Primordial black holes as a novel probe of primordial gravitational waves

RESCEU-20/15

Primordial black holes as a novel probe of primordial gravitational waves Tomohiro Nakama1,2 and Teruaki Suyama2 1

arXiv:1506.05228v1 [gr-qc] 17 Jun 2015

2

Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan and Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan (Dated: June 18, 2015)

We propose a novel method to probe primordial gravitational waves by means of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations on comoving scales much smaller than those relevant to Cosmic Microwave Background is very large, it induces scalar perturbations due to second-order effects substantially. If the amplitude of resultant scalar perturbations becomes too large, then PBHs are overproduced to a level that is inconsistent with a variety of existing observations constraining their abundance. This leads to upper bounds on the amplitude of initial tensor perturbations on super-horizon scales. The resultant PBH upper bounds turn out be tighter than other bounds obtained from Big Bang Nucleosynthesis and Cosmic Microwave Background. PACS numbers:

Introduction.- Stochastic gravitational wave background (SGWB) on a wide range of scales is thought to have been generated in the early universe. SGWB on largest observable scales have been investigated by Planck [1] and BICEP2 [2]. SGWB on smaller scales can be constrained by inferring the value of Neff , the effective number of extra degrees of freedom of relativistic species, at Big Bang Nucleosynthesis (BBN) through the current abundance of light elements [3], or at photon decoupling through the anisotropy of Cosmic Microwave Background (CMB) [4, 5]. More recently SGWB on small scales has been constrained by CMB spectral distortions as well [6, 7]. SGWB can also be directly constrained by gravitational wave detectors (see e.g. [8]). BBN and CMB have played a major role in constraining SGWB since they are applicable on a wide range of scales, noting ground-based laser interferometers tend to target gravitational waves (GWs) on a relatively limited frequency range with high sensitivity. However, it would be worthwhile to note that upper bounds obtained through Neff needs an assumption about the number of relativistic species in the early universe, as is discussed later. In addition, in obtaining BBN or CMB bounds we implicitly assume that any physical mechanisms, both known and unknown, increase Neff , making Neff larger than the standard value Neff = 3.046 [9]. However, in principle it is possible to decrease Neff (see e.g. [10–12]). Therefore, it would be desirable to have another independent cosmological method to probe SGWB on a wide range of scales, which does not depend on the assumptions mentioned above.

nisms is the direct collapse of radiation overdensity during the radiation-dominated era, which happens when the density perturbation becomes order unity at the moment of the horizon crossing of the perturbation [14–16]. (See also [17, 18] for updated discussions of the formation condition and see also [19–25] for numerical simulations of the formation of PBHs.) There is no conclusive evidence for the existence of PBHs and so upper bounds on the abundance of PBHs on various mass scales have been obtained by various kinds of observations (see e.g. [13] and references therein). These upper bounds can be translated into upper bounds on the power spectrum of the curvature perturbation on small scales [26, 27], namely, they can be used to exclude models of the early universe which predict too many PBHs. (Other methods to constrain primordial perturbations on small scales include CMB spectral distortions [28], acoustic reheating [29, 30], ultracompact minihalos [31, 32] etc.) In this Letter we use PBHs to constrain primordial tensor fluctuations. Large amplitude tensor perturbations induce scalar perturbations (induced scalar perturbations) due to their second-order effects, though tensor and scalar perturbations are decoupled at the linear level. If the typical amplitude of primordial GWs is too large, then the typical amplitude of resultant induced scalar perturbations becomes also too large and there appear too many regions where the amplitude of density perturbation becomes order unity at the horizon crossing to form PBHs. Thus we can obtain upper bounds on the amplitude of primordial tensor modes requiring PBHs are not overproduced as they enter the Hubble radius [57].

The very physical mechanism we employ to probe SGWB is the formation of primordial black holes (PBHs), black holes formed in the early universe well before the cosmic structure formation. There are a number of possible mechanisms to create PBHs (see e.g. [13] and references therein), but one of the simple and natural mecha-

Due to our current ignorance of the correct model of inflation and the subsequent thermal history of the universe, new upper limits on tensor perturbations on small scales in themselves are worthwhile. In addition, it makes our new upper limits still more valuable that there are models of the early universe which can predict

2

FIG. 1: Comparison of the constraints on the amplitude of the delta-function power spectrum eq.(2), obtained from PBH, BBN, CMB, PTAs, LIGO and Virgo. For more details see [44].

