We present a complete computation of the scalar power spectrum in the inflation without inflaton (IWI) framework, where the inflationary expansion is driven solely by a de Sitter (dS) background and scalar fluctuations arise as second-order effects sourced by tensor perturbations. By explicitly deriving and numerically integrating the full second-order kernel of the Einstein equations, we obtain a scale-invariant scalar spectrum without invoking a fundamental scalar field. In this framework, the amplitude of the scalar fluctuations is directly linked to the scale of inflation. More precisely, we show that matching the observed level of scalar fluctuations Delta 2 phi (k*) approximate to 10-9 at cosmic microwave background scales fixes the inflationary energy scale Hinf as a function of the number of observed e-folds Nobs. For Nobs 30-60, we find Hinf 5 x 1013 GeV-2 x 1010 GeV, corresponding to a tensor-to-scalar ratio r 0.01-5 x 10-9. In particular, requiring consistency with instantaneous reheating, we predict a number of e-foldings O(50) and an inflationary scale Hinf 1011 GeV. We also incorporate in our framework the quantum break time of the dS state, and we show that it imposes an upper bound on the number of particle species. Specifically, using laboratory constraints on the number of species limits the duration of inflation to Nobs less than or similar to 126 e-folds. These results establish the IWI scenario as a predictive and falsifiable alternative to standard inflaton-driven models, linking the observed amplitude of primordial fluctuations directly to the quantum nature and finite lifetime of dS space.

Current and forthcoming galaxy surveys will map the observable Universe with unprecedented depth, sky coverage, and precision. These maps are affected by relativistic redshift-space distortions (RSDs), which become increasingly relevant on ultralarge scales. Accurate modeling of these relativistic RSDs is essential to avoid systematic biases in key cosmological measurements, such as primordial non-Gaussianity (PNG). To address this, we introduce an updated implementation of the LIGER method, LIGER4GAL, which incorporates all linear-order relativistic RSDs directly at the tracer level of high-resolution N-body simulations. We demonstrate that LIGER4GAL improves upon previous iterations of the LIGER method by reproducing the expected nonlinear clustering while maintaining accuracy for relativistic RSDs on large scales. We use the updated code to generate a DESI-like sample of luminous red galaxies from the Huge Multidark Planck Simulation. By measuring the power spectrum multipoles of this sample with and without the imprint of relativistic RSDs, we assess the impact of relativistic effects on measurements of the local PNG signal (fnl). We find that the omission of the "finger-of-the-observer" (sourced by the peculiar velocity of the observer) effect in the power spectrum modeling can bias measurements of fnl by more than 1 (0.25) sigma fnl in 40% (80%) of the possible realizations of the universe if scales down to kmin 1/4 0.0015 h/Mpc are included.

We extend the Schrödinger approach to large-scale structure formation beyond the Newtonian regime by working at first post-Friedmann (1PF) order. The standard Schrödinger–Poisson system gives a useful reformulation of the dynamics of a self-gravitating pressureless fluid, but it corresponds to the leading post-Friedmann, or Newtonian, limit. It therefore misses the relativistic corrections that enter at next-to-leading order and become relevant on horizon scales and for high-precision cosmological surveys. Starting from the 1PF continuity and Euler equations in a flat ΛCDM background, we identify the conserved density variable associated with covariant mass conservation. In terms of this variable, the continuity equation takes a Newtonian-like conservative form. However, even for vanishing covariant vorticity, the spatial velocity field in the cosmological frame contains a transverse 1PF component. Thus the full 1PF mass flux cannot be represented solely by the gradient of a scalar phase. We show that the Schrödinger-like formulation at 1PF order requires an effective vector potential fixed by this transverse velocity component. This vector potential contains the post-Friedmann metric vector perturbation, related to relativistic frame-dragging effects, together with nonlinear scalar terms required by the zero-vorticity condition. Equivalently, when the equation is written in scalar form, these corrections appear as an imaginary contribution to the effective potential. At leading order our system reduces to the usual Schrödinger–Poisson formulation, while at 1PF order it provides a relativistic extension of the Schrödinger description of cold matter dynamics.

