Big Bang nucleosynthesis with a stiff fluid

Models that lead to a cosmological stiff fluid component, with a density $\rho_S$ that scales as $a^{-6}$, where $a$ is the scale factor, have been proposed recently in a variety of contexts. We calculate numerically the effect of such a stiff fluid on the primordial element abundances. Because the stiff fluid energy density decreases with the scale factor more rapidly than radiation, it produces a relatively larger change in the primordial helium-4 abundance than in the other element abundances, relative to the changes produced by an additional radiation component. We show that the helium-4 abundance varies linearly with the density of the stiff fluid at a fixed fiducial temperature. Taking $\rho_{S10}$ and $\rho_{R10}$ to be the stiff fluid energy density and the standard density in relativistic particles, respectively, at $T = 10$ MeV, we find that the change in the primordial helium abundance is well-fit by $\Delta Y_p = 0.00024(\rho_{S10}/\rho_{R10})$. The changes in the helium-4 abundance produced by additional radiation or by a stiff fluid are identical when these two components have equal density at a "pivot temperature", $T_*$, where we find $T_* = 0.55$ MeV. Current estimates of the primordial $^4$He abundance give the constraint on a stiff fluid energy density of $\rho_{S10}/\rho_{R10} < 30$.