YITP Uji Research Center YITP/U-95-26

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1 YITP Uji Research Center YITP/U-95-6 July, 995 Formation of MACHO-Primordial Black Holes in Inflationary Cosmology Jun'ichi Yokoyama Uji Research Center, Yukawa Institute for Theoretical Physics Kyoto University, Uji 6, Japan Abstract As a nonbaryonic explanation of massive compact halo objects, a phenomenological model is presented which predicts formation of primordial black holes at a desired mass scale. The required feature of initial density uctuation is realized making use of the primordially isocurvature uctuation generated in an inationary universe model with multiple scalar elds.

2 Introduction If overdensity of order of unity exists in the hot early universe, a black hole can be formed when the perturbed region enters the cosmological horizon. The primordial black holes (hereafter PBHs) thus produced was a subject of active research decades ago (Zel'dovich & Novikov 967; Hawking 97) and various observational constraints have been obtained against their mass spectrum with no observational evidence of their existence at that time (Novikov et al. 979). Recently, however, several independent projects reported observation of massive compact halo objects (MACHOs) through gravitational microlensing (Alcock etal. 99; Aubourg et al. 99). It is estimated that their mass is around 0:0 0:M and that they occupy 0% of the galactic halo mass which makes up about O(0 ) of the critical density (Griest et al. 995). While the primary candidate of MACHOs is substellar baryonic objects such as brown dwarfs, it is dicult to reconcile such a large amount of these objects with the observed mass function of low mass stars (Richer & Fahlman 99) and with the infrared observation of dwarf component (Boughn & Uson 995), unless the mass function is extrapolated to the lower masses in an extremely peculiar manner. Therefore it is also an interesting and potentially important theoretical issue to consider nonbaryonic explanation of the origin of MACHOs. In the present paper we consider the possibility that MACHOs consist of PBHs produced in the early universe and present a simple model which generates a desired spectrum of primordial density perturbations in the context of inationary cosmology (Guth 98; Sato 98; for a review see, e.g. Olive 990). In the simplest models of ination with one inaton scalar eld, the predicted adiabatic density uctuation has an almost scale-invariant spectrum (Hawking 98; Starobinsky 98; Guth & Pi 98), unless the inaton has a peculiar potential. Hence they do not predict PBH formation in general. In models with multiple scalar elds, on the other hand, not only adiabatic but also isocurvature uctuations are generated during ination. The latter can be cosmologically important if energy density ofits carrier becomes signicant in a later epoch (Linde 985; Kofman & Linde 987). Furthermore it is relatively easier to imprint a nontrivial feature on the spectral shape of the isocurvature uctuations. Making use of this property here we construct a model which possesses a peak in the spectrum of total density uctuation at the horizon crossing. Then a signicant amount of PBHs can be produced around the horizon mass scale when the uctuation at the horizon crossing becomes maximal. While our goal is to produce PBHs of mass 0:M with the

3 abundance of BH 0, one can easily see that our model can also be applied to PBH formation of dierent masses and abundances as well by choosing dierent values of model parameters. The rest of the paper is organized as follows. In x we review basics of PBH formation and discuss necessary initial condition of density uctuations to obtain adequate PBHs. Then in x possibility of generating the necessary uctuations is considered in the context of inationary cosmology and a model Lagrangian is proposed. xx4-6 are devoted to detailed description of the evolution of the universe in this model and in x7 constraints on the model parameters are obtained. x8 is the conclusion. Formation processes of the PBHs. PBHs are formed if initial density uctuations grow suciently and a high density region collapses within its gravitational radius. First let us review its formation process. The background spacetime of the early universe dominated by radiation is satisfactorily described by the spatially-at Friedmann universe, ds = dt + a (t) h dr + r d + sin d' i ; () whose expansion rate is given by _a H (t) = 8G (t); () a with (t) being the background energy density and a dot denotes time derivation. Following Carr (975), let us consider a spherically symmetric high density region with its initial radius, R(t 0 ), larger than the horizon scale t 0. The assumption of spherical symmetry will be justied below. Then the perturbed region locally constitutes a spatially closed Friedmann universe with a metric, ds = dt 0 + R (t 0 ) Then the Einstein equation reads " dr r + r (d + sin d' ) # ; >0; () H 0 (t 0 ) R dr dt 0! = 8G +(t 0 ) R (t 0 ) ; (4) there, where + is the local energy density. One can choose the coordinate so that both the background and the perturbed region have the same expansion rate