large tensor perturbations on small scales [33–43]. If a model predicts large tensor perturbations on small scales, and even larger scalar perturbations at the same time, then such a model would be more severely constrained by PBHs generated from first-order scalar perturbations. Here we consider PBH formation only from second-order tensor perturbations, but if scalar perturbations are also large, PBHs are formed more, and so our bounds on tensor perturbations are conservative or model-independent, in the sense that those bounds do not depend on the amplitude of first-order scalar perturbations on small scales. Note also that some of the above previous works can (or possibly can) predict not only large tensor perturbations on small scales, but also large tensor-to-scalar ratio on small scales, and our method is particularly useful to constrain these types of models. To calculate induced scalar perturbations one may solve the Einstein equations for scalar perturbations including the source terms which are second-order in tensor perturbations, assuming some form of the initial power spectrum of tensor perturbations. In general, the statistics of induced scalar perturbations are highly nonGaussian, since they are generated by second-order tensor perturbations. If we assume some statistical properties of initial tensor perturbations, e.g. Gaussianity, in principle one can determine the probability density function (PDF) of induced scalar perturbation, which is necessary to calculate the abundance of PBHs given some initial tensor power spectrum to obtain PBH constraints on primordial GWs. All the details of those analyses will be presented elsewhere [44], and here we present simple analytic estimations, which correctly reproduce the main results of [44], as is discussed shortly. PBH upper bounds on tensor perturbation.- For simplicity we assume the typical amplitude of tensor fluctuations of only a limited range of wavelength around λp (say, in the range (e−1/2 λp , e1/2 λp )) is large, namely we

consider a peak in the power spectrum around λp . Let us denote by σh the root-mean-square amplitude (RMS) or the typical amplitude of tensor fluctuations when the relevant modes under consideration are on super-horizon scales. Note that σh2 roughly corresponds to the dimensionless power spectrum Ph of primordial GWs. Suppose these large amplitude tensor perturbations whose wavelength is around λp reenter the horizon during the radiation-dominated era. It generates radiation density perturbations δr whose wavelength is around λp and whose RMS σδr at around the horizon crossing is roughly given by σδr ∼ σh2 , simply because δr is generated by second-order GWs. As confirmed in [44], this relation indeed holds well at least for the delta-function power spectrum. If δr > δr,th ∼ 0.4 [17], the induced radiation density perturbations collapse to form PBHs. The probability of formation of PBHs has to be extremely small to be consistent with various observations [13]. Therefore, in order not to overproduce PBHs, one may require Ph ∼ σh2 ∼ σδr . 0.1δr,th ∼ 0.04. More comments will be made later regarding this inequality. This upper bound is applicable on all the scales which reenter the horizon during the radiation-dominated era. Furthermore, this bound is probably conservative on scales which reenter the horizon during the (early as well as late) matterdominated era, since formation of black holes by direct collapse of primordial perturbations is easier when the universe is (literally or effectively) dominated by pressureless dust [58] . To summarize, we have Ph . 0.04 (PBH),

(1)

and this is applicable from the comoving wavelength of ∼ 1Gpc all the way down to ∼ 1Gpc × e−60 ∼ 0.3m, assuming the total number of e-foldings during inflation is sixty [59]. See Discussion for some comments regarding the constraints on smallest scales. Our analytic estimations can reproduce the results of more rigorous calculations in [44], where the deltafunction initial tensor power spectrum is assumed: Ph (k) = A2 kp δ(k − kp ).

(2)

The PBH abundance for this power spectrum is calculated and constraints on A2 as a function of kp are obtained, which is shown in Fig.1, along with other constraints. One can see that our analytic estimations (eqs. (1), (13) and (14)) indeed explain Fig.1 well. Comparison with other methods.- First note that the total energy density of radiation without the presence of gravitational waves can be written as follows; ρrad (T ) = (π 2 /30)g∗ T 4 .

(3)

Here g∗ is the effective number of degrees of freedom of relativistic species and is given by, before the electronpositron annihilation, [3, 45] 7 g∗ = 2 + (4 + 2Nν ), 8

(4)

3 where Nν is the effective number of degrees of freedom of neutrinos Nν = 3.046. It is convenient to denote the presence of GWs (or possibly dark radiation) by ∆Neff , as a correction to Nν above [3]. In the following we use ∆NGW as a contribution of GWs and derive the expression of it in terms of Ph , enabling observational upper limits on ∆Neff to be translated into upper limits on Ph . When GWs are present, the critical density ρcrit during the radiation-dominated era is ρcrit ≃ ρrad + ρGW ,

(5)

where ρrad and ρGW are the energy density of radiation and GWs respectively. Note that ρGW /ρrad ∼ Ph ≪ 1 by definition of tensor ”perturbations”, or this is marginally guaranteed at least thanks to our PBH constraint. Therefore in the following we assume this inequality, but it turns out that upper limits other than PBHs’ are mostly too weak to be compatible with this condition, which may appear to invalidate our analysis below. However, our purpose here is to show that other constraints are mostly weaker and so the above condition can be assumed. After the horizon crossing of GWs, their energy density starts to scale as ∝ a−4 , while, denoting the effective degrees of freedom of relativistic species in terms of entropy at temperature T by gS (T ), the photon temperature evolves following gS (T )T 3 a3 =const. (i.e. 1/3 4/3 constant entropy) and so ρrad ∝ g∗ /(a4 gS ) ∼ 1/(a4 gS ) (see e.g. [45]). Let us define ΩGW ≡ ρGW /ρcrit ≃ ρGW /ρrad . Then, ΩGW (T ) =



gS (T ) gS (Tin )

