Second Order Perturbations of a Macroscopic String; Covariant Approach

Using a world-sheet covariant formalism, we derive the equations of motion for second order perturbations of a generic macroscopic string, thus generalizing previous results for first order perturbations. We give the explicit results for the first and second order perturbations of a contracting near-circular string; these results are relevant for the understanding of the possible outcome when a cosmic string contracts under its own tension, as discussed in a series of papers by Vilenkin and Garriga. In particular, second order perturbations are necessaary for a consistent computation of the energy. We also quantize the perturbations and derive the mass-formula up to second order in perturbations for an observer using world-sheet time $\tau$. The high frequency modes give the standard Minkowski result while, interestingly enough, the Hamiltonian turns out to be non-diagonal in oscillators for low-frequency modes. Using an alternative definition of the vacuum, it is possible to diagonalize the Hamiltonian, and the standard string mass-spectrum appears for all frequencies. We finally discuss how our results are also relevant for the problems concerning string-spreading near a black hole horizon, as originally discussed by Susskind.Comment: New discussion about the quantum mass-spectrum in chapter

Zero-th law in structural glasses: an example

We investigate the validity of a zeroth thermodynamic law for non-equilibrium systems. In order to describe the thermodynamics of the glassy systems, it has been introduced an extra parameter, the effective temperature which generalizes the fluctuation-dissipation theorem (FDT) to off-equilibrium systems and supposedly describes thermal fluctuations around the aging state. In particular we analyze two coupled systems of harmonic oscillators with Monte Carlo dynamics. We study in detail two types of dynamics: sequential dynamics, where the coupling between the subsystems comes only from the Hamiltonian; and parallel dynamics where there is another source of coupling: the dynamics. We show how in the first case the effective temperatures of the two interacting subsystems are different asymptotically due to the smallness of the thermal conductivity in the aging regime. This explains why, in structural glasses, different interacting degrees of freedom can stay at different effective temperatures, and never thermalize.Comment: 10 pages. Contribution to the Proceedings of the ESF SPHINX meeting `Glassy behaviour of kinetically constrained models' (Barcelona, March 22-25, 2001). To appear in a special issue of J. Phys. Cond. Mat

Stable and Unstable Circular Strings in Inflationary Universes

It was shown by Garriga and Vilenkin that the circular shape of nucleated cosmic strings, of zero loop-energy in de Sitter space, is stable in the sense that the ratio of the mean fluctuation amplitude to the loop radius is constant. This result can be generalized to all expanding strings (of non-zero loop-energy) in de Sitter space. In other curved spacetimes the situation, however, may be different. In this paper we develop a general formalism treating fluctuations around circular strings embedded in arbitrary spatially flat FRW spacetimes. As examples we consider Minkowski space, de Sitter space and power law expanding universes. In the special case of power law inflation we find that in certain cases the fluctuations grow much slower that the radius of the underlying unperturbed circular string. The inflation of the universe thus tends to wash out the fluctuations and to stabilize these strings.Comment: 15 pages Latex, NORDITA 94/14-

Solutions to the cosmological constant problems

We critically review several recent approaches to solving the two cosmological constant problems. The "old" problem is the discrepancy between the observed value of $\Lambda$ and the large values suggested by particle physics models. The second problem is the "time coincidence" between the epoch of galaxy formation $t_G$ and the epoch of $\Lambda$-domination t_\L. It is conceivable that the "old" problem can be resolved by fundamental physics alone, but we argue that in order to explain the "time coincidence" we must account for anthropic selection effects. Our main focus here is on the discrete-$\Lambda$ models in which $\Lambda$ can change through nucleation of branes. We consider the cosmology of this type of models in the context of inflation and discuss the observational constraints on the model parameters. The issue of multiple brane nucleation raised by Feng {\it et. al.} is discussed in some detail. We also review continuous-\L models in which the role of the cosmological constant is played by a slowly varying potential of a scalar field. We find that both continuous and discrete models can in principle solve both cosmological constant problems, although the required values of the parameters do not appear very natural. M-theory-motivated brane models, in which the brane tension is determined by the brane coupling to the four-form field, do not seem to be viable, except perhaps in a very tight corner of the parameter space. Finally, we point out that the time coincidence can also be explained in models where $\Lambda$ is fixed, but the primordial density contrast $Q=\delta\rho/\rho$ is treated as a random variable.Comment: 30 pages, 3 figures, two notes adde