We propose a possible quantum signature of the early Universe that could lead to observational imprints of the quantum nature of the inflationary period. Graviton production from the presence of a classical, coherent state of the inflaton scalar field results in entangled states in the gravitons' polarizations. At horizon crossing, interactions between the gravitons and (lower scale) inflatons, together with the gathering of “which-path information” from the cosmological horizon, perform the required Bell experiments leading to a definitive measure, which can be imprinted in the scalar correlation four-point function. This is because of a non-trivial effect due to the derivatives on two scalar fluctuations, and it provides a fingerprint that depends on the polarization of the graviton that Alice and/or Bob measured in their patch. We hint how this signature could be measured in the high-order correlation function of galaxies, in particular on the halo bias and the intrinsic alignment.

We investigate static, spherically symmetric halo configurations within Unified Dark Matter (UDM) scalar-field models, developing a systematic mapping between standard cold dark matter (CDM) density profiles and their UDM counterparts. Exploiting the equivalence-class structure of UDM models, we show that, in principle, different Lagrangian realisations can share the same weak-field rotation curve while exhibiting distinct field properties. We reconstruct the effective energy density, radial and tangential pressures from a phenomenological circular velocity profile, ensuring the absence of ghosts and instabilities and the preservation of the Null Energy Condition (NEC). Applying our procedure to several commonly used CDM halo profiles – including Persic, Salucci & Stel, NFW, and Burkert models – we demonstrate that their phenomenological success can be retained within a relativistic UDM framework, reproducing the observed flatness of rotation curves without introducing separate dark matter and dark energy components.

We propose a method that provides an observational signature of the quantum origin of primordial fluctuations generated during inflation. The method gives a prescription for testing a Bell inequality constructed exclusively from the standard scalar and tensor perturbations of minimal single-field inflation. We consider an inflationary spacetime populated by pairs of gravitons entangled in their polarization states. Third-order interactions between two scalars and one graviton transfer polarization information to the scalar sector through the product of spatial derivatives of scalars with the tensor polarization factors. Rather than performing the full multidimensional momentum integrations, we isolate and compute the tensor polarization structure of the primordial scalar eight-point correlation function. This eight-point correlation function factorizes into the product of four scalar two-point functions associated with opposite (mirrored) momentum configurations in Fourier space. This factorization falls from the fact that the two gravitons are spatially well-separated within the cosmological horizon of inflation, replicating the setup of standard Bell experiments. Through this interactions, we track how non-local correlations between both gravitons from polarization entanglement are imprinted on the scalar sector. We show that, for specific configurations of the scalar momenta after the end of inflation (detailed in the text), this observable can be used to construct a Bell-violating quantity in a way that matches the well-known Clauser-Horne-Shimony-Holt inequality definition. In principle, this offers a route to probe the quantum nature of primordial fluctuations through observables accessible today.

The presence of a gravitational wave background can be established not only via exquisitely precise pulsar timing array (PTA) measurements, but also via astrometric observations. In fact, the very same background responsible for the delay in the arrival time of pulse is also responsible of an apparent displacement of galactic objects as stars and asteroids. In this chapter we explore the natural synergy between the SKA Observatory, and current/future astrometric probes of the position of Milky Way objects. On top of presenting the potential of SKAO alone in terms of detecting a gravitational wave background, we also demonstrate the increased sensitivity that is actually achievable when SKAO measurements are used in combination with astrometric ones. In particular, we observe an approximate improvement ranging from 10% up to 50% in terms of forecast sensitivity for a PTA-astrometry joint-analysis.

