Measuring the Equation-of-state of the Universe: Pitfalls and Prospects

TL;DR

Maor, Brustein, McMahon, and Steinhardt analyze the feasibility of determining the dark-energy equation of state $w_Q(z)$ from supernova distance measurements and assess the incremental value of combining SN data with CMB and Alcock-Paczyński tests. They identify a fundamental degeneracy between $w_Q(z)$ and its redshift derivative $dw_Q/dz$, showing that luminosity-distance measurements predominantly constrain a weighted average of the EOS ($w_T(z)=\Omega_Q(z) w_Q(z)$) and that independent knowledge of density parameters is required to isolate $w_Q(z)$. The study demonstrates that common priors (constant $w_Q$ or $w_Q\ge -1$) can severely distort inferences and that time variation is more easily detected when $w_Q(z)$ increases with $z$, though often only for large derivatives. Even optimistic joint analyses of SN, AP, and CMB data leave substantial degeneracies, underscoring the need for a new, precision probe to resolve $w_Q(z)$ and its evolution. The authors discuss potential directions, including measurements of the Hubble parameter derivative $H'$ and growth- or lensing-based observables, while emphasizing the challenges posed by model dependence and degeneracy.

Abstract

We explore various pitfalls and challenges in determining the equation-of-state (w) of dark energy component that dominates the universe and causes the current accelerated expansion. We demonstrated in an earlier paper the existence of a degeneracy that makes it impossible to resolve well the value of w or its time-derivative with supernovae data. Here we consider standard practices, such as assuming priors that w is constant or greater than -1, and show that they also can lead to gross errors in estimating the true equation-of-state. We further consider combining measurements of the cosmic microwave background anisotropy and the Alcock-Paczynski test with supernovae data and find that the improvement in resolving the time-derivative of w is marginal, although the combination can constrain its present value perhaps to 20 percent uncertainty.

Figures (10)

• Figure 1: $w_T (z)$ for three best fit models of three fits under three different assumptions: constant $w_Q$ (dashed), linear $w_Q$ (solid), and quadratic $w_T$ (dotted), to data generated from a single fiducial model: $(w_Q,\Omega_m)=(-0.7-0.8z, \, .3)$. All fits prefer $w_T^*\equiv w_T(z^*\simeq .15)\simeq -.52$, but diverge for other values of $z$.
• Figure 2: Models within 95% CL region of a fit to data generated from the fiducial model $(w_Q,\Omega_m)=(-1,0.3)$ assuming $w_Q=w_0+ w_1 z$. Top: The total EOS $w_T(z)$, for three different linear models. Middle: $w_Q(z)$ assuming $\Omega_m=0.3$ exactly, for the same linear models. Bottom: $w_Q(z)$ for nine models, assuming that $0.2<\Omega_m<0.4$ (no relation between the dashed, dotted and solid lines of the bottom panel to those of the middle and top ones).
• Figure 3: Magnitude differences between pairs of degenerate models and a flat pure matter ($\Omega_m=1$) Universe. Each pair consists of simulated data points generated from one constant $w_Q$ model (open circles) and one linear $w_Q$ model with a large (positive or negative) derivative (full squares). The pairs are well separated but it is hard to separate between "members" of each pair.
• Figure 4: 95% CL contours of fits to data generated from two fiducial models. The curvature of degeneracy contours is positive for a positive $w_Q$ fiducial model (right) and negative for a negative $w_Q$ fiducial model (left).
• Figure 5: Constrained (small) and unconstrained (larger and more negative) 68% and 95% confidence contours of a fit to data generated from a fiducial model with linear $w_Q$$(w_Q,\Omega_m)=(-.7+.8 z,.3)$. The fit is done under the (wrong) assumption that $w_Q$ is constant.