### Topological phantom AdS black holes in $F(R)$ gravity

In this paper, we obtain exact phantom (A)dS black hole solutions in the context of $F(R)$ gravity with topological spacetime in four dimensions. Then, we study the effects of different parameters on the event horizon. In the following, we calculate the conserved and thermodynamic quantities of the system and check the first law of thermodynamics for these kinds of black holes. Next, we evaluate the local stability of the topological phantom (A)dS black holes in $F(R)$ gravity by studying the heat capacity and the geometrothemodynamic, where we show that the two approaches agrees. We extend our study and investigate global stability by employing the Gibbs potential and the Helmholtz free energy. In addition, the effects of different parameters on local and global stabilities will be highlighted.

### Dynamical Henneaux-Teitelboim Gravity

We consider a modified gravity model which we call "dynamical Henneaux-Teitelboim gravity" because of its close relationship with the Henneaux-Teitelboim formulation of unimodular gravity. The latter is a fully diffeomorphism-invariant formulation of unimodular gravity, where full diffeomorphism invariance is achieved by introducing two additional non-dynamical fields: a scalar, which plays the role of a cosmological constant, and a three-form whose exterior derivative is the spacetime volume element. Dynamical Henneaux-Teitelboim gravity is a generalization of this model that includes kinetic terms for both the scalar and the three-form with arbitrary couplings. We study the field equations for the cases of spherically symmetric and homogeneous, isotropic configurations. In the spherically symmetric case, we solve the field equations analytically for small values of the coupling to obtain an approximate black hole solution. In the homogeneous and isotropic case, we perturb around de Sitter space to find an approximate cosmological background for our model.

### Topological classes of black holes in de-Sitter spacetime

In this paper, we investigate the topological number of de-Sitter black hole solutions with different charges $(q)$ and rotational $(a)$ parameters. By using generalized free energy and Duan's $\phi$-mapping topological current theory, we find that the topological numbers of black holes can still be classified as three types. In addition, we interestingly found the topological classes for de-Sitter $($dS$)$ spacetime with distinct horizon, i.e, black hole event horizon and cosmological horizon, will be different. Moreover, we also investigate topological classifications of dS black hole solutions in higher dimensions with or without Gauss-Bonnet term.

### Two fluid cosmological models in $f(R, T)$ theory of gravity

Present work deals with the two fluid Bianchi Type-V cosmological models consisting of matter and radiating source in the $f(R, T)$ theory of gravity studied by Harko et al. (2011). In this paper, we developed a new idea about $f(R, T)$ gravity with the help of two fluids: one fluid is matter field modeling material content of the Universe and other fluid is radiation field modeling the CMB. We have determined the solution of the two fluid gravitational field equations with the systematic structure in $f(R, T)$ gravity. Here we have deliberated four types of universe such as dust universe, radiation universe, hard universe and Zeldovich universe and also extended our work to observe the big rip and big bang singularity. We have also tested the cosmological parameters.

### Relativistic quantum communication between harmonic oscillator detectors

We propose a model of communication employing two harmonic oscillator detectors interacting through a scalar field in a background Minkowski spacetime. In this way, the scalar field plays the role of a quantum channel, namely a Bosonic Gaussian channel. The classical and quantum capacities of the communication channel are found, assuming that the detectors' spatial dimensions are negligible compared to their distance. In particular, we study the evolution in time of the classical capacity after the detectors-field interaction is switched on for various detectors' frequencies and coupling strengths with the field. As a result, we find a finite value of these parameters optimizing the communication of classical messages. Instead, a reliable communication of quantum messages turns out to be always inhibited.

### Tunneling and entropy analysis of parameterized black hole with rotating case

In this work, we study the parameterized black hole solution by applying the Newman-Janis approach and also examine the Hawking temperature. We consider a Lagrangian field equation associated with the generalized uncertainty principle to study the motion of boson particles. By using semi-classical phenomenon, we analyze the modified Hawking temperature and graphically check the effects of deformation, rotation and correction parameter on black hole geometry. Furthermore, we investigate the logarithmic corrected entropy and also analyze the graphical behavior of deformation and quantum gravity parameter on the logarithmic corrected entropy of black hole.

