Observational constraints on a unified dark matter and dark energy model based on generalized Chaplygin gas

We study a generalized version of Chaplygin gas as unified model of dark matter and dark energy. Using realistic theoretical models and the currently available observational data from the age of the universe, the expansion history based on the type Ia supernovae, the matter power spectrum, the cosmic microwave background radiation anisotropy power spectra, and the perturbation growth factor we put the unified model under observational test. As the model has only two free parameters in the flat Friedmann background [$\Lambda \mathrm{CDM}$ (cold dark matter) model has only one free parameter] we show that the model is already tightly constrained by currently available observations. The only parameter space extremely close to the $\Lambda \mathrm{CDM}$ model is allowed in this unified model.

Figure 1

Likelihood ($1-3\sigma$) contours in $\alpha -w_{X0}$ plane based on the SNIa Union sample [49]. The locations of $\Lambda \mathrm{CDM}$ ($\alpha =0$) based on the WMAP (${\Omega }_{\Lambda 0}=0.726$) and Union data (${\Omega }_{\Lambda 0}=0.713$) are indicated as $\times$. The maximum likelihood point is indicated by a bullet (●). Locations indicated by ★ are studied in details in Figs. 2, 3, 4. Locations near $\alpha =0$ at $w_{X0}=-0.7607$ are studied in details in Figs. 5, 6, 7. We marginalized over the Hubble constant. Short-and-long-dashed lines are age lines in the unit of $(h/0.705{)}^{-1}\mathrm{Gyrs}$. The age lines together with the SNIa data already show that parameters with $\alpha >0$ are largely excluded by currently available observations. A short-dashed line shows parameters which reach $c_{X0}^2=1$ at the present epoch; the right side of the line has $c_{X0}^2>1$ thus becoming superluminal.

Figure 2

Top panels: Evolution of ${\Omega }_i$ and ${\mu }_i$ as a function of scale factor $a(t)$ for values of $(\alpha ,w_{X0})$ indicated as ★ in Fig. 1; $i=\mathrm{r}$, $\mathrm{b}$, $X$ indicates radiation, baryon, and GCG, respectively. Middle and bottom panels: Evolution of ${\Omega }_X$, $w_X$, $H_X(z)/H_{\Lambda \mathrm{CDM}}$ ($H_X$ is the Hubble parameter in our GCG model), and the relative distance modulus $\Delta \mu (z)={\mu }_X(z)-{\mu }_{\Lambda \mathrm{CDM}}(z)$ for the same set of GCG parameters. In all panels, $\Lambda \mathrm{CDM}$ predictions are shown as thick black curves. In the $\Delta \mu$-plot, the grey open squares with error bars represent the deviation of SNIa data points from the fiducial $\Lambda \mathrm{CDM}$ model considered here. The binned SNIa data are based on the Union sample [49].

Figure 3

The matter (baryon) power spectrum (top-left), and CMB TT (top-right), EE (bottom-left), TE (bottom-right) power spectra of GCG models with parameters used in Fig. 2, and the same colored code. All calculations are made in three different gauge conditions (SG, UEG, and UCG), and the results in the three gauges coincide exactly. The matter and CMB power spectra of the $\Lambda \mathrm{CDM}$ model have been normalized with ${\sigma }_8$ and COBE spectrum, respectively. For comparison, all the GCG power spectra have been normalized with the $\Lambda \mathrm{CDM}$ ones at $k=0.01h{\mathrm{Mpc}}^{-1}$ for matter power spectrum and $\ell =700$ for CMB ones. The ratios of GCG powers to $\Lambda \mathrm{CDM}$ predictions are also shown in the bottom region of top panels. The dashed curves indicate changed power spectra due to the reionization for two GCG models; the corresponding models without reionization are placed just below the dashed curves (with the same color codes). For matter and CMB TT power spectra, recent measurements from SDSS DR7 luminous red galaxies (LRG) [48] and WMAP 5-year [41] data (including the cosmic variance) have been added (grey dots with error bars) together with fractional errors of observed spectra. For a correct comparison with the LRG band powers, the model power spectrum should include the convolution effect caused by LRG band-power window functions and the nonlinear clustering information.

