The bispectrum of Large Scale Structures: modelling, prediction and estimation

In this thesis we study the higher-order statistics of Large Scale Structures (LSS). In particular, we examine the potential of the bispectrum (Fourier transform of the three-point correlator) of galaxies for both probing the non-linear regime of structure growth and setting constraints on primordial non-Gaussianity. The starting step is to construct accurate models for the power spectrum (Fourier transform of the two-point correlator) and bispectrum of galaxies by using the predictions of perturbation methods. In addition, the recent developments on the relation between dark matter and galaxy distributions (i.e. bias) are discussed and incorporated into the modelling, in order to have an accurate theoretical formalism on the galaxy formation. In order to build models that are as realistic as possible, we take into account additional non-linear effects, such as redshift space distortions. The analysis is mainly restricted to the large and intermediate scales, where the available perturbation theories have been heavily tested and give predictions that are in agreement with simulation and past LSS surveys. The reasoning for constructing accurate models for the non-linear gravitational evolution of galaxies is that, it is crucial to distinguish the primordial non-Gaussian (PNG) signal from the late time non-linearities. Furthermore, we investigate forecasted constraints on primordial non-Gaussianity and bias parameters from measurements of galaxy power spectrum and bispectrum in future radio continuum (EMU and SKA) and optical surveys (Euclid, DESI, LSST and SPHEREx). In the galaxy bispectrum modelling, we consider the bias expansion for non-Gaussian initial conditions up to second order, including trispectrum (Fourier transform of the four-point correlator) scale-dependant contributions, originating from the galaxy bias expansion, where for the first time we extend such correction to redshift space. We study the impact of uncertainties in the theoretical modelling of the bispectrum expansion and of redshift space distortions (theoretical errors), showing that they can all affect the final predicted bounds. We find that the bispectrum generally has a strong constraining power and can lead to improvements up to a factor ~5 over bounds based on the power spectrum alone. Our results show that constraints for local-type PNG can be significantly improved compared to current limits: future radio (e.g. SKA) and photometric surveys could obtain a measurement error on $f_{NL}^\text{loc}$, $\sigma(f_{NL}^\text{loc}) \approx 0.2-0.3$. More specifically, near future optical spectroscopic surveys, such as Euclid, will also improve over Planck by a factor of a few, while LSST will provide competitive constraints to radio continuum. In the case of equilateral PNG, galaxy bispectrum constraints are very weak, and current constraints could be tightened only if significant improvements in the redshift determinations of large volume surveys could be achieved. For orthogonal non-Gaussianity, expected constraints are comparable to the ones from Planck, e.g. $\sigma(f_{NL}^\text{ortho})\approx18$ for radio surveys. In the last part of the thesis we development a pipeline that measures the bispectrum from N-body simulations or galaxy surveys, which is based on the modal estimation formalism. This computationally demanding task is reduced from $O(N^6)$ operations to $O(N^3)$, where N is the number of modes per dimension inside the said simulation box or survey. The main idea of the modal estimator is to construct a suitable basis (``modes'') on the domain defined by the triangle condition and decompose on it the desired theoretical or observational bispectrum. This allows for massive data compression, making it an extremely useful tool for future LSS surveys. We show the results of tests performed to improve the performance of the pipeline and the convergence of the modal expansion. In addition, we present the measured bispectrum from a set of simulations with Gaussian initial condition, where the small amount of modes needed to accurately reconstruct the matter bispectrum shows the power of the modal expansion. The effective $f_{NL}$ value, corresponding to the bispectrum of the non-linear gravitational evolution, comes at no computational cost. In order to further test the pipeline, we proceed to measure the bispectrum of a few realisations with non-Gaussian initial conditions of the local type. We show that the modal decomposition can accurately separate the primordial signal, from the late-time non-Gaussianity, and put tight constraints on its amplitude.