The cosmic equation of state

Abstract

The cosmic spacetime is often described in terms of the Friedmann-Robertson-Walker (FRW) metric, though the adoption of this elegant and convenient solution to Einstein's equations does not tell us much about the equation of state, $p=w\rho$, in terms of the total energy density $\rho$ and pressure $p$ of the cosmic fluid. $\Lambda$CDM and the $R_{\rm h}=ct$ Universe are both FRW cosmologies that partition $\rho$ into (at least) three components, matter $\rho_{\rm m}$, radiation $\rho_{\rm r}$, and a poorly understood dark energy $\rho_{\rm de}$, though the latter goes one step further by also invoking the constraint $w=-1/3$. This condition is apparently required by the simultaneous application of the Cosmological principle and Weyl's postulate. Model selection tools in one-on-one comparisons between these two cosmologies favor $R_{\rm h}=ct$, indicating that its likelihood of being correct is $\sim 90\%$ versus only $\sim 10\%$ for $\Lambda$CDM. Nonetheless, the predictions of $\Lambda$CDM often come quite close to those of $R_{\rm h}=ct$, suggesting that its parameters are optimized to mimic the $w=-1/3$ equation-of-state. In this paper, we explore this hypothesis quantitatively and demonstrate that the equation of state in $R_{\rm h}=ct$ helps us to understand why the optimized fraction $\Omega_{\rm m}\equiv \rho_m/\rho$ in $\Lambda$CDM must be $\sim 0.27$, an otherwise seemingly random variable. We show that when one forces $\Lambda$CDM to satisfy the equation of state $w=(\rho_{\rm r}/3-\rho_{\rm de})/\rho$, the value of the Hubble radius today, $c/H_0$, can equal its measured value $ct_0$ only with $\Omega_{\rm m}\sim0.27$ when the equation-of-state for dark energy is $w_{\rm de}=-1$. (We also show, however, that the inferred values of $\Omega_{\rm m}$ and $w_{\rm de}$ change in a correlated fashion if dark energy is not a cosmological constant, so that $w_{\rm de}\not= -1$.) This peculiar value of $\Omega_{\rm m}$ therefore appears to be a direct consequence of trying to fit the data with the equation of state $w=(\rho_{\rm r}/3-\rho_{\rm de})/\rho$ in a Universe whose principal constraint is instead $R_{\rm h}=ct$ or, equivalently, $w=-1/3$.