Figure 4

Evolution of baryon density perturbation (top-left), and the normalized perturbation growth factor $g\equiv ({\delta }_b/a)$ in three different scales for the same parameters used in Figs. 2, 3. We add 1% error bar expected from future X-ray and weak lensing observations [50].

Figure 5

The same as in Fig. 3 for values near $\alpha =0$ (indicated in the Figure) with $w_{X0}=-0.7607$.

Figure 6

The same as in Fig. 4 for the same parameters used in Fig. 5.

Figure 7

The baryonic matter power spectrum in Fig. 5, together with the GCG power spectrum plotted in dashed lines. Although the baryon power spectra are normalized at $k=0.01h{\mathrm{Mpc}}^{-1}$, the GCG power spectra are normalized at $k=0.001h{\mathrm{Mpc}}^{-1}$ due to heavy oscillation of the latter ones. Compared with the GCG power spectra which oscillate significantly, baryon power spectra relatively behave mildly. The grey and blue dots with error bars represent the SDSS DR7 LRG and the $\Lambda \mathrm{CDM}$ mock power spectrum data, respectively. For the latter, we define 50 data points in the logarithmic interval between 0.02 and $0.3h{\mathrm{Mpc}}^{-1}$, and perturb the linear $\Lambda \mathrm{CDM}$ matter power spectrum at each point by adding a Gaussian noise with 10% of the power spectrum amplitude as standard deviation. This $\Lambda \mathrm{CDM}$-motivated mock data will be used to constrain our GCG model in Fig. 8 and Table .

Figure 8

Likelihood ($1-3\sigma$) contours in $\alpha -w_{X0}$ plane based on the matter power spectrum measurements from SDSS DR7 LRG [48] (red, yellow, green curves). For the method used, see Appendix . Notice that besides the region around $\Lambda \mathrm{CDM}$ near $\alpha =0$ (see the inner panel), the matter power spectrum favors another island with positive $\alpha$. This island, however, can be excluded by the CMB power spectra in Fig. 9 and the age limit in Fig. 1. In the both panels, we add similar constraint based on $\Lambda \mathrm{CDM}$-motivated mock power spectrum data in Fig. 7 (blue contours). GCG model parameters used in Figs. 2, 3, 4 are indicated by ★; power spectra for several parameters consistent with LRG power spectrum indicated by $+$ are presented in Fig. 9.

Figure 9

The same as in Fig. 3 for parameters indicated. We present parameters which show similar behavior as the $\Lambda \mathrm{CDM}$ in the matter power spectra. All these models, however, show severe deviations in the CMB power spectra, thus are excluded.

Figure 10

Top-left: SNIa likelihood contours and age lines. The same as in Fig. 1 for large $\alpha$. The solid curves indicate $3\sigma$ level (99.7% CL) in the likelihood. Top-right: the same as left figure with $w_{X0}$ closer to $-1$. The 6, 7, $8\sigma$ levels are indicated by black solid curves. Regions indicated by $\times$ are the same as in the next figure. Middle-left: Matter power spectrum likelihood contours. The same as in Fig. 8 for large $\alpha$. Middle-right: relative distance modulus with respect to that of fiducial $\Lambda \mathrm{CDM}$ model for regions indicated by $\times$ with the same color code. Bottom-left and bottom-right: the matter and the CMB power spectra, respectively, for regions indicated by $\times$. For $\alpha \gg 1$ there exist narrow regions with $w_{X0}$ extremely close to $-1$ allowed by the supernovae data and the matter power spectrum. As the top-right and middle-left figures show, however, these two regions do not overlap. Moreover, the region allowed by the matter power spectrum, shows severe deviations in the CMB power spectra. Thus, this region is excluded.