Phenomenological reconstruction of $f\mathbf{(}T\mathbf{)}$ teleparallel gravity

Phenomenological reconstruction of $f (T)$ teleparallel gravity

W. El Hanafy and G. G. L. Nashed

Phys. Rev. D 100, 083535 – Published 23 October 2019

Abstract

We present a novel reconstruction method of $f (T)$ teleparallel gravity from phenomenological parametrizations of the deceleration parameter or other alternatives. This can be used as a toolkit to produce viable modified gravity scenarios directly related to cosmological observations. We test two parametrizations of the deceleration parameter considered in recent literatures in addition to one parametrization of the effective (total) equation of state (EoS) of the universe. We use the asymptotic behavior of the matter density parameter as an extra constraint to identify the viable range of the model parameters. One of the tested models shows how tiny modification can produce viable cosmic scenarios quantitatively similar to $Λ CDM$ but qualitatively different whereas the dark energy (DE) sector becomes dynamical and fully explained by modified gravity not by a cosmological constant.

Received 5 July 2019

DOI:https://doi.org/10.1103/PhysRevD.100.083535

Physics Subject Headings (PhySH)

Alternative gravity theories Gravitation

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

W. El Hanafy ^*,† and G. G. L. Nashed ^‡

Centre for Theoretical Physics, the British University in Egypt, El Sherouk City 11837, Egypt

^*waleed.elhanafy@bue.edu.eg
^†Also at Egyptian Relativity Group, Cairo University, Giza 12613, Egypt.
^‡nashed@bue.edu.eg

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Vol. 100, Iss. 8 — 15 October 2019