### A test of Einstein's equivalence principle in ngEHT observations

We show that Event Horizon Telescope (EHT) observations allow us to test the fundamental principles of General Relativity (GR). GR is based on the universality of gravity and Einstein's equivalence principle (EEP). However, EEP is not a basic principle of physics but an empirical fact. Non-Minimal Coupling (NMC) of electromagnetic fields violates EEP, and their effects manifest in the strong-gravity regime. Hence, EHT provides an opportunity to test NMC in the strong-gravity regime. We show that, to the leading order in the spin parameter, NMC of the electromagnetic field modifies the black hole image in two ways: First, for one polarization mode, the horizon casts a shadow of radius \emph{greater than} $\sqrt{27} GM/c^2$ on the image of the source. For the other polarization mode, it is \emph{smaller than} $\sqrt{27} GM/c^2$. Second, the brightness and the position of the lensing ring are affected by the non-minimal coupling. The lensing ring is more prominent for one polarization mode than the other. Finally, we discuss the constraints on the NMC constant from future ngEHT observations.

### Extremal Black Holes as Relativistic Systems with Kepler Dynamics

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrization to the Kepler problem, and that all Hamiltonians conserving a Laplace-Runge-Lenz-like vector are related in this way to Kepler.

### Spinor wave function of the Universe in non-minimally coupled varying constants cosmologies

In this paper, we introduce a non-minimally coupled varying speed of light and varying gravitational constant cosmological toy model. Using the Eisenhart-Duval lifting method, we extend the original minisuperspace of the model and depict the evolution of the system in the presence of the potential term as a geometrical flow associated with the lifted metric. We write the Dirac-Wheeler-DeWitt equation, which solution is a spinor wave function of the Universe. Then we find the solution of the Dirac-Wheeler-DeWitt equation, which describes the emergence of two early universe-antiuniverse pairs that differ with the conserved quantity, which is an analog of the spin.

### Quantum space-time marginal problem: global causal structure from local causal information

Spatial and temporal quantum correlations can be unified in the framework of the pseudo-density operators, and quantum causality between the involved events in an experiment is encoded in the corresponding pseudo-density operator. We study the relationship between local causal information and global causal structure. A space-time marginal problem is proposed to infer global causal structures from given marginal causal structures where causal structures are represented by the pseudo-density operators; we show that there almost always exists a solution in this case. By imposing the corresponding constraints on this solution set, we could obtain the required solutions for special classes of marginal problems, like a positive semidefinite marginal problem, separable marginal problem, etc. We introduce a space-time entropy and propose a method to determine the global causal structure based on the maximum entropy principle, which can be solved effectively by using a neural network. The notion of quantum pseudo-channel is also introduced and we demonstrate that the quantum pseudo-channel marginal problem can be solved by transforming it into a pseudo-density operator marginal problem via the channel-state duality.

### Impact of symmetron screening on the Hubble tension: new constraints using cosmic distance ladder data

Fifth forces are ubiquitous in modified theories of gravity. To be compatible with observations, such a force must be screened on solar-system scales but may still give a significant contribution on galactic scales. If this is the case, the fifth force can influence the calibration of the cosmic distance ladder, hence changing the inferred value of the Hubble constant $H_0$. In this paper, we analyze symmetron screening and show that it generally increases the Hubble tension. On the other hand, by doing a full statistical analysis, we show that cosmic distance ladder data are able to constrain the theory to a level competitive with solar-system tests -- currently the most constraining tests of the theory. For the standard coupling case, the constraint on the symmetron Compton wavelength is $\lambda_{\rm C} \lesssim 2.5 \, \mathrm{Mpc}$. Thus, distance ladder data constitutes a novel and powerful way of testing this, and similar, types of theories.

