Gravitational Wave Anisotropies from Primordial Black Holes
N. Bartolo, D. Bertacca, V. De Luca, G. Franciolini, S. Matarrese, M. Peloso, A. Ricciardone, A. Riotto, G. Tasinato
PPrepared for submission to JCAP
Gravitational Wave Anisotropies fromPrimordial Black Holes
N. Bartolo, a,b,c
D. Bertacca, a V. De Luca, d G. Franciolini, d S.Matarrese, a,b,c,e
M. Peloso, a,b
A. Ricciardone, b A. Riotto, d,f
G. Tasinato g a Dipartimento di Fisica e Astronomia Galileo Galilei Universita‘ di Padova, 35131 Padova,Italy b INFN, Sezione di Padova, 35131 Padova, Italy c INAF - Osservatorio Astronomico di Padova, I-35122 Padova, Italy d Department of Theoretical Physics and Center for Astroparticle Physics (CAP), CH-1211Geneva 4, Switzerland e Gran Sasso Science Institute, I-67100 L’Aquila, Italy f CERN,Theoretical Physics Department, Geneva, Switzerland g Department of Physics, Swansea University, Swansea, SA2 8PP, UK
Abstract.
An observable stochastic background of gravitational waves is generated wheneverprimordial black holes are created in the early universe thanks to a small-scale enhancementof the curvature perturbation. We calculate the anisotropies and non-Gaussianity of suchstochastic gravitational waves background which receive two contributions, the first at for-mation time and the second due to propagation effects. The former contribution can begenerated if the distribution of the curvature perturbation is characterized by a local andscale-invariant shape of non-Gaussianity. Under such an assumption, we conclude that a size-able magnitude of anisotropy and non-Gaussianity in the gravitational waves would suggestthat primordial black holes may not comply the totality of the dark matter. a r X i v : . [ a s t r o - ph . C O ] M a r ontents Following the first measurements of the gravitational waves (GWs) generated by ∼ O (10) M (cid:12) black-hole mergers [1], the past few years have witnessed a renewed interest in PrimordialBlack Holes (PBHs) [2–5]. Bounds of various origins exist on the PBHs abundance for awide range of PBHs masses [4], leaving also open the possibility that PBHs in certain massranges could be identified with a substantial fraction or, possibly, the totality of the darkmatter of the universe. This is particularly true for PBH of masses of ∼ O (cid:0) − (cid:1) M (cid:12) ,for which previously expected limits from femtolensing [6] and dynamical constraints fromWhite Dwarves [7] have been shown to be invalid.A simple mechanism for PBHs generation is from enhanced density perturbations δρ produced during inflation. If, using standard arguments from (nearly) scale invariance, weextrapolate the power of the perturbations P ζ ∼ ( δρ/ρ ) = O (cid:0) − (cid:1) measured at CMBscales to small scales, we obtain a completely negligible fraction of PBHs. On the otherhand, an increase of this power can strongly increase the portion of the universe which, athorizon re-entry, have an energy density above the threshold that leads to the collapse andPBH formation [8–11]. This increase would be associated to a breaking of scale invarianceat some given scale, which in turn reflects some specific dynamical mechanism taking placeduring inflation [12–15] (see [4] for a review and references therein).The increased density perturbations unavoidably lead to GWs production due to theintrinsic nonlinear nature of gravity [16–22] . This GW emission can be used to constrainthe PBH abundance [21]. In fact, let us assume that the power P ζ of the primordial scalar It is important to stress that we do not refer here to the GWs produced only in the regions that collapseto form the PBH, but from everywhere in the universe, due to the general increased of the power of the densityperturbations [23]. – 1 –erturbations has an enhancement at some give scale k ∗ , leading to a significant fraction f PBH of dark matter and also to an observable amount of GWs. Then, even a small decrease of P ζ from this level would lead to a completely negligible value for f PBH with a very minorchange of the amount of GWs. Therefore GW observations are sensitive even to peaks in P ζ that are associated to a very small (possibly, otherwise unobservable) amount of PBHs.The characteristic frequency of the GWs emitted by the production of PBH of mass M is f (cid:39) × − Hz (
M/M (cid:12) ) − / [21].Given the potential relevance of these observations in upcoming experiments like LISA[25] and DECIGO [26], it is important to characterize the stochastic background of gravita-tional waves (SGWB) produced with this mechanism [34–37] . Is it homogeneous in space?Does its spatial distribution obey a Gaussian statistics? To our knowledge, these questionshave not yet been addressed for the SGWB studied in this paper. This is the purpose of thiswork.Even assuming a completely homogeneous and isotropic SGWB at its production, theseGWs propagate in a perturbed universe. As a consequence, the GW signal arriving to Earthhas angular anisotropies [38–44] which are non-Gaussian [45]. In addition, as we show andquantify in this work, the GW production itself has some degree of anisotropy and non-Gaussianity. A necessary condition for large scale anisotropies and non-Gaussianity is thepresence of large-scale perturbations that are needed to produce correlations on cosmolog-ical scales, much greater than the scale k − ∗ associated to the typical regions forming thePBHs. The GW formation is a local event, that, by the equivalence principle, cannot belocally affected by modes of wavelength much greater than the PBH horizon. However, non-Gaussianity of the primordial density perturbations can lead to small-long scale correlations,so that long modes can lead to a large-scale modulation of the local power of the densityperturbations and, consequently, on the amount of GWs produced within each region.We show here that an amount of (local) non-Gaussianity of the scalar perturbations com-patible with the current upper bounds from Planck [46] can lead to an amount of anisotropiesand non-Gaussianity of the GWs distribution greater than that due to the propagation [45].On the other hand, if the PBHs constitute a significant fraction of the dark matter, additionallimits from isocurvature apply, leading to much stronger limits on the scalar non-Gaussianity[47]. This significantly limits the SGWB anisotropy and non-Gaussianity imprinted at theSWGB production. Therefore, our prediction is that a significant amount of PBH dark mat-ter is associated with a SGWB that is isotropic and Gaussian, up to propagation effects. Astronger amount of anisotropy and non-Gaussianity of the SGWB would signify the existenceof a local enhancement of the density perturbations, and of a PBH population that is wellbelow the dark matter abundance. These conclusions hold under the strict assumption oflocal, scale-invariant primordial non-Gaussianity in the curvature perturbations, extendingfrom CMB scales down to the small scales relevant for PBH formation. On the other hand,given the huge range of scales involved, different conditions for structure formation mighthold, especially on the smallest scales, that might break the assumption of scale-invariantnon-Gaussianity. This would leave open the possibility of relaxing our constraints, and toallow for PBHs to be the totality of the observed dark matter, with an accompanying SGWB This is due to the fact that, assuming Gaussian primordial perturbations, only the rare regions with δρ (cid:29) σ , being σ the square root of the variance, have an energy density above the threshold for PBHformation. A change of the variance have a strong impact on the area of the tail of the distribution above thePBH threshold. For an example of a case with non-Gaussian primordial perturbations, see, e.g., [24]. Other mechanisms to generate a SGWB from the early universe can be found in Refs. [22, 27–33]. – 2 –hat might still be anisotropic and non-Gaussian.The paper is organized as follows. In Section 2 we review the mechanism of GWsproduction at second order from scalar density perturbations. In Section 3 we compute theamount of anisotropy and non-Gaussianity of the SGWB produced by this mechanism, in thecase in which the primordial density perturbations are non-Gaussian. In Section 4 we reviewthe additional limits on the scalar non-Gaussianity that are present if the PBHs constitute asignificant portion of the dark matter. In Section 5 we present a summary of our results andof the existing constraints. Finally, in Section 6 we provide some final remarks. The paperis concluded by two appendices where we present some technical steps of our computations.
