Duality invariance of cosmological perturbation spectra

Abstract

I show that cosmological perturbation spectra produced from quantum fluctuations in massless or self-interacting scalar fields during an inflationary era remain invariant under a two parameter family of transformations of the homogeneous background fields. This relates slow-roll inflation models to solutions which may be far from the usual slow-roll limit. For example, a scale-invariant spectrum of perturbations in a minimally coupled, massless field can be produced by an exponential expansion with a ∝ e Ht , or by a collapsing universe with a ∝ (−t) 2/3 . to appear in Phys Rev D PACS numbers: 98.80.Cq Preprint PU-RCG/98-15v3, gr-qc/9809062The spectrum of perturbations on large scales is a key test of any models of the early universe. During an inflationary era quantum fluctuations on small scales become stretched beyond the horizon generating inhomogeneities on super-horizon scales that are otherwise inexplicable in the standard big bang model [1][2][3]. Conventional models of slow-roll inflation generally predict an almost scaleinvariant spectrum of adiabatic density perturbations [4]. As major observational programs are now underway to produce detailed maps of these perturbations, it is important to investigate whether one can uniquely reconstruct the inflationary history of the universe from the spectrum of inhomogeneities. This question has received considerable attention in recent years in the context of slow-roll inflation [5] where it has been realised that there is a degeneracy in the spectrum of adiabatic density perturbations. This could be removed by a detection of the gravitational wave background on the same scale which, in the slow-roll approximation, gives a direct record of the evolution of the scale factor, and hence of the inflaton potential.By contrast there has been relatively little study of the reconstruction of the evolution in the non-slow-roll case [6]. Recently it has been discovered that the spectra of perturbations produced in so-called pre-big bang models of the early universe [7], based on solutions of the low energy string effective action, are invariant under SL(2,R) symmetry transformations (including S-duality transformations) of the background fields [8]. This raises the interesting question of what is the most general type of cosmological evolution that yields a given perturbation spectrum.I will consider linear perturbations, δφ(η, x i ), about a homogeneous background, φ(η), in a homogeneous cosmology. For a minimally coupled massless field we can neglect any back-reaction upon the spacetime curvature to first order, and so perturbations obey the wave equationδwhere a dot denotes derivatives with respect to cosmic time t, and 3H corresponds to the expansion rate of the homogeneous hypersurfaces. One can decompose the perturbations into independent wavemodes δφ k (η)Q k (x i ), where Q k (x i ) is the eigenfunction of the spatial Laplacian ∇ 2 with eigenvalue −k 2 . The canonically normalised quantum field corresponds to the conformal field perturbation u = aδφ, where the sc...

New duality relating density perturbations in expanding and contracting Friedmann cosmologies

“…In section V we consider tensor perturbations. In section VI, we contrast our duality with other kinds of cosmological dualities that have been studied in the literature [16,17,18,19,20]. In section VII, we interpret our duality as a relation between the scale factor and the Hubble parameter, discuss its observational significance, and mention some open questions.…”

“…One duality, due to Wands [16] (see also [17]), pairs models that share the same "v" perturbations. By contrast, our duality pairs models that share the same "u" perturbations.…”

“…By contrast, our duality pairs models that share the same "u" perturbations. (Note: the variable called "u" in [16] is called "v" in our paper, in agreement with Mukhanov's convention [23,24].) Thus, whereas our duality connects expanding and contracting models through the substitution ǫ → ǫ = 1/ǫ (which leaves θ ′′ /θ, and hence u, invariant), Wands' duality instead uses the substitution ǫ → ǫ = (2ǫ − 3)/(ǫ − 2) (which leaves z ′′ /z, and hence v, invariant).…”

“…Other dualities have been identified in the literature [16,17,18,19,20] that relate cosmological solutions and perturbations. See Section VI for a comparison.…”

An extended matter bounce scenario: current status and challenges

“…These results are obtained upon the standard Lambda Cold Dark Matter (ΛCDM) model with a parameterized power spectrum and thus, are general to any inflation and alternative models. It is well-known that the ways to obtain a nearly scale invariant power spectrum of primordial perturbations are either a quasi de Sitter phase in the expanding phase or a nearly matter dominated phase in the contracting phase [14]. However, in an exact MBS with the background equation of state being w = 0, one finds n s = 1 for the single field model and therefore, conflicts with observations (which is similar to the fact that an exact de Sitter inflation was ruled out by the data).…”