We investigate primordial non-Gaussianity in the Inflation without an Inflaton (IWI) framework, where scalar perturbations are generated at second order by primordial gravitational waves in Einstein gravity on an exact de Sitter (dS) background. Since scalar modes are produced nonlinearly from tensor modes, non-Gaussianity is an intrinsic prediction of the mechanism. We compute the corresponding scalar bispectrum, derive the relevant contribution to the three-point function of the scalar potential, and evaluate its shape numerically. We find that, unlike the scalar power spectrum, the bispectrum depends logarithmically on the ultraviolet cutoff set by the end of inflation, indicating a structural difference between the two- and three-point statistics in this scenario. Its shape is enhanced toward squeezed configurations, but its amplitude becomes strongly suppressed once the scalar power spectrum is normalized to the observed value. The resulting non-Gaussianity at CMB scales is therefore negligibly small, well below present observational sensitivity.

The presence of a gravitational wave background (GWB) can be established not only via exquisitely precise pulsar timing array (PTA) measurements, but also via astrometric observations. Indeed, the very same background responsible for the delay in the arrival time of pulse causes an apparent displacement of galactic objects as stars and asteroids. In this work we provide a framework that allows to derive the displacement of sources overcoming the usually adopted ``infinite distance'' approximation. We also present how this formalism can be used to study the displacements of objects at distances comparable to the GW wavelength, as asteroids, and of objects with a non-trivial three-dimensional distribution, as stars in the Milky Way. Thus, it can be used to probe frequencies beyond PTA experiments, reaching the mHz GWs, also detectable by LISA. We forecast the capability of observing the astrometric deflection induced by a GWB evaluating the harmonic signal-to-noise ratio including correlations between different probes. We find an SNR greater than one for the relevant cases considered and as a consequence a promising Fisher forecast, suggesting a constraining power up to the percent level for a flat background.

The nature of the gravitational wave background (GWB) is a key question inmodern astrophysics and cosmology, with significant implications for understanding of thestructure and evolution of the Universe. We demonstrate how cross-correlating large-scalestructure (LSS) tracers with the GWB spatial anisotropies can extract a clear astrophysicalimprint from the GWB signal. Focusing on the unresolved population of supermassive blackhole binaries (SMBHBs) as the primary source for the GWB at nanohertz frequencies, weconstruct full-sky maps of galaxy distributions and characteristic strain of the GWB toexplore the relationship between GWB anisotropies and the LSS. We find that at currentpulsar timing array (PTA) sensitivities, very few loud SMBHBs act as Poisson-like noise.This results in anisotropies dominated by a small number of sources, making GWB mapswhere SMBHBs trace the LSS indistinguishable from a GWBs from a uniform distribution ofSMBHBs. In contrast, we find that the bulk of the unresolved SMBHBs produce anisotropieswhich mirror the spatial distribution of galaxies, and thus trace the LSS. Importantly, we showthat cross-correlations are required to retrieve a clear LSS imprint in the GWB. Specifically,we forecast the distinguishability of this LSS signature at a 3 sigma level in near-future PTAexperiments that probe angular scales of & ell;max >= 42, and 5 sigma for & ell;max >= 72 in optimisticsettings. These values assume that anisotropic GWB maps can be reconstructed at thecorresponding angular resolution and that loud sources above a resolvability threshold can beidentified and removed; actual sensitivities will depend on instrumental noise, map-makinguncertainties, and source-subtraction accuracy. Our approach opens new avenues to employ the GWB as an LSS tracer, providing unique insights into SMBHB population models andthe nature of the GWB itself. Our results motivate further exploration of potential synergiesbetween next-generation PTA experiments and cosmological tracers of the LSS.