### Fast and accurate collapse-time predictions for collisionless matter

We consider the gravitational collapse of collisionless matter seeded by three crossed sine waves with various amplitudes, also in the presence of a linear external tidal field. We explore two theoretical methods that are more efficient than standard Lagrangian perturbation theory (LPT) for resolving shell-crossings, the crossing of particle trajectories. One of the methods completes the truncated LPT series for the displacement field far into the UV regime, thereby exponentially accelerating its convergence while at the same time removing pathological behavior of LPT observed in void regions. The other method exploits normal-form techniques known from catastrophe theory, which amounts here to replacing the sine-wave initial data by its second-order Taylor expansion in space at shell-crossing location. This replacement leads to a speed-up in determining the displacement field by several orders of magnitudes, while still achieving permille-level accuracy in the prediction of the shell-crossing time. The two methods can be used independently, but the overall best performance is achieved when combining them. Lastly, we find accurate formulas for the nonlinear density and for the triaxial evolution of the fluid in the fundamental coordinate system, as well as report a newly established exact correspondence between perfectly symmetric sine-wave collapse and spherical collapse.

### de Sitter State in Heterotic String Theory

Recent no-go theorems have ruled out four-dimensional classical de Sitter vacua in heterotic string theory. On the other hand, the absence of a well-defined Wilsonian effective action and other related phenomena also appear to rule out such time-dependent vacua with de Sitter isometries, even in the presence of quantum corrections. In this note, we argue that a four-dimensional de Sitter space can still exist in SO(32) heterotic string theory as a Glauber-Sudarshan state, i.e. as a coherent state, over a supersymmetric Minkowski background, albeit within a finite temporal domain. Borel resummation and resurgence play a crucial role in constructing such a state in the Hilbert space of heterotic theory governed entirely by the IR degrees of freedom.

### Universal aspects of holographic quantum critical transport with self-duality

We prove several universal properties of charge transport in generic CFTs holographic to nonminimal extensions of four-dimensional Einstein-Maxwell theory with exact electromagnetic duality invariance. First, we explicitly verify that the conductivity of these theories at zero momentum is a universal frequency-independent constant. Then, we derive their analytical expressions for non-zero momentum in any holographic duality-invariant theory for large frequencies and in the limit of small frequencies and momenta. Next, in the absence of terms that couple covariant derivatives of the curvature to gauge field strengths, two universal features are proven. On the one hand, it is shown that for a general-relativity neutral black-hole background the conductivities at any frequency and momentum are independent of the choice of duality-invariant theory, thus coinciding with those in the Einstein-Maxwell case. On the other hand, if higher-curvature terms affect the gravitational background, the conductivities get modified, but the contributions from nonminimal couplings of the gauge field to gravity are subleading. We illustrate this feature with an example.

### Studies of Critical Phenomena in Causal Dynamical Triangulations on a Torus

This is a Ph.D. thesis that presents the author's findings in the area of Causal Dynamical Triangulations. In compliance with Jagiellonian University of Krak\'ow regulations, the document consists of six publications and a general summary, which serves as a guide to assist readers in navigating through the publications. Although the six publications that constitute the main content of the thesis are not included in this version of the text on arXiv, they are referred to frequently throughout the document. The original document is available at the following link: https://fais.uj.edu.pl/documents/41628/150115897/Thesis_DN-skompresowany.pdf

### Minkowski vacuum in Rindler spacetime and Unruh thermal state for Dirac fields

We consider a free Dirac field in flat spacetime and we derive the representation of the Minkowski vacuum as an element of the Rindler-Fock space. We also compute the statistical operator obtained by tracing away the left wedge. We detail the resulting thermal state for fermionic particles.

### Black Holes Up Close

Recent developments have ushered in a new era in the field of black hole astrophysics, providing our first direct view of the remarkable environment near black hole event horizons. These observations have enabled astronomers to confirm long-standing ideas on the physics of gas flowing into black holes with temperatures that are hundreds of times greater than at the center of the Sun. At the same time, the observations have conclusively shown that light rays near a black hole experience large deflections which cause a dark shadow in the center of the image, an effect predicted by Einstein's theory of General Relativity. With further investment, this field is poised to deliver decades of advances in our understanding of gravity and black holes through new and stringent tests of General Relativity, as well as new insights into the role of black holes as the central engines powering a wide range of astronomical phenomena.

