Reconstructing the Cosmic Equation of State from Supernova distances

TL;DR

A model-independent method for estimating the form of the potential V(phi) of the scalar field driving this acceleration, and the associated equation of state w(phi), based on a versatile analytical form for the luminosity distance D(L).

Abstract

Observations of high-redshift supernovae indicate that the universe is accelerating. Here we present a {\em model-independent} method for estimating the form of the potential $V(φ)$ of the scalar field driving this acceleration, and the associated equation of state $w_φ$. Our method is based on a versatile analytical form for the luminosity distance $D_L$, optimized to fit observed distances to distant supernovae and differentiated to yield $V(φ)$ and $w_φ$. Our results favor $w_φ\simeq -1$ at the present epoch, steadily increasing with redshift. A cosmological constant is consistent with our results.

Figures (5)

• Figure 1: The maximum deviation $\Delta D_L/D_L$ between the actual value and that calculated from the ansatz (\ref{['eqn:star']}) in the redshift range $z=$0--10, as a function of $\Omega_{\rm M}\equiv 1-\Omega_{\phi}$. The three curves plotted are for constant values of the equation of state parameter (as defined in Eq. \ref{['eqn:wzed']}) $w_\phi=-1$ (solid line), $-2/3$ (dotted line) and $-1/3$ (dashed line).
• Figure 2: The distance modulus $(m-M)$ of the SNe Ia relative to an $\Omega_{\rm M}\to 0$ Milne Universe (dashed line), together with the best-fit model of our ansatz (\ref{['eqn:star']}), plotted as the solid line. The extreme cases of the ($\Omega_{\rm M}$, $\Omega_{\phi}$)= (0, 1) and (1, 0) universes are plotted as dotted lines. Also plotted as the dot-dashed line is the best fit Perlmutter et al.perlmutter99 model ($\Omega_{\rm M}$, $\Omega_{\phi}$)= (0.28, 0.72). The filled circles are the 54 SNe of the "primary fit" of perlmutter99. The high-$z$ SNe of riess98 (not used in this analysis) are plotted as open circles.
• Figure 3: The effective potential $V(z)$, and the kinetic energy term ${\dot\phi}^2$, are shown in units of $\rho_{\rm cr}=3H_0^2/8\pi G$. Also plotted are the two forms of $V(\phi)$ for this $V(z)$, where the errors do not reflect errors in the $z$-$\phi$ relation. The value of $\phi$ (known up to an additive constant) is plotted in units of the Planck mass $m_{\rm P}$. The solid line corresponds to the best-fit values of the parameters. In each case, the shaded area covers the range of 68% errors, and the dotted lines the range of 90% errors. The hatched area represents the unphysical region ${\dot\phi}^2\!< \! 0$.
• Figure 4: The equation of state parameter $w_\phi(z)\!=\! P/\rho$ as a function of redshift. The solid line corresponds to the best-fit values of the parameters. The shaded area covers the range of 68% errors, and the dotted lines the range of 90% errors. The hatched area represents the region $w_\phi\!\le\! -1$, which is disallowed for a minimally coupled scalar field.
• Figure 5: The age of the Universe at a redshift $z$, given in units of $H_0^{-1}$ (left vertical axis) and in Gyr, for the value of $H_0\!=\!61.3$ km s$^{-1}$ Mpc$^{-1}$ (right vertical axis). The shaded region represents the range of 68% errors, and the dotted lines the range of 90% errors. The three high-redshift objects for which age-dating has been published dunlop-yoshii are plotted as lower limits to the age of the Universe at the corresponding redshifts. The dashed curve shows the same relation for an ($\Omega_{\rm M}, \Omega_{\phi}$)=(1,0) Universe for the same $H_0$.