1/3

ΩGW (Tin ),

(6)

where Tin is the temperature of radiation at the moment of the horizon crossing of GWs whose wavelength is ∼ λp , and T < Tin . The contribution of GWs is characterized by ∆NGW as follows; ρrad (T ) + ρGW (T )   7 = 2σSB 2 + {4 + 2(Nν + ∆NGW )} T 4 , 8

(7)

which leads to ρGW (T ) = ρrad (T ) × ≃

  7 7 × 2 × ∆NGW (T )/ 2 + (4 + 2Nν ) 8 8

7 ρrad (T )∆NGW (T ). 43

(8)

Noting [60] ΩGW (Tin ) ∼ Ph ,

(9)

∆NGW (T ) can be written as 43 43 ΩGW (T ) ∼ Ph ∆NGW (T ) = 7 7



gS (T ) gS (Tin )

1/3

. (10)

In the literature an upper bound on ∆Neff < ∆Nupper is translated into an upper bound on ∆NGW , ∆NGW < ∆Nupper. As is already mentioned, here it is assumed that any physical mechanisms, both known and unknown, contribute positively to Neff . However, it would be worthwhile to note that at least there are examples where Neff decreases [10–12]. With this in mind, the requirement above, ∆NGW < ∆Nupper, is translated into an upper bound on Ph from (10) as follows:  1/3 7 gS (Tin ) Ph < ∆Nupper . (11) 43 gS (T ) Note that this constraint depends on gS , which one may regard as a drawback of these methods since gS is uncertain especially at high temperatures. It also depends on other potential entropy productions [46]. On the other hand, the PBH constraint does not depend on gS nor other entropy productions. Assuming the standard model of particle physics, we use the following for simplicity:  (150MeV . T)  100, (1MeV . T . 150MeV) (12) gS (T ) ∼ 11,  4, (T . 1MeV)

Note that T ∼ 150MeV corresponds to the QCD phase transition, and T ∼ 1MeV to the electron-positron annihilation. In order not to spoil BBN, we set, following [46], ∆Nupper = 1.65 as a 95% C.L. upper limit, which can constrain GWs only of the scales smaller than the comoving horizon at the time of BBN, namely, 70kpc−1 . k. Noting the comoving scales which enter the horizon at T ∼ 150MeV and T ∼ 1MeV correspond to k ∼ 4pc−1 and k ∼ 1kpc−1 respectively, we obtain the following constraints  0.6, (4pc−1 . k) Ph . (13) 0.3, (70kpc−1 . k . 4pc−1 )

As for the CMB bound we follow [5] to set ∆Nupper = 1 and this is applicable for 10Gpc−1 . k.  (4pc−1 . k)  0.5, (1kpc−1 . k . 4pc−1 ) Ph . 0.2, (14)  0.2, (10Gpc−1 . k . 1kpc−1 )

One may not regard these constraints as meaningful, because upper limits correspond to the amplitude of GWs which is (almost) non-linear. The current energy density of SGWB, ΩGW,0 , is also constrained by LIGO and Virgo, most severely in the band 41.5 − 169.25Hz as ΩGW,0 . 5.6 × 10−6 × log(169.25/41.5) ≃ 8 × 10−6 [8]. Noting ΩGW,0 ∼ −1 zeq (4/100)1/3 Ph ∼ 10−4 Ph (zeq ∼ 3000 is the redshift −1 reat the matter-radiation equality, and the factor zeq flects ΩGW ∝ (1+z)/(1+zeq) during a matter-dominated era), we have Ph . 0.08 [61] . Pulsar timing arrays (PTAs) have also been used to constrain GWs. Following [46] we use the most

