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The Cosmic Gravitational Wave Background in a Cyclic Universe

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The Cosmic Gravitational Wave Background in a Cyclic Universe The Cosmic Gravitational-Wave Background in a Cyclic Universe Latham A. Boyle1, Paul J. Steinhardt1, and Neil Turok2 1Department of Physics, Princeton University, Princeton, New Jersey 08544 2Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, United Kingdom (Dated: July 2003) Inflation predicts a primordial gravitational wave spectrum that is slightly “red,” i.e. nearly scale-invariant with slowly increasing power at longer wavelengths. In this paper, we compute both the amplitude and spectral form of the primordial tensor spectrum predicted by cyclic/ekpyrotic models. The spectrum is exponentially suppressed compared to inflation on long wavelengths, and the strongest constraint emerges from the requirement that the energy density in gravitational waves should not exceed around 10 per cent of the energy density at the time of nucleosynthesis. The recently-proposed cyclic model[1, 2] differs radi- V(φ) cally from standard inflationary cosmology [3, 4], while retaining the inflationary predictions of homogeneity, 4 5 φ 6 flatness, and nearly scale-invariant density perturbations. end It has been suggested that the cosmic gravitational wave 3 1 φ background provides the best experimental means for dis- tinguishing the two models. Inflation predicts a nearly 2 scale-invariant (slightly red) spectrum of primordial ten- sor perturbations, whereas the cyclic model predicts a blue spectrum.[1] The difference arises because inflation involves an early phase of hyper-rapid cosmic accelera- tion, whereas the cyclic model does not. In this paper, we compute the gravitational wave spec- trum for cyclic models to obtain both the normalization and spectral shape as a function of model parameters, im- V proving upon earlier heuristic estimates. We find that the end spectrum is strongly blue. The amplitude is too small to be observed by currently proposed detectors on all scales. Hence, the discovery of a stochastic background of grav- FIG. 1: Schematic of cyclic potential with numbers represent- itational waves would be evidence in favor of inflation, ing the stages described in the text. To the left of φend, where the scalar kinetic energy dominates, we approximate V with and would rule out the cyclic model. a Heaviside function, jumping to zero as shown by the dashed Readers unfamiliar with the cyclic model may consult line. [5] for an informal tour, and [6] for a recent analysis of arXiv:hep-th/0307170v1 18 Jul 2003 phenomenological constraints. Cyclic cosmology draws strongly on earlier ideas associated with the “ekpyrotic (2) φ intermediate and decreasing: the universe is dom- universe” scenario. [7, 8, 9] Briefly, the scenario can be inated by a combination of scalar kinetic and potential described in terms of the periodic collision of orbifold energy, leading to slow contraction and to the generation planes moving in an extra spatial dimension, or, equiv- of fluctuations; (3) φ negative and decreasing (beginning alently, in terms of a four-dimensional theory with an at conformal time τend < 0): the generation of fluctua- evolving (modulus) field φ rolling back and forth in an tions ends, φ rolls past φend and, in the four-dimensional effective potential V (φ). The field theory description is description, the universe contracts rapidly, dominated by the long wavelength approximation to the brane picture scalar field kinetic energy, to the bounce (τ = 0) at which in which the potential represents the interbrane interac- matter and radiation are generated; (4) φ increasing from tion and the modulus field determines the distance be- minus infinity: the universe remains dominated by scalar tween branes. For the purposes of this paper, the field field kinetic energy, which decreases rapidly compared to theoretic description is more useful. the radiation energy; (5) φ large and increasing (begin- The potential (Fig. 1) is small and positive for large φ, ning at τr > 0): the scalar field kinetic energy red-shifts falling steeply negative at intermediate φ, and increasing to a negligible value and the universe begins the radiation again for negative φ. Each cycle consists of the following dominated expanding phase; (6) φ large and nearly sta- stages: (1) φ large and decreasing: the universe expands tionary: the universe undergoes the transitions to matter at an accelerated rate as V (φ) > 0 acts as dark energy; and