International Astronomy and Astrophysics Research Journal

Spatially Closed FRW Model Leads to Einstein’s Universe with Cosmological Constant

A. S. M. Mohiul Islam

Department of Mathematics, University of Chittagong, 4331, Bangladesh.

Md. Habibur Rahman *

Department of Mathematics, University of Chittagong, 4331, Bangladesh.

*Author to whom correspondence should be addressed.

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Abstract

Einstein revisited his equations and altered by introducing something known as a universal constant. Einstein's 1917 concept has been with us ever since in various variants and incarnations, including the broader concept of 'Dark Energy’. Einstein’s field equation has been studying under the assumption of a power law time variation of the expansion factor. The Hubble’s parameter and distance modulus in our descended model are in good concordance with recent data of astrophysical observations under appropriate conditions. Theoretically, the cosmological constant Λ and the density parameter  p are determined using dynamical tests like as density and velocity profiles around clusters and virialization. This paper, we discuss and briefly other classes of models that have a close relationship with the Freidmann models, and these are models derived from modified Einstein equations containing the cosmological constant. Thus the cosmological constant is not any old value but rather simply the inverse of the scale factor squared, where the scale factor is a fixed value in static closed universe. Ultimately we derived to describe the final radius of a virialized cluster, in which a repulsive cosmological constant lambda (Λ) gives a smaller value. Based on the results, two scenarios for the universe are proposed, one with a huge proportion of nonbaryonic matter and a zero cosmological constant, and the other with all matter being baryonic. The cosmological constant is added to save inflation and build up a static universe model.

Keywords: Friedmann model, cosmological constant, Einstein’s universe, Ricci tensor, energy momentum tensor, Hubble’s parameter

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References

Sahni V. The cosmological constant problem and quintessence. Classical and Quantum Gravity. 2002;19(13):3435.

Dyson L, Kleban M, Susskind L. Disturbing implications of a cosmological constant. Journal of High Energy Physics. 2002;10:011.

Padmanabhan T. Cosmological constant—the weight of the vacuum. Physics Reports. 2003;380(5-6):235-320.

Das P, Pan S, Ghosh S, Pal P. Cosmological time crystal: Cyclic universe with a small cosmological constant in a toy model approach. Physical Review D. 2018;98(2):024004.

Davidson A, Rubin S. Normalized general relativity: Nonclosed universe and a zero cosmological constant. Physical Review D. 2014;89(2):024036.

Altaie MB, Setare MR. Finite-temperature scalar fields and the cosmological constant in an Einstein universe. Physical Review D. 2003;67(4):044018.

Chand A, Mishra RK, Pradhan A. FRW cosmological models in Brans-Dicke theory of gravity with variable q $ q $ and dynamical Λ $varLambda $-term. Astrophysics and Space Science. 2016;361(2):1-12.

Barrow JD, Hervik S. Anisotropically inflating universes. Physical Review D. 2006;73(2):023007.

Kim SP. Canonical quantization of a closed Euclidean universe with a cosmological constant. arXiv preprint gr-qc/9909002; 1999.

Helbig P. Can’t get there from here? Curious logic in the famous paper by Einstein and de Sitter. Observatory. 2021;141(1282).

Tangherlini FR. The cosmological constant and the pulsating universe. Il Nuovo Cimento B. (1971-1996). 1991;106(2):123-146.

Harvey A. Cosmological models. American Journal of Physics. 1993;61(10):901-906.

Ellis GF, Maartens R. The emergent universe: Inflationary cosmology with no singularity. Classical and Quantum Gravity. 2003;21(1):223.

Dechant PP, Lasenby AN, Hobson MP. Anisotropic, nonsingular early universe model leading to a realistic cosmology. Physical Review D. 2009;79(4):043524.

Kranas D, Tsagas CG, Barrow JD, Iosifidis D. Friedmann-like universes with torsion. The European Physical Journal C. 2019;79(4):1-12.

Banerjee N, Sen S. Einstein pseudotensor and total energy of the universe. Pramana. 1997;49(6):609-615.

Coley AA, Lim WC. Cosmic microwave background limits on spatially homogeneous cosmological models with a cosmological constant. Classical and Quantum Gravity. 2007;24(4):889.

Padmanabhan T, Padmanabhan H. Cosmological constant from the emergent gravity perspective. International Journal of Modern Physics D. 2014;23(06): 1430011.

Sorkin RD. Is the cosmological “constant” a nonlocal quantum residue of discreteness of the causal set type?. In AIP Conference Proceedings. American Institute of Physics. 2007;957(1):142- 153.

Pradhan A, Garg P, Dixit A. FRW cosmological models with cosmological constant in f (R, T) theory of gravity. Canadian Journal of Physics. 2021;99(999):741-753.

Gibbons GW. Tunnelling with a negative cosmological constant. Nuclear Physics B. 1996;472(3):683-708.

Esposito-Farese G, Polarski D. Scalar-tensor gravity in an accelerating universe. Physical Review D. 2001; 63(6):063504.

Einstein A. Cosmological considerations in the general theory of relativity. Sitzungsber. Preuss. Akad. Wiss, Berlin (Math. Phys.). 1917;142-152.