Primordial Black Holes in an Accelerating Universe

General expressions are given for the generation of Primordial Black Holes (PBH) in a universe with a presently accelerated expansion due to a(n effective) cosmological constant. We give expressions both for a powerlaw scalefree primordial spectrum and for spectra which are not of that type. Specializing to the case of a pure cosmological constant $\Lambda$ and assuming flatness, we show that a comological constant with $\Omega_{\Lambda,0}=0.7$ will decrease the mass variance at the PBH formation time by about 15% compared with a critical density universe.


Introduction
The generation of a spectrum of primordial fluctuations in the very Early Universe is the crucial ingredient of all inflationary scenarios. These fluctuations can explain the generation of all (classical) inhomogeneities that can be seen in our universe, from the Cosmic Microwave Background (CMB) anisotropies to the Large Scale Structures (LSS) in the form of galaxies and clusters of galaxies. The inflationary paradigm therefore reconciles Big Bang cosmology with the appearance of an inhomogeneous universe [1]. In addition, each inflationary scenario makes accurate predictions allowing for the observations of ever increasing variety and quality to discriminate between the various model candidates. One such prediction is the possible formation of Primordial Black Holes (PBH). Indeed, it was realized already some time ago that a spectrum of primordial fluctuations would lead to the production of PBH [2]. For this generation mechanism to be efficient, one typically needs a "blue" spectrum [3]. In this way, one can hope that the density contrast averaged over the Hubble radius is sufficiently large that the resulting PBH production is not unsignificant and can be used as a powerful constraint on the spectrum of inflationary primordial fluctuations and the underlying high-energy physics model [4,5]. The production of PBH takes place on scales much smaller than those probed by the CMB anisotropy and LSS formation. In this sense, it is analogous if less spectacular, to the generation of a primordial gravitational wave background in inflationary models. Of course in the latter case, its discovery would be a remarkable prediction of inflation while the existence of PBH is a confirmation of the existence of the primordial fluctuations spectrum itself, irrespective of the way it was generated. In a recent paper [6], it was shown that the mass variance σ H (t k ) at the PBH formation time t k , was significantly overestimated due essentially to an incorrect relationship between σ H (t k ) and δ H (t k ) or equivalently k 3 2 Φ(k, t k ) which quantifies the power at Hubble radius scale (these quantities will be carefully introduced below). This must be corrected if one is to make accurate predictions. In [6], only a critical density universe was considered and expressions were derived for scalefree powerlaw spectra. Recent Supernovae observations strongly suggest that we live in a presently accelerating universe with Ω m,0 ≈ 0.3, Ω Λ,0 ≈ 0.7, the inclusion of a(n effective) cosmological constant seems further to make all observations converge into a consistent picture. Hence, it is important to derive general expressions in our formalism valid for a flat universe with an effective cosmological constant like in quintessence models. In the particular important case of a pure cosmological constant Λ, also considered in [7], the dependence on Λ can be quantified accurately and the relative decrease we find is in agreement with part of the analysis done in [7]. Finally we also generalize our results to primordial perturbations spectra which are not scalefree, an interesting possibility to consider in view of the wide range of scales probed by PBH formation. We first review the formalism describing PBH formation.

