A Selberg integral for the Lie algebra A_n

A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g.Comment: 32 page

The Impact of Non-Equipartition on Cosmological Parameter Estimation from Sunyaev-Zel'dovich Surveys

The collisionless accretion shock at the outer boundary of a galaxy cluster should primarily heat the ions instead of electrons since they carry most of the kinetic energy of the infalling gas. Near the accretion shock, the density of the intracluster medium is very low and the Coulomb collisional timescale is longer than the accretion timescale. Electrons and ions may not achieve equipartition in these regions. Numerical simulations have shown that the Sunyaev-Zel'dovich observables (e.g., the integrated Comptonization parameter Y) for relaxed clusters can be biased by a few percent. The Y-mass relation can be biased if non-equipartition effects are not properly taken into account. Using a set of hydrodynamical simulations, we have calculated three potential systematic biases in the Y-mass relations introduced by non-equipartition effects during the cross-calibration or self-calibration when using the galaxy cluster abundance technique to constraint cosmological parameters. We then use a semi-analytic technique to estimate the non-equipartition effects on the distribution functions of Y (Y functions) determined from the extended Press-Schechter theory. Depending on the calibration method, we find that non-equipartition effects can induce systematic biases on the Y functions, and the values of the cosmological parameters Omega_8, sigma_8, and the dark energy equation of state parameter w can be biased by a few percent. In particular, non-equipartition effects can introduce an apparent evolution in w of a few percent in all of the systematic cases we considered. Techniques are suggested to take into account the non-equipartition effect empirically when using the cluster abundance technique to study precision cosmology. We conclude that systematic uncertainties in the Y-mass relation of even a few percent can introduce a comparable level of biases in cosmological parameter measurements.Comment: 10 pages, 3 figures, accepted for publication in the Astrophysical Journal, abstract abridged slightly. Typos corrected in version

Cocycle categories

Suppose that G is a sheaf of groups on a space X and that Uα ⊂ X is an open covering. Then a cocycle for the covering is traditionally defined to be a family of elements gαβ ∈ G(Uα ∩ Uβ) such that gαα = e and gαβgβγ = gαγ when all elements are restricted to the group G(Uα ∩ Uβ ∩ Uγ)