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Register a new userFundamental solutions of the Dirac operator in the Friedmann-Lema^itre-Robertson-Walker spacetime
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Karen Yagdjian
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The equation of the spin-$frac{1}{2}$ particles in the Friedmann-Lema^itre-Robertson-Walker spacetime is investigated. The retarded and advanced fundamental solutions to the Dirac operator and generalized Dirac operator as well as the fundamental solutions to the Cauchy problem are written in explicit form via the fundamental solution of the wave equation in the Minkowski spacetime.
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Scalar field cosmologies with a generalized harmonic potential and matter with energy density $rho_m$, pressure $p_m$, and barotropic equation of state (EoS) $p_m=(gamma-1)rho_m, ; gammain[0,2]$ in Kantowski-Sachs (KS) and closed Friedmann--Lema^itre--Robertson--Walker (FLRW) metrics are investigated. We use methods from non--linear dynamical systems theory and averaging theory considering a time--dependent perturbation function $D$. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late--time attractors of full and time--averaged systems are two anisotropic contracting solutions, which are non--flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for $0leq gamma leq 2$, and flat FLRW matter--dominated universe if $0leq gamma leq frac{2}{3}$. For closed FLRW metric late--time attractors of full and averaged systems are a flat matter--dominated FLRW universe for $0leq gamma leq frac{2}{3}$ as in KS and Einstein-de Sitter solution for $0leqgamma<1$. Therefore, time--averaged system determines future asymptotics of full system. Also, 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 $0leq gamma< 1$) too. However, for $gammageq 1$ closed FLRW solutions of the full system depart from the solutions of the averaged system as $D$ is large. Our results are supported by numerical simulations.
Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic Equation of State (EoS) with barotropic index $gamma$ for Locally Rotationally Symmetric (LRS) Bianchi III metric and open Friedmann-Lema^itre-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-averag
Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic Equation of State (EoS) with barotropic index $gamma$ for the Locally Rotationally Symmetric (LRS) Bianchi I and flat Friedmann-Lema^itre-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-averag
We present the fundamental solutions for the spin-1/2 fields propagating in the spacetimes with power type expansion/contraction and the fundamental solution of the Cauchy problem for the Dirac equation. The derivation of these fundamental solutions is based on formulas for the solutions to the generalized Euler-Poisson-Darboux equation, which are obtained by the integral transform approach.
In a recent paper [arXiv:1206.4916] by T. Padmanabhan, it was argued that our universe provides an ideal setup to stress the issue that cosmic space is emergent as cosmic time progresses and that the expansion of the universe is due to the difference between the number of degrees of freedom on a holographic surface and the one in the emerged bulk. In this note following this proposal we obtain the Friedmann equation of a higher dimensional Friedmann-Robertson-Walker universe. By properly modifying the volume increase and the number of degrees of freedom on the holographic surface from the entropy formulas of black hole in the Gauss-Bonnet gravity and more general Lovelock gravity, we also get corresponding dynamical equations of the universe in those gravity theories.
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