4 initially at t = t 0 = t 0. Then the initial density contrast, 0, satises = ; (5) 0 H0 R 0 where a subscript 0 implies values at t 0. It can be shown that the two time variables are related by (Harrison 970) 0= 8G 0( + 0 ) ( + 0 ) 4 R0 R c 4 dt 0 R(t 0 ) = R c dt a(t) : (6) The perturbed region will eventually stop expanding at R R c, which is obtained from = H0( R ) H 0 0 ; (7) as corresponding to the epoch R c 4 R s +0 R c = R 0 ' 0 R 0 ; (8) 0 R c t c ' t 0 0 : (9) The perturbed region must be larger than the Jeans scale, R J, in order to contract further against the pressure gradient, while it should be smaller than the horizon scale to avoid formation of a separate universe. We thus require or R J ' c s t c < R c < t c ; (0) R c s < c ' R 0 t c t < 0 0 ; () where c s is the sound velocity equal to = p in the radiation dominated era. Since R 0 = 0 =t 0 is time independent, it suces to calculate the constraint on at a specic epoch, say, when the region enters the Hubble radius, R =t. We nd that the amplitude should lie in the range < (R = t) < : () The gravitational radius, R g, of the perturbed region in the beginning of contraction with R c ' c s t c is given by R g =GM ' H R c ' R c t c ' c s R c < R c: () 4

5 This is somewhat smaller than R c but it also implies that a black hole will be formed soon after the high density region starts contraction. Thus we expect that a black hole with a mass around a horizon mass at t = t c will result and it has in fact been shown by numerical calculations (Nadezhin et al. 978; Bicknell et al. 979) that the nal mass of the black hole is about O(0 0:5 ) times the horizon mass at that time. It has been discussed that these black holes do not accrete surrounding matter very much and that their mass do not increase even one order of magnitude (Carr 975). Note also that evaporation due to the Hawking radiation is unimportant for M 0 5 g (Hawking 974). Since the horizon mass at the time t is given by t M hor =0 5 M ; (4) sec what is required in order to produce a signicant number of PBHs with mass M 0:M is the sucient amplitude of density uctuations on the horizon scale at t 0 6 sec. Because the initial mass fraction of PBHs,, is related with the present fraction BH as = a(0 6 sec) BH = 0 8 BH ; (5) a(t eq ) where t eq 0 0 sec is the equality time, only an extremely tiny fraction of the universe, ' 0 should collapse into black holes. That is, the probability ofhaving a density contrast = < < on the horizon scale at t 0 6 sec should be equal to. Let us assume density uctuations on the relevant scale obey the Gaussian statistics with the dispersion BH, which would be the case in the model introduced in the next sections. Then the probability of PBH formation is estimated as = ' ' Z p BH = Z =+O( BH ) = BH which implies that we should have 0 0 BH A d 0 p BH 8 BH BH A d A ; (6) BH ' 0:05; (7) 5

6 to produce appropriate amount of PBHs. See gure. Although it is true that in principle an exponential accuracy is required on the amplitude of uctuations in order to produce the desired amount of PBHs, we have not been able to obtain the correspondence between BH and BH with such an accuracy because numerical coecients appearing in the above expressions, such as 8 in (6), have been calculated based on a rather qualitative argument. We therefore will not attempt exceedingly quantitative analysis in what follows. Note also that for BH =0:05 the threshold of PBH formation, ==, corresponds to 6.4 standard deviation. It has been argued by Doroshkevich (970) that such a high peak has very likely a spherically symmetric shape. Thus the assumption of spherical symmetry in the above discussion is justied and it is also expected that gravitational wave produced during PBH formation is negligibly small. Non-at perturbation in inationary cosmology Since the amplitude of density perturbations on large scales probed by the anisotropy of the background radiation (Smoot et al. 99) is known to be ' 0 5, the primordial uctuations must have such a spectral shape that it has an amplitude of 0 5 on large scales, sharply increases by a factor of 0 4 on the mass scale of PBHs, and decreases again on smaller scales at the time of horizon crossing. It is dicult to produce such a spectrum of uctuations in inationary cosmology with a single component. In generic inationary models with a single scalar eld, which drives ination with a potential V [], the root-mean-square amplitude of adiabatic uctuations generated is given by (r) (r()) = 8 p 6V [] ; (8) V 0 []MPl on the comoving scale r() when that scale reenters the Hubble radius (Hawking 98; Starobinsky 98; Guth & Pi 98). The right-hand-side is evaluated when the same scale leaves the horizon during ination. Because of the slow variation of and rapid cosmic expansion during ination, (8) implies an almost scaleinvariant spectrum in general. Nonetheless one could in principle obtain various shapes of uctuation spectra making use of the nontrivial dependence of (r()) on V [] (Hodges & Blumenthal 990). In order to obtain a desirable spectrum for 6