This chapter explores theoretical and observational strategies to use the stochastic gravitational-wave background detectable by Square Kilometre Array Observatory (SKAO) as a probe of precision cosmology. We detail the critical phenomenon of scalar-induced gravitational waves, demonstrating their unique features and their sensitivity to primordial non-Gaussianity on scales much smaller than those probed by the cosmic microwave background and large-scale structure. We investigate the phenomenology of parity violation in the early Universe through the chirality imprinted in the stochastic gravitational-wave background, demonstrating that a parity-odd primordial trispectrum can generate a detectable scale-dependent helicity. On the observational side, we point out that the standard gravitational-wave angular auto-correlation analysis is significantly limited by astrophysical shot noise. We show that cross-correlating the gravitational wave signal with independent large-scale structure tracers enhances the signal-to-noise ratio and allows us to isolate the astrophysical and cosmological components in the background. These results can be achieved only thanks to the enhanced sensitivity of SKAO, extensive sky coverage, and high angular resolution, which together make such targets observationally feasible.

We present the Flagship galaxy mock, a simulated catalogue of billions of galaxies designed to support the scientific exploitation of the Euclid mission. Euclid is a medium-class mission of the European Space Agency optimised to determine the properties of dark matter and dark energy on the largest scales of the Universe. It probes structure formation over more than 10 billion years primarily from the combination of weak gravitational lensing and galaxy clustering data. The breath of Euclid's data will also foster a wide variety of scientific analyses. The Flagship simulation was developed to provide a realistic approximation to the galaxies that will be observed by Euclid and used in its scientific analyses. We ran a state-of-the-art N-body simulation with four trillion particles, producing a lightcone on the fly. From the dark matter particles, we produced a catalogue of 16 billion haloes in one octant of the sky in the lightcone up to redshift z=3. We then populated these haloes with mock galaxies using a halo occupation distribution and abundance matching approach, calibrating the free parameters of the galaxy mock against observed correlations and other basic galaxy properties. Modelled galaxy properties include luminosity and flux in several bands, redshifts, positions and velocities, spectral energy distributions, shapes and sizes, stellar masses, star formation rates, metallicities, emission line fluxes, and lensing properties. We selected a final sample of 3.4 billion galaxies with a magnitude cut of H_E<26, where we are complete. We have performed a comprehensive set of validation tests to check the similarity to observational data and theoretical models. In particular, our catalogue is able to closely reproduce the main characteristics of the weak lensing and galaxy clustering samples to be used in the mission's main cosmological analysis. (abridged)

Context. The Euclid mission of the European Space Agency will deliver weak gravitational lensing and galaxy clustering surveys that can be used to constrain the standard cosmological model and extensions thereof. Aims. We present forecasts from the combination of the Euclid photometric galaxy surveys (weak lensing, galaxy clustering, and their crosscorrelations) and its spectroscopic redshift survey with respect to their sensitivity to cosmological parameters. We include the summed neutrino mass, Sigma m (v), and the e ffective number of relativistic species, N-e ff, in the standard Lambda alpha CDM scenario and in the dynamical dark energy (w (0) w(alpha)CDM) scenario. Methods. We compared the accuracy of di fferent algorithms predicting the non-linear matter power spectrum for such models. We then validated several pipelines for Fisher matrix and Markov chain Monte Carlo (MCMC) forecasts, using di fferent theory codes, algorithms for numerical derivatives, and assumptions on the non-linear cut-o ff scale. Results. The Euclid primary probes alone will reach a sensitivity of sigma(Sigma m (v) = 60 meV) = 56 meV in the Lambda CDM +Sigma m (v) model, whereas the combination with cosmic microwave background (CMB) data from Planck is expected to achieve sigma(Sigma m (v)) = 23 meV, o ffering evidence of a non-zero neutrino mass to at least the 2:6 sigma level. This could be pushed to a 4 sigma detection if future CMB data from LiteBIRD and CMB Stage-IV were included. In combination with Planck, Euclid will also deliver tight constraints on Delta N-e ff < 0:144 (95%CL) in the Lambda CDM +Sigma m (v)+N-e ff model or even Delta N-e ff < 0:063 when future CMB data are included. When floating the dark energy parameters, we find that the sensitivity to Ne ff remains stable, but for Sigma m (v), it gets degraded by up to a factor of 2, at most. Conclusions. This work illustrates the complementarity among the Euclid spectroscopic and photometric surveys and among Euclid and CMB constraints. Euclid will o ffer great potential in measuring the neutrino mass and excluding well-motivated scenarios with additional relativistic particles.