4 stringent upper bound around f = 5.72×10−9Hz (∼ −1 4 × 106 Mpc−1 ), ΩGW ∼ zeq (4/11)1/3 Ph . 5 × 10−8 , −4 which leads to Ph . 2 × 10 . Note that GW detectors or PTA experiments usually target GWs on a relatively limited frequency range, while cosmological methods like PBHs probe primordial GWs on a wide range of frequencies, and this is another advantage of PBHs in constraining primordial GWs (see Fig.1). Discussion.- As already mentioned PBH constraints are applicable from ∼ Gpc all the way down to ∼ 0.3m if we assume the number of e-foldings during inflation is sixty. Note that the exclusion of an overproduction of smallest PBHs (MPBH . 105 g) depends on the assumption that stable Planck mass relics are left over at the end of Hawking evaporation, which contribute to cold dark matter (see [47], [13] and references therein). The range of comoving scales corresponding to MPBH . 105 g is roughly . 50m, and so PBH upper bounds in this range depend on this assumption. If Planck mass relics are not formed, to what extent an overproduction of PBHs with MPBH . 105 g is cosmologically problematic is uncertain. Such an overproduction of smallest PBHs may lead to an early matter-dominated era, during which PBH binaries are formed and emit GWs, or larger PBHs may form due to merger taking place after the collapse of perturbations of PBHs’ density, thereby leaving observable traces [48]. Therefore, in principle one may still exclude such an over production of smallest PBHs even without the left over of Planck mass relics to fully validate our upper bounds on smallest scales, though we do not discuss it in detail here. We can also constrain a blue tensor power spectrum of the following form: nT  k , (k < kmax ), (15) Ph (k) = rPζ (kref ) kref where Pζ is the dimensionless power spectrum of the curvature perturbation on uniform-density hypersurfaces and r is the so-called tensor-to-scalar ratio, here defined at kref = 0.01Mpc−1 . Let us assume r = 0.2 and P(kref ) = 2.2 × 10−9 following [46] and also kmax ∼ 1Gpc−1 e60 ∼ 3m−1 . If we simply require Ph (kmax ) . 0.04, we obtain nT . 0.3, which is tighter than other constraints such as BBN constraints shown in [46]. Once more, this PBH bound does not depend on the details of a potential early matter-dominated era phase nor an entropy production, on which constraints other than PBHs are sensitive [46]. If we use a more secure but weaker constraint on PBHs of around 105 g by their entropy production [13], we may require Ph (20km−1 ) . 0.4, leading to nT . 0.4, still tighter than other constraints. To conclude, PBHs can provide important constraints also on a

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blue spectrum, though a more careful analysis would be merited. We have imposed σδr . 0.1δr,th to obtain PBH constraints, eq.(1). The factor of 0.1 here is to ensure the sufficient rareness of formation of PBHs. Note that to discuss the probability of PBH formation one needs to know the PDF of induced radiation density perturbation, which is in general highly non-Gaussian since it is generated by second-order tensor perturbations. In [44] we assume Gaussianity of tensor perturbations on top of the delta-function power spectrum, and numerically confirmed the factor of roughly 0.1 is indeed necessary to avoid an overproduction of PBHs (compare eq.(1) and the PBH constraints in Fig.1). Strictly speaking, observational constraints on PBHs depend on their mass [13] and so how rare PBH formation has to be also depends on their mass, hence the scale dependence of PBH constraints in Fig.1 [44]. This argument implies PBH constraints on tensor perturbations depend on the statistics of tensor perturbations, determining the statistics of induced density perturbation, just as PBH constraints on scalar perturbations depend on the statistics of scalar perturbations [49]. If high-σ realizations of tensor perturbations are suppressed (enhanced) in comparison to a Gaussian case, PBH constraints on tensor perturbations are tighter (weaker). One may conjecture the dependence of PBH constraints of tensor perturbations on the statistics of tensor perturbations is weaker than the dependence of PBH constraints of scalar perturbations on the statistics of scalar perturbations. This is because in the former case the probability of PBHs is determined by the statistics of induced radiation perturbations, generated from secondorder tensor perturbations, and so the PDF of induced radiation perturbations is something similar to a χ2 distribution [44]. Namely, the probability of high-σ peaks of radiation leading to PBH formation is determined by lower-σ realizations of tensor perturbations, for which non-Gaussianity is less important than it is for higherσ realizations. A more careful analysis on this matter would be merited. Acknowledgement.- We thank Jun’ichi Yokoyama for reading the manuscript, useful comments and continuous encouragement. We also thank Kazunari Eda, Yuki Watanabe, Yousuke Itoh, Daisuke Yamauchi, Tsutomu Kobayashi, Masahide Yamaguchi, Takahiro Tanaka for helpful comments. This work was supported by JSPS Grant-in-Aid for Young Scientists (B) No.15K17632 (T.S.), MEXT Grant-in-Aid for Scientific Research on Innovative Areas ”New Developments in Astrophysics Through Multi-Messenger Observations of Gravitational Wave Sources” No.15H00777 (T.S.) and Grant-in-Aid for JSPS Fellow No.25.8199 (T.N.).

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6 [61] Note that they also obtained weaker upper bounds on a few other frequency ranges other than the one around ∼ 100Hz. Strictly speaking in [8] some power low frequency dependence is assumed in each band, and so their results may not be directly translated into constraints

on a narrow peak in the power spectrum we consider. Namely, our comparison here is only a rough one, but it is sufficient for our purposes. The same applies to the comparison with PTA.