A simple mechanism for the production of a distribution of PBHs peaked at a given mass isto assume that some inflationary mechanism has produced a peak of the primordial densityperturbations at some given scale. This enhancement reflects some specific dynamical mech-anism that took place at some given moment during inflation, thus breaking the approximatescale invariance for modes that exited the horizon at that specific moment. This enhance-ment increases the amount of regions where, at horizon re-entry of this mode (in the radiationdominated era, well after inflation) the energy density is above the necessary threshold toproduce PBHs, thus increasing the PBH density. We introduce the power spectrum for theprimordial density perturbations as (cid:68) ζ ( (cid:126)k ) ζ ( (cid:126)k (cid:48) ) (cid:69) = 2 π k P ζ ( k ) (2 π ) δ (3) ( (cid:126)k + (cid:126)k (cid:48) ) , P ζ ( k ) = P ζ s ( k ) + P ζ L ( k ) , (2.1)where P ζ L ( k ) is the power spectrum of the standard (nearly) scale-invariant perturbationsgenerated during inflation. The suffix “ L ” in this term stands for long-wavelength modes,relevant at cosmological scales, which are much greater than the scale k − ∗ of the modesforming the PBH, which are labelled with the suffix “ s ” and are related to the small-scalepower spectrum P ζ s ( k ). At the short scale k − ∗ these long modes are completely subdominantwith respect to those contributing to the first term in (2.1). So they play no role in the localPBH formation and in the local production of GWs that we discuss next. However, as wesee in the next Section, in presence of primordial non-Gaussianity these modes can add along-scale modulation to this production, thus resulting in anisotropies of the SGWB.The necessity of local non-Gaussianity to create anisotropies of the SGWB is crucialdue to the generation of a cross-talk between the short scale k − ∗ , of the order of the horizonscale at PBH production, and the long wavelength scale q − , associated to ζ L . If absent,the long scalar modes of wavelength of cosmological size do not change the local physics ineach patch of size k − ∗ , and so the amount of PBHs and the induced GWs is locally the samein any patch. This is simply due to the Equivalence Principle which also dictates that theanisotropies in the SGWB should decay like ( q/k ∗ ) .To have a confirmation of such a general result we present, in the following, the calcu-lation of the contribution from the enhanced scalar modes in (2.1) to the production of GWsat second-order, without primordial non-Gaussianity. This leads to the GW energy density– 3 –perator [48, 49] ρ GW ( η, (cid:126)x ) = M p η a (cid:90) d k d k d p d p (2 π ) k k e i(cid:126)x · ( (cid:126)k + (cid:126)k ) T [ˆ k , ˆ k , (cid:126)p , (cid:126)p ] × ζ ( (cid:126)p ) ζ ( (cid:126)k − (cid:126)p ) ζ ( (cid:126)p ) ζ ( (cid:126)k − (cid:126)p ) (cid:68) (cid:89) i =1 (cid:104) I s ( (cid:126)k i , (cid:126)p i ) cos( k i η ) − I c ( (cid:126)k i , (cid:126)p i ) sin( k i η ) (cid:105) (cid:69) T . (2.2)This expression is valid during radiation domination, and the function T (cid:104) ˆ k , ˆ k , (cid:126)p , (cid:126)p (cid:105) is obtained from a contraction between the internal momenta and the GW polarizationoperators (we provide the expression in Appendix A, where we also outline our conventions.In that Appendix, we also provide the analytic expressions for I c,s [49, 50]). The angularbrackets in the second line denote a time average, that is necessary for the definition of theenergy density in GW [51–53], and it is performed on a timescale T much greater than theGW phase oscillations ( T k i (cid:29)
1) but much smaller than the cosmological time (
T H (cid:28) ζ in this expression. Thisis due to the fact that the GW energy density is a bilinear in the GW field (see Eq. (A.7),and the GW field sourced at second-order is a bilinear in ζ (see Eq. (A.3)).The one-point expectation value of the operator (2.2) is the expected GW energy densityfrom this mechanism. As already mentioned, we assume that the scalar perturbations ζ areGaussian (this assumption will be relaxed in the next section) so that the four point function (cid:10) ζ (cid:11) emerging from the expectation value of (2.2) can be written as sum of three terms, eachcontaining two products (cid:10) ζ (cid:11) . Schematically,Gaussian ζ ⇒ (cid:10) ζ (cid:11) = (cid:10) ζ (cid:11) (cid:10) ζ (cid:11) + (cid:10) ζ (cid:11) (cid:10) ζ (cid:11) + (cid:10) ζ (cid:11) (cid:10) ζ (cid:11) , (2.3)with all possible permutations of the four operators. One contraction gives a vanishing con-tribution at finite momentum, while the other two contractions give an identical contribution,and (using the definition of the power spectrum in (2.1)) lead to (cid:104) ρ GW ( η, (cid:126)x ) (cid:105) ≡ ρ c ( η ) (cid:90) d ln k Ω GW ( η, k )= 2 π M p η a (cid:90) d k d p (2 π ) k (cid:104) p − ( (cid:126)k · (cid:126)p ) /k (cid:105) p (cid:12)(cid:12)(cid:12) (cid:126)k − (cid:126)p (cid:12)(cid:12)(cid:12) P ζ ( p ) P ζ ( | (cid:126)k − (cid:126)p | ) (cid:104) I c ( (cid:126)k , (cid:126)p ) + I s ( (cid:126)k , (cid:126)p ) (cid:105) . (2.4)The contraction forced k = k in Eq. (2.2) using Eq. (2.1). In this case the time averageprocedure in (2.2) became straightforward, namely (cid:10) sin ( k η ) (cid:11) T = (cid:10) cos ( k η ) (cid:11) T = 1 / , (cid:104) sin ( k η ) cos ( k η ) (cid:105) T = 0 . (2.5)Following the standard convention, in the first line of (2.4) we defined the fractional energydensity in the GW for log interval. The quantity ρ c = 3 H M p denotes the critical energy One additional contribution to the GWs abundance, which is not considered in this paper, is related to thecontraction of peaks in the density fluid, generated by the same curvature perturbations which are responsiblefor the production of GWs at second-order, which are not high enough to collapse into PBHs (see Ref. [23]for details). – 4 –ensity of a spatially flat universe with Hubble rate H . The notation in the second lineof (2.4) exploits the fact that the integral over the two angles d Ω k can be made trivial byexploiting that the only angular dependence of the integrand is on the angle between (cid:126)k and (cid:126)p (this is a consequence of the statistical isotropy of the background). By introducing therescaled magnitudes x ≡ p /k and y ≡ | (cid:126)k − (cid:126)p | /k , the expression (2.4) reduces to [49]Ω GW ( k, η ) = 1972 a H η (cid:90) (cid:90) S dxdy x y (cid:34) − (cid:0) x − y (cid:1) x (cid:35) P ζ ( kx ) P ζ ( ky ) I ( x, y ) , (2.6)where the integration region S extends to x > | − x | ≤ y ≤ x and where wedefined I ≡ I c + I s . For a Dirac delta power spectrum of the scalar curvature perturbationon small scales, P ζ s ( k ) = A s k ∗ δ ( k − k ∗ ), this expression then becomes Ω GW ( k, η ) = 1 a H η A s k k ∗ (cid:20) k ∗ k − (cid:21) θ (2 k ∗ − k ) I (cid:18) k ∗ k , k ∗ k (cid:19) (2.7)where θ is the Heaviside step function, and I (cid:18) k ∗ k , k ∗ k (cid:19) ≡ I c (cid:18) k ∗ k , k ∗ k (cid:19) + I s (cid:18) k ∗ k , k ∗ k (cid:19) = 72916 (cid:18) kk ∗ (cid:19) (cid:18) − k ∗ k (cid:19) (cid:34) (cid:18) − k k ∗ (cid:19) − − log (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − k ∗ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:35) + π θ (cid:18) k ∗ √ k − (cid:19) . (2.8)We note that the result (2.7) for the one-point expectation value of the GW energy density isindependent of position. This follows from statistical homogeneity of the FLRW backgrounduniverse (at the technical level, it is due to the fact that the contraction of the four ζ operators in Eq. (2.2) forces (cid:126)k + (cid:126)k = 0). However, as explained in the introduction, onedoes not expect that the sourced GWs are perfectly homogeneous across the universe. Asa consequence, the SGWB reaching us from different directions will present some angularanisotropies.To quantify the level of these anisotropies one needs to compute the two-point correlationfunction (cid:104) ρ GW ( (cid:126)x ) ρ GW ( (cid:126)y ) (cid:105) . This correlator depends on space only through its dependence on | (cid:126)x − (cid:126)y | as a consequence of statistical isotropy and homogeneity.In computing (cid:10) ρ GW (cid:11) we need to evaluate the correlator (cid:10) ζ (cid:11) . The resulting contractionsare given in Eq. (B.2). The first line of that equation represents the case in which allthe ζ s emerging from the same ρ are contracted among each other. This gives rise to thedisconnected diagram shown in Figure 1, which is evaluated to (cid:104) ρ GW ( (cid:126)x ) ρ GW ( (cid:126)y ) (cid:105) (cid:12)(cid:12)(cid:12) disconnected = (cid:104) ρ GW (cid:105) , (2.9)and is homogeneous.The other lines of Eq. (B.2) are represented by the different topologies of connecteddiagrams shown in Figure 5. As we show in Appendix B these contributions are completelynegligible at the distances | (cid:126)x − (cid:126)y | of our interest. Our goal is to compute the large scale This expression is valid during radiation domination; we see that it is costant, and independent of thenormalization of the scale factor. – 5 –
1. ForGaussian scalar perturbations, the enhanced scalar modes (the first term in Eq. (2.1)) do notlead to statistical correlations on these cosmological scales. It is also easy to check that, asimposed by the Equivalence Principle, the anisotropies decay at large distances as ( q/k ∗ ) .Furthermore, when we measure the GW energy density at some given angular scalewe effectively coarse grain the GW energy density with a resolution related to that scale.This results in averaging an extremely large number of patches of size k − ∗ , and the resultingenergy density becomes extremely homogeneous due to the central limit theorem.We conclude that the effects that we have discussed so far lead to a homogeneous andisotropic distribution of ρ GW , up to the completely negligible contributions from the termsevaluated in Appendix B. There are however two additional effects of the long-scale modesthat can lead to sizeable anisotropy. The first effect is a propagation effect [39, 45]. Even ifproduced isotropically, GWs coming from different regions travel through disconnected andeffectively different realizations of the large scales density perturbations. This makes thearriving GWs anisotropic. The second effect, that we study in this work, is that the scalarperturbations are not perfectly Gaussian. Most shapes of non-Gaussianity, starting from themost common local-type, give rise to correlations between short and long scales. Due to this,long wavelength modes can modulate the