FLRW Cosmology with Horava-Lifshitz Gravity: Impacts of Equations of State

Abstract

Inspired by Lifshitz theory for quantum critical phenomena in condensed matter, Horava proposed a theory for quantum gravity with an anisotropic scaling in ultraviolet. In Horava-Lifshitz gravity (HLG), we have studied the impacts of six types of equations of state on the evolution of various cosmological parameters such as Hubble parameters and scale factor. From the comparison of the general relativity gravity with the HLG with detailed and without with non-detailed balance conditions, remarkable differences are found. Also, a noticeable dependence of singular and non-singular Big Bang on the equations of state is observed. We conclude that HLG explains various epochs in the early universe and might be able to reproduce the entire cosmic history with and without singular Big Bang.

Appendices

Appendix A: Einstein Gravity

For EoS p = ω ρ and Friedmann equations with finite cosmological constant, the continuity equation reads

$$\begin{array}{@{}rcl@{}} \dot{a}^{2} = \frac{8\, \pi\, G}{3} \rho_{0}\, a^{-(1+3\, \omega)} + \frac{1}{3}\, {\Lambda}_{\omega}\, a^{2}. \end{array} $$

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Accordingly, we get following solutions

$$\begin{array}{@{}rcl@{}} \text{at}\; \omega=0,\;\;\; t &=& \frac{2 \mathcal{C}_{1} \log\left[2\left( \sqrt{B}\mathcal{C}_{1} + B a^{3/2}\right)\right]}{3\, \sqrt{B}\, \mathcal{C}_{1}} , \end{array} $$

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$$\begin{array}{@{}rcl@{}} \text{at}\; \omega=\frac{1}{3},\;\;\; t &=& \frac{1}{\sqrt{B}} \ln \left[a^{2} - \frac{2A}{B} \right], \end{array} $$

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$$\begin{array}{@{}rcl@{}} \text{at}\; \omega=-\frac{1}{3},\;\;\; a(t) &=& c_{1} \exp\left( \sqrt{B}\, t\right) + c_{2} \exp\left( -\sqrt{B}\, t\right), \end{array} $$

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$$\begin{array}{@{}rcl@{}} \text{at}\; \omega=-1:\;\;\; a(t) &=& c_{3} \exp\left( \sqrt{C}\, t\right) + c_{4} \exp\left( -\sqrt{C}\, t\right), \end{array} $$

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$$\begin{array}{@{}rcl@{}} \text{ and for generalized Chaplugin gas, }\; a(t) &=& c_{5} \exp\left( \sqrt{D}\, t\right) + c_{6} \exp\left( -\sqrt{D}\, t\right), \end{array} $$

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where $\mathcal {C}_{1}=\sqrt {B\, a^{3} - 2 A}$, A = −4 π G ρ ₀/3, B =Λ _ω /3, and C = −2A + B, $D=8\, \pi \, G_{N}\, A^{\frac {1}{1+\alpha }}/3$. c ₁⋯c ₆ are integration arbitrary constants.

Appendix B: QCD Equation of State

From the recent lattice QCD simulations [41], we illustrate in Fig. 3, the pressure in dependence on the energy density. As discussed in Ref. [42], the ultimate goal of the high-energy experiments is first-principle determination of the underlying dynamics in the strongly interacting matter. Determining the thermodynamical quantities, which are well-known tools to describe nature, degrees of freedom, decomposition, size and even the overall dynamics controlling evolution of the medium from which they are originating, is therefore very essential. For the thermodynamical pressure, an approximate attempt utilizing the higher-order moments of the particle multiplicity seems to be promising [42]. On the other hand, determining the thermodynamical energy density and even relating the Bjorken energy density to the lattice energy density depends on lattice QCD at finite baryon chemical potential and first-principle estimation of the formation time of the quark-gluon plasma (QGP). The energy density can be deduced from the derivative of the free energy with respect to inverse temperature. This would explain the need to implement another variable different than the one responsible for the derivatives of the high-order moments.

Fig. 3

In units of GeV/fm ³, the pressure is given as function of the energy density (symbols). The curves represent fits for hadron (long-dashed) and parton phase (dashed curves), separately. Both phases are fitted by the dotted curve

In Fig. 3, the lattice QCD results on pressure vs. energy density are fitted as follows.

• Hadron:

  $$\begin{array}{@{}rcl@{}} p &=& (0.157\pm 0.0007)\, \rho^{1.008\pm 0.07} \approx \frac{1}{6}\, \rho. \end{array} $$

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• Parton:

  $$\begin{array}{@{}rcl@{}} p &=& (0.222\pm 0.002)\, \rho^{1.069\pm 0.0024} \approx \frac{1}{5}\, \rho. \end{array} $$

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• Hadron-parton:

  $$\begin{array}{@{}rcl@{}} p &=& -0.01757 + (0.2413\pm 0.0007)\, \rho^{1.053\pm 0.008} \approx \frac{1}{4}\, \rho. \end{array} $$

  (80)