Dark energy models from a parametrization of H: a comprehensive analysis and observational constraints

Abstract

The presented paper is a comprehensive analysis of two dark energy (DE) cosmological models wherein exact solutions of the Einstein field equations (EFEs) are obtained in a model-independent way (or by cosmological parametrization). A simple parametrization of Hubble parameter (H) is considered for the purpose in the flat Friedmann–Lemaitre–Robertson–Walker background. The parametrization of H covers some known models for some specific values of the model parameters involved. Two models are of special interest which show the behavior of cosmological phase transition from deceleration in the past to acceleration at late times. The model parameters are constrained with 57 points of Hubble datasets together with the 580 points of Union 2.1 compilation supernovae datasets and baryonic acoustic oscillation datasets. With the constrained values of the model parameters, both the models are analyzed and compared with the standard $\varLambda$CDM model and showing nice fit to the datasets. Two different candidates of DE are considered, cosmological constant $\varLambda$ and a general scalar field $\phi$, and their dynamics are discussed on the geometrical base built. The geometrical and physical interpretations of the two models in consideration are discussed in detail, and the evolution of various cosmological parameters is shown graphically. The age of the Universe in both models is also calculated. Various cosmological parametrization schemes used in the past few decades to find exact solutions of the EFEs are also summarized at the end which can serve as a unified reference for the readers.

Appendix

A brief list of various parametrization schemes of parametrization of geometrical and physical parameters used in the past few decades to find exact solutions of Einstein field equations is given below.

1.1 Parametrizations of geometrical parameters

Scale factor a(t)

Given below a list of different expansion laws of the scale factor those have been extensively studied in different contexts.

• $a(t)=constant$ [33] (Static model)

• $a(t)=ct$ [44, 103] (Milne model or Linear expansion)

• $a(t)\sim \exp (H_{0}t)$ [104] ($\varLambda$CDM model or Exponential expansion)

• $a(t)\sim \exp \left[ -\alpha t\ln \left( \frac{t}{t_{0}}\right) +\beta t \right]$ [105] (Inflationary model)

• $a(t)\sim \exp \left[ -\alpha t-\beta t^{n}\right]$ [105] (Inflationary model)

• $a(t)\sim \left[ \exp (\alpha t)-\beta \exp (-\alpha t)\right] ^{n}$ [105] (Inflationary model)

• $a(t)\sim \exp \left( \frac{t}{M}\right) \left[ 1+\cos \left( \frac{ \varsigma (t)}{N}\right) \right]$ [106] (quasi-steady-state cosmology, Cyclic Universe)

• $a(t)\sim t^{\alpha }$ [107] (Power law Cosmology)

• $a(t)\sim t^{n}\exp (\alpha t)$ [108] (Hybrid expansion)

• $a(t)\sim \exp \left[ n(\log t)^{m}\right]$ [109] (Logamediate expansion)

• $a(t)\sim \cosh \alpha t$ [104] (Hyperbolic expansion)

• $a(t)\sim \left( \sinh \alpha t\right) ^{\frac{1}{n}}$ [110] (Hyperbolic expansion)

• $a(t)\sim \left( \frac{t}{t_{s}-t}\right) ^{n}$ [111] (Singular model)

• $a(t)\sim t^{n}\exp \left[ \alpha (t_{s}-t)\right]$ [111] (Singular model)

• $a(t)\sim \exp \left( \alpha \frac{t^{2}}{t_{*}^{2}}\right)$ [112] (Bouncing Model)

• $a(t)\sim \exp \left( \frac{\beta }{\alpha +1}(t-t_{s})^{\alpha +1}\right)$ [112] (Bouncing Model)

• $a(t)\sim \left( \frac{3}{2}\rho _{cr}t^{2}+1\right) ^{\frac{1}{3}}$ [112] (Bouncing Model)

• $a(t)\sim \left( \frac{t_{s}-t}{t_{*}}\right)$ [112] (Bouncing Model)

• $a(t)\sim \sin ^{2}\left( \alpha \frac{t}{t_{*}}\right)$ [112] (Bouncing Model)

Hubble parameter H(t) or H(a)

• $H(a)=Da^{-m}$ [113]

• $H(a)=e^{\frac{1-\gamma a^{2}}{\alpha a}}$ [114]

