Designing Cyclic Universe Models

Designing Cyclic Universe Models
Justin Khoury1 , Paul J. Steinhardt2 and Neil Turok3
2

ISCAP, Columbia University, New York, NY 10027, USA Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA 3 DAMTP, CMS, Wilberforce Road, Cambridge, CB3 0WA, UK

1

The phenomenological constraints on the scalar field potential in cyclic models of the universe are presented. We showthat cyclic models require a comparable degree of tuning to that needed for inflationary models. The constraints are reduced to a set of simple design rules including “fastroll” parameters analogous to the “slow-roll” parameters in inflation.

arXiv:hep-th/0307132v4 18 Jan 2004

In conventional big bang/inflationary cosmology [1], the universe begins with the big bang and expands forever. Thecyclic model [2] is an alternative in which the bang is replaced by a transition to an earlier phase of evolution. The history of the universe is periodic, and the key events that shape the large scale structure of the observable universe occurred a cycle ago. Each cycle consists of: (i) a hot big bang phase during which large-scale structures form, (ii) a phase of slow, accelerated expansion [3],as observed today, which dilutes inhomogeneities and flattens the universe,(iii) a phase of contraction during which nearly scale invariant density perturbations are generated, and (iv) a big crunch/big bang transition at which matter and radiation are created and the next cycle is triggered. The cyclic model thus addresses the homogeneity, flatness and monopole problems of the standard hot big bangpicture, and also provides a nearly scale invariant spectrum of density fluctuations, without invoking a period of high energy inflation. The distinctive, non-inflationary mechanism for generating density perturbations in cyclic models results in a key observational difference: whereas inflation predicts a nearly scale invariant spectrum of gravitational waves, the cyclic model does not. The cyclicuniverse model is an extension of the ekpyrotic scenario [4, 5, 6], in which the hot big bang is viewed as the result of a collision between two brane worlds, in the simplest case between two orbifold fixed planes. The theory can be described by an effective 4d action in which the size of the orbifold is represented by a scalar field φ and the force between the boundary planes is represented by aneffective potential V (φ). The cyclic model corresponds to regularly repeating collisions with a period of dark energy domination during each cycle. The condition that cycles repeat and that the resulting solution is an attractor requires that V (φ) takes the general form shown in Fig. 1. The purpose of this Letter is to summarize the constraints for designing successful cyclic models. We show that a widerange of scalar field effective potentials are phenomenologically viable. The constraints on the steepness of the potential turn out to be remarkably similar to those on the flatness of the potential in inflation. The constraints depend strongly on the amplitude of the growing mode density perturbation propagating across the bounce into the hot big bang phase. We employ here the recent treatment ofthe transition as a collision

V(φ)
(c) (b) (a)

V0

φ

Vend

FIG. 1: Examples of cyclic potentials.

between branes in five dimensions [7], which resolves ambiguities present in earlier treatments [5, 8]. The gravity wave spectrum constraints quoted here are derived in [9]. At all times except around the big crunch/big bang transition, the dynamics of the cyclic model are welldescribed by the 4d effective Einstein-frame Lagrangian L= √ −g R (∂φ)2 − − V (φ) − β 4 (φ)(ρM + ρR ) 2 2 , (1)

where g is the determinant of the metric gµν , R the corresponding Ricci scalar, and we use units where 8πG = 1. The coupling β(φ) depends on the details of the theory, but, when the branes approach each other, the warping of the extra dimension becomes irrelevant and one finds √ a...