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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.

We investigate how the cosmological equation of state can be used for scrutinizing extended theories of gravity, in particular, the Palatini f ( R ) gravity. Specifically, the approach consists, at first, in investigating the effective equation of state produced by a given model. Then, the inverse problem can also be considered in view of determining which models are compatible with a given effective equation of state. We consider and solve some cases and show that, for example, power-law models are (the only models) capable of transforming barotropic Equations of State into effective barotropic ones. Moreover, the form of equation of state is preserved (only) for f ( R ) = R , as expected. In this perspective, modified Equations of State are a feature capable of distinguishing extended gravity with respect to general relativity. We investigate also quadratic and non-homogeneous effective Equations of State showing, in particular, that they contain the Starobinsky model and other ones.

The angular power spectrum is a natural tool to analyse the observed galaxy number count fluctuations. In a standard analysis, the angular galaxy distribution is sliced into concentric redshift bins and all correlations of its harmonic coefficients between bin pairs are considered—a procedure referred to as ‘tomography’. However, the unparalleled quality of data from oncoming spectroscopic galaxy surveys for cosmology will render this method computationally unfeasible, given the increasing number of bins. Here, we put to test against synthetic data a novel method proposed in a previous study to save computational time. According to this method, the whole galaxy redshift distribution is subdivided into thick bins, neglecting the cross-bin correlations among them; each of the thick bin is, however, further subdivided into thinner bins, considering in this case all the cross-bin correlations. We create a simulated data set that we then analyse in a Bayesian framework. We confirm that the newly proposed method saves computational time and gives results that surpass those of the standard approach.

New data from ongoing galaxy surveys, such as the \\(Euclid\\) satellite and the Dark Energy Spectroscopic Instrument (DESI), are expected to unveil physics on the largest scales of our universe. Dramatically affected by cosmic variance, these scales are of interest to large-scale structure studies as they exhibit relevant corrections due to general relativity (GR) in the \\(n\\)-point statistics of cosmological random fields. We focus on the relativistic, sample-dependent Doppler contribution to the observed clustering of galaxies, whose detection will further confirm the validity of GR in cosmological regimes. Sample- and scale-dependent, the Doppler term is more likely to be detected via cross-correlation measurements, where it acts as an imaginary correction to the power spectrum of fluctuations in galaxy number counts. We present a method allowing us to exploit multi-tracer benefits from a single data set, by subdividing a galaxy population into two sub-samples, according to galaxies' luminosity/magnitude. To overcome cosmic variance we rely on a multi-tracer approach, and to maximise the detectability of the relativistic Doppler contribution in the data, we optimise sample selection. As a result, we find the optimal split and forecast the relativistic Doppler detection significance for both a DESI-like Bright Galaxy Sample and a \\(Euclid\\)-like H\\(\\alpha\\) galaxy population.

We present a method to obtain a high-significance detection of relativistic effects on cosmological scales. Measurements of such effects would be instrumental for our understanding of the Universe, as they would provide a further confirmation of the validity of general relativity as the correct description of the gravitational interaction, in a regime very far from that of strong gravity, where it has been tested to exquisite accuracy. Despite its relevance, the detection of relativistic effects has hitherto eluded us, mainly because they are stronger on the largest cosmic scales, plagued by cosmic variance. Our work focuses on the cosmological probe of galaxy clustering, describing the excess probability of finding pairs of galaxies at a given separation due to them being part of the same underlying cosmic large-scale structure. We focus on the two-point correlation function of the distribution of galaxies in Fourier space -- the power spectrum -- where relativistic effects appear as an imaginary contribution to the real power spectrum. By carefully tailoring cuts in magnitude/luminosity, we are able to obtain two samples (bright and faint) of the same galaxy population, whose cross-correlation power spectrum allows for a detection of the relativistic contribution. In particular, we optimise the definition of the samples to maximise the detection significance of the relativistic Doppler term for both a low-\\(z\\) Bright Galaxy Sample and a high-\\(z\\) H\\(\\alpha\\) emission line galaxy population.

We present a new, model-independent measurement of the clustering amplitude of galaxies and the growth of cosmic large-scale structures from the Baryon Oscillation Spectroscopic Survey (BOSS) 12th data release (DR12). This is achieved by generalising harmonic-space power spectra for galaxy clustering to measure separately the magnitudes of the density and the redshift-space distortion terms, respectively related to the clustering amplitude of structures, \\(b\\sigma_8(z)\\), and their growth, \\(f\\sigma_8(z)\\). We adopt a tomographic approach with 15 redshift bins in \\(z\\in[0.15,0.67]\\). We restrict our analysis to strictly linear scales, implementing a redshift-dependent maximum multipole for each bin. The measurements do not appear to suffer from systematic effects and show excellent agreement with the theoretical predictions from the Planck cosmic microwave background analysis assuming a \\(\\Lambda\\)CDM cosmology. Our results also agree with previous analyses by the BOSS collaboration. Furthermore, our method provides the community with a new tool for data analyses of the cosmic large-scale structure complementary to state-of-the-art approaches in configuration or Fourier space. Amongst its merits, we list: it being more agnostic with respect to the underlying cosmological model; its roots in a well-defined and gauge-invariant observable; the possibility to account naturally for wide-angle effects and even relativistic corrections on ultra-large scales; and the capability to perform an almost arbitrarily fine redshift binning with little computational effort. These aspects are all the more relevant for the oncoming generation of cosmological experiments such as Euclid, the Dark Energy Spectroscopic Instrument (DESI), the Legacy Survey of Space and Time (LSST), and the SKA Project.

