Towards conformal cosmology

Abstract

Approximate de Sitter symmetry of inflating Universe is responsible for the approximate flatness of the power spectrum of scalar perturbations. However, this is not the only option. Another symmetry which can explain nearly scale-invariant power spectrum is conformal invariance. We give a short review of models based on conformal symmetry which lead to the scale-invariant spectrum of the scalar perturbations. We discuss also potentially observable features of these models.Observational data show that primordial scalar perturbations in the Universe must have been generated at some early cosmological stage, preceding the hot epoch. They are nearly Gaussian and have nearly flat power spectrum [1]. The first property suggests that these perturbations originate from amplified vacuum fluctuations of weakly coupled quantum field(s). Indeed, the defining property of Gaussian random field ζ(x) is that it obeys the Isserlis-Wick theorem, which holds also for any free quantum field in its vacuum state, while linear evolution in classical background does not induce non-Gaussianity.The second property is also very suggestive. The power spectrum P(k) defined asgives the fluctuation in logarithmic interval of momenta,where n s is a spectral index. The flat or scale invariant spectrum corresponds to n s = 1. In early 70's Harrison, Zeldovich and Peebles and Yu [2] conjectured that the spectrum is flat, to avoid large deviation from homogeneity and isotropy of the observed Universe on large scales and black hole formation on small scales. Current observational data [1] show that the power spectrum is indded nearly flat and give n s − 1 ≈ −0.032. The flatness of the power spectrum may be due to some symmetry. The best known candidate is the symmetry SO(4, 1) of the de Sitter metric ds 2 = 1)

Perturbations on and off de Sitter brane in anti–de Sitter bulk

“…Some time ago it has been pointed out that conformal symmetry SO(4, 2) broken down to de Sitter SO(4, 1) in the early Universe may be responsible for the generation of the (nearly) flat spectrum of scalar cosmological perturbations [1,2,3,4] (see Ref. [5] for a review). The main ingredient of the (pseudo)conformal scenarios is the expectation value of a scalar operator O of non-zero conformal weight △ which depends on time τ and gives rise to symmetry breaking,…”

Primordial scalar perturbations via conformal mechanisms: statistical anisotropy

“…Namely, the conformal scalar field with negative quartic potential may drive the evolution of the Universe, which undergoes the stage of the slow contraction in that case [6]. Though several distinctions, predictions of the dynamical model about the non-Gaussianity and the statistical anisotropy coincide with ones of the spectator model [11] If cosmologically interesting modes are still subhorizon by the end of the roll, relevant phase perturbations proceed to evolve at the intermediate stage, which takes place between the end of the conformal rolling and the beginning of the hot epoch [12]. Provided this evolution is long enough, one results with fairly non-trivial predictions for the properties of primordial scalar perturbations: negative scalar tilt, non-Gaussianity of the peculiar shape and the statistical anisotropy of all even Remarkably, conformal rolling scenario is just the particular case in the myriad of models representing the idea of the conformal Universe [6].…”

“…Furthermore, one makes use ot the Eqs. (11) and (10) to generate anisotropic maps. We compare values of the quantity C q 2 estimated from these maps with one obtained from the seven-year WMAP data.…”

Holographic CFTs on maximally symmetric spaces: Correlators, integral transforms, and applications

“…The vacuum expectation value can then be read off from coefficient of the normalizable term 16) which is of the form required by the unbroken subgroup of the conformal group, namely,…”

“…The boundary t = 0 now preserves a de Sitter subgroup O(1, d) ⊂ O(2, d) and the most general vacuum expectation values for scalar operators of dimension ∆ can evolve in time as 1/(−t) ∆ . Our proposal can be considered as the hard-wall version of [14][15][16].…”