power of scales k − ∗ , thus giving rise to long-scalecorrelations in the initial GWs distribution. We study this effect in the next section. The non-Gaussianity of the primordial scalar perturbations is parametrized by ζ ( (cid:126)k ) = ζ g ( (cid:126)k ) + 35 f NL (cid:90) d p (2 π ) ζ g ( (cid:126)p ) ζ g ( (cid:126)k − (cid:126)p ) , (3.1)namely it is assumed (as verified experimentally) that the perturbations are very close to begaussian, so that a mode can be expanded as a large Gaussian contribution ζ g plus the squareof a Gaussian term. The specific shape in (3.1) is known as local shape, as it correspondsto the local expansion ζ = ζ g + f NL ζ g in real space. This is the most studied shapeof non-Gaussianity, and it leads to significant correlation between large and small scales.Other shapes could also be considered, corresponding to a momentum-dependent non-linearparameter in the convolution (3.1). For simplicity, in this work we consider only the local– 6 – h
35 ˜ f NL ( k ) (cid:90) d q (2 π ) ζ L ( (cid:126)q ) Y ∗ (cid:96)m (ˆ q ) j (cid:96) ( q ( η − η in )) . (3.8)To keep into account all the possible sources of anisotropy in the GW background, one shalladd to this term the contribution from the propagation across the universe,Γ S ( η , (cid:126)q ) = T S ( q, η , η in ) ζ L ( (cid:126)q ) , (3.9)where T S ( q, η , η in ) = (cid:90) η η in dη (cid:48) e − i ˆ k · ˆ qq ( η − η (cid:48) ) (cid:20) T Φ (cid:0) η (cid:48) , q (cid:1) δ (cid:0) η (cid:48) − η in (cid:1) + ∂ [ T Ψ ( η (cid:48) , q ) + T Φ ( η (cid:48) , q )] ∂η (cid:48) (cid:21) , (3.10)and where Φ( η, (cid:126)k ) ≡ T Φ ( η, k ) ζ ( (cid:126)k ) , Ψ( η, (cid:126)k ) ≡ T Ψ ( η, k ) ζ ( (cid:126)k ) . (3.11)The large scale modes of our interest entered the horizon during matter domination, andso the transfer functions become T Φ ( η in , q ) = T Ψ ( η in , q ) = 3 /
5. Eq. (3.9) represents thecontribution of the scalar sources ( S ) when the signal travels across the universe towards us,and we see that it is composed by two pieces equivalent to the Sachs-Wolfe and integratedSachs-Wolfe effect, respectively. We are using the spherical harmonics normalised as (cid:82) d ˆ n Y (cid:96)m Y ∗ (cid:96) (cid:48) m (cid:48) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) . – 8 –herefore adding these two contributions one gets the full source of anisotropy asΓ (cid:96)m,I + S ( k ) = 4 π ( − i ) (cid:96) (cid:90) d q (2 π ) ζ L ( (cid:126)q ) Y ∗ (cid:96)m (ˆ q ) T I + S(cid:96) ( k, q, η , η in ) (3.12)where we defined the quantity T I + S(cid:96) ( k, q, η , η in ) ≡ (cid:104) f NL ( k ) (cid:105) j (cid:96) ( q ( η − η in ))+ (cid:90) η η in dη ∂ [ T Ψ ( η, q ) + T Φ ( η, q )] ∂η j (cid:96) ( q ( η − η )) . (3.13)Before going into the details of the computation of the correlators, one can have a deeperlook of the ISW contribution to estimate its value with respect to the other. Introducing thevariable η (cid:48) = η/η and parametrising the scalar transfer functions as T Φ ( η, q ) = T Ψ ( η, q ) = 35 g ( η ) , (3.14)one gets T S(cid:96) ( k, q, η , η in ) = 35 (cid:20)(cid:104) f NL ( k ) (cid:105) j (cid:96) ( qη ) + 2 (cid:90) dη (cid:48) ∂g ( η (cid:48) ) ∂η (cid:48) j (cid:96) (cid:0) qη (1 − η (cid:48) ) (cid:1)(cid:21) , (3.15)where we neglected the term qη in in the Bessel function of the first term. Starting from theexpression of g ( η ), see for example Ref. [55, 56], one can use the analytical fit given by [57] ∂g ( η (cid:48) ) ∂η (cid:48) = − . η (cid:48) (3.16)to perform the integral numerically, finding that the ISW effect is subdominant. Thereforeone can approximate the total contribution of the long mode, at leading order in the non-linear parameter, through the quantityΓ (cid:96)m,I + S ( k ) (cid:39) π ( − i ) (cid:96) (cid:90) d q (2 π ) ζ L ( (cid:126)q ) Y ∗ (cid:96)m (ˆ q ) 35 (cid:104) f NL ( k ) (cid:105) j (cid:96) ( q ( η − η in )) . (3.17)In the following subsections we will compute the two-point and three-point functions of therescaled energy density as a function of the long modes power spectra and the local non-linearparameter. We start with the computation of the two-point function (cid:10) Γ (cid:96) m ,I + S ( k ) Γ ∗ (cid:96) m ,I + S ( k ) (cid:11) = (4 π ) ( − i ) (cid:96) − (cid:96) (cid:90) d q (2 π ) d q (2 π ) Y ∗ (cid:96) m (ˆ q ) Y (cid:96) m (ˆ q ) × (cid:18) (cid:19) (cid:104) f NL ( k ) (cid:105) j (cid:96) ( q ( η − η in )) j (cid:96) ( q ( η − η in )) (cid:104) ζ L ( (cid:126)q ) ζ ∗ L ( (cid:126)q ) (cid:105) . (3.18)Using the orthonormality of the spherical harmonics and for the choice of a scale invariantpower spectra of the long modes P ζ L ( q ) = P ζ L , the previous expression becomes (cid:10) Γ (cid:96) m ,I + S ( k ) Γ ∗ (cid:96) m ,I + S ( k ) (cid:11) = δ (cid:96) (cid:96) δ m m π (cid:18) (cid:19) (cid:104) f NL ( k ) (cid:105) (cid:96) ( (cid:96) + 1) P ζ L . (3.19)– 9 –ollowing the notation of [45], one can define the two-point function as (cid:10) Γ (cid:96) m ,I + S ( k ) Γ ∗ (cid:96) m ,I + S ( k ) (cid:11) = δ (cid:96) (cid:96) δ m m C (cid:96),I + S ( k ) (3.20)such that one finally gets (cid:114) (cid:96) ( (cid:96) + 1)2 π C (cid:96),I + S ( k ) (cid:39) (cid:12)(cid:12)(cid:12) f NL ( k ) (cid:12)(cid:12)(cid:12) P / ζ L (cid:39) . · − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f