dark energy domination, and the cycle begins anew. 2 We model the scalar field potential as: THE PRIMORDIAL SPECTRUM, ∆h(k,τr) − V (φ)= V (1 e cφ/Mpl )Θ(φ φ ) (1) 0 − − end A quasi-stationary, isotropic, stochastic background of gravitational waves is characterized by the quantity where Mpl is the reduced Planck mass and Θ(φ) the ∆h(k, τ), the rms dimensionless strain per unit logarith- Heaviside step function. A potential of this form, with an mic wavenumber at time τ (i.e. the δL/L that would be exponentially steep form, is required by the cyclic model measured by a detector with sensitivity band centered on in order to produce an acceptable spectrum of cosmolog- mode k and bandwidth ∆k = k). Accounting for both ical perturbations. [2, 6] Choosing c = 10 for example polarizations, it is given by ∆h(k, τ) = k3/2 h (τ) /π, | k | results in a scalar spectral index ns = .96 which is com- where the Fourier amplitude hk(τ) satisfies patible with current constraints. The Heaviside function ′ marks the end of the steeply decreasing part of the poten- ′′ a ′ 2 hk +2 hk + k hk = 0 (7) tial; for φ < φend the potential is small and the universe a is dominated by scalar field kinetic energy. and the boundary condition that the solution approaches Our calculation begins in the “ekpyrotic phase,” stage the Minkowski vacuum at short distances (2), with the Einstein-frame scale factor contracting: e−ikτ α h (τ) as τ . (8) τ τek k a(τ)= a − , τ<τ , (2) → a(τ)Mpl√2k → −∞ end τ τ end end − ek To solve equation (7), it is useful to define fk(τ) 2 ≡ where α 2/(c 2) 1 and τek (1 2α)τend, being a(τ) hk(τ) and rewrite (7) as the conformal≡ time− the≪ potential would≡ − have diverged to ′′ minus infinity had the exponential form continued. At ′′ 2 a (fk) + (k )fk =0 . (9) τ = τend, the ekpyrotic phase ends and the “contracting − a kinetic phase,” stage (3), begins: During the ekpyrotic phase, a(τ) is given by (2), and the general solution of (9) is τ 1/2 a(τ)= − , τend <τ< 0 . (3) (1 + χ)τr (1) (2) fk(τ)= √y A1(k)Hn (y)+ A2(k)Hn (y) , (10) At τ = 0, the universe bounces and the “expanding ki- 1 netic phase,” stage (4), begins: where A1,2(k) are arbitrary constants, n 2 α, y (1,2) ≡ − ≡ k(τ τ ), and Hn are the Hankel functions. The − − ek τ 1/2 boundary condition (8) implies a(τ)= , 0 <τ<τ . (4) τ r r 1 π A (k)= , A (k)=0, (11) 1 2 k 2 Radiation is produced at the bounce, but is less than r the scalar kinetic energy until, at τ = τ , the expanding r where we have dropped a physically irrelevant phase. In kinetic phase ends, and standard radiation-dominated, the contracting kinetic phase, stage (4), a(τ) is given by matter-dominated, and dark-energy-dominated epochs (3), and the general solution of (9) is ensue. The transition times, τr and τend, are given by (1) (2) −1 fk(τ)=√ kτ B1(k)H0 ( kτ)+B2(k)H0 ( kτ) (12) τr = (√2Hr) , τend = τr/Γ, (5) − − − − where B1,2(k) are arbitrary constants. Then, continuity and ′ of hk and hk at τ = τend implies 1/3 τr 1 2α Vend Γ = , (6) iπ πα (2,1) (1) ≡ τ 1+ χ 1 2α H2M 2 B1,2(k) = xe H1 (xe)Hn (2αxe)+ end " − r pl !# ∓ 4 2k r h +H(2,1)(x )H(1) (2αx ) (13) where H H(τ ) is the Hubble constant at τ , V = 0 e n−1 e r ≡ r r end V (φend) is the depth of the potential at its minimum, i − where x k τ . Finally, in the expanding kinetic and χ 1 is a small positive constant that measures the e ≡ | end| amount≪ of radiation created at the bounce. Note that phase, a(τ) is given by (4), and the general solution of a(τ) and a′(τ) are both continuous at the transition time (9) is τ = τend, and we have chosen to normalize a(τ) to unity √ (1) (2) at the start of radiation domination (a(τr) = 1). fk(τ)= kτ C1(k)H0 (kτ)+ C2(k)H0 (kτ) . (14) 3 To fix C1,2(k), we need to match the solution across τ = log ∆h 0. At the level of quantum field theory in curved space- time, the choice is essentially unique[11], and amounts -2+ (n /2) k T to analytically continuing the positive (negative) fre- quency part of h f /a around the origin in the lower k ≡ k (1,2) -1+(n /2) (upper) half of the complex τ-plane, so H0 ( kτ) k T (2,1) − → inflati H0 (kτ). This yields − on α C (k)= 1+ χB (k) . (15) (1/2)+ 1,2 2,1 -1+α α k − k k p The pre-factor arises because a(τ) differs by a factor of cyclic √1+ χ between the kinetic contracting and expanding phases; see Eqs. (3) and (4). Combining our results, we arrive at the “primordial” dimensionless strain spectrum log k at the beginning of the radiation dominated epoch: ko keq kr kend 2 ∆h(k, τr) = (k /πMpl) 2(1 + χ )τr (1) (2) (16) B2(k)H0 (pxr)+ B1(k)H0 (xr) FIG.
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