In this work we provide a detailed derivation of the observed galaxy number over-density obtained by computing cosmological perturbations up to third order in redshift space and on very large scales. We compute all the relativistic and projection effects, arising from the observation of galaxies on the past light cone, including all redshift effects, i.e. peculiar velocities, Sachs-Wolfe (SW) effects, integrated SW effects, gravitational lensing and time delay terms. Moreover, we have considered all post- and post-post-Born contributions from the photon geodesic equations in order to take into account all possible effects due to the lensing distortions. The derivation is performed in the Poisson gauge. This work largely follows the formalism used in (Bertacca et al. 2014a, Bertacca et al. 2014b, Bertacca 2015, Bertacca et al. 2020), pushing it for the first time up to the third perturbative order. This result will be important for a variety of applications, such as a complete estimation of projection effects and the investigation of possible parity violation signatures in the 3- and 4-point galaxy correlation functions.

Our peculiar velocity imprints a dipole on galaxy density maps derived from redshift surveys. The dipole gives rise to an oscillatory signal in the multipole moments of the observed power spectrum which we indicate as the finger-of-the-observer (FOTO) effect. Using a suite of large mock catalogues mimicking ongoing and future Hα- and HI-selected surveys, we demonstrate that the oscillatory features can be measured with a signal-to-noise ratio of up to 7 (depending on the sky area coverage and provided that observational systematics are kept under control on large scales). We also show that the FOTO effect cannot be erased by correcting the individual galaxy redshifts. On the contrary, by leveraging the power of the redshift corrections, we propose a novel method to determine both the magnitude and the direction of our peculiar velocity. After applying this technique to our mock catalogues, we conclude that it can be used to either test the kinematic interpretation of the temperature dipole in the cosmic microwave background or to extract cosmological information such as the matter density parameter and the equation of state of dark energy.

We study the constraint on f(R) gravity that can be obtained by photometric primary probes of the Euclid mission. Our focus is the dependence of the constraint on the theoretical modelling of the nonlinear matter power spectrum. In the Hu-Sawicki f(R) gravity model, we consider four different predictions for the ratio between the power spectrum in f(R) and that in Lambda cold dark matter (Lambda CDM): a fitting formula, the halo model reaction approach, ReACT, and two emulators based on dark matter only N-body simulations, FORGE and e-Mantis. These predictions are added to the MontePython implementation to predict the angular power spectra for weak lensing (WL), photometric galaxy clustering, and their cross-correlation. By running Markov chain Monte Carlo, we compare constraints on parameters and investigate the bias of the recovered f(R) parameter if the data are created by a different model. For the pessimistic setting of WL, one-dimensional bias for the f(R) parameter, log10|fR0|, is found to be 0.5 sigma when FORGE is used to create the synthetic data with log10|fR0| = -5.301 and fitted by e-Mantis. The impact of baryonic physics on WL is studied by using a baryonification emulator, BCemu. For the optimistic setting, the f(R) parameter and two main baryonic parameters are well constrained despite the degeneracies among these parameters. However, the difference in the nonlinear dark matter prediction can be compensated for the adjustment of baryonic parameters, and the one-dimensional marginalised constraint on log10|fR0| is biased. This bias can be avoided in the pessimistic setting at the expense of weaker constraints. For the pessimistic setting, using the Lambda CDM synthetic data for WL, we obtain the prior-independent upper limit of log10|fR0| < -5.6. Finally, we implement a method to include theoretical errors to avoid the bias due to inaccuracies in the nonlinear matter power spectrum prediction.

