General Relativity And Quantum Cosmology

We define an attractive gravity probe surface (AGPS) as a compact 2-surface S α with positive mean curvature k satisfying r a D a k/ k 2 ?��?(for a constant α>??/2 ) in the local inverse mean curvature flow, where r a D a k is the derivative of k in the outward unit normal direction. For asymptotically flat spaces, any AGPS is proved to satisfy the areal inequality A α ???[(3+4α)/(1+2α) ] 2 (Gm ) 2 , where A α is the area of S α and m is Arnowitt-Deser-Misner (ADM) mass. Equality is realized when the space is isometric to the t= constant hypersurface of the Schwarzschild spacetime and S α is an r=constant surface with r a D a k/ k 2 =α . We adapt the two methods, the inverse mean curvature flow and the conformal flow. Therefore, our result is applicable to the case where S α has multiple components. For anti-de Sitter (AdS) spaces, the similar inequality is derived, but the proof is performed only by using the inverse mean curvature flow. We also discuss the cases with asymptotically locally AdS spaces.

In this thesis, we try to resolve the alleged problem of non-unitarity for various anisotropic cosmological models. Using Wheeler-DeWitt formulation, we quantized the anisotropic models with variable spatial curvature, namely Bianchi II and Bianchi VI. We showed that Hamiltonian of respective models admits self-adjoint extension, thus unitary evolution. We further extended the unitary evolution for higher dimensional anisotropic cosmological models. We also showed that unitarity of the model preserves the Noether symmetry but loses the scale invariance. In later part of this thesis, we showed the equivalence of Jordan and Einstein frames at the quantum level for the flat FRW model. Obtained expressions for wave packet matched exactly in both the frames indicating the equivalence of frames. We also showed that equivalence holds true for various anisotropic quantum cosmological models, i.e., Bianchi I, V, X, LRS Bianchi-I and Kantowski-Sachs models.

Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic Equation of State (EoS) with barotropic index γ for Locally Rotationally Symmetric (LRS) Bianchi III metric and open Friedmann-Lemaître-Robertson-Walker (FLRW) metric are investigated. Methods from the theory of averaging of nonlinear dynamical systems are used to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, simple time-averaged systems determine the future asymptotic behavior. Depending on values of barotropic index γ late-time attractors of physical interests for LRS Bianchi III metric are Bianchi III flat spacetime, matter dominated FLRW universe (mimicking de Sitter, quintessence or zero acceleration solutions) and matter-curvature scaling solution. For open FLRW metric late-time attractors are a matter dominated FLRW universe and Milne solution. With this approach, oscillations entering nonlinear system through Klein-Gordon (KG) equation can be controlled and smoothed out as the Hubble factor H - acting as a time-dependent perturbation parameter - tends monotonically to zero. Numerical simulations are presented as evidence of such behaviour.

Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic Equation of State (EoS) with barotropic index γ for the Locally Rotationally Symmetric (LRS) Bianchi I and flat Friedmann-Lemaître-Robertson-Walker (FLRW) metrics are investigated. Methods from the theory of averaging of nonlinear dynamical systems are used to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, the simplest time-averaged system determines the future asymptotic behavior. Depending on the values of γ , the late-time attractors of physical interests are flat quintessence dominated FLRW universe and Einstein-de Sitter solution. With this approach, the oscillations entering the system through the Klein-Gordon (KG) equation can be controlled and smoothed out as the Hubble parameter H - acting as time-dependent perturbation parameter - tends monotonically to zero. Numerical simulations are presented as evidence of such behavior.

Scalar field cosmologies for Kantowski-Sachs (KS) and closed Friedmann-Lemaître-Robertson-Walker (FLRW) metrics with generalized harmonic potential and matter with energy density ? m , pressure p m and barotropic equation of state (EoS) p m =(γ??) ? m where γ?�[0,2] are investigated. Using methods from the theory of averaging of nonlinear dynamical systems it is proved that solutions of full time-depending equations and their corresponding time-averaged versions remain close as a time-dependent perturbation parameter D is small. When D becomes monotonic increasing we define a regular dynamical system over a compact phase space; obtaining global results. That is, for KS metric global late-time attractors of full and time-averaged systems are: two anisotropic contracting solutions which are non-flat locally rotationally symmetric (LRS) Kasner Bianchi I and Taub (flat LRS Kasner) for 0?��?2 and flat FLRW matter dominated universe if 0?�γ≤ 2 3 . For closed FLRW metric late-time attractors of the full and averaged systems are: a flat matter dominated FLRW universe for 0?�γ≤ 2 3 as in KS and the Einstein-de-Sitter solution for 0?��?1 . Therefore, the time-averaged system determines future asymptotics of the full system and the oscillations entering the system through Klein-Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally, for the whole D -range for KS and for closed FLRW (if 0?�γ≤1 ) too. However, for γ?? closed FLRW's solutions of the full system depart from the solutions of the averaged system as D become large. Our results are supported by numerical simulations.

