The main aim of this paper is to provide a qualitative introduction to the cosmological inflation theory and its relationship with current cosmological observations. The inflationary model solves many of the fundamental problemsthat challenge the Standard Big Bang cosmology such as the Flatness, Horizon and the magnetic Monopole problems. Additionally it provides an explanation for the initial conditions observed throughout the Large-Scale Structure of the Universe, such as galaxies. In this review we describe general solutions to the problems in the Big Bang cosmology carry out by a single scalar eld. Then, with the use of current surveys, we show the constraints imposed on the inflationary parameters (ns; r) which allow us to make the connection between theoretical and observational cosmology. In this way, with the latest results, it is possible to select or at least to constrain the right inflationary model, parameterized by a single scalar eld potential V (\phi).

Bayesian statistics and Markov Chain Monte Carlo (MCMC) algorithms have found their place in the field of Cosmology. They have become important mathematical and numerical tools, especially in parameter estimation and model comparison. In this paper, we review some fundamental concepts to understand Bayesian statistics and then introduce MCMC algorithms and samplers that allow us to perform the parameter inference procedure. We also introduce a general description of the standard cosmological model, known as the $\mathsf{\Lambda }$CDM model, along with several alternatives, and current datasets coming from astrophysical and cosmological observations. Finally, with the tools acquired, we use an MCMC algorithm implemented in python to test several cosmological models and find out the combination of parameters that best describes the Universe.

Scalar-field dark matter (SFDM) halos exhibit a core-envelope structure with soliton-like cores and cold-dark matter(CDM)-like envelopes. Simulations without self-interaction (free-field case) have reported a core-halo mass relation of the form $M_c\propto M_h^{\beta }$, with either $\beta =1/3$ or $\beta =5/9$. These results can be understood if the core and halo follow some special energy or velocity scaling relations. We extend these core-halo mass relations here to include the case of SFDM with self-interaction, either repulsive or attractive, and investigate its implications for the possible gravitational instability and collapse of solitonic cores, leading to the formation of supermassive black holes (SMBHs). Core sizes are set by the larger of two length scales, the de Broglie wavelength (in the free-field limit) or the radius $R_{\mathrm{TF}}$ of the ($n=1$)-polytrope for repulsive SFDM (in the Thomas-Fermi regime), depending upon particle mass $m$ and interaction strength $\lambda$. For parameters selected by previous literature to make approximately Kpc-sized cores and CDM-like structure formation on large scales but suppressed on small scales, we find that cores are stable for all galactic halos of interest, from the free-field to the repulsive Thomas-Fermi limit. For attractive self-interaction in this regime, however, halos of mass $M_h\sim {10}^{10}–{10}^{12}{M}_{\odot }$ have cores that collapse to form seed SMBHs with $M_{\mathrm{SMBH}}\sim {10}^6–{10}^8{M}_{\odot }$, as observations seem to require, while smaller-mass halos have stable cores, for particle masses $m=2.14\times {10}^{-22}–9.9\times {10}^{-20}\mathrm{eV}/c^2$, if the free-field limit has $\beta =1/3$, or $m=2.23\times {10}^{-21}–1.7\times {10}^{-18}\mathrm{eV}/c^2$, if $\beta =5/9$. We also place bounds on $\lambda$ for this case. For free-field and repulsive cases, if previous constraints on particle parameters are relaxed to allow much smaller (subgalactic scale) cores, then halos can also form SMBHs, for the same range of halo and black hole masses, as long as $\beta =5/9$ is correct for the free-field limit. In that case, structure formation in SFDM would be largely indistinguishable from that in CDM. As such, while these SFDM models might not help to resolve the small-scale structure problems of CDM, they would explain the formation of SMBHs quite naturally, which is otherwise not a direct feature of CDM. Since CDM, itself, has not yet been ruled out, such SFDM models must also be viable.

Nowadays cosmological inflation is the most accepted mechanism to explain the primordial seeds that led to the structure formation observed in the Universe. Current observations are in well agreement to initial adiabatic conditions, which imply that single-scalar-field inflation may be enough to describe the early Universe. However, there are several scenarios where the existence of more than a single field could be relevant during this period, for instance, the situation where the so-called spectator is present. Within the spectator scenario we can find the possibility that an ultra-light scalar field dark matter candidate could coexist with the inflaton. In this work we study this possibility where the additional scalar field could be free or self-interacting. We use isocurvature observations to constrain the free parameters of the model.

In the scalar field dark matter model virialized halos present condensed central cores called boson stars. Considering the equivalent process during reheating, we look at the formation of primordial black holes (PBHs) through the gravitational collapse of structures virialized in this era. We present the criteria necessary for collapse of either the whole structure, or that of the central core, in terms of the threshold amplitude for the primordial density contrast. This is computed for both the free and the self-interacting scalar fields. We discuss the relevance of our results for the abundance of PBHs.

Several observations suggest the existence of supermassive black holes (SMBH) at the centres of galaxies. However, the mechanism under which these objects form remains non-completely understood. In this work, we review an alternative mechanism of formation of galactic SMBHs from the collapse of a fraction of a dark matter (DM) halo made of an ultra-light scalar field (SF) whose critical mass of collapse is ∼10¹³ M_⊙. Once the BH is formed, a long-living quasi-resonant SF configuration survives and plays the role of a central fraction of the galactic DM halo. In this work, we construct a model with an ultra-light SF configuration laying in a Schwarzschild space–time to describe the centre of the DM halo hosting an SMBH in equilibrium, in the limit where self-gravitating effects can be neglected. We compute the induced stellar velocity dispersion in order to investigate the influence of the BH on to the velocity field of visible matter at the central galactic regions. We fit the empirical correlation between stellar velocity dispersions and masses of SMBHs considering two instances: the idealized case of DM-dominated (DMD) systems, where the gravitational influence of baryons is neglected, and cases of real luminous galaxies (LGAL). In the DMD case, we found it is possible to reproduce the observed stellar velocity dispersions at the effective radius of systems hosting SMBHs of at most 10⁸ M_⊙. In the LGAL case, we found that the baryons are crucial to reproduce the observed velocity dispersion.