The current standard model of cosmology successfully describes a variety of measurements, but the nature of its main ingredients, dark matter and dark energy, remains unknown. Euclid is a medium-class mission in the Cosmic Vision 2015–2025 programme of the European Space Agency (ESA) that will provide high-resolution optical imaging, as well as near-infrared imaging and spectroscopy, over about 14 000 deg² of extragalactic sky. In addition to accurate weak lensing and clustering measurements that probe structure formation over half of the age of the Universe, its primary probes for cosmology, these exquisite data will enable a wide range of science. This paper provides a high-level overview of the mission, summarising the survey characteristics, the various data-processing steps, and data products. We also highlight the main science objectives and expected performance.

We implement the effective field theory (EFT) approach to dark energy and modified gravity in the public Einstein-Boltzmann solver CAMB. The resulting code, which we dub EFTCAMB, is a powerful and versatile tool that can be used for several objectives. It can be employed to evolve the full dynamics of linear scalar perturbations of a broad range of single field dark energy and modified gravity model, once the model of interest is mapped into the EFT formalism. It offers a numerical implementation of EFT as a model-independent framework to test gravity on cosmological scales. EFTCAMB has a built-in check for the fulfillment of general stability conditions such as the absence of ghost and superluminal propagation of perturbations. It handles phantom-divide crossing models and does not contain any quasistatic approximation, but rather evolves the full dynamics of perturbations on linear scales. As we will show, the latter is an important feature in view of the accuracy and scale range of upcoming surveys. We show the reliability and applicability of our code by evolving the dynamics of linear perturbations and extracting predictions for power spectra in several models. In particular we perform a thorough analysis of $f(R)$ theories, comparing our outputs with those of an existing code for $\mathrm{$\Lambda${}}\mathrm{CDM}$ backgrounds, and finding an agreement that can reach 0.1% for models with a Compton wavelength consistent with current cosmological data. We then showcase the flexibility of our code studying two different scenarios. First we produce new results for designer $f(R)$ models with a time-varying dark energy equation of state. Second, we extract predictions for linear observables in some parametrized EFT models with a phantom-divide crossing equation of state for dark energy.

We introduce EFTCAMB/EFTCosmoMC as publicly available patches to the commonly used camb/CosmoMC codes. We briefly describe the structure of the codes, their applicability and main features. To illustrate the use of these patches, we obtain constraints on parametrized pure effective field theory and designer $f(R)$ models, both on $\mathrm{\Lambda }\mathrm{CDM}$ and $w\mathrm{CDM}$ background expansion histories, using data from Planck temperature and lensing potential spectra, WMAP low-$\ell$ polarization spectra (WP), and baryon acoustic oscillations (BAO). Upon inspecting the theoretical stability of the models on the given background, we find nontrivial parameter spaces that we translate into viability priors. We use different combinations of data sets to show their individual effects on cosmological and model parameters. Our data analysis results show that, depending on the adopted data sets, in the $w\mathrm{CDM}$ background case these viability priors could dominate the marginalized posterior distributions. Interestingly, with $\mathrm{Planck}+\mathrm{WP}+\mathrm{BAO}+\mathrm{lensing}$ data, in $f(R)$ gravity models, we get very strong constraints on the constant dark energy equation of state, $w_0\in (-1,-0.9997)$ (95% C.L.).

We investigate the impact on cosmological observables of $f(Q)$-gravity, a specific class of modified gravity models in which gravity is described by the nonmetricity scalar, $Q$. In particular we focus on a specific model which is indistinguishable from the $\mathrm{\Lambda }$-cold-dark-matter ($\mathrm{\Lambda }\mathrm{CDM}$) model at the background level, while showing peculiar and measurable signatures at linear perturbation level. These are attributed to a time-dependent Planck mass and are regulated by a single dimensionless parameter, $\alpha$. In comparison to the $\mathrm{\Lambda }\mathrm{CDM}$ model, we find for positive values of $\alpha$ a suppressed matter power spectrum and lensing effect on the cosmic microwave background radiation (CMB) angular power spectrum and an enhanced integrated-Sachs-Wolfe tail of CMB temperature anisotropies. The opposite behaviors are present when the $\alpha$ parameter is negative. We also investigate the modified gravitational waves (GWs) propagation and show the prediction of the GWs luminosity distance compared to the standard electromagnetic one. Finally, we infer the accuracy on the free parameter of the model with standard sirens at future GWs detectors.