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.

Brazil ranks second worldwide in total number of COVID-19 cases and deaths. Understanding the possible socioeconomic and ethnic health inequities is particularly important given the diverse population and fragile political and economic situation. We aimed to characterise the COVID-19 pandemic in Brazil and assess variations in mortality according to region, ethnicity, comorbidities, and symptoms. We conducted a cross-sectional observational study of COVID-19 hospital mortality using data from the SIVEP-Gripe (Sistema de Informação de Vigilância Epidemiológica da Gripe) dataset to characterise the COVID-19 pandemic in Brazil. In the study, we included hospitalised patients who had a positive RT-PCR test for severe acute respiratory syndrome coronavirus 2 and who had ethnicity information in the dataset. Ethnicity of participants was classified according to the five categories used by the Brazilian Institute of Geography and Statistics: Branco (White), Preto (Black), Amarelo (East Asian), Indígeno (Indigenous), or Pardo (mixed ethnicity). We assessed regional variations in patients with COVID-19 admitted to hospital by state and by two socioeconomically grouped regions (north and central-south). We used mixed-effects Cox regression survival analysis to estimate the effects of ethnicity and comorbidity at an individual level in the context of regional variation. Of 99 557 patients in the SIVEP-Gripe dataset, we included 11 321 patients in our study. 9278 (82·0%) of these patients were from the central-south region, and 2043 (18·0%) were from the north region. Compared with White Brazilians, Pardo and Black Brazilians with COVID-19 who were admitted to hospital had significantly higher risk of mortality (hazard ratio [HR] 1·45, 95% CI 1·33–1·58 for Pardo Brazilians; 1·32, 1·15–1·52 for Black Brazilians). Pardo ethnicity was the second most important risk factor (after age) for death. Comorbidities were more common in Brazilians admitted to hospital in the north region than in the central-south, with similar proportions between the various ethnic groups. States in the north had higher HRs compared with those of the central-south, except for Rio de Janeiro, which had a much higher HR than that of the other central-south states. We found evidence of two distinct but associated effects: increased mortality in the north region (regional effect) and in the Pardo and Black populations (ethnicity effect). We speculate that the regional effect is driven by increasing comorbidity burden in regions with lower levels of socioeconomic development. The ethnicity effect might be related to differences in susceptibility to COVID-19 and access to health care (including intensive care) across ethnicities. Our analysis supports an urgent effort on the part of Brazilian authorities to consider how the national response to COVID-19 can better protect Pardo and Black Brazilians, as well as the population of poorer states, from their higher risk of dying of COVID-19. None.

The determination of the Hubble constant $H_0$ from the cosmic microwave background by the Planck Collaboration (N. Aghanim et al., arXiv:1807.06209) is in tension at $4.2\sigma$ with respect to the local determination of $H_0$ by the SH0ES collaboration [M. J. Reid et al., Astrophys. J. Lett. 886, L27 (2019)]. Here we improve upon the local determination, which fixes the deceleration parameter to the standard $\mathrm{\Lambda }\mathrm{CDM}$ model value of $q_0=-0.55$, that is, uses information from observations beyond the local universe. First, we derive the effective calibration prior on the absolute magnitude $M_B$ of type Ia supernovae, which can be used in cosmological analyses in order to avoid the double counting of low-redshift supernovae. We find $M_B=-19.2334\pm 0.0404$ mag. Then we use the above $M_B$ prior in order to obtain a determination of the local $H_0$ which uses only local observations and assumes only the cosmological principle, that is, large-scale homogeneity and isotropy. This is achieved by adopting an uninformative flat prior for $q_0$ in the cosmographic expansion of the luminosity distance. We use the latest Pantheon sample and find $H_0=75.35\pm 1.68\text{km}{\text{s}}^{-1}{\mathrm{Mpc}}^{-1}$, which features a 2.2% uncertainty, close to the 1.9% error obtained by the SH0ES Collaboration. Our determination is at the higher tension of $4.5\sigma$ with the latest results from the Planck Collaboration that assume the $\mathrm{\Lambda }\mathrm{CDM}$ model. Furthermore, we also constrain the deceleration parameter to $q_0=-1.08\pm 0.29$, which disagrees with the Planck Collaboration at the $1.9\sigma$ level. These estimations only use supernovae in the redshift range $0.023\le z\le 0.15$.

A dark-energy, which behaves as the cosmological constant until a sudden phantom transition at very low redshift (z < 0.1), seems to solve the >4σ disagreement between the local and high-redshift determinations of the Hubble constant, while maintaining the phenomenological success of the Λ cold dark matter model with respect to the other observables. Here, we show that such a hockey-stick dark energy cannot solve the H₀ crisis. The basic reason is that the supernova absolute magnitude M_B that is used to derive the local H₀ constraint is not compatible with the M_B that is necessary to fit supernova, baryon acoustic oscillation, and cosmic microwave background data, and this disagreement is not solved by a sudden phantom transition at very low redshift. We make use of this example to show why it is preferable to adopt in the statistical analyses the prior on M_B as an alternative to the prior on H₀. The three reasons are: (i) one avoids potential double counting of low-redshift supernovae, (ii) one avoids assuming the validity of cosmography, in particular, fixing the deceleration parameter to the standard model value q₀ = −0.55, (iii) one includes in the analysis the fact that M_B is constrained by local calibration, an information which would otherwise be neglected in the analysis, biasing both model selection and parameter constraints. We provide the priors on M_B relative to the recent Pantheon and DES-SN3YR supernova catalogs. We also provide a Gaussian joint prior on H₀ and q₀ that generalizes the prior on H₀ by Supernova H0 for the Equation of State.

There is an approximately 9% discrepancy, corresponding to $2.4\sigma$, between two independent constraints on the expansion rate of the Universe: one indirectly arising from the cosmic microwave background and baryon acoustic oscillations and one more directly obtained from local measurements of the relation between redshifts and distances to sources. We argue that by taking into account the local gravitational potential at the position of the observer this tension—strengthened by the recent Planck results—is partially relieved and the concordance of the Standard Model of cosmology increased. We estimate that measurements of the local Hubble constant are subject to a cosmic variance of about 2.4% (limiting the local sample to redshifts $z>0.010$) or 1.3% (limiting it to $z>0.023$), a more significant correction than that taken into account already. Nonetheless, we show that one would need a very rare fluctuation to fully explain the offset in the Hubble rates. If this tension is further strengthened, a cosmology beyond the Standard Model may prove necessary.

Photon geodesics are calculated in a Swiss-cheese model, where the cheese is made of the usual Friedmann-Robertson-Walker (FRW) solution and the holes are constructed from a Lemaître-Tolman-Bondi solution of Einstein’s equations. The observables on which we focus are the changes in the redshift, in the angular-diameter-distance relation, in the luminosity-distance-redshift relation, and in the corresponding distance modulus. We find that redshift effects are suppressed when the hole is small because of a compensation effect acting on the scale of half a hole resulting from the special case of spherical symmetry. However, we find interesting effects in the calculation of the angular distance: strong evolution of the inhomogeneities (as in the approach to caustic formation) causes the photon path to deviate from that of the FRW case. Therefore, the inhomogeneities are able to partly mimic the effects of a dark-energy component. Our results also suggest that the nonlinear effects of caustic formation in cold dark matter models may lead to interesting effects on photon trajectories.