with the tensor-to-scalar ratio \\(r\\), so that small values of \\(r\\) can be compensated by negative constant values of \\(\\alpha_M\\). We quantify this degeneracy and its dependence on the epoch at which \\(\\alpha_{M}\\) is different from the standard, zero, value and on the speed of gravitational waves \\(c_{T}\\). In the particular case of scalar-tensor theories, \\(\\alpha_{M}\\) is constant and strongly constrained by background and scalar perturbations, \\(0\\le \\alpha_{M}< 0.01\\) and the degeneracy with \\(r\\) is removed. In more general cases however such tight bounds are weakened and the B modes can provide useful constraints on early-time modified gravity.