Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests

Abstract

We review different dark energy cosmologies. In particular, we present the ΛCDM cosmology, Little Rip and Pseudo-Rip universes, the phantom and quintessence cosmologies with Type I, II, III and IV finite-time future singularities and non-singular dark energy universes. In the first part, we explain the ΛCDM model and well-established observational tests which constrain the current cosmic acceleration. After that, we investigate the dark fluid universe where a fluid has quite general equation of state (EoS) [including inhomogeneous or imperfect EoS]. All the above dark energy cosmologies for different fluids are explicitly realized, and their properties are also explored. It is shown that all the above dark energy universes may mimic the ΛCDM model currently, consistent with the recent observational data. Furthermore, special attention is paid to the equivalence of different dark energy models. We consider single and multiple scalar field theories, tachyon scalar theory and holographic dark energy as models for current acceleration with the features of quintessence/phantom cosmology, and demonstrate their equivalence to the corresponding fluid descriptions. In the second part, we study another equivalent class of dark energy models which includes F(R) gravity as well as F(R) Hořava-Lifshitz gravity and the teleparallel f(T) gravity. The cosmology of such models representing the ΛCDM-like universe or the accelerating expansion with the quintessence/phantom nature is described. Finally, we approach the problem of testing dark energy and alternative gravity models to general relativity by cosmography. We show that degeneration among parameters can be removed by accurate data analysis of large data samples and also present the examples.

Appendix: Inertial force and w DE

In this Appendix, we check if the occurrence of inertial force in the universe with rip (Little rip or the finite-time future singularities) may somehow constrain the w _DE.

As the universe expands, the relative acceleration between two points separated by a distance l is given by $l \ddot{a}/a$. If there is a particle with mass m at each of the points, an observer at one of the masses will measure an inertial force on the other mass of

$$ F_\mathrm{inert}=m l \ddot{a}/a = m l \bigl( \dot{H} + H^2 \bigr). $$

(706)

By using the deceleration parameter $q_{\mathrm{dec}} \equiv-\ddot{a} a^{-1} H^{-2}$ in Eq. (136), we may express the inertial force F _inert as

$$ F_\mathrm{inert}= - m l H^2 q_\mathrm{dec}\, . $$

(707)

The observational constraint of the value q _dec=q _dec(0) in the present universe is given by (Amanullah et al. 2010)

$$ -0.60<q_{\mathrm{dec} (0)}< -0.30 . $$

(708)

Here, we mean the present value by the suffix 0. The present value of the Hubble rate H=H ₀ could be

$$ H_0 \sim70\, \mathrm{km}/ (\mathrm{s}\,\mathrm{Mpc} ) = 2.3 \times10^{-18}~\mathrm{s}^{-1}. $$

(709)

We now rewrite the expression in (707) by using the gravitational field equations in the FLRW space-time,

$$ \frac{3}{\kappa^2}H^2 = \rho,\qquad - \frac{1}{\kappa^2} \bigl(2 \dot{H} + 3 H^2 \bigr) = P, $$

(710)

in Eqs. (68) and (69). Since the pressures of the usual matters and cold dark matter are negligible, if we neglect the contribution from the radiation in the present universe, we may identify the pressure P with that of the dark energy: P=P _DE=w _DE ρ _DE. Thus, we obtain

$$ w_{\mathrm{DE}} \varOmega_{\mathrm{DE}} = - \frac{1}{\kappa^2} \bigl(2 \dot{H}_0 + 3 H_0^2 \bigr), $$

(711)

where $\varOmega_{\mathrm{DE}} \equiv\rho_{\mathrm{DE}}/\rho_{\mathrm{crit}}^{(0)}$ with $\rho_{\mathrm{crit}}^{(0)} \equiv(3/\kappa^{2})H_{0}^{2}$ in the first relation in Eq. (6). Then, the inertial force F _inert has the following form:

$$ F_{\mathrm{inert}}= - \frac{m l H_0^2}{2} ( 1 + 3 w_{\mathrm{DE}} \varOmega_{\mathrm{DE}} ). $$

(712)

The galaxy group has a size of 10²³ m and the galaxy is moving with the speed of 10⁵ m/s. Hence, the acceleration by the central force can be estimated to be a _central∼[(10⁵)²/10²³] m/s²=10⁻¹³ m/s². On the other hand, $l H_{0}^{2} \sim10^{23}\times ( 10^{-18})^{2} = 10^{-13}~\mathrm{m}/\mathrm{s}^{2}$. Therefore, we find $a_{\mathrm{central}} \sim l H_{0}^{2}$, which informs that the precise measurement of the sizes of galaxy groups and/or galaxy clusters and the rotation speeds of galaxies may give a constraint on the value of w _DE Ω _DE.

Conversely, provided that w _DE∼−1, Ω _DE∼0.73, Eq. (712) implies

$$ F_\mathrm{inert} \sim0.6 m l H_0^2 \sim \bigl(3.2\times 10^{-36}~\mathrm{s}^{-2} \bigr) \times m l. $$

(713)

On the other hand, if we choose l as the size of a galaxy group or galaxy cluster and let v a rotational speed of a galaxy in the galaxy group or galaxy cluster, the central force by the gravity F _gravity minus the inertial force could be estimated as

$$ F_{\mathrm{cent}} \sim\frac{m v^2}{l}. $$

(714)

Thus, since F _cent=F _gravity−F _inert, by combining Eqs. (713) and (714), we acquire

$$ \frac{F_{\mathrm{gravity}}}{lm} - \frac{v^2}{l^2} \sim3.2\times 10^{-36}~\mathrm{s}^{-2}. $$

(715)

If F _cent∼F _gravity with a difference by a factor, we may find

$$ \frac{v^2}{l^2} \sim10^{-36}~\mathrm{s}^{-2}. $$

(716)

We may also consider the constraint coming from the energy density of galaxy clusters. By using the gravitational field equations (710), the inertial force (706) can be rewritten as F _inert=−(mlκ ²/6)(ρ+3P) in the first equality in Eq. (165). Now, we assume the energy density ρ _cluster in a galaxy cluster is almost homogeneous. In this case, the total mass inside the sphere with a radius l whose center is the center of cluster is given by

$$ M=\frac{4\pi}{3}l^3 \rho_{\mathrm{cluster}}. $$

(717)

Accordingly, the Newton gravity which the point particle with mass m suffers is given by

$$ F_{\mathrm{grav.}}= G\frac{mM}{l^2} = \frac{4\pi G}{3}m l \rho _{\mathrm{cluster}} = \frac{m l \kappa^2}{6} \rho_{\mathrm{cluster}}. $$

(718)

Therefore, if $F_{\mathrm{inert}}>F_{\mathrm{grav.}}$, that is, if −(ρ+3P)>ρ _cluster, the point particle is separated from the cluster. We here define w ₀≡P/ρ. Since ρ _cluster=200ρ, we find the bound for the EoS parameter w ₀ as w ₀>−67. As a result, this is not so strong constraint.