A primordial cosmological magnetic field induces Faraday rotation of the cosmic microwave background polarization. This rotation produces a curl-type polarization component even when the unrotated polarization possesses only gradient-type polarization, as expected from scalar density perturbations. We compute the angular power spectrum of curl-type polarization arising from small Faraday rotation due to a weak stochastic primordial magnetic field with a power-law power spectrum. The induced polarization power spectrum peaks at arcminute angular scales. Faraday rotation is one of the few cosmological sources of curl-type polarization, along with primordial tensor perturbations, gravitational lensing, and the vector and tensor perturbations induced by magnetic fields; the Faraday rotation signal peaks on significantly smaller angular scales than any of these, with a power spectrum amplitude which can be comparable to that from gravitational lensing. Prospects for detection are briefly discussed.

We present a numerical classification of the spherically symmetric, static solutions to the Einstein-Yang-Mills equations with cosmological constant Λ. We find three qualitatively different classes of configurations, where the solutions in each class are characterized by the value of Λ and the number of nodes, n, of the Yang-Mills amplitude.For sufficiently small, positive values of the cosmological constant, Λ < Λ crit (n), the solutions generalize the Bartnik-McKinnon solitons, which are now surrounded by a cosmological horizon and approach the deSitter geometry in the asymptotic region. For a discrete set of values Λ reg (n) > Λ crit (n), the solutions are topologically 3-spheres, the ground state (n = 1) being the Einstein Universe. In the intermediate region, that is for Λ crit (n) < Λ < Λ reg (n), there exists a discrete family of global solutions with horizon and "finite size".

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We evaluate the one-loop prefactor in the false vacuum decay rate in a theory of a self interacting scalar field in 3 + 1 dimensions. We use a numerical method, established some time ago, which is based on a well-known theorem on functional determinants. The proper handling of zero modes and of renormalization is discussed. The numerical results in particular show that quantum corrections become smaller away from the thin-wall case. In the thin-wall limit the numerical results are found to join into those obtained by a gradient expansion.

It is proven that there are precisely n odd-parity sphaleron-like unstable modes of the n-th Bartnik-McKinnon soliton and the n-th non-abelian black hole solution of the Einstein-Yang-Mills theory for the gauge group SU(2).