• $H(a)=\alpha (1+a^{-n})$ [115]

• $H(t)=\frac{m}{\alpha t+\beta }$ [116]

• $H(t)=\frac{16\alpha t}{15\left[ 1+(8\alpha t^{2})/5\right] }$ [117]

• $H(t)=m+\frac{n}{t}$ [118]

• $H(t)=\frac{\alpha t_{R}}{t(t_{R}-t)}$ [119]

• $H(t)=\frac{\alpha }{3}\left( t+T_{0}\right) ^{3}-\beta \left( t+T_{0}\right) +\gamma$ [120]

• $H(t)=\alpha e^{\lambda t}$ [121]

• $H(t)=\alpha +\beta (t_{s}-t)^{n}$ [121]

• $H(t)=\alpha -\beta e^{-nt}$ [122]

• $H(t)=f_{1}(t)+f_{2}(t)(t_{s}-t)^{n}$ [123]

• $H(t)=\frac{\beta t^{m}}{\left( t^{n}+\alpha \right) ^{p}}$ [64]

• $H(t)=n\alpha \tanh (m-nt)+\beta$ [124]

• $H(t)=\alpha \tanh \left( \frac{t}{t_{0}}\right)$ [18]

• $H(z)=\left[ \alpha +\left( 1-\alpha \right) \left( 1+z\right) ^{n}\right] ^{ \frac{3}{2n}}$ [125]

Deceleration parameter q(t)  or q(a), q(z)

• $q(t)=m-1$ [126]

• $q(t)=-\alpha t+m-1$ [127]

• $q(t)=\alpha \cos (\beta t)-1$ [128]

• $q(t)=-\frac{\alpha t}{1+t}$ [129]

• $q(t)=-\frac{\alpha (1-t)}{1+t}$ [129]

• $q(t)=-\frac{\alpha }{t^{2}}+\beta -1$ [130]

• $q(t)=(8n^{2}-1)-12nt+3t^{2}$ [131]

• $q(a)=-1-\frac{\alpha a^{\alpha }}{1+a^{\alpha }}$ [132]

• $q(z)=q_{0}+q_{1}z$ [133]

• $q(z)=q_{0}+q_{1}z(1+z)^{-1}$ [134]

• $q(z)=q_{0}+q_{1}z(1+z)(1+z^{2})^{-1}$ [135]

• $q(z)=\frac{1}{2}+q_{1}(1+z)^{-2}$ [136]

• $q(z)=q_{0}+q_{1}[1+\ln (1+z)]^{-1}$ [137]

• $q(z)=\frac{1}{2}+(q_{1}z+q_{2})(1+z)^{-2}$ [138]

• $q(z)=-1+\frac{3}{2}\left( \frac{\left( 1+z\right) ^{q_{2}}}{ q_{1}+(1+z)^{q_{2}}}\right)$ [139]

• $q(z)=-\frac{1}{4}\left[ 3q_{1}+1-3(q_{1}+1)\left( \frac{q_{1}e^{q_{2}\left( 1+z\right) }-e^{-q_{2}\left( 1+z\right) }}{q_{1}e^{q_{2}\left( 1+z\right) }+e^{-q_{2}\left( 1+z\right) }}\right) \right]$ [140]

• $q(z)=-\frac{1}{4}+\frac{3}{4}\left( \frac{q_{1}e^{q_{2}\frac{z}{\sqrt{1+z}} }-e^{-q_{2}\frac{z}{\sqrt{1+z}}}}{q_{1}e^{q_{2}\frac{z}{\sqrt{1+z}} }+e^{-q_{2}\frac{z}{\sqrt{1+z}}}}\right)$ [140]

• $q(z)=q_\mathrm{f}+\frac{q_{i}-q_\mathrm{f}}{1-\frac{q_{i}}{q_\mathrm{f}}\left( \frac{1+z_{t}}{ 1+z }\right) ^{\frac{1}{\tau }}}$ [141]

• $q(z)=q_{0}-q_{1}\left( \frac{(1+z)^{-\alpha }-1}{\alpha }\right)$ [142]

• $q(z)=q_{0}+q_{1}\left[ \frac{\ln (\alpha +z)}{1+z}-\beta \right]$ [143]