PBH formation
We assume for simplicity that a PBH is formed when the density contrast averaged over a volume of the (linear) size of the Hubble radius satisfies δ min ≤ δ ≤ δ max , and further that the PBH mass, M P BH , is of the order of the "horizon mass" M H , the mass contained inside the Hubble volume. Relyimg on semianalytic considerations it is common to take δ min = 1 3 , δ max = 1 but recent numerical simulations suggest rather δ min ≈ 0.7 [8] and show that M P BH can span a certain range, around M H though, at a given formation time. More accurately, when some scale defined by the wavenumber k reenters the Hubble radius after inflation at some time t k with k = (aH)| t k , it can lead to the production of PBH with M P BH ≈ M H (t k ). Obviously, there is a one-to-one correspondence between For Gaussian primordial fluctuations, the probability density p R (δ), where δ is the density contrast averaged over a sphere of radius R, is given by Here, the dispersion (mass variance) is computed using a top-hat window function, where P (k) is the power spectrum (we assume isotropy of the ensemble). From a point of view of principles, the averages are quantum averages; however, an effective quantum-to-classical transition is achieved during inflation [9]. For PBHs produced by inflationary perturbations, this quantum-to-classical transition is guaranteed for all masses of interest to us (see [10]). The expression W T H (kR) stands for the Fourier transform of the top-hat window function divided by the probed volume V W = 4 3 πR 3 , Hence the probability β(M H ) that a region of comoving size R = H −1 (t k ) a(t k ) has an averaged density contrast at horizon crossing t k in the range δ min ≤ δ ≤ δ max , is given by where σ 2 H (t k ) ≡ σ 2 (R)| t k , and the last approximation is valid for δ min ≫ σ H (t k ), and (δ max − δ min ) ≫ σ H (t k ). Important conclusions can be drawn from (4). Let us consider first the value of β(M H ) today. Today we have σ 2 H (t 0 ) ≃ 10 −8 , so clearly the probability of forming a black hole today is extraordinarily small. This probability can increase in the primordial universe if the power is increased when we go backwards in time, but the probability will remain very small, β(M H ) ≪ 1, at all times due to the magnitude of δ 2 3 Mass variance in the presence of Λ When the universe contains a cosmological constant Λ, this must be taken into account for a correct accurate calculation of the mass variance at early times. In this section, we will extend the formulas derived in [6]. As stressed there, one should distinguish the behaviour of the quantity σ 2 H (t k ), which is ultimately the quantity of interest, from the quantity where t k is the PBH formation time of interest, deep in the radiation dominated stage. However, when dealing with a flat universe with Ω m,0 < 1, we have today where f 0 stands for any quantity f evaluated today (at time t 0 ), Ω m,0 = ρ m,0 ρ cr,0 is the present energy density of dust-like matter relative to the critical density and Ω Λ,0 ≡ Λ We first relate the quantities appearing in (5,6) at the formation time t k and at the present time t 0 for arbitrary evolution of the universe after radiation domination and for a scalefree powerlaw spectrum.
General expressions with powerlaw spectrum: assuming a scalefree powerlaw primordial spectrum of the type k 3 Φ 2 (k) = A(t) k n−1 on super Hubble radius ("superhorizon") scales, we then have [11] and analogously The lower limit of integration in (7,8) can be safely taken to be zero. In (7,8), we have used a radiation dominated stage followed by some arbitrary evolution of the scale factor. In earlier work, we considered a radiation dominated stage followed by a matter dominated stage which constitutes a special case of (7,8).
An effective cosmological constant as in quintessence models would also be a particular case of (7,8). However, in contrast to a pure cosmological constant Λ, the time evolution of the scale factor a(t) is model-dependent and cannot be given in full generality. The quantity k 3 0 Φ 2 (k 0 , t 0 ), or equivalently δ 2 H (k 0 , t 0 ), at the present Hubble radius scale can be derived using the large angular scale CMB anisotropy data. It is that quantity that comes from observations which fixes the overall amplitude of the fluctuations spectrum. The COBE data show that it is of the following order of magnitude [12] where A 2 0 ({n i }) parametrizes the amplitude variations and is chosen such that The exact amplitude depends on the cosmological parameters {n i }, referring to the background as well as to the inflationary perturbations, and this model dependence is encoded in the quantity of order unity A 0 ({n i }). Eq.(10) assumes a powerlaw spectrum with spectral index n at least on large scales. For fixed n = 1, while the absolute values in (10) are modified, the ratio between them is unaltered [12]. Note that for a quintessence model, A 2 0 ({n i }) is model dependent. Finally, we must relate all our results to the quantity of interest for the computation of the PBH abundance, the mass variance σ 2 H (t k ) on the Hubble radius scale at horizon crossing time t k . As stressed in [6], one has (with k = (aH)| t k ) It is crucial to distinguish both quantities σ H (t k ) and δ H (k, t k ). As seen from (11), the quantity σ 2 H (t k ) which depends on the averaging procedure through eq.(2) is correctly related to the (non-averaged) quantity δ 2 H (k, t k ) in a non-trivial way. The quantity δ H (k, t k ) can be reconstructed at the time t k from its present value δ H (k 0 , t 0 ) using (8). But this is not the case for the quantity σ H (t k ) because the deformation of the power spectrum is different at the time t k and today. In other words, where the transfer function T (k, t) is defined through Here, t i is some initial time when all scales are outside the Hubble radius, k ≪ aH, we can take for example t i = t e , the end of inflation. For a powerlaw scalefree spectrum we have [6] where k e corresponds to the shortest fluctuations wavelengths with the size of the Hubble radius at the end of inflation. The transfer function T (k ′ , t) in the integrand of (14) must be taken at the time t k , not today. The accurate value of α(k) requires numerical calculations but estimates made in [6] show a significant overestimation of the mass variance σ H (t k ) when it is not computed correctly using the right quantity α(k). One then gets that (10/9) 2 α 2 (k) ≪ 25 (the value usually taken in the literature for a critical density universe) for all the mass range of PBH produced in the radiation era. If one is willing to use PBH formation as a precision tool in cosmology, it is important to check in how far the presence of a cosmological constant with Ω Λ,0 = 0.7 brings further modifications. An important conclusion can be immediately drawn by inspection of the integrand in (14) without accurate knowledge of the transfer function at time t k . Indeed, as we are interested in times t k ≪ t eq and in universes where Ω Λ domination occurs late, it is clear that at the time t k , neither the long-wave nor the short-wave fluctuation modes are affected in any way by the presence of (an effective) Λ. As can be seen from (13), this implies that the transfer function at time t k does not depend on Λ and the same must apply therefore to the quantity α(k). We conclude that the influence of a(n effective) cosmological constant on the probability β(M H ) comes solely from its influence on the quantity δ 2 H (k, t k ), or k 3 Φ 2 (k, t k ). It is this influence that we will quantify in the next subsection. We now consider a powerlaw spectrum and specialize to a universe with a cosmological constant Λ.
