Perturbations in a Chaplygin gas Cosmology

In the last two decades, it has become apparent that the standard cosmological model ($\Lambda$CDM) fails to explain the accelerating expansion of the Universe [1], the existence of dark matter in galaxy clusters [2], the formation of the large-scale structure [3], the inherent inhomogeneity and anisotropy of the Universe on the small scales [4] and so forth. To solve the first two problems, scholars have proposed an exotic fluid, the so-called Chaplygin gas (CG), which acts as a cosmological fluid with an equation of state (EoS) of the form $p_{ch}=-A/\rho_{ch}^{\alpha}$, where $p_{ch}$ and $\rho_{ch}$ are the pressure and energy density of the CG, and $A$ and $\alpha$ are constants such that $A>0$ and $0<\alpha\leq 1$ [5]. From this EoS we can conclude that the EoS parameter for CG can be written as $w_{ch}=-A/{\rho_{ch}^{1+\alpha}}=-A/(A+B(1+z)^{3(1+\alpha)})$, where $z$ is the redshift. The purpose of this CG fluid is to substitute the dark matter and dark-energy components of the Universe. This model acts as dark-matter in the early universe and dark-energy in the late times of the cosmos. In a universe in which we assume the energy density to consist of matter and CG fluids, the conservation equation reads as

where $\rho_{t}$ is the energy density and $p_{t}$ is the pressure for the total fluid¹¹1When referring to matter, we mean either radiation or dust, and total fluid implies matter and Chaplygin gas., $a$ is the cosmological scale factor and the subscript $t$ stands for total (matter+CG) fluids. From this equation, the energy density becomes $\rho_{t}(z)=\left[A+{B}{(1+z)^{3(1+\alpha)}}\right]^{\frac{1}{1+\alpha}}+\rho_{m}(1+z)^{3(1+w_{m})}$, where $B=e^{C(1+\alpha)}$ with $C$ a constant of integration.²²2$w_{m}$ is the equation of state parameter for the matter fluid which takes the value of $1/3$ for the radiation component ($w_{r}$) and $0$ for the dust component ($w_{d}$), and $w_{t}=p_{t}/\rho_{t}=p_{t}/(\rho_{ch}+\rho_{m})$. Whereas much has been said of the Chaplygin gas as an alternative to a unified dark matter and dark energy model that mimics the cosmic expansion history of a Friedmann-Lemaître-Robertson-Walker (FLRW) background, to our knowledge there is no work in the literature that considered the cosmological perturbations of this fluid model in the 1+3 covariant formalism [6, 7, 8]. Henceforth we derive the density perturbation equations and present the solutions both analytically and numerically. From the results, we analyse cosmological implications as far as large-scale structure formation is concerned on both sub- and super-horizon scales [8]. The outline of the manuscript is as follows: in the next section we review the basic spatial gradient variables and we also derive the linear evolution equations. We apply scalar and harmonic decomposition techniques and obtain the wave-number dependent energy density fluctuations for both CG and matter fluids. In Sec. 3, we present the analytical and numerical solutions of density perturbations by considering radiation-CG, dust-CG and CG like fluids for both wave-length ranges. We then devote Sec. 4 to discussions of our results and the conclusions.

For a perfect-fluid system, the following fluid equations hold: $\dot{\rho}_{t}=-\Theta(\rho_{t}+p_{t})\;,$ and $(\rho_{t}+p_{t})\dot{u}_{a}+\tilde{\nabla}_{a}p_{t}=0\;,$ from which one can conclude that the 4-acceleration is $\dot{u}_{a}=\alpha w_{ch}\rho_{ch}D^{ch}_{a}/(a(\rho_{t}+p_{t}))-w_{m}\rho_{m}D^{m}_{a}/(a(\rho_{t}+p_{t}))\;.$ Another key equation for a general fluid is the so-called Raychaudhuri equation which can be expressed as $\dot{\Theta}=-\Theta^{2}/3-(\rho_{t}+3p_{t})/2-\tilde{\nabla}^{a}\dot{u}_{a}\;,$ where $\Theta$ is the cosmological expansion scalar. After finding the comoving gradients of the cosmological expansion and the comoving fractional density gradient of the matter components in a covariant and gauge-invariant way, and taking the harmonically decomposed scalar parts, it can be shown that the scalar perturbations for the matter and CG energy densities, respectively, evolve according to

where $k=2\pi a/\lambda,$ $k$ being the co-moving wave-number and $\lambda$ the wavelength of the perturbations. In GR, with normal forms of matter, one would obtain a closed second-order equation of the density fluctuations and the equations are generally easier to solve. But here we get a coupled system of second-order equations for the density fluctuations of both matter and CG, the solutions to which are more complicated to compute analytically. As such, we consider short-wavelength ($k/aH\gg 1$) and long-wavelength ($k/aH\ll 1$) limits of the perturbations [8] and analyse large-scale structure implications of the resulting solutions.

In redshift space, the corresponding equations can respectively be found to be ³³3A single dot ( $\dot{}$ ) and double dots ( $\ddot{}$ ) represent the first- and second-order derivatives, respectively, with respect to time, $t$, and a single prime ( ^′ ) and double prime ( ^′′ ) represent the first- and second-order derivatives, respectively, with respect to redshift, $z$.

For further discussion of the growth of the energy density fluctuations with redshift, we assume non-interacting fluids, namely radiation-CG, dust-CG, and only CG fluids in the following subsections.⁴⁴4For all analysis, we consider the original CG model, $\alpha=1$.

In this work, we explored the solutions of cosmological perturbations in a multi-fluid cosmic medium where one of the fluids is a Chaplygin gas. We applied the $1+3$ covariant and gauge-invariant formalism to define the spatial gradient variables and applied scalar and harmonic decomposition methods to analyse the scalar perturbations of the different energy densities involved. We considered different systems such as radiation-CG, dust-CG and CG fluids in both short- and long-wavelength modes to present the numerical and analytical solutions to the perturbation equations. Our results show that, at least in the simplest CG model, the formation of large-scale structures is enhanced, rather than discouraged (as one would expect from dark energy fluid models), since all our preliminary calculations show the growth of density fluctuations with time.

SS gratefully acknowledges financial support from WKU, EORC and ESSTI, as well as the hospitality of the Physics Department of North-West University (NWU) during the preparation of part of this manuscript. HS acknowledges the financial support from the Mwalimu Nyerere African Union scholarship and the National Research Foundation (NRF) free-standing scholarship. TT has been funded through the National Research Foundation (NRF) free-standing scholarship. AS acknowledges funding through the National Astrophysics and Space Science Program (NASSP) scholarship. ME acknowledges that this work is supported by an NWU/NRF postdoctoral fellowship. AA acknowledges that this work is based on the research supported in part by the NRF of South Africa with grant numbers 109257 and 112131.