• $q(z)=q_{0}-(q_{0}-q_{1})(1+z)\exp \left[ z_{c}^{2}-(z+z_{c})^{2}\right]$ [144]

Jerk parameter j(z)

$j(z)=-1+j_{1}\frac{f(z)}{E^{2}(z)},$ where $f(z)=z$, $\frac{z}{1+z}$, $\frac{z}{(1+z)^{2}}$, $\log (1+z)$ and $E(z)=\frac{H(z)}{H_{0}}$ [145]

$j(z)=-1+j_{1}\frac{f(z)}{h^{2}(z)},$ where $f(z)=1$, $1+z$, $(1+z)^{2}$, $(1+z)^{-1}$ and $h(z)=\frac{H(z)}{H_{0}}$ [146]

1.2 Parametrizations of physical parameters

Pressure $p(\rho )$, p(z)

The matter content in the Universe is not properly known but it can be categorized with its equations of states $p=p(\rho )$. Following is a list of some cosmic fluid considerations with their EoS. Also, some dark energy pressure parametrization is listed.

• $p(\rho )=w\rho$ (Perfect fluid EoS)

• $p{(\rho )}=w\rho -f(H)$ [147] (Viscous fluid EoS)

• $p{ (\rho )}=w\rho +k\rho ^{1+\frac{1}{n}}$ [148] (Polytropic gas EoS)

• $p{(\rho )}=\frac{8w\rho }{3-\rho }-3\rho ^{2}$ [149] (Vanderwaal gas EoS)

• $p{(\rho )}=-(w+1)\frac{\rho ^{2}}{\rho _{P}}+w\rho +(w+1)\rho _{\varLambda }$ [150] (EoS in quadratic form)

• $p{(\rho )}=-\frac{B}{\rho }$ [151] (Chaplygin gas EoS)

• $p{(\rho )}=-\frac{B}{\rho ^{\alpha }}$ [152] (Generalized Chaplygin gas EoS)

• $p{(\rho )}=A\rho -\frac{B}{\rho ^{\alpha }}$ [153] (Modified Chaplygin gas EoS)

• $p{(\rho )}=A\rho -\frac{B(a)}{\rho ^{\alpha }}$ [154] (Variable modified Chaplygin gas EoS)

• $p{(\rho )}=A(a)\rho -\frac{B(a)}{\rho ^{\alpha }}$ [155] (New variable modified Chaplygin gas EoS)

• $p{(\rho )}=-\rho -\rho ^{\alpha }$ [156] (DE EoS)

• $p(z)=\alpha +\beta z$ [157] (DE EoS)

• $p(z)=\alpha +\beta \frac{z}{1+z}$ [157] (DE EoS)

• $p(z)=\alpha +\beta \left( z+\frac{z}{1+z}\right)$ [158] (DE EoS)

• $p(z)=\alpha +\beta \ln (1+z)$ [159] (DE EoS)

Equation of state parameter w(z)

• $w(z)=w_{0}+w_{1}z$ [160] (Linear parametrization)

• $w(z)=w_{0}+w_{1}\frac{z}{\left( 1+z\right) ^{2}}$ [161] (JBP parametrization)

• $w(z)=w_{0}+w_{1}\frac{z}{\left( 1+z\right) ^{n}}$ [162] (Generalized JBP parametrization)

• $w(z)=w_{0}+w_{1}\frac{z}{1+z}$ [163] (CPL parametrization)

• $w(z)=w_{0}+w_{1}\left( \frac{z}{1+z}\right) ^{n}$ [162] (Generalized CPL parametrization)

• $w(z)=w_{0}+w_{1}\frac{z}{\sqrt{1+z^{2}}}$ [164] (Square-root parametrization)

• $w(z)=w_{0}+w_{1}\sin (z)$ [165] (Sine parametrization)

• $w(z)=w_{0}+w_{1}\ln (1+z)$ [166] (Logarithmic parametrization)

• $w(z)=w_{0}+w_{1}\ln \left( {\small 1+\frac{z}{1+z}}\right)$ [167] (Logarithmic parametrization)

• $w(z)=w_{0}+w_{1}\frac{z(1+z)}{1+z^{2}}$ [168] (BA parametrization)

• $w(z)=w_{0}+w_{1}\left( \frac{\ln (2+z)}{1+z}-\ln 2\right)$ (MZ parametrization)