NL ( k )10 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) P ζ L . · − (cid:19) / (3.21)which has been evaluated for value of the non-linear parameter close to its upper bound (3.2)and using the CMB value for the power spectrum of the long modes. For the computation of the three-point function we need to go to the next-to-leading orderin the non-linear parameter f NL , such that the expression of the initial condition term Γ I inthe (cid:96), m space becomesΓ (cid:96)m,I ( k ) (cid:39) π ( − i ) (cid:96) (cid:90) d q (2 π ) Y ∗ (cid:96)m (ˆ q ) j (cid:96) ( q ( η − η in )) 35 ˜ f NL ( k ) (cid:20) ζ L ( (cid:126)q ) + 95 f NL (cid:90) d p (2 π ) ζ L ( (cid:126)p ) ζ L ( (cid:126)q − (cid:126)p ) (cid:21) . (3.22)At this order in the long perturbations ζ L , also the propagation term gets a contributionproportional to the non-linear parameter as 3 / f NL ζ L , so that the total term becomesΓ (cid:96)m,I + S ( k ) (cid:39) π ( − i ) (cid:96) (cid:90) d q (2 π ) Y ∗ (cid:96)m (ˆ q ) j (cid:96) ( q ( η − η in )) (cid:40) (cid:104) f NL ( k ) (cid:105) ζ L ( (cid:126)q )+ 925 f NL (cid:104) f NL ( k ) (cid:105) (cid:90) d p (2 π ) ζ L ( (cid:126)p ) ζ L ( (cid:126)q − (cid:126)p ) (cid:41) (3.23)where we stress once again that all the long modes ζ L in this expression are Gaussian fields.We can now start the evaluation of the three-point function (cid:42) (cid:89) i =1 Γ (cid:96) i m i ,I + S ( k ) (cid:43) = (4 π ) ( − i ) (cid:96) + (cid:96) + (cid:96) f NL (cid:104) f NL ( k ) (cid:105) (cid:104) f NL ( k ) (cid:105) (cid:90) d q (2 π ) (cid:90) d q (2 π ) × (cid:90) d q (2 π ) (cid:90) d p (2 π ) (cid:34) (cid:89) i =1 Y ∗ (cid:96) i m i (ˆ q i ) j (cid:96) i ( q i ( η − η in )) (cid:35) (cid:104) ζ L ( (cid:126)q ) ζ L ( (cid:126)q ) ζ L ( (cid:126)p ) ζ L ( (cid:126)q − (cid:126)p ) (cid:105) + 2 perm . (3.24)After having performed the contractions of the long modes with the Wick theorem, onecan introduce the bispectrum of the modes in momentum space B Γ , such that the previousexpression becomes (cid:42) (cid:89) i =1 Γ (cid:96) i m i ,I + S ( k ) (cid:43) = (4 π ) ( − i ) (cid:96) + (cid:96) + (cid:96) (cid:90) d q (2 π ) (cid:90) d q (2 π ) (cid:90) d q (2 π ) B Γ ( k, q , q , q ) × (cid:34) (cid:89) i =1 Y ∗ (cid:96) i m i (ˆ q i ) j (cid:96) i ( q i ( η − η in )) (cid:35) (2 π ) δ (3) ( (cid:126)q + (cid:126)q + (cid:126)q ) (3.25)– 10 –ith B Γ ( k, q , q , q ) = 162625 f NL (cid:104) f NL ( k ) (cid:105) (cid:104) f NL ( k ) (cid:105) (cid:20) π q P ζ L ( q ) 2 π q P ζ L ( q ) + 2 perm . (cid:21) . (3.26)Using the representation of the Dirac δ -function in terms of the spherical harmonics, andusing their orthonormality, one gets after some algebra (cid:42) (cid:89) i =1 Γ (cid:96) i m i ,I + S ( k ) (cid:43) = G m m m (cid:96) (cid:96) (cid:96) (cid:90) ∞ dr r (cid:89) i =1 (cid:20) π (cid:90) dq i q i j (cid:96) i ( q i ( η − η in )) j (cid:96) i ( q i r ) (cid:21) B Γ ( k, q , q , q )(3.27)where one could recognize the Gaunt integral G m m m (cid:96) (cid:96) (cid:96) = (cid:90) d Ω y Y ∗ (cid:96) m (Ω y ) Y ∗ (cid:96) m (Ω y ) Y ∗ (cid:96) m (Ω y ) . (3.28)In the limit of one sufficiently large (cid:96) , it is then possible to evaluate one of the q integral, foreach of the three permutations in the bispectrum, by using the approximation2 π (cid:90) dq q j (cid:96) ( q η ) j (cid:96) ( q r ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) (cid:29) = δ ( η − r ) η (3.29)and then use the resulting Dirac delta to integrate over r . The result of this computation istherefore (cid:42) (cid:89) i =1 Γ (cid:96) i m i ,I + S ( k ) (cid:43) = G m m m (cid:96) (cid:96) (cid:96) f NL (cid:104) f NL ( k ) (cid:105) (cid:104) f NL ( k ) (cid:105) × (cid:18) π (cid:90) dq q j (cid:96) ( q η ) P ζ L ( q ) (cid:19) (cid:18) π (cid:90) dq q j (cid:96) ( q η ) P ζ L ( q ) (cid:19) + 2 perm . (3.30)Finally, one can factorize the tensorial structures following from statistical isotropy to definethe three-point function as [58] (cid:104) Γ (cid:96) m ,I + S ( k )Γ (cid:96) m ,I + S ( k )Γ (cid:96) m ,I + S ( k ) (cid:105) = G m m m (cid:96) (cid:96) (cid:96) b (cid:96) (cid:96) (cid:96) ,I + S ( k ) , (3.31)where, in terms of the two-point functions found in Eq. (3.21), the expression becomes b (cid:96) (cid:96) (cid:96) ,I + S ( k ) (cid:39) f NL (cid:104) f NL ( k ) (cid:105)(cid:104) f NL ( k ) (cid:105) [ C (cid:96) ,I + S C (cid:96) ,I + S + C (cid:96) ,I + S C (cid:96) ,I + S + C (cid:96) ,I + S C (cid:96) ,I + S ] . (3.32)We dedicate the next sections to the discussion of these results. The presence of such a non-Gaussianity in the comoving curvature perturbation has a smalleffect on the value of the threshold which is necessary to the overdensity to collapse into PBHs,– 11 –ee Ref. [59] for details, while it induces a significant large-scale variation of the primordialblack holes abundance through the modulation of the power on small scales induced bythe long modes. If all or a part of the dark matter is composed by PBHs, then this non-Gaussianity is responsible for the production of isocurvature modes in the DM density fluid,which are strongly constrained by the CMB observations.The present bounds provided by the Planck experiments on the relative abundance ofthe isocurvature modes are, at 95% CL, [60]100 β iso < .