I present a new technique for the measurement of the growth of cosmic structures via the power spectrum of weak lensing cosmic shear. It is based on a template-fitting approach, where a redshift-dependent amplitude of lensing modulates a fixed template power spectrum. Such an amplitude, which is promoted to a free parameter and fit against tomographic cosmic shear data, reads \\(D(z)\\,\\sigma_8\\,\\Omega_{{\\rm m},0}=:\\Omega\\sigma_8(z)\\), with \\(D(z)\\) the growth factor, \\(\\sigma_8\\) a proxy for the overall amplitude of the matter power spectrum, and \\(\\Omega_{{\\rm m},0}\\) the present-day matter abundance. I show that this method is able to correctly reconstruct \\(\\Omega\\sigma_8\\) at the per cent level across redshift, thus allowing us to measure the growth of structures unbiased by observing discrete tracers. Moreover, I only makes use of measurements on linear scales. The method is highly complementary to measurements of the bias and growth, \\(b\\sigma_8(z)\\) and \\(f\\sigma_8(z)\\), from galaxy clustering analysis. I also demonstrate that the method is robust against an incorrect choice of cosmological parameters in the template, thanks to the inclusion of an Alcock-Paczyński parameter.

The chemical composition of the highest-energy cosmic rays, namely the atomic number \\(Z\\) of rays with energies \\(E\\gtrsim40~\\mathrm{EeV}\\), remains to date largely unknown. Some information on the composition can be inferred from the deflections that charged ultra-high-energy cosmic rays experience while they traverse intervening magnetic fields. Indeed, such deflections distort and suppress the original anisotropy in the cosmic ray arrival directions; thus, given a source model, a measure of the anisotropy is also a measurement of the deflections, which in turn informs us on the chemical composition. In this work, we show that, by quantifying ultra-high-energy cosmic ray anisotropies through the angular cross-correlation between cosmic rays and galaxies, we would be able to exclude iron fractions \\(f_{\\rm Fe}\\geq{\\cal O}(10\\%)\\) assuming a fiducial hydrogen map at \\(2\\,\\sigma\\) level, and even smaller fractions in the reverse case of hydrogen on an iron map, going well below \\(f_{\\rm H}\\approx10\\%\\) when we mask the Galactic Centre up to latitudes of \\(40^\\circ\\). This is an improvement of a factor of a few compared to our previous method, and is mostly ascribable to a new test statistics which is sensitive to each harmonic multipole individually. Our method can be applied to real data as an independent test of the recent claim that current cosmic-ray data can not be reproduced by any existing model of the Galactic magnetic field, as well as an additional handle to compare any realistic, competing, data-driven composition models.

The astrophysical engines that power ultra-high-energy cosmic rays (UHECRs) remain to date unknown. Since the propagation horizon of UHECRs is limited to the local, anisotropic Universe, the distribution of UHECR arrival directions should be anisotropic. In this paper we expand the analysis of the potential for the angular, harmonic cross-correlation between UHECRs and galaxies to detect such anisotropies. We do so by studying simulations performed assuming proton, oxygen and silicon injection models, each simulation containing a number of events comparable to a conservative estimate of currently available datasets, as well as by extending the analytic treatment of the magnetic deflections. Quantitatively, we find that, while the correlations for each given multipole are generally weak, (1) the total harmonic power summed over multipoles is detectable with signal-to-noise ratios well above~5 for both the auto-correlation and the cross-correlation (once optimal weights are applied) in most cases studied here, with peaks of signal-to-noise ratio around between~8 and~10 at the highest energies; (2) if we combine the UHECR auto-correlation and the cross-correlation we are able to reach detection levels of (3\\sigma) and above for individual multipoles at the largest scales, especially for heavy composition. In particular, we predict that the combined-analysis quadrupole could be detected already with existing data.

We present a method that allows us for the first time to estimate the signal-to-noise ratio (SNR) of the harmonic-space galaxy bispectrum induced by gravity, a complementary probe to already well established Fourier-space clustering analyses. We show how to do it considering only \\(\\sim1000\\) triangle configurations in multipole space, corresponding to a computational speedup of a factor \\(\\mathcal{O}(10^2)-\\mathcal{O}(10^3)\\), depending on the redshift bin, when including mildly non-linear scales. Assuming observational specifications consistent with forthcoming spectroscopic and photometric galaxy surveys like the Euclid satellite and the Square Kilometre Array (phase 1), we show: that given a single redshift bin, spectroscopic surveys outperform photometric surveys; and that -- due to shot-noise and redshift bin width balance -- bins at redshifts \\(z\\sim1\\) bring higher cumulative SNR than bins at lower redshifts \\(z \\sim 0.5\\). Our results for the largest cumulative SNR \\(\\sim 15\\) suggest that the harmonic-space bispectrum is detectable within narrow (\\(\\Delta z \\sim 0.01\\)) spectroscopic redshift bins even when including only mildly non-linear scales. Tomographic reconstructions and inclusion of highly non-linear scales will further boost detectability with upcoming galaxy surveys. In addition, we discuss how, using the Karhunen-Loève transform, a detection analysis only requires a \\(1 \\times 1\\) covariance matrix for a single redshift bin.