The different structures we observe in the Universe result from cosmic evolution starting from primordial seeds. These are believed to have originated from quantum fluctuations. During the inflationary epoch, they were stretched and frozen just after the Big Bang, giving rise to an almost homogeneous, closely Gaussian-distributed Universe. However, these fluctuations were slightly larger than the mean in some regions, causing gravitational instabilities, thus, density peaks were attractors of surrounding matter. In this way, a complex non-Gaussian Cosmic Web emerged, composed of galaxies clustering into knots, filaments, sheets and cosmic voids, where the mean density is very low.

Galaxy surveys provide a necessary cosmological probe for studying dark energy and structure formation in the Universe. Big Data techniques are crucial to managing the vast amount of data from galaxy surveys and improving our cosmological knowledge of the Universe. Moreover, the cosmological content is encoded in biased tracers of the non-linearly evolved cosmic density field and requires the sampling of non-Gaussian posterior distribution functions (PDFs) to obtain reconstructions of the primordial dark matter density field. In this context, we choose a Bayesian statistics framework with a Hamiltonian Monte Carlo (HMC) sampling algorithm, an efficient technique when we need to sample non-Gaussian posterior distribution functions and when we have a high number of dimensions in our parameter space. Owing to the high computational cost associated with this method, we investigate higher-order symplectic integration schemes with Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of the Hamiltonian equations of motion (EoM), implemented in the COSMIC BIRTH code, an algorithm for reconstructing the primordial and evolved cosmic density fields from galaxy surveys on the light-cone. This is achieved by recursively applying the basic second-order Leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward-time integration steps with a single intermediate backward step. This approach vastly reduces the number of evaluations and random gradient computations required in the usual second-order case for high-dimensional cases. We first restrict this study to the log-normal Poissonian model, applied to a total volume halo catalogue in real space on a cubical mesh of side 1250 Mpc/h and 256^3 cells. Hence, we neglect selection effects, redshift-space distortions, and displacements. Those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps. We find that going from the usual second to fourth-order in the Leap-frog scheme shortens the burn-in phase by a factor of at least 30. This implies that ¿ 280 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3 fewer evaluations of the Hamiltonian equations of motion for meshes of 256^3 cells. In the cosmological scenario considered, the traditional Leap-frog scheme outperforms higher-order integration techniques only when considering lower dimensional problems, e.g. meshes with 64^3 cells. This gain in computational efficiency can contribute towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

The algorithm has also been applied to a realistic case, where the primordial and evolved dark matter density fields were obtained with the COSMIC BIRTH code. This includes a non-linear Lagrangian bias, survey geometry, radial selection functions and redshift-space distortions modelling, yielding accurate reconstructions of the initial and evolved density fields. This code is generally applicable to any structure formation model. In particular, the Augmented Lagrangian Perturbation Theory (ALPT) has been chosen for this purpose. Additionally, COSMIC BIRTH with the fourth-order Leap-frog algorithm has been applied to a multi-tracer and multi-survey Bayesian analysis combining the data of five spectroscopic redshift surveys, revealing a complete view of the known proto-clusters in the Cosmic Evolution Survey (COSMOS) field, and the growth of the Cosmic Web towards lower redshifts.

The last study considered in this thesis was to perform, for the first time, constrained hydrodynamic simulations with  PKDGRAV3 code from the initial conditions obtained from the COSMOS surveys through COSMIC BIRTH at redshift z = 100, up to redshift z = 2.3, where the Hyperion proto-cluster has been identified. We have performed the simulation at redshift z = 0, obtaining the future evolution of the known proto-clusters seen at z = 2.3, and where Hyperion proto-cluster has collapsed into a large filamentary structure. Finally, we have computed  a Friends-of-Friends (FoF) algorithm, we obtained the group mass function in the volume of the simulation, and the volume of the data, where we could observe an excess of halos in the latter respect to the average. Moreover, selecting exclusively the volume that contains the Hyperion proto-cluster we could estimate its total mass in 3.46·10^15M¿ for z = 2.3, and 1.36·10^16M¿ for z= 0.