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.

Phantom dark energy ($w<-1$) can produce amplified cosmic acceleration at late times, thus increasing the value of $H_0$ favored by CMB data and releasing the tension with local measurements of $H_0$. We show that the best fit value of $H_0$ in the context of the CMB power spectrum is degenerate with a constant equation-of-state parameter $w$, in accordance with the approximate effective linear equation $H_0+30.93w-36.47=0$ ($H_0$ in $\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$). This equation is derived by assuming that both ${\mathrm{\Omega }}_{0m}{h}^2$ and $d_A={\int }_0^{z_{\mathrm{rec}}}\frac{dz}{H(z)}$ remain constant (for an invariant CMB spectrum) and equal to their best fit Planck/$\mathrm{\Lambda }\mathrm{CDM}$ values as $H_0$, ${\mathrm{\Omega }}_{0m}$, and $w$ vary. For $w=-1$, this linear degeneracy equation leads to the best fit $H_0=67.4\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$ as expected. For $w=-1.22$, the corresponding predicted CMB best fit Hubble constant is $H_0=74\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$, which is identical with the value obtained by local-distance ladder measurements, while the best fit matter density parameter is predicted to decrease, since ${\mathrm{\Omega }}_{0m}{h}^2$ is fixed. We verify the above $H_0-w$ degeneracy equation by fitting a $w\mathrm{CDM}$ model with fixed values of $w$ to the Planck TT spectrum, showing also that the quality of fit (${\chi }^2$) is similar to that of $\mathrm{\Lambda }\mathrm{CDM}$. However, when including SnIa, baryon acoustic oscillation, or growth data, the quality of fit becomes worse than $\mathrm{\Lambda }\mathrm{CDM}$ when $w<-1$. Finally, we generalize the $H_0-w(z)$ degeneracy equation for the parametrization $w(z)=w_0+w_{1}z/(1+z)$ and identify analytically the full $w_0-w_1$ parameter region (straight line) that leads to a best fit $H_0=74\mathrm{km}{\sec }^{-1}{\mathrm{Mpc}}^{-1}$ in the context of the Planck CMB spectrum. This exploitation of $H_0-w(z)$ degeneracy can lead to immediate identification of all parameter values of a given $w(z)$ parametrization that can potentially resolve the $H_0$ tension.

A rapid phantom transition of the dark energy equation of state parameter $w$ at a transition redshift $z_t<0.1$ of the form $w(z)=-1+\mathrm{\Delta }w\mathrm{\Theta }(z_t-z)$ with $\mathrm{\Delta }w<0$ can lead to a higher value of the Hubble constant while closely mimicking a $\mathrm{Planck}18/\mathrm{\Lambda }\mathrm{CDM}$ form of the comoving distance $r(z)={\int }_0^z\frac{d{z}^{\prime }}{H(z^{\prime })}$ for $z>z_t$. Such a transition however would imply a significantly lower value of the SnIa absolute magnitude $M$ than the value $M_C$ imposed by local Cepheid calibrators at $z<0.01$. Thus, in order to resolve the $H_0$ tension it would need to be accompanied by a similar transition in the value of the SnIa absolute magnitude $M$ as $M(z)=M_C+\mathrm{\Delta }M\mathrm{\Theta }(z-z_t)$ with $\mathrm{\Delta }M<0$. This is a late $w-M$ phantom transition ($LwMPT$). It may be achieved by a sudden reduction of the value of the normalized effective Newton constant $\mu =G_{\mathrm{eff}}/G_{\mathrm{N}}$ by about 6% assuming that the absolute luminosity of SnIa is proportional to the Chandrasekhar mass which varies as ${\mu }^{-3/2}$. We demonstrate that such an ultra low $z$ abrupt feature of $w-M$ provides a better fit to cosmological data compared to smooth late time deformations of $H(z)$ that also address the Hubble tension. For $z_t=0.02$ we find $\mathrm{\Delta }w\simeq -4$, $\mathrm{\Delta }M\simeq -0.1$. This model also addresses the growth tension due to the predicted lower value of $\mu$ at $z>z_t$. A prior of $\mathrm{\Delta }w=0$ (no $w$ transition) can still resolve the $H_0$ tension with a larger amplitude $M$ transition with $\mathrm{\Delta }M\simeq -0.2$ at $z_t\simeq 0.01$. This implies a larger reduction of $\mu$ for $z>0.01$ (about 12%). The $LwMPT$ can be generically induced by a scalar field nonminimally coupled to gravity with no need of a screening mechanism since in this model $\mu =1$ at $z<0.01$.