095 for fully correlated , β iso < .
107 for fully anti-correlated , (4.1)where by fully correlated (fully anti-correlated) we mean a positive (negative) f NL .Following the results obtained in [47], one can express the PBH mass fraction in thepresence of non-Gaussianity. It reads (see also [61])¯ β ≡ ρ PBH ( η in ) ρ c ( η in ) = (cid:115) πσ s (cid:20)(cid:90) ∞ ζ + dζ exp (cid:18) − ζ σ s (cid:19) + (cid:90) ζ − −∞ dζ exp (cid:18) − ζ σ s (cid:19)(cid:21) for f NL > , (cid:115) πσ s (cid:20)(cid:90) ∞ ζ + dζ exp (cid:18) − ζ σ s (cid:19) − (cid:90) ∞ ζ − dζ exp (cid:18) − ζ σ s (cid:19)(cid:21) for f NL < , (4.2)where [47] ζ ± = − ± (cid:112)
25 + 60 ζ c f NL + 36 f NL σ s f NL (4.3)and ζ c is the threshold for collapse of PBH in the presence of non-Gaussianity recentlycalculated in Ref. [62], and σ s is the variance of the short modes.The corresponding mass fraction perturbation with respect to the average value ¯ β atleading order in the long modes is δ β ≡ β − ¯ β ¯ β = (cid:32)
25 + 30 ζ c f NL + 36 f NL σ s − (cid:112)
25 + 60 ζ c f NL + 36 f NL σ s f NL σ s (cid:112)
25 + 60 ζ c f NL + 36 f NL σ s (cid:33) ζ L ≡ b ζ L . (4.4)One can express the relative abundance of the isocurvature modes in terms of the bias b induced by the long mode as β iso ≡ P iso P iso + P ζ L = b f PBH b f PBH + 1 , (4.5)where we used the fact that local non-Gaussianity induces the bias P iso = b f PBH P ζ L , where f PBH is the fraction of dark matter in PBH. Once written in terms of b and f PBH , the bounds(4.1) become − . < bf PBH < . . (4.6)One can finally relate the bias to the parameter of non-Gaussianity as done in [47], givingthe colored allowed region in Fig. 4. In making the plot, we are assuming that the value oflocal f NL has no scale dependence, as explained in the Introduction. From the plot it is clearthat a large value of the non-linear parameter implies that only a small fraction of DM canbe composed by PBHs. We remind to the reader the fact that the non-linear parameter hasa lower bound due to the inadequacy of the perturbative approach in the computation of thePBH abundance [63, 64], because of which we decided to cut the allowed region in the plotat f NL ≥ − /
3. – 12 –
Results
To have a more physical intuition of the amount of anisotropy in the GWs abundance, weexpress the above results in terms of the GW density contrast δ GW rather than of Γ. We thusdefine the two and three point functions as (cid:10) δ GW ,(cid:96)m δ ∗ GW ,(cid:96) (cid:48) m (cid:48) (cid:11) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) ˆ C (cid:96) ( k ) , (cid:104) δ GW ,(cid:96) m δ GW ,(cid:96) m δ GW ,(cid:96) m (cid:105) = G m m m (cid:96) (cid:96) (cid:96) ˆ b (cid:96) (cid:96) (cid:96) ( k ) , (5.1)where we have again factorised the tensorial structures dictated by statistical isotropy, suchthat the above results then become (cid:114) (cid:96) ( (cid:96) + 1)2 π ˆ C (cid:96) ( k ) (cid:39) (cid:12)(cid:12)(cid:12) f NL ( k ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) − ∂ ln ¯Ω GW ( η, k ) ∂ ln k (cid:12)(cid:12)(cid:12)(cid:12) P / ζ L , ˆ b (cid:96) (cid:96) (cid:96) ( k ) (cid:39) ˜ f NL (cid:104) f NL ( k ) (cid:105) (cid:104) f NL ( k ) (cid:105) (cid:16) ˆ C (cid:96) ˆ C (cid:96) + ˆ C (cid:96) ˆ C (cid:96) + ˆ C (cid:96) ˆ C (cid:96) (cid:17) . (5.2)We can now discuss the limits in which the anisotropies are dominated by the propagationterm or by the initial condition term.In the case in which the propagation term dominates, one can formally consider thelimit ˜ f NL →
0, and thus findDominated by propagation : (cid:113) (cid:96) ( (cid:96) +1)2 π ˆ C (cid:96) ( k ) (cid:39) (cid:12)(cid:12)(cid:12) − ∂ ln ¯Ω GW ( η, k ) ∂ ln k (cid:12)(cid:12)(cid:12) P / ζ L , ˆ b (cid:96) (cid:96) (cid:96) ( k ) (cid:39) ˜ f NL (cid:104) ˆ C (cid:96) ˆ C (cid:96) + ˆ C (cid:96) ˆ C (cid:96) + ˆ C (cid:96) ˆ C (cid:96) (cid:105) , (5.3)which agrees with the results of the previous paper [45].In the case in which the initial condition term dominates, one can instead consider thelimit ˜ f NL → ∞ . The correlators for δ GW then becomeDominated by initial condition : (cid:113) (cid:96) ( (cid:96) +1)2 π ˆ C (cid:96) ( k ) (cid:39) | f NL | P / ζ L , ˆ b (cid:96) (cid:96) (cid:96) ( k ) (cid:39) (cid:16) ˆ C (cid:96) ˆ C (cid:96) + ˆ C (cid:96) ˆ C (cid:96) + ˆ C (cid:96) ˆ C (cid:96) (cid:17) , (5.4)where we note that f NL has disappeared from the last expression, since Γ I is maximallynon-Gaussian (as opposite to Γ S , that is Gaussian up to O ( f NL ) non-Gaussianity).In Fig. 4 we show the two-point function anisotropy ˆ C (cid:96) for the density contrast, forthe choice of a Dirac delta and gaussian power spectrum of the curvature perturbation onsmall scales. The peak frequency of this signal has been chosen as the one correspondingto PBH masses given by M PBH = 10 − M (cid:12) for which PBHs can represent all the DM, alsocoinciding with the frequency of maximum sensitivity at LISA. The dot-dashed lines identifythe corresponding GWs abundance computed at present time and at the peak frequency.Finally, the results for different masses of PBH do not change significantly.