Constraining the equation of state of the Universe from Distant Type Ia Supernovae and Cosmic Microwave Background Anisotropies

TL;DR

The paper investigates how distant Type Ia supernovae (SN) and CMB anisotropies jointly constrain the Universe’s equation of state and geometry. It uses a likelihood/Fisher-matrix framework to forecast constraints on the matter density $\Omega_m$, the cosmological constant $\Omega_\Lambda$, and a quintessence-like component with equation of state $w_Q = p/\rho$, examining both constant $w_Q$ and slowly evolving tracker scenarios. Results show that SN data alone suffer from strong $w_Q$–$\Omega_m$ degeneracies, but incorporating CMB information breaks these degeneracies, yielding $w_Q$ constraints such as $w_Q< -0.6$ in flat universes and $w_Q< -0.4$ when curvature is allowed; the combined SN+CMB analyses favor a nearly flat Universe with $\Omega_m\approx 0.12$ and $\Omega_Q\approx 0.73$, consistent with a cosmological-constant-like component. The study further demonstrates that extending SN observations to higher redshift is valuable for narrowing $\Omega_m$ and $\Omega_\Lambda$, and discusses the potential of future missions to tighten these constraints and test quintessence models against simple potentials, with implications for the viability and tuning of such models.

Abstract

We analyse the constraints that can be placed on a cosmological constant or quintessence-like component by combining observations of Type Ia supernovae with measurements of anisotropies in the cosmic microwave background. We use the recent supernovae sample of Perlmutter et al and observations of the CMB anisotropies to constraint the equation of state (w_Q = p/rho) in quintessence-like models via a likelihood analysis. The 2 sigma upper limits are w_Q < -0.6 if the Universe is assumed to be spatially flat, and w_Q < -0.4 for universes of arbitrary spatial curvature. The upper limit derived for a spatially flat Universe is close to the lower limit (w_Q approx -0.7) allowed for simple potentials, implying that additional fine tuning may be required to construct a viable quintessence model.

Figures (8)

• Figure 1: The dashed lines in each panel show $1$, $2$ and $3 \sigma$ likelihood contours in the $\Omega_\Lambda$--$\Omega_m$ plane for the SCP distant supernova sample as analysed by E99. The solid contours are derived from the Fisher matrix (equation 4) for the SCP sample supplemented by 20 SN with a mean redshift of $\langle z \rangle =1$ (Figure 1a) and for twice the SCP sample and $40$ SN with $\langle z \rangle =1.5$ (Figure 1b). The points show maximum likelihood values of $\Omega_\Lambda$ and $\Omega_m$ for Monte-Carlo realizations of these samples, as described in the text.
• Figure 2: Distributions along the major and minor axes of the likelihood contours shown in Figure 1. The histograms show the distributions derived from the Monte-Carlo simulations and the dotted lines show Gaussian distributions with variances determined from the Fisher matrix after marginalizing over the parameter ${\cal M}$.
• Figure 3: Fisher matrix constraints for a sample of SN extending to redshifts $z>3$ (see text) illustrating that by extending the redshift range one can determine $\Omega_m$ independently of $\Omega_\Lambda$.
• Figure 4: As Figure 2, but for an arbitrary constant equation of state in a spatially flat Universe. The dashed lines in each panel show $1$, $2$ and $3 \sigma$ likelihood contours in the $w_Q$--$\Omega_m$ plane for the SCP distant supernova sample as analysed in Section 4 (assuming a constant equation of state). The solid contours are derived from the Fisher matrix for enhanced samples of high redshift supernovae and the points show maximum likelihood derived from Monte-Carlo realizations of these samples.
• Figure 5: The evolution of the equation of state $w_Q$ and its contribution to the cosmic density parameter $\Omega_Q$ as a function of redshift derived from solutions to the tracker equation (8) for three potentials: $V(Q) = M^4({\rm exp}(1/Q) - 1)$ (figures 5a and 5b); $V(Q) = M^4(M/Q)^2$ (figures 5c and 5d); $V(Q) = M^4(M/Q)^6$ (figures 5e and 5f). The curves in each figure are computed by varying the parameter $M$, with more negative values of $w_Q$ corresponding to higher values of $\Omega_Q$.