The 2-point correlation function of the galaxy spatial distribution is a major cosmological observable that enables constraints on the dynamics and geometry of the Universe. The Euclid mission aims at performing an extensive spectroscopic survey of approximately 20–30 million Hα-emitting galaxies up to about redshift two. This ambitious project seeks to elucidate the nature of dark energy by mapping the 3-dimensional clustering of galaxies over a significant portion of the sky. This paper presents the methodology and software developed for estimating the 3-dimensional 2-point correlation function within the Euclid Science Ground Segment. The software is designed to overcome the significant challenges posed by the large and complex Euclid data set, which involves millions of galaxies. Key challenges include efficient pair counting, managing computational resources, and ensuring the accuracy of the correlation function estimation. The software leverages advanced algorithms, including kd-tree, octree, and linked-list data partitioning strategies, to optimise the pair-counting process. The implementation also includes parallel processing capabilities using shared-memory open multi-processing to further enhance performance and reduce computation times. Extensive validation and performance testing of the software are presented. The results indicate that the software is robust and can reliably estimate the 2-point correlation function, which is essential for deriving cosmological parameters with high precision. Furthermore, the paper discusses the expected performance of the software during different stages of the Euclid Wide Survey observations and forecasts how the precision of the correlation function measurements will improve over the mission's timeline, highlighting the software's capability to handle large data sets efficiently.

We propose a novel scenario in which scalar perturbations, which seed the large-scale structure of the universe, are generated without relying on a scalar field (the inflaton). In this framework, inflation is driven by a de Sitter space time, where tensor metric fluctuations (i.e., gravitational waves) naturally arise from quantum vacuum oscillations, and scalar fluctuations are generated via second-order tensor effects. We compute the power spectrum of such scalar fluctuations and show it to be consistent with near scale invariance. We derive the necessary conditions under which scalar perturbations become significant and much larger than the tensor modes, and we identify a natural mechanism to end inflation via a transition to a radiation-dominated phase. Our proposed mechanism could remove the need for a model-dependent scenario: the choice of a scalar field, as the inflaton, to drive inflation.

The Euclid mission will measure cosmological parameters with unprecedented precision. To distinguish between cosmological models, it is essential to generate realistic mock observables from cosmological simulations that were run in both the standard Lambda-cold-dark-matter (Lambda CDM) paradigm and in many non-standard models beyond Lambda CDM. We present the scientific results from a suite of cosmological N-body simulations using non-standard models including dynamical dark energy, k-essence, interacting dark energy, modified gravity, massive neutrinos, and primordial non-Gaussianities. We investigate how these models affect the large-scale-structure formation and evolution in addition to providing synthetic observables that can be used to test and constrain these models with Euclid data. We developed a custom pipeline based on the Rockstar halo finder and the nbodykit large-scale structure toolkit to analyse the particle output of non-standard simulations and generate mock observables such as halo and void catalogues, mass density fields, and power spectra in a consistent way. We compare these observables with those from the standard Lambda CDM model and quantify the deviations. We find that non-standard cosmological models can leave large imprints on the synthetic observables that we have generated. Our results demonstrate that non-standard cosmological N-body simulations provide valuable insights into the physics of dark energy and dark matter, which is essential to maximising the scientific return of Euclid.

We present a formalism for analyzing galaxy clustering on the light cone with the two-point correlation in the spherical Fourier-Bessel formalism, which is a natural choice to account for all wide-angle and relativistic (general relativity, GR) effects. We extend previous studies by including all projection and GR effects, developing an efficient numerical implementation that avoids the use of the Limber approximation, includes multibins correlations and a full nondiagonal covariance. Using this formalism, we investigate the impact of neglecting GR corrections, and in particular how much this could bias measurements of the nonGaussianity parameter f NL . Our results show that not including relativistic projection terms can systematically and non-negligibly bias estimates of f NL . The exact results depend on survey specifications and galaxy population properties, but we stress that a bias will generally be present. Finally, we develop a novel prescription for cross-bin correlations that allow to search for a clean signal of relativistic corrections, and show that this requires the use of the 3D full-sky formalism.