Cosmology with the galaxy bispectrum

Abstract

The last decades witnessed huge progress in understanding the large-scale structure of the Universe. While homogeneous and isotropic on the largest scales, the matter and galaxy distributions display complex patterns on smaller scales where we observe elongated filaments, compact clusters and volume-filling underdense regions. These features are not captured by studies of two-point statistics like the power spectrum that does not retain information on the phases of the Fourier modes of the density field. Therefore, higher-order statistics like the bispectrum should provide additional information. However, the exact gain has never been measured convincingly. <br /> Current and forthcoming galaxy redshift surveys, such as $textit{Euclid}$, cover large enough volumes to provide robust measurements of the bispectrum. For this reason, it is a perfect time to develop the tools to interpret these measurements and extract cosmological information out of them. The main goals of this work are to explore this field, to study the properties of the bispectrum, discover and demonstrate advantages and difficulties of making the bispectrum a useful and applicable tool to learn more about the Universe. <br /> Historically the bispectrum has been considered a useful tool to learn about the statistical properties of the primordial density perturbations that seeded structure formation, and to study non-linear processes like gravitational dynamics and galaxy biasing. Since these processes generate different functional dependencies on the triangular configurations, they can be disentangled by fitting bispectrum measurements with theoretical templates. This will ultimately remove the degeneracy between the linear bias coefficient and the amplitude of the dark matter perturbations found in power-spectrum studies.<br /> However, the potential of the bispectrum as a means to extract additional cosmological information has never been explored. One part of my thesis has been devoted to quantifying this potential using the Fisher-matrix formalism.<br /> Firstly, it is necessary to understand the power of the bispectrum as a statistic for extracting cosmological information and its advantages with respect to the classic power spectrum analysis. For this research we developed a code for evaluating the Fisher matrix and the covariance matrix for the power spectrum, the bispectrum and their combination with redshift space distortions considering $Lambda$CDM, $w$CDM and $w_0w_a$CDM models and using tree-level perturbation theory. As the result, the Fisher matrix forecasts for all these cosmological parameters are presented. Our study shows that there is a clear advantage in combining the power spectrum and the bispectrum to infer the galaxy bias parameters and constrain the dark-energy equation of state. <br /> Another significant problem is compressing the information contained in the bispectrum. In this work, we are using all possible variants of triangles and it is necessary to combine them in a sensible way, which from one hand will be possible to compare with observational data and from the other hand provide a good statistics. The bispectrum is described by five variables: three of them fix the shape of a triangle and two others fix the orientation in space. We explored the symmetry of the bispectrum with respect to its triangular bin spatial orientation. Results demonstrate that with a one quarter of the original parameter space it is possible to describe all configurations. That helps saving computation time and resources. <br /> Finally, there is another issue related to compression of the bispectrum data. It is unclear how to optimally bin the bispectrum and how to compute theoretical models for the binned data. Different strategies affect accuracy and computational cost. These research provide the golden mean between computational resources and accuracy. We have developed several ways of calculating the bispectrum for a given theoretical model by averaging differently the variables. The advantages and disadvantages of this approach will depend on the specific purpose of the required task. The ultimate purpose of this research is to demonstrate an optimal way of compressing the information to compare it against actual observational data.