In this paper we investigate a class of phenomenologically viable F(R,G) theories that are able to avoid the cosmological constant issue. While the absence of ghosts and other kinds of instability issues is of prime importance, other reasonable requirements such as vanishing effective (low curvature) cosmological constant, including the flat space as a stable vacuum solution, are also imposed on the viable models. These are free of the cosmological constant problem thanks to the following outstanding feature: the de Sitter space is an attractor of the asymptotic cosmological dynamics, with the resulting constant Hubble rate being unrelated both to the energy density of vacuum and to the low-curvature effective cosmological constant.

The axial and polar modes for the ring down of a Schwarzschild black hole are calculated, by first deriving the Regge-Wheeler and Zerilli equations, respectively, and finally applying the Asymptotic Iteration Method (AIM). We were able to reach up to 500 iterations, obtaining for the first time convergence for a wide range of large damping modes. The General Relativity (GR) and a particular version of an extended model with an r-dependent mass-function are compared. This mass-function allows an analytical solution for the Tortoise coordinate. The example of the mass-function corresponds to the leading correction for extended theories and serves as a starting point to treat other r-dependent parameter mass-functions.

f(Q,T) gravity is a novel extension of the symmetric teleparallel gravity where the Lagrangian L is represented through an arbitrary function of the nonmetricity Q and the trace of the energy-momentum tensor T \cite{fqt}. In this work, we have constrained a widely used f(Q,T) gravity model of the form f(Q,T)= Q n+1 +mT from the primordial abundances of the light elements to understand its viability in Cosmology. We report that the f(Q,T) gravity model can elegantly explain the observed abundances of Helium and Deuterium while the Lithium problem persists. From the constraint on the expansion factor in the range 0.9425?�Z??.1525 , we report strict constraints on the parameters m and n in the range ??.13?�n?��?1.08 and ??.86?�m??2.52 respectively.

Inspired by the Covid- 19 virus structure, Barrow argued that quantum-gravitational effects may introduce intricate, fractal features on the black hole horizon [Phys. Lett. B {\bf808} (2020) 135643]. In this viewpoint, black hole entropy no longer obeys the area law and instead it can be given by S??A 1+δ/2 , where the exponent δ ranges 0?�δ≤1 , and indicates the amount of the quantum-gravitational deformation effects. Based on this, and using the deep connection between gravity and thermodynamics, we disclose the effects of the Barrow entropy on the cosmological equations. For this purpose, we start from the first law of thermodynamics, dE=TdS+WdV , on the apparent horizon of the Friedmann-Robertson-Walker (FRW) Universe, and derive the corresponding modified Friedmann equations by assuming the entropy associated with the apparent horizon has the form of Barrow entropy. We also examine the validity of the generalized second law of thermodynamics for the Universe enclosed by the apparent horizon. Finally, we employ the emergence scenario of gravity and extract the modified Friedmann equation in the presence of Barrow entropy which coincide with one obtained from the first law of thermodynamics. When δ=0 , the results of standard cosmology are deduced.

We have analyzed the Barrow holographic dark energy (BHDE) in the framework of the flat FLRW Universe by considering the various estimations of Barrow exponent ??. Here we define BHDE, by applying the usual holographic principle at a cosmological system, for utilizing the Barrow entropy rather than the standard Bekenstein-Hawking. To understand the recent accelerated expansion of the universe, considering the Hubble horizon as the IR cut-off. The cosmological parameters, especially the density parameter ( Ω D ), the equation of the state parameter ( ? D ), energy density ( ? D ) and the deceleration parameter( q ) are studied in this manuscript and found the satisfactory behaviors. Moreover, we additionally focus on the two geometric diagnostics, the statefinder (r,s) and O m (z) to discriminant BHDE model from the ?CDM model. Here we determined and plotted the trajectories of evolution for statefinder (r,s) , (r,q) and O m (z) diagnostic plane to understand the geometrical behavior of the BHDE model by utilizing Planck 2018 observational information. Finally, we have explored the new Barrow exponent ??, which strongly affects the dark energy equation of state that can lead it to lie in the quintessence regime, phantom regime, and exhibits the phantom-divide line during the cosmological evolution.