• $w(z)=w_{0}+w_{1}\left( \frac{\sin (1+z)}{1+z}-\sin 1\right)$ [169] (MZ parametrization)

• $w(z)=w_{0}+w_{1}\frac{z}{1+z^{2}}$ (FSLL parametrization)

• $w(z)=w_{0}+w_{1}\frac{z^{2}}{1+z^{2}}$ [170] (FSLL parametrization)

• $w(z)=-1+\frac{1+z}{3}\frac{\alpha +2\beta (1+z)}{\gamma +2\alpha (1+z)+\beta (1+z)^{2}}$ [171] (ASSS parametrization)

• $w(z)=\frac{1+\left( \frac{1+z}{1+z_{s}}\right) ^{\alpha }}{ w_{0}+w_{1}\left( \frac{1+z}{1+z_{s}}\right) ^{\alpha }}$ [172] (Hannestad–Mortsell parametrization)

• $w(z)=-1+\alpha (1+z)+\beta (1+z)^{2}$ [173] (Polynomial parametrization)

• $w(z)=-1+\alpha \left[ 1+f(z)\right] +\beta \left[ 1+f(z)\right] ^{2}$ [174] (Generalized Polynomial parametrization)

• $w(z)=w_{0}+z\left( \frac{\mathrm{d}w}{\mathrm{d}z}\right) _{0}$ [175]

• $w(z)=\frac{-2(1+z)d_{c}^{^{\prime \prime }}-3d_{c}^{^{\prime }}}{3\left[ d_{c}^{^{\prime }}-\varOmega _{M}(1+z)^{3}\left( d_{c}^{^{\prime }}\right) ^{3} \right] }$ where $d_{c}^{^{\prime }}=\int \limits _{0}^{z}\frac{H_{0}\mathrm{d}z}{H(z)}$ [176]

• $w_{x}(a)=w_{0}\exp (a-1)$ [177]

• $w_{x}(a)=w_{0}a(1-\log a)$ [177]

• $w_{x}(a)=w_{0}a\exp (1-a)$ [177]

• $w_{x}(a)=w_{0}a(1+\sin (1-a))$ [177]

• $w_{x}(a)=w_{0}a(1+\arcsin (1-a))$ [177]

• $w_\mathrm{de}(z)=w_{0}+w_{1}q$ [178]

• $w_\mathrm{de}(z)=w_{0}+w_{1}q(1+z)^{\alpha }$ [178]

• $w_\mathrm{de}(z)=\frac{w_{0}}{\left[ 1+b\ln (1+z)\right] ^{2}}$ [179]

• $w_{x}(z)=w_{0}+b\left\{ 1-\cos \left[ \ln (1+z)\right] \right\}$ [180]

• $w_{x}(z)=w_{0}+b\sin \left[ \ln (1+z)\right]$ [180]

• $w_{x}(z)=w_{0}+b\left[ \frac{\sin (1+z)}{1+z}-\sin 1\right]$ [180]

• $w_{x}(z)=w_{0}+b\left( \frac{z}{1+z}\right) \cos (1+z)$ [180]

• $w(z)=w_{0}+w_{a}\left[ \frac{\ln (2+z)}{1+z}-\ln 2\right]$ [181]

• $w(z)=w_{0}+w_{a}\left[ \frac{\ln (\alpha +1+z)}{\alpha +z}-\frac{\ln (\alpha +1)}{\alpha }\right]$ [182]

Energy density $\rho$

• $\rho =\rho _{c}$ [183, 184]

• $\rho \sim \theta ^{2}$ [185]

• $\rho =\frac{A}{a^{4}}\sqrt{{a}^{2}{\small +b}}$ [186]

• $\left( {\small \rho +3p}\right) a^{3}=A$ [187]

• $\rho +p=\rho _{c}$ [188]

• $\rho _\mathrm{de}(z)=\rho _\mathrm{de}(0)\left[ 1+\alpha \left( \frac{z}{1+z}\right) ^{n} \right]$ [189]

• $\rho _\mathrm{de}(z)=\frac{1}{\rho _{\phi }}\left( \frac{\mathrm{d}\rho _{\phi }}{\mathrm{d}\phi } \right) =-\frac{\alpha a}{\left( \beta +a\right) ^{2}}$ [190]

