Constraints on the asymptotic baryon fractions of galaxy clusters at large radii

While X-ray measurements have so far revealed an increase in the volume-averaged baryon fractions fb(r) of galaxy clusters with cluster radii r, fb(r) should asymptotically reach a universal value fb(∞) = fb, provided that clusters are representative of the Universe. In the framework of hydrostatic equilibrium for intracluster gas, we have derived the necessary conditions for fb(∞) = fb: The X-ray surface brightness profile described by the β model and the temperature profile approximated by the polytropic model should satisfy γ ≈ 2(1 − 1/3β) and γ ≈ 1 + 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 (30rc) where hydrostatic equilibrium has probably become inapplicable. We further explore how to obtain the asymptotic value fb(∞) 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 fb(∞): 0.094±0.035≤ fb(∞) ≤ 0.41± 0.18, corresponding to a cosmological density parameter 0.122± 0.069 ≤ ΩM ≤ 0.53± 0.28 for H0 = 50 km s−1 Mpc−1. An optimal estimate of fb(∞) based on three cooling flow clusters with β < 1/2 in our lensing cluster sample yields 〈fb(∞)〉 = 0.142± 0.007 or ΩM = 0.35± 0.09.