Erratum: Cosmological perturbations seeded by topological defects: Setting the initial conditions [Phys. Rev. D 56, 7608 (1997)]

Abstract

We consider a perfectly homogeneous and isotropic universe which undergoes a sudden phase transition. If the transition produces topological defects, which we assume, perturbations in the geometry and the cosmic fluid also suddenly appear. We apply the standard general relativistic junction conditions to match the pre-and post-transition eras and thus set the initial conditions for the perturbations. We solve their evolution equations analytically in the case when the defects act as a coherent source and their density scales like the background density. We show that isocurvature as well as adiabatic perturbations are created, in a ratio which is independent of the detailed properties of the defects. We compare our result to the initial conditions currently used in the literature and show how the cosmic fluid naturally "compensates" for the presence of the defects.1

CMB temperature anisotropy at large scales induced by a causal primordial magnetic field

“…The solutions are completely determined once the initial conditions have been specified. To set the initial conditions, we follow what has been done for the topological defect case [13,14]. First of all, we assume a sudden phase transition, as for example the electroweak phase transition.…”

“…Before the phase transition, the universe is the usual perturbed FLRW universe with only the standard inflationary adiabatic perturbations; after the phase transition, the magnetic field energy momentum tensor contributes at first order to the metric perturbations. In order to connect these two stages, we match the geometry and the fluid variables on the surface of constant density, so that the induced three metric and the extrinsic curvature are continuous [13]. This implies the conservation of the total energy momentum tensor at the magnetic field generation time.…”

Conservation laws and cosmological perturbations in curved universes

“…Provided that the defect network is coherent, uncorrelated on scales larger than the Hubble radius and scales with the background (the precise definition of these properties is given in e.g. [1]), it can be modelled by (18), (see e.g. [4], [10]).…”

“…The question is therefore to determine which quantities are indeed continuous through the phase transition. This has been studied in detail in [1]. For perturbations varying on scales larger than the Hubble radius the phase transition which is at their origin looks instantaneous and the matching conditions between the pre-and post-transition eras are the standard ones : the induced threemetric on, and the extrinsic curvature of the surface of transition, which is taken to be a surface of constant density, must be continuous.…”

“…Finally, the two constants Ψ 0 and Ψ 1 correspond to the two physical degrees of freedom and are determined by the matching conditions. They turn out not to be zero, but the terms proportional to them in (42), (43) become nevertheless quickly negligible (see [1]). Hence the assumptions that h and E be continuous are therefore asymptotically valid.…”