Constraints on the Asymptotic Baryon Fractions of Galaxy Clusters at Large Radii

Abstract

While X-ray measurements have so far revealed an increase in the volume-averaged baryon fractions $f_b(r)$ of galaxy clusters with cluster radii $r$, $f_b(r)$ should asymptotically reach a universal value $f_b(\infty)=f_b$, provided that clusters are representative of the Universe. In the framework of hydrostatic equilibrium for intracluster gas, we have derived the necessary conditions for $f_b(\infty)=f_b$: The X-ray surface brightness profile described by the $β$ model and the temperature profile approximated by the polytropic model should satisfy $γ\approx2(1-1/3β)$ and $γ\approx1+1/3β$ for $β<1$ and $β>1$, respectively, which sets a stringent limit to the polytropic index: $γ<4/3$. In particular, a mildly increasing temperature with radius is required if the observationally fitted $β$ parameter is in the range $1/3<β<2/3$. It is likely that a reliable determination of the universal baryon fraction can be achieved in the small $β$ clusters because the disagreement between the exact and asymptotic baryon fractions for clusters with $β>2/3$ breaks down at rather large radii ($\ga30r_c$) where hydrostatic equilibrium has probably become inapplicable. We further explore how to obtain the asymptotic value $f_b(\infty)$ of baryon fraction from the X-ray measurement made primarily over the finite central region of a cluster. We demonstrate our method using a sample of 19 strong lensing clusters, which enables us to place a useful constraint on $f_b(\infty)$: $0.094\pm0.035 \leq f_b(\infty) \leq 0.41\pm0.18$. An optimal estimate of $f_b(\infty)$ based on three cooling flow clusters with $β<1/2$ in our lensing cluster sample yields $ = 0.142\pm0.007$ or $Ω_M = 0.35\pm0.09$.