Powerlaw spectrum with Λ: in order to account for the presence of a cosmological constant Λ, we must replace the evolution of the scale factor a(t) after the radiation dominated stage. The scale factor for this stage of the universe evolution is very well approximated by [13] a(t) = a 1 sinh 2/3 (βt) , where 2 3 β = Λ/3 = H 0 Ω Λ,0 , a 1 =const. The evolution (15) smoothly interpolates between a pure (flat) dust-like matter dominated stage, with a(t) ∝ t 2 3 , for βt ≪ 1 which is of course the case right after t eq , and a Λ dominated universe in the asymptotic future. In particular for scales for which z eq > z(t k ) ≫ 1, k . It is this evolution (15) which must be used in (7,8). It is physically appealing to express the results in terms of the quantity M H (t k ). Then the following result is obtained The evolution (15) must now be substituted in (16). Actually it is slightly more presence of a cosmological constant as in (22). This is because the correct factor α(k) does not depend on Λ. We finally note that equation (18) makes use of the observed amplitude today on the present Hubble radius scale and, as far as perturbations are concerned, combines it with an assumed (powerlaw) behaviour towards small scales on a very broad range of scales. Other behaviours are certainly possible however. This is reminiscent of the primordial gravitational wave background generated during inflation extending up to frequencies as high as 10 10 Hz. There too, one can imagine a behaviour towards large frequencies departing from a simple scalefree law (see e.g. [14] for such a model with a jump in the tensorial spectral index n T ). Hence if the assumption of a scalefree spectrum does not hold, a more general expression will be needed. For this reason, we now generalize our results also to these cases.
Spectrum with a characteristic scale: A further important generalization concerns the primordial fluctuations spectrum itself. Indeed, the equations written in the previous subsection assume a scalefree spectrum. However, this needs not be the case especially in view of the large range of scales that are probed by PBH formation.
Let us therefore define in full generality where F (k) can be any complicated function of k. The expression (23) represents the primordial spectrum on "super-Hubble radius" (superhorizon) scales. For example, the spectrum of double inflation considered in [11,15] is of this general type. The corresponding generalization of (7), or (8), leads to a more complicated equation, viz.
where α 1 =constant. In case the function F (k) is actually of the form F ( k ks ), where k s defines the characteristic scale -an example of such a spectrum was found in [16] and considered in [17] -eq.(24) can be recast into a slightly simpler form with M s ≡ M H (t ks ) and t ks < t eq . Specializing to the particular case Ω Λ,0 = 0.7 just requires the substitution, like in (18), of the corresponding numbers into (24,25). Finally, we come to the calculation of σ H (t k ) itself. We now have the corresponding generalization to primordial spectra of arbitrary shape In particular, the general expression for α 2 (k) is given by Again, for the case of interest to us, t k ≪ t eq , the transfer function at the time t k and therefore also α(k) are independent of Λ. We conclude that the same decrease found in (22) will apply here too. Note that (27) extends the result (14) derived in [6] for a powerlaw scalefree spectrum which just corresponds to F (k) ∝ k n−1 . A characteristic scale in the primordial spectrum is an interesting possibility with respect to PBH formation in view of the large range of scales involved, much larger than CMB anisotropy or LSS formation. We have already considered some simple toy models in [6] and interesting results were obtained. The detailed numerical study of more sophisticated spectra and their possible relevance to observations is under progress [18].