Holographic multiverse and the measure problem

We discuss the duality, conjectured in earlier work, between the wave function of the multiverse and a 3D Euclidean theory on the future boundary of spacetime. In particular, we discuss the choice of the boundary metric and the relation between the UV cutoff scale xi on the boundary and the hypersurfaces Sigma on which the wave function is defined in the bulk. We propose that in the limit of xi going to 0 these hypersurfaces should be used as cutoff surfaces in the multiverse measure. Furthermore, we argue that in the inflating regions of spacetime with a slowly varying Hubble rate H the hypersurfaces Sigma are surfaces of constant comoving apparent horizon (CAH). Finally, we introduce a measure prescription (called CAH+) which appears to have no pathological features and coincides with the constant CAH cutoff in regions of slowly varying H.Comment: A minor change: the discussion of unitarity on p.9 is clarifie

Covariant perturbations of domain walls in curved spacetime

A manifestly covariant equation is derived to describe the perturbations in a domain wall on a given background spacetime. This generalizes recent work on domain walls in Minkowski space and introduces a framework for examining the stability of relativistic bubbles in curved spacetimes.Comment: 15 pages,ICN-UNAM-93-0

Is there the radion in the RS2 model ?

We analyse the physical boundary conditions at infinity for metric fluctuations and gauge functions in the RS2 model with matter on the brane. We argue that due to these boundary conditions the radion field cannot be gauged out in this case. Thus, it represents a physical degree of freedom of the model.Comment: 9 page

Braneworld Cosmological Perturbation Theory at Low Energy

Homogeneous cosmology in the braneworld can be studied without solving bulk equations of motion explicitly. The reason is simply because the symmetry of the spacetime restricts possible corrections in the 4-dimensional effective equations of motion. It would be great if we could analyze cosmological perturbations without solving the bulk. For this purpose, we combine the geometrical approach and the low energy gradient expansion method to derive the 4-dimensional effective action. Given our effective action, the standard procedure to obtain the cosmological perturbation theory can be utilized and the temperature anisotropy of the cosmic background radiation can be computed without solving the bulk equations of motion explicitly.Comment: 10 pages, Based on a talk presented at ACRGR4, the 4th Australasian Conference on General Relativity and Gravitation, Monash University, Melbourne, January 2004. To appear in the proceedings, in General Relativity and Gravitatio

Circular String-Instabilities in Curved Spacetime

We investigate the connection between curved spacetime and the emergence of string-instabilities, following the approach developed by Loust\'{o} and S\'{a}nchez for de Sitter and black hole spacetimes. We analyse the linearised equations determining the comoving physical (transverse) perturbations on circular strings embedded in Schwarzschild, Reissner-Nordstr\"{o}m and de Sitter backgrounds. In all 3 cases we find that the "radial" perturbations grow infinitely for $r\rightarrow 0$ (ring-collapse), while the "angular" perturbations are bounded in this limit. For $r\rightarrow\infty$ we find that the perturbations in both physical directions (perpendicular to the string world-sheet in 4 dimensions) blow up in the case of de Sitter space. This confirms results recently obtained by Loust\'{o} and S\'{a}nchez who considered perturbations around the string center of mass.Comment: 24 pages Latex + 2 figures (not included). Observatoire de Paris, Meudon No. 9305

Geometry of Deformations of Relativistic Membranes

A kinematical description of infinitesimal deformations of the worldsheet spanned in spacetime by a relativistic membrane is presented. This provides a framework for obtaining both the classical equations of motion and the equations describing infinitesimal deformations about solutions of these equations when the action describing the dynamics of this membrane is constructed using {\it any} local geometrical worldsheet scalars. As examples, we consider a Nambu membrane, and an action quadratic in the extrinsic curvature of the worldsheet.Comment: 20 pages, Plain Tex, sign errors corrected, many new references added. To appear in Physical Review