Analytic results for a flat universe dominated by dust and dark energy

We find the solution for the scale factor in a flat universe driven by dust plus a component characterized by a constant parameter of state which dominates in the asymptotic future. We also present an analytic formula (in terms of hypergeometric functions) for the past light cone in such a universe. As applications of this result, we give analytic expressions for the luminosity distance and for the acoustic scale, where the latter determines the peaks positions in the pattern of cosmic microwave background (CMB) anisotropies.

Figure 1

Left Panel: Exact luminosity distance $H_0{d}_L$ (solid line) and approximated luminosity distance $H_0{d}_{\mathrm{app}}$ (up to third order, dotted line), both vs redshift $z$. Right Panel: $p$ vs $z$ where $p$ is implicitly defined by $d_L=(1+p)d_{\mathrm{app}}$ where $d_L$ is the exact luminosity distance and $d_{\mathrm{app}}$ is the approximated (up to third order) luminosity distance. In both panels ${\Omega }_C=0.3$ and $\alpha =0$.

Figure 2

Exact luminosity distance (solid line) and three approximated luminosity distances for ${\Omega }_C=0.3$ and $\alpha =0$ vs redshift. Long, short-dashed, and dotted lines are second, third, and fourth order in $z$ expansion, respectively. Future cosmological interpretations of high-redshift data [9] need the accuracy of the exact solution.

Figure 3

Left Panel: Exact luminosity distance vs redshift for ${\Omega }_C=0.3$ and $\alpha =-2$ (solid line), $\alpha =-1$ (dotted line), $\alpha =0$ (short-dashed line), $\alpha =1$ (dashed line), and $\alpha =2$ (long-dashed line). Right Panel: Exact luminosity distance vs redshift for $\alpha =0$ and for ${\Omega }_X/{\Omega }_C=9$ (solid line), ${\Omega }_X/{\Omega }_C=7/3$ (dotted line), ${\Omega }_X/{\Omega }_C=1$ (short dashed-line), ${\Omega }_X/{\Omega }_C=3/7$ (dashed line), ${\Omega }_X/{\Omega }_C=1/9$ (long-dashed line).

Figure 4

Acoustic scale vs $\alpha$. Our analytic evaluation of the acoustic scale lies between the dashed lines. The higher-dashed curve is computed with Eq. (21) while the lower is the 99.5% of the previous one. In this way we give an estimate of the order of magnitude of the error it is made, neglecting the contribution of the radiation in Eq. (12). Horizontal lines represent the $1\sigma$ contour to the value given by 10.