A general equation, describing the lowest order corrections coming from quantum gravitational effects to the spectrum of cosmological scalar fluctuations is obtained. These corrections are explicitly estimated for the case of a de Sitter evolution. * Alexander.Kamenshchik@bo.infn.it † Alessandro.Tronconi@bo.infn.it ‡ Giovanni.Venturi@bo.infn.it

Inflation is studied in the context of induced gravity (IG) γσ 2 R, where R is the Ricci scalar, σ a scalar field and γ a dimensionless constant. We study in detail cosmological perturbations in IG and examine both a Landau-Ginzburg (LG) and a Coleman-Weinberg (CW) potential toy models for small field and large field (chaotic) inflation and find that small field inflationary models in IG are constrained to γ 3 × 10 −3 by WMAP 5 yrs data. Finally we describe the regime of coherent oscillations in induced gravity by an analytic approximation, showing how the homogeneous inflaton can decay in its short-scale fluctuations when it oscillates around a non-zero value σ0. PACS numbers:Beginning with the association of the gravitational coupling with a scalar field [1] many attempts have been made to relate the gravitational constant to dynamics. Indeed starting from an attempt to relate it to one loop effects in some fundamental interaction [2] induced gravity (IG) theories γσ 2 R have been developed [3,4,5]. In such theories σ acquires a non zero vacuum expectation value through the spontaneous breaking of scale invariance arising through the presence of a condensate [3] or quantum effects (radiative corrections) [6]. Further such theories can be generalized [7] leading to a viable dark energy model [8].In IG one may also use σ to achieve inflation. "Old inflation" in such a context is not satisfactory [9] and small field and large field (chaotic) inflation appear more promising. In particular for the last case with γ ≫ 1 the constraints on σ are such that it may even be compatible with spontaneous symmetry breaking in the usual particle physics context [10,11,12].In this Letter we analyze the slow-roll predictions for single-field inflation in IG and compare them with the recent WMAP 5-yrs data [14]. We also give an analytic approximate solution for the coherent oscillation regime during which reheating in IG takes place.Let us start by considering the IG actionwhere γ is a dimensionless and positive definite parameter and we assume a spatially flat Robertson-Walker background. The variation of the above action leads to the following set of independent equationswhere V eff,σ = dV /dσ−4V /σ and the dot is the derivative with respect to cosmic time.Inflation generically occurs during the slow-rolling of σ, for which Eqs. (2,3) reduce to:The slow-roll parameter ǫ 1 ≡ −Ḣ/H 2 obtains contributions not only from the square of the velocity of the scalar field as in EG but also from other terms:As in Einstein Gravity (EG), it is useful to introduce the hierarchy of Hubble flow functions [15]: d ln ǫ n /dN = ǫ n+1 with n ≥ 0, ǫ 0 = H i /H and N being the number of e-folds. In scalar-tensor gravity it is also necessary to introduce another hierarchy associated with σ: d ln δ n /dN = δ n+1 with n ≥ 0, δ 0 = σ/σ i . Thus in IG the cosmological perturbations depend on these two hierarchies which completely specify the background evolution. As usual, one can replace equations (2,3) with an equivalent set written in terms of the two ...

We study the cosmological evolution of an induced gravity model with a self-interacting scalar field σ and in the presence of matter and radiation. Such model leads to Einstein Gravity plus a cosmological constant as a stable attractor among homogeneous cosmologies and is therefore a viable dark-energy (DE) model for a wide range of scalar field initial conditions and values for its positive γ coupling to the Ricci curvature γσ 2 R.PACS numbers:

We derive the semiclassical evolution of massless minimally coupled scalar matter in the de Sitter space-time from the Born-Oppenheimer reduction of the Wheeler-DeWitt equation. We show that the dynamics of trans-Planckian modes can be cast in the form of an effective modified dispersion relation and that high energy corrections in the power spectrum of the cosmic microwave background radiation produced during inflation remain very small if the initial state is the Bunch-Davies vacuum.Comment: 6 pages, no figures, final version to appear in PR

We develop a technique for the reconstruction of the potential for a scalar field in cosmological models based on induced gravity. The potentials reproducing cosmological evolutions driven by barotropic perfect fluids, a cosmological constant, a Chaplygin gas and a modified Chaplygin gas are constructed explicitly.

We study a mechanism for the amplification of the inflationary scalar perturbation when the inflaton field action is non-canonical, i.e. the inflaton kinetic term has a non-standard form. For such a case the speed of sound of the perturbations generated during inflation is less than one and in general changes with time. Furthermore in such models, even when the scalar field potential is negligible, diverse inflationary attractors may exist. The possible effects of a speed of sound approaching zero during some stage of inflation may lead to a large amplification for the amplitude of the scalar spectrum which, on horizon re-entry during the radiation dominated phase, can collapse and form primordial black holes (PBH) of a mass M BH ∼ 10 −15 M which may constitute a large fraction of the total Dark Matter (DM) today. * Alexander.Kamenshchik@bo.infn.it † Alessandro.Tronconi@bo.infn.it ‡ Tereza.Vardanyan@bo.infn.it § Giovanni.Venturi@bo.infn.it

We solve a general equation describing the lowest order corrections arising from quantum gravitational effects to the spectrum of cosmological fluctuations. The spectra of scalar and tensor perturbations are calculated to first order in the slow roll approximation and the results are compared with the most recent observations. The slow roll approximation gives qualitatively new quantum gravitational effects with respect to the pure de Sitter case. * Alexander.Kamenshchik@bo.infn.it † Alessandro.Tronconi@bo.infn.it ‡ Giovanni.Venturi@bo.infn.it

The emergence of time in the matter-gravity system is addressed within the context of the inflationary paradigm. A quantum minisuperspace-homogeneous minimally coupled inflaton system is studied with suitable initial conditions leading to inflation and the system is approximately solved in the limit for large scale factor. Subsequently normal matter (either non homogeneous inflaton modes or lighter matter) is introduced as a perturbation and it is seen that its presence requires the coarse averaging of a gravitational wave function (which oscillates at trans-Planckian frequencies) having suitable initial conditions. Such a wave function, which is common for all types of normal matter, is associated with a "time density" in the sense that its modulus is related to the amount of time spent in a given interval (or the rate of flow of time). One is then finally led to an effective evolution equation (Schrödinger Schwinger-Tomonaga) for "normal" matter. An analogy with the emergence of a temperature in statistical mechanics is also pointed out.