• $\rho _\mathrm{de}(z)=\alpha H(z)$ [191]

• $\rho _\mathrm{de}(z)=\alpha H(z)+\beta H^{2}(z)$ [191]

• $\rho _\mathrm{de}(z)=\frac{3}{\kappa ^{2}}\left[ \alpha +\beta H^{2}(z)\right]$ [191]

• $\rho _\mathrm{de}(z)=\frac{3}{\kappa ^{2}}\left[ \alpha +\frac{2}{3}\beta {\dot{H}} (z)\right]$ [191]

• $\rho _\mathrm{de}(z)=\frac{3}{\kappa ^{2}}\left[ \alpha H^{2}(z)+\frac{2}{3}\beta {\dot{H}}(z)\right]$ [191]

• $\rho _\mathrm{de}(z)=\frac{3}{\kappa ^{2}}\left[ \alpha +\beta H^{2}(z)+\frac{2}{3} \gamma {\dot{H}}(z)\right]$ [191]

• $\rho _\mathrm{de}(z)=\rho _{\phi 0}\left( 1+z\right) ^{\alpha }e^{\beta z}$ [192]

Cosmological constant ($\varLambda$)

In order to resolve the long standing cosmological constant problem, authors have considered some variation laws for the cosmological constant in the past forty years, commonly known as “$\varLambda$-varying cosmologies” or “Decaying vacuum cosmologies” . Later, the idea was adopted to explain the accelerated expansion of the Universe considering varying $\varLambda$. Following is list of such decay laws of $\varLambda$.

$\varLambda \sim a^{-n}$ [193]

$\varLambda \sim H^{n}$ [49]

$\varLambda \sim \rho$ [193]

$\varLambda \sim t^{n}$ [49]

$\varLambda \sim q^{n}$ [49]

$\varLambda \sim e^{-\beta a}$ [194]

$\varLambda =\varLambda (T)$ [195] T is Temperature

$\varLambda \sim C+e^{-\beta t}$ [196]

$\varLambda =3\beta H^{2}+\alpha a^{-2}$ [197]

$\varLambda =\beta \frac{{{\ddot{a}}}}{a}$ [198]

$\varLambda =3\beta H^{2}+\alpha \frac{{{\ddot{a}}}}{a}$ [199]

$\frac{\mathrm{d}\varLambda }{\mathrm{d}t}\sim \beta \varLambda -\varLambda ^{2}$ [200]

Scalar field Potentials $V(\phi )$

$V(\phi )=V_{0}\phi ^{n}$ [17] (Power law)

$V(\phi )=V_{0}\exp \left[ -\frac{\alpha \phi }{M_{pl}}\right]$ [17] (exponential)

$V(\phi )=\frac{V_{0}}{\cosh \left[ \phi /\phi _{0}\right] }$ [17]

$V(\phi )=V_{0}\left[ \cosh \left( \alpha \phi /M_{pl}\right) \right] ^{-\beta }$ (hyperbolic) [17]

$V(\phi )=\frac{\alpha }{\phi ^{n}}$ (Inverse power law) [201]

$V(\phi )=\frac{V_{0}}{1+\beta \exp (-\alpha \kappa \phi )}$ (Woods–Saxon potential) [202]

$V(\phi )=\alpha c^{2}\left[ \tanh \frac{\phi }{\sqrt{6}\alpha }\right] ^{2}$ ($\alpha$-attractor) [203]

$V(\phi )=V_{0}(1+\phi ^{\alpha })^{2}$ [204]

$V(\phi )=V_{0}\exp (\alpha \phi ^{2})$ [204]

$V(\phi )=\frac{1}{4}(\phi ^{2}-1)^{2}$ [205].

Note: All the parametrization listed above contain some arbitrary constants such as $\alpha$, $\beta$, $\gamma$, m, n, p, $q_{0}$, $q_{1}$, $q_{2}$, $w_{0}$, $w_{1}$, A, B are model parameters which are generally constrained through observational datasets or through any analytical methods and also some arbitrary functions $f_{1}(t)$, $f_{2}(t)$. $t_{s}$ denotes the bouncing time or future singularity time and $t_{*}$ some arbitrary time.