– 13 – .2. × - × - × - × - × - × - × - × - × - × - Figure 4 . Contour plot of (cid:113) (cid:96) ( (cid:96) + 1) ˆ C (cid:96) ( k ∗ ) / π in the region permitted by the constraints of Planckon f PBH and f NL for the choice of a Dirac delta and gaussian power spectrum of the short modes,respectively. The peak frequency has been chosen to correspond to M PBH = 10 − M (cid:12) . The dot-dashed lines identify the corresponding GWs abundance. Notice that the results shown here onlyhold for a local, scale-invariant primordial non-Gaussianity of scalar perturbations. The measurement of a SGWB is one of the main goals of future experiments devoted to thedetection of sources of GWs. One possible and well-motivated source of GWs from the earlyuniverse is associated to the birth of PBHs when enhanced scalar perturbations created duringinflation re-enter the horizon and collapse into BHs. This phenomenon is accompanied by thegeneration of GWs at second-order in perturbation theory. In particular, it turns out that forPBHs of masses around 10 − M (cid:12) , which can still play the role of dark matter in its totality,the frequency of the GWs is located in the mHz range where the LISA mission happens tohave the maximum sensitivity. In the positive case of a detection of the SGWB, the nextstep will be to identify the source and therefore any characterisation of the background willbe extremely useful. In this sense, its anisotropies will bring important information.In this paper we have studied in detail the strength of the GW anisotropies associatedto the production of the PBHs. There are two contributions to the anisotropy, the first oneis created at the generation epoch and the second one is due to the propagation effects fromthe time of production down to the detection time. In order to have the first source on largescales a non-vanishing squeezed type of non-Gaussianity must be present in the curvatureperturbation in order to create a cross-talk between the PBH short wavelengths and thelarge scales at which the anisotropies are tested. At the same time, the amount of primordialnon-Gaussianity is constrained by the requirement of not generating a too large amount ofisocurvature perturbations, in the case in which PBHs compose a sizeable fraction of thedark matter.We have considered the simplest possibility, namely a primordial scale-invariant localnon-Gaussianity for the curvature perturbations. Under such an assumption, our resultsare summarised in Fig. 4 out of which we conclude that the typical anisotropies are of theorder of ζ L ∼ − . Correspondingly, the reduced bispectrum is of the order of ζ L ∼ − .Our findings show also that, if the PBHs compose a large fraction of the dark matter, theSGWB must be highly isotropic and Gaussian, up to propagation effects. A large amount ofanisotropy and non-Gaussianity would imply, within our mechanism, a PBH population wellbelow the measured dark matter abundance.– 14 –uch conclusions hold only in the case of our working hypothesis, namely a local modelof primordial non-Gaussianity with f NL = const for the curvature perturbations. In this case,one is directly using the Planck constraints on f NL (and the isocurvature limits discussed inSec. 4) down to the scales typical of PBH formation. However, if that is not the case, thenour constraints shown in Fig. 4 can be relaxed, with PBHs that might constitute all of themeasured dark matter. For example, one possibility might be to extend our computation byconsidering a running (local) non-Gaussianity [65], which is presently constrained by CMBtemperature measurements [66] and might possibly avoid the isocurvature bounds. We leaveit for further studies.The next step is of course understanding if such small anisotropies can be detected bythe current and future experiments and, if so, at which angular resolution [67]. In particular,for a SGWB of cosmological origin only anisotropies at low multipoles, (cid:96) (cid:46)
10, can beresolved. To resolve the angular features of the SGWB at larger multipoles, a gravitationalwave telescope characterised by a ∼ AU effective baseline seems to represent the best option[67].
Acknowledgements
We thank C. Byrnes for useful discussions. N.B., D.B. and S.M. acknowledge partial financialsupport by ASI Grant No. 2016-24-H.0. V.DL., G.F. and A.R. are supported by the SwissNational Science Foundation (SNSF), project
The Non-Gaussian Universe and CosmologicalSymmetries , project number: 200020-178787. The work of G.T. is partially supported bySTFC grant ST/P00055X/1.
