Exact solutions of Einstein's equations with ideal gas sources

Sussman, Roberto A.; Triginer, Josep

doi:10.1088/0264-9381/16/1/012

Cited by 14 publications

(39 citation statements)

References 32 publications

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“…In addition to the growth, it is worth plotting the energy density evolution in order to verify that, after the expected initial singularity, it has no other divergences during the matter-dominated era of interest here. We use equations (5) and (20) and some of the steps above to write the energy density as follows:…”

Section: B Growth Rate Equations and Integration In Curved Szekeres mentioning

confidence: 99%

Growth of structure in the Szekeres class-II inhomogeneous cosmological models and the matter-dominated era

Ishak

Peel

2012

Phys. Rev. D

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This study belongs to a series devoted to using the Szekeres inhomogeneous models in order to develop a theoretical framework where cosmological observations can be investigated with a wider range of possible interpretations. While our previous work addressed the question of cosmological distances versus redshift in these models, the current study is a start at looking into the growth rate of large-scale structure. The Szekeres models are exact solutions to Einstein's equations that were originally derived with no symmetries. We use here a formulation of the Szekeres models that is due to Goode and Wainwright, who considered the models as exact perturbations of a Friedmann-Lemaître-Robertson-Walker (FLRW) background. Using the Raychaudhuri equation we write, for the two classes of the models, exact growth equations in terms of the under/overdensity and measurable cosmological parameters. The new equations in the overdensity split into two informative parts. The first part, while exact, is identical to the growth equation in the usual linearly perturbed FLRW models, while the second part constitutes exact non-linear perturbations. We integrate numerically the full exact growth rate equations for the flat and curved cases. We find that for the matter-dominated cosmic era, the Szekeres growth rate is up to a factor of three to five stronger than the usual linearly perturbed FLRW cases, reflecting the effect of exact Szekeres non-linear perturbations. We also find that the Szekeres growth rate with an Einstein-de Sitter background is stronger than that of the well-known non-linear spherical collapse model, and the difference between the two increases with time. This highlights the distinction when we use general inhomogeneous models where shear and a tidal gravitational field are present and contribute to the gravitational clustering. Additionally, it is worth observing that the enhancement of the growth found in the Szekeres models during the matter-dominated era could suggest a substitute to the argument that dark matter is needed when using FLRW models to explain the enhanced growth and resulting large-scale structures that we observe today.PACS numbers: 98.80. Es,95.30.Sf

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Section: B Growth Rate Equations and Integration In Curved Szekeres mentioning

confidence: 99%

Growth of structure in the Szekeres class-II inhomogeneous cosmological models and the matter-dominated era

Ishak

Peel

2012

Phys. Rev. D

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“…B. Conditions (31b), (31c) and (32) For the examination of these conditions we will assume that (33) holds for y ≥ 1, |∆…”

Section: A Conditions (24) and (31a)mentioning

confidence: 99%

“…We consider only the case that would be equivalent to spacelike sections of zero curvature. A generalization of this class of exact solutions to the more general Szekeres-Szafron metrics admitting no isometries has been published recently [32], while the study of a non-relativistic ideal gas is considered in [33].…”

Section: Introductionmentioning

confidence: 99%

Exact inhomogeneous cosmologies whose source is a radiation-matter mixture with consistent thermodynamics

Sussman

Pavón

1999

Phys. Rev. D

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We derive a new class of exact solutions of Einstein's equations providing a physically plausible hydrodynamical description of cosmological matter in the radiative era, between nucleosynthesis and decoupling. The solutions are characterized by the Lemaître-Tolman-Bondi metric with a viscous fluid source, subjected to the following conditions: ͑a͒ the equilibrium state variables satisfy the equation of state of a mixture of an ultrarelativistic and a nonrelativistic ideal gases, where the internal energy of the latter has been neglected, ͑b͒ the particle numbers of the mixture components are independently conserved, ͑c͒ the viscous stress is consistent with the transport equation and entropy balance law of extended irreversible thermodynamics, with the coefficient of shear viscosity provided by kinetic theory. The satisfaction of ͑a͒, ͑b͒, and ͑c͒ restricts initial conditions in terms of an initial value function ⌬ i (s) , which in the limit of small density contrasts becomes the average of spatial gradients of the fluctuations of photon entropy per baryon in the initial hypersurface. For ⌬ i (s) 0 and choosing the phenomenological coefficients of the ''radiative gas'' model, we have an interactive photon-baryon mixture under local thermal equilibrium, with radiation dominance and temperatures characteristic of the radiative era (10 6 KϾTϾ10 3 K). Constraints on the observed anisotropy of the microwave cosmic radiation and the condition that decoupling occurs at TϭT D Ϸ4ϫ10 3 K yield an estimated value ͉⌬ i (s) ͉Ϸ10 Ϫ8 which can be associated with a bound on promordial entropy fluctuations. The Jeans mass at decoupling is of the same order of magnitude as that of baryon dominated perturbation models (Ϸ10 16 M ᭪ ).

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“…In astrophysics, for example, these are used in studying (1) the image generation formulae in a Kerr spacetime (Viergutz 1993), (2) the exact solutions of Einstein's field equation (Maharaj et al 1996;Nolan 1998;Sussman and Triginer 1999;Conway 2000;Halburd 2001;Gair 2001), (3) the solution of an axisymmetric Poisson equation (Hure 2005;Pierens and Hure 2005), (4) the gravitational potential of power-law disks (Hure et al 2008), (5) the self-gravitation of a mass element in a disk (Hure et al 2009), and (6) the magnetic field of interplanetary coronal mass ejections (Nieves-Chinchilla et al 2009). …”

Section: Complete Elliptic Integrals Of First and Second Kindsmentioning

confidence: 99%

Fast computation of complete elliptic integrals and Jacobian elliptic functions

Fukushima

2009

Celest Mech Dyn Astr

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As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K (m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9-19 obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, K (1 − m) and E(1 − m), are first computed by a pair of the approximate polynomials and then transformed to K (m) and E(m) by means of Jacobi's nome, q, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on q-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, sn(u|m), cn(u|m), and dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, u, after its domain is reduced to the standard range, 0 ≤ u < K (m)/4, with the help of the new method to compute K (m). The new procedure is 25-70% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of K (m) is not taken into account.

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Exact solutions of Einstein's equations with ideal gas sources

Cited by 14 publications

References 32 publications

Growth of structure in the Szekeres class-II inhomogeneous cosmological models and the matter-dominated era

Growth of structure in the Szekeres class-II inhomogeneous cosmological models and the matter-dominated era

Exact inhomogeneous cosmologies whose source is a radiation-matter mixture with consistent thermodynamics

Fast computation of complete elliptic integrals and Jacobian elliptic functions

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