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Figure 1
The best fit values of Model 1 parameters ( $q_{0}$ , $q_{1}$ ) are taken from [23] according to the datasets combination used in that analysis, whereas ( $- 0.82$ , 0.98) using $H (z)$ dataset, ( $- 0.57$ , 0.70) using SN Ia dataset, ( $- 0.59$ , 0.67) using SN $Ia + H (z)$ datasets combination and ( $- 0.5$ , 0.78) using SN $Ia + H (z) + BAO / CMB$ datasets combination, respectively; (a) Evolution of $f (T (z))$ gravity (47) normalized to $Λ CDM$ (43). For a viable theory, one expects it to oscillate about $Λ CDM$ . As clear the theory is not in agreement with $Λ CDM$ , therefore in practice one do not expect a viable thermal history; (b) The effective (total) EoS does not match the standard cold dark matter (sCDM), $w_{eff} \to 0$ , at redshifts $z ≳ 3$ ; (c) For $H (z)$ , SN Ia and SN $Ia + H (z)$ datasets, The matter density parameter crosses the unit boundary at redshifts $1 ≲ z ≲ 2$ , while for SN $Ia + H (z) + BAO / CMB$ it occurs at $z \sim 10.3$ ; (d) For $H (z)$ , SN Ia and SN $Ia + H (z)$ datasets, the torsion (DE) EoS shows phase transition at redshifts $1 ≲ z ≲ 2$ as $w_{T} \to \pm \infty$ , for SN $Ia + H (z) + BAO / CMB$ it diverges at $z \sim 10.3$ . However, the theory shows better results when the CMB/BAO datasets are added, but it still cannot produce a thermal history compatible with the standard cosmology.
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Figure 2
The best fit values of Model 1 parameters ( $q_{0}$ , $q_{1}$ ) are taken according to the constraint (51). The model parameter $q_{0}$ is kept fixed to its value as measured in [23], since it is compatible with the present values of other cosmological parameters. However, the model parameter $q_{1}$ is recalculated to fulfill (51) as given on the plots; (a) The $f (T)$ gravity matches $Λ CDM$ at large redshifts on the contrary to the corresponding plots in Fig. 1. (b) The effective (total) EoS shows that the universe can effectively produce sCDM dominant era at large redshifts as $w_{eff} \to 0$ ; (c) The matter density parameter does not cross the uint boundary line anymore and consequently the torsional density parameter does not have negative values; (d) The torsion (DE) EoS has finite values at all redshifts.
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Figure 3
The best fit values of Model 2 parameters ( $q_{0}$ , $q_{1}$ ) are taken from [16] according to the datasets combination used in that analysis, whereas ( $N = 2$ , $q_{0} = - 0.45$ , $q_{1} = - 2.56$ ), ( $N = 3$ , $q_{0} = - 0.54$ , $q_{1} = - 1.35$ ), ( $N = 4$ , $q_{0} = - 0.56$ , $q_{1} = - 1.03$ ) and ( $N = 5$ , $q_{0} = - 0.56$ , $q_{1} = - 0.85$ ); (a) Evolution of $f (T (z))$ gravity (47) normalized to $Λ CDM$ (43), as clear the theory is not in agreement with $Λ CDM$ so in practice one do not expect a viable thermal history; (b) The effective (total) EoS does not match the sCDM, $w_{eff} \to 0$ , at redshifts $z ≳ 3$ ; (c) The matter density parameter crosses the unit boundary at redshifts $1 ≲ z ≲ 2$ ; (d) The torsion (DE) EoS show phase transition at redshifts $1 ≲ z ≲ 2$ as $w_{T} \to \pm \infty$ . However, the theory shows better results when the CMB/BAO datasets are added, but it still cannot produce a thermal history compatible with the standard cosmology.
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Figure 4
The best fit values of Model 2 parameters ( $q_{0}$ , $q_{1}$ ) are taken according to the constraint (58). The model parameter $q_{0}$ is kept fixed to it value as measured in [16], since it is compatible with the present values of other cosmological parameters. However, the model parameter $q_{1}$ is recalculated to fulfill (58) as given on the plots; (a) The $f (T)$ gravity matches $Λ CDM$ at large redshifts on the contrary to the corresponding plots in Fig. 3. (b) The effective (total) EoS shows that the universe can effectively produce sCDM dominant era at large redshifts as $w_{eff} \to 0$ ; (c) The matter density parameter does not cross the unit boundary line anymore and consequently the torsional density parameter does not have negative values; (d) The torsion (DE) EoS has finite values at all redshifts.
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Figure 5
Evolution of $f (T (z))$ gravity, (62), vs $Λ CDM$ , (43): The model parameters ( $α$ , $n$ ) are taken from [24] according to the datasets combinations: $SN + OHD$ (0.445, 2.8), $SN + OHD + BAO$ (0.409, 3.13) and $SN + OHD + BAO + CMB$ (0.444, 2.907), while the $α_{\min}$ is taken for the value $n = 3$ with the constraint (65).
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Figure 6
In subfigs. (a)–(c), the best fit values of the model parameters ( $α$ , $n$ ) are taken according to the dataset combination as measured in [24]; (0.445, 2.8) for $SN + OHD$ , (0.409, 3.13) for $SN + OHD + BAO$ , (0.444, 2.907) for $SN + OHD + BAO + CMB$ and ( $α = \frac{Ω_{m, 0}}{1 - Ω_{m, 0}}$ , $n = 3$ ) for $Λ CDM$ with Planck parameters. In subfigs. (d)–(f), we use the constraint (65) to determine $α_{\min}$ for different choices of $n$ , while the additive constant is taken as $δ α = 10^{- 3}$ .
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Physical Review D

covering particles, fields, gravitation, and cosmology

Phenomenological reconstruction of $f (T)$ teleparallel gravity

W. El Hanafy and G. G. L. Nashed

Phys. Rev. D 100, 083535 – Published 23 October 2019

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Physical Review D

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Phenomenological reconstruction of f(T) teleparallel gravity

W. El Hanafy and G. G. L. Nashed

Phys. Rev. D 100, 083535 – Published 23 October 2019

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Phenomenological reconstruction of $f (T)$ teleparallel gravity