A Conventions and computational details on the SGWB energy density
In this Appendix we list our conventions and some explicit expressions that are relevant forthe GW energy density in eq. (2.2). We introduce the GW field through the line element ds = a ( η ) (cid:2) − dη + ( δ ij + h ij ) dx i dx j (cid:3) , (A.1)and we decompose it as h ij ( η, (cid:126)x ) = (cid:90) d k (2 π ) (cid:88) λ = R,L h λ ( η, (cid:126)k ) e ij,λ (ˆ k ) e i(cid:126)k · (cid:126)x , (A.2)where the circular polarization operators are transverse and traceless, and satisfy the nor-malization condition e ij,λ ( (cid:126)k ) e ∗ ij,λ (cid:48) ( (cid:126)k ) = δ λλ (cid:48) . The second-order production from the scalarperturbations then gives, in the radiation dominated era [49], h λ ( η, (cid:126)k ) = 12 49 k η (cid:90) d p (2 π ) e ∗ λ ( (cid:126)k, (cid:126)p ) ζ ( (cid:126)p ) ζ ( (cid:126)k − (cid:126)p ) (cid:104) I c ( (cid:126)k, (cid:126)p ) cos ( kη ) + I s ( (cid:126)k, (cid:126)p ) sin ( kη ) (cid:105) , (A.3) We note an additional factor 1 / – 15 –here e λ ( (cid:126)k, (cid:126)p ) ≡ e ij,λ (ˆ k ) (cid:126)p i (cid:126)p j , and where the two functions I c,s have been computed analyti-cally in [49, 50] I c ( x, y ) = − π ( s + d − ( s − d ) θ ( s − , (A.4) I s ( x, y ) = −
36 ( s + d − s − d ) (cid:20) ( s + d − s − d ) log (1 − d ) | s − | + 2 (cid:21) , (A.5)with d ≡ √ | x − y | , s ≡ √ x + y ) , ( d, s ) ∈ [0 , / √ × [1 / √ , + ∞ ) . (A.6)We insert these expressions into the GW energy density [51] ρ GW = M p (cid:68) ˙ h ab ( t, (cid:126)x ) ˙ h ab ( t, (cid:126)x ) (cid:69) T , (A.7)where the dots denote differentiation with respect to physical time, and we obtain the expres-sion (2.2) in the main text. The GW polarization operators enter in this expression throughthe combination T (cid:104) ˆ k , ˆ k , (cid:126)p , (cid:126)p (cid:105) ≡ (cid:88) λ ,λ e ij,λ (ˆ k ) e ∗ ab,λ (ˆ k ) e ij,λ (ˆ k ) e ∗ cd,λ (ˆ k ) (cid:126)p a (cid:126)p b (cid:126)p c (cid:126)p d . (A.8)Using the identity2 (cid:88) λ e ij,λ (ˆ k ) e ∗ ab,λ (ˆ k ) = (cid:16) δ ia − ˆ k i ˆ k a (cid:17) (cid:16) δ jb − ˆ k j ˆ k b (cid:17) + (cid:16) δ ib − ˆ k i ˆ k b (cid:17) (cid:16) δ ja − ˆ k j ˆ k a (cid:17) − (cid:16) δ ij − ˆ k i ˆ k j (cid:17) (cid:16) δ ab − ˆ k a ˆ k b (cid:17) , (A.9)we obtain, after some algebra, T (cid:104) ˆ k , ˆ k , (cid:126)p , (cid:126)p (cid:105) = (cid:104) (cid:126)p · (cid:126)p − ˆ k · (cid:126)p ˆ k · (cid:126)p − ˆ k · (cid:126)p ˆ k · (cid:126)p + ˆ k · ˆ k ˆ k · (cid:126)p ˆ k · (cid:126)p (cid:105) − (cid:20) p − (cid:16) ˆ k · (cid:126)p (cid:17) (cid:21) (cid:20) p − (cid:16) ˆ k · (cid:126)p (cid:17) − (cid:16) ˆ k · (cid:126)p (cid:17) + 2 ˆ k · ˆ k ˆ k · (cid:126)p ˆ k · (cid:126)p − (cid:16) ˆ k · ˆ k (cid:17) (cid:16) ˆ k · (cid:126)p (cid:17) (cid:21) − (cid:20) p − (cid:16) ˆ k · (cid:126)p (cid:17) (cid:21) (cid:20) p − (cid:16) ˆ k · (cid:126)p (cid:17) − (cid:16) ˆ k · (cid:126)p (cid:17) + 2 ˆ k · ˆ k ˆ k · (cid:126)p ˆ k · (cid:126)p − (cid:16) ˆ k · ˆ k (cid:17) (cid:16) ˆ k · (cid:126)p (cid:17) (cid:21) + 14 (cid:20) (cid:16) ˆ k · ˆ k (cid:17) (cid:21) (cid:20) p − (cid:16) ˆ k · (cid:126)p (cid:17) (cid:21) (cid:20) p − (cid:16) ˆ k · (cid:126)p (cid:17) (cid:21) . (A.10) B Connected contributions to GW energy density two-point function
In this appendix we give a sketch of the contribution of the connected diagrams of theenergy density two-point function, giving rise to an anisotropy at extremely small scales. Wecompute the two-point function starting from the definition of the energy density operator– 16 – i h i + h i h i + h i h i ⇢ GW
1) 1 | ˆ p − ˆ s | | q ˆ e z + ˆ s − ˆ p | T (cid:20) ˆ p − ˆ s | ˆ p − ˆ s | , q ˆ e z + ˆ s − ˆ p | q ˆ e z + ˆ s − ˆ p | , ˆ p , − ˆ p (cid:21) × (cid:34) I s (cid:18) | ˆ p − ˆ s | , | ˆ p − ˆ s | (cid:19) I s (cid:18) | q ˆ e z + ˆ s − ˆ p | , | q ˆ e z + ˆ s − ˆ p | (cid:19) + I c (cid:18) | ˆ p − ˆ s | , | ˆ p − ˆ s | (cid:19) I c (cid:18) | q ˆ e z + ˆ s − ˆ p | , | q ˆ e z + ˆ s − ˆ p | (cid:19) (cid:35) (cid:20) sin (∆ T )∆ T (cid:21) (B.5)with ∆ T = ∆ T = ( k − k ) T = {| ˆ p − ˆ s | − | q ˆ e z + ˆ s − ˆ p |} k ∗ T, (B.6)where we have redefined (cid:126)q = k ∗ (cid:126)q (cid:48) and dropped the prime.The spherical Bessel function plays a role of a window function that forces its argumentto be of order one, and so q ∼ k ∗ | (cid:126)x − (cid:126)y | (cid:28) | (cid:126)x − (cid:126)y | (cid:29) k ∗ . In the limit of small externalmomentum q , the time averaged term goes to 1 and we have (cid:104) ρ GW ( η, (cid:126)x ) ρ GW ( η, (cid:126)y ) (cid:105) B (cid:39) π ) (cid:32) M p A s a η (cid:33) × π (cid:18) k ∗ | (cid:126)x − (cid:126)y | (cid:19) × S B (B.8)where we defined S B = (cid:90) d Ω s (cid:90) d Ω p (cid:90) d Ω p δ ( | ˆ s − ˆ p + ˆ p | −
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