Many late-time approaches for the solution of the Hubble tension use late time smooth deformations of the Hubble expansion rate H(z) of the Planck18/ΛCDM best fit to match the locally measured value of H₀ while effectively keeping the comoving distance to the last scattering surface and Ω_0mh² fixed to maintain consistency with Planck CMB measurements. A well-known problem of these approaches is that they worsen the fit to low z distance probes. Here, we show that another problem of these approaches is that they worsen the level of the Ω_0m − σ₈ growth tension. We use the generic class of CPL parametrizations corresponding to evolving dark energy equation of state parameter |$w(z)=w_0+w_1\frac{z}{1+z}$| with local measurements H₀ prior and identify the pairs (w₀, w₁) that satisfy this condition. This is a generic class of smooth deformations of H(z) that are designed to address the Hubble tension. We show that for these models the growth tension between dynamical probe data and CMB constraints is worse than the corresponding tension of the standard Planck18/ΛCDM model. We justify this feature using a full numerical solution of the growth equation and fit to the data, as well as by using an approximate analytic approach. The problem does not affect recent proposed solutions of the Hubble crisis involving a SnIa intrinsic luminosity transition at z_t ≃ 0.01.

Gravitational transitions at low redshifts ($z_t<0.1$) have been recently proposed as a solution to the Hubble and growth tensions. Such transitions would naturally lead to a transition in the absolute magnitude $M$ of type Ia supernovae (SNIa) at $z_t$ (late $M$ transitions—$LMT$) and possibly in the dark energy equation of state parameter $w$ (late $w-M$ transitions). Here, we compare the quality of fit of this class of models to cosmological data, with the corresponding quality of fit of the cosmological constant model ($\mathrm{\Lambda }\mathrm{CDM}$) and some of the best smooth $H(z)$ deformation models [$w\mathrm{CDM}$ (cold dark matter), Chevallier-Polarski-Linder, phenomenologically emergent dark energy]. We also perform model selection via the Akaike information criterion (AIC) and the Bayes factor. We use the full cosmic microwave background temperature anisotropy spectrum data, the baryon acoustic oscillations data, the Pantheon SNIa data, the SNIa absolute magnitude $M$ as determined by Cepheid calibrators and the value of the Hubble constant $H_0$ as determined by local SNIa calibrated using Cepheids. We find that smooth $H(z)$ deformation models perform worse than transition models for the following reasons: (1) they have a worse fit to low-$z$ geometric probes (baryon acoustic oscillations and SNIa data); (2) they favor values of the SNIa absolute magnitude $M$ that are lower as compared to the value $M_c$ obtained with local Cepheid calibrators at $z<0.01$; (3) They tend to worsen the ${\mathrm{\Omega }}_{\mathrm{m},0}-{\sigma }_{8,0}$ growth tension. We also find that the $w-M$ transition model does not provide a better quality of fit to cosmological data than a pure $M$ transition model ($LMT$), where $w$ is fixed to the $\mathrm{\Lambda }\mathrm{CDM}$ value $w=-1$ at all redshifts. We conclude that the $LMT$ model has significant statistical advantages over smooth late-time $H(z)$ deformation models in addressing the Hubble crisis.

We use a particular machine learning approach, called the genetic algorithms (GAs), in order to place constraints on deviations from general relativity (GR) via a possible evolution of Newton’s constant $\mu \equiv G_{\mathrm{eff}}/G_{\mathrm{N}}$ and of the dark energy anisotropic stress $\eta$, both defined to be equal to one in GR. Specifically, we use a plethora of background and linear-order perturbations data, such as Type Ia supernovae, baryon acoustic oscillations, cosmic chronometers, redshift space distortions, and $E_g$ data. We find that although the GA is affected by the lower quality of the currently available data, especially from the $E_g$ data, the reconstruction of Newton’s constant is consistent with a constant value within the errors. On the other hand, the anisotropic stress deviates strongly from unity due to the sparsity and the systematics of the $E_g$ data. Finally, we also create synthetic data based on a next-generation survey and forecast the limits of any possible detection of deviations from GR. In particular, we use two fiducial models: one based on the cosmological constant $\mathrm{\Lambda }\mathrm{CDM}$ model and another on a model with an evolving Newton’s constant, dubbed $\mu \mathrm{CDM}$. We find that the GA reconstructions of $\mu (z)$ and $\eta (z)$ can be constrained to within a few percent of the fiducial models and in the case of the $\mu \mathrm{CDM}$ mocks, they can also provide a strong detection of several $\sigma \mathrm{s}$, thus demonstrating the utility of the GA reconstruction approach.