### On a Torsion/Curvature Analogue of Dual Connections and Statistical Manifolds

In analogy to the concept of a non-metric dual connection, which is essential in defining statistical manifolds, we develop that of a torsion dual connection. Consequently, we illustrate the geometrical meaning of such a torsion dual connection and show how the use of both connections preserves the cracking of parallelograms in spaces equipped with a connection and its torsion dual. The coefficients of such a torsion dual connection are essentially computed by demanding a vanishing mutual torsion among the two connections. For this manifold we then prove two basic Theorems. In particular, if both connections are metric-compatible we show that there exists a specific $3$-form measuring how the connection and its torsion dual deviate away from the Levi-Civita one. Furthermore, we prove that for these torsion dual manifolds flatness of one connection does not necessary impose flatness on the other but rather that the curvature tensor of the latter is given by a specific divergence. Finally, we give a self-consistent definition of the mutual curvature tensor of two connections and subsequently define the notion of a curvature dual connection.

### Holographic torus correlators of stress tensor in $AdS_3/CFT_2$

In the context of $\rm AdS_3/CFT_2$, we investigate holographic correlators of the stress tensor of a conformal field theory (CFT) on a torus in this work. To calculate the correlators of the stress tensor, we employ the Einstein-Hilbert theory of gravity and perturbatively solve Einstein's equation in the bulk. We offer an explicit prescription to develop a recurrence relation that makes it simple to compute higher point correlators. The correlators and the recurrence relation are found to be consistent with what is known in CFTs. Following the spirit of the proposed cutoff $\rm AdS$/$T\bar{T}$ CFT holography, we then expand our computation program to investigate holographic torus correlators at a finite cutoff in the $\rm AdS_3$. A parallel recurrence relation associated with higher point correlators can be obtained.

### The extremal Kerr entropy in higher-derivative gravities

We investigate higher derivative corrections to the extremal Kerr black hole in the context of heterotic string theory with $\alpha'$ corrections and of a cubic-curvature extension of general relativity. By analyzing the near-horizon extremal geometry of these black holes, we are able to compute the Iyer-Wald entropy as well as the angular momentum via generalized Komar integrals. In the case of the stringy corrections, we obtain the physically relevant relation $S(J)$ at order $\alpha'^2$. On the other hand, the cubic theories, which are chosen as Einsteinian cubic gravity plus a new odd-parity density with analogous features, possess special integrability properties that enable us to obtain exact results in the higher-derivative couplings. This allows us to find the relation $S(J)$ at arbitrary orders in the couplings and even to study it in a non-perturbative way. We also extend our analysis to the case of the extremal Kerr-(A)dS black hole.

### The Ternary Structure of Lie Algebroid Connections

We examine the abelian heap of linear connections on anchored vector bundles and Lie algebroids. We show how the ternary structure on the set of linear connections `interacts' with the torsion and curvature tensors. The endomorphism truss of linear connections is constructed.

### On the origin of extreme trans-Neptunian objects within Modified Newtonian Dynamics

In this work, we investigate the dynamical origin of extreme trans-Neptunian objects (ETNOs) under the action of the External Field Effect (EFE), which is a consequence of Modified Newtonian Dynamics (MOND) applied to gravity around the Sun embedded in the gravitational field of the Galaxy. We perform N-body integrations of known ETNOs treated as massless particles and perturbed by four giant planets and EFE. Backward integrations show that these objects originated in the giant planet region, from where they were scattered and then evolved to their current orbits. A striking example of such evolution is Sedna, which may have been temporarily in a horseshoe orbit with Jupiter and Saturn only $30$~Myr ago. Another interesting example is the newly discovered retrograde ETNOs, whose dynamical connection with prograde ETNOs and Centaurs is shown. The EFE is considered as an alternative to Planet Nine in explaining the anomalous distribution of ETNO orbits, namely the orbital plane clustering and apsidal confinement. We also analyse the effect of MOND on the obliquity of the solar spin with respect to the invariant plane of the solar system. Finally, we discuss the significance of trans-Neptunian solar system in the context of the dark matter hypothesis.

### Superradiance and quantum states on black hole space-times

We consider the definition of the Boulware and Hartle-Hawking states for quantum fields on black hole space-times. The properties of these states on a Schwarzschild black hole have been understood for many years, but neither of these states has a direct analogue on a Kerr black hole. We show how superradiant modes play an important role in the definition of quantum states on Kerr. Superradiance is also present on static black hole space-times, in particular for a charged scalar field on a Reissner-Nordstrom black hole. We explore whether analogues of the Boulware and Hartle-Hawking states exist in this situation.