7 PBH formation with a mountain on a particular scale, we must employ a scalar potential with two breaks and a plateau in between (Ivanov et al. 994). Such a solution is not aesthetically appealing. Here we instead consider an ination model with multiple scalar elds in which not only adiabatic but also primordially isocurvature uctuations are produced. In fact it is much easier to imprint nontrivial structure on the isocurvature spectrum as mentioned in the beginning. We introduce three scalar elds,, and in order to generate the desired spectrum of density uctuation. is the inaton eld which induces the new ination (Linde 98; Albrecht & Steinhardt 98) with a double-well potential, starting its evolution near the origin where its potential is approximated as U[ ] = V : (9) See Linde (994) and Vilenkin (994) for the natural realization of its initial condition. The Hubble parameter during ination, H I, is given by H I = 8V 0 : MPl On the other hand, is a long-lived scalar eld which induces primordially isocurvature uctuations that contribute to black hole formation later. Finally is an auxiliary eld coupled to both and and it changes the eective mass of the latter to imprint a specic feature on the spectrum of its initial uctuations. We adopt the following model Lagrangian. L = (@ ) (@ ) (@ ) V [ ; ; ]+L int ; (0) where L int represents interaction of j 's with other elds. Here V [ ; ; ]isthe eective scalar potential governing the dynamics of the elds, V [ ; ; ]=U[ ]+ ( c ) m ; () where j,,, c, and m are positive constants. m is assumed to be much smaller than the scale of ination, H I, and it does not aect the dynamics of during ination. Hence we ignore it for the moment. Let us briey outline how the system evolves before presenting its detailed description. In the early inationary stage is smaller than c, and has its potential minimum o the origin. Then also settles down to a nontrivial minimum, where it can have an eective mass larger than H I so that its quantum 7

8 uctuation is suppressed. As becomes larger than c, rolls down to its origin. Then the potential of also becomes convex and its amplitude gradually decreases due to its quartic term. However, since its potential is now nearly at, its motion is extremely slow with its eective mass smaller than H I well until the end of ination. In this stage quantum uctuations are generated to with a nearly scale-invariant spectrum. Thus the initial spectrum of the isocurvature uctuations due to has a scale-invariant spectrum with a cut-o on a large scale. After ination, the Hubble parameter starts to decrease in the reheating processes. As it becomes smaller than the eective mass of, the latter starts rapid coherent oscillation. dissipates its energy in the same way as radiation in the beginning when its oscillation is governed by the quartic term. But later on when becomes smaller than m, its energy density decreases more slowly in the same manner as nonrelativistic matter. Thus contributes to the total energy density more and more later, which implies that the total density uctuation due to the primordially isocurvature uctuations or grows with time. Since what is relevant for PBH formation is the magnitude of uctuations at the horizon crossing, we thus obtain a spectrum with a larger amplitude on larger scale until the cut-o scale in the initial spectrum is reached, that is, it has a single peak on the mass scale of PBH formation. In the above scenario we have assumed that survives until after the PBH formation. On the other hand, were stable, it would soon dominate the total energy density of the universe in conict with the successful nucleosynthesis. As a natural possibility we assume that decays through gravitational interaction, so that it does not leave anyunwanted relics with its only trace being the tiny amount of PBHs produced. In the subsequent sections we describe the detailed evolution of the above model and obtain constraints on the model parameters to produce the right amount of PBHs on the right scale. 4 Background evolution First we consider the evolution of the homogeneous part of the elds. During ination, the behavior of the inaton is governed by the U[ ] part of the potential. Solving the equation of motion with the slow-roll approximation, H I _ = U 0 [ ] = ; () 8

9 we nd (t) = i + ih(t t H i ) ; () I where i is the eld amplitude at some initial epoch t i. The above approximate solution remains valid until ju 00 [ ]j becomes as large as 9HI at t t f, when inationary expansion is terminated and we nd (t f)=hi =. Then () can also be written as (t) = H I H I (t f t)+ H I (t)+ = H I (t) ; (4) where (t) is the e-folding number of exponential expansion after t (< t f ) and the last approximation is valid when. From now on, we often use (t) as a new time variable or to refer to the comoving scale leaving the Hubble radius at t. Note that it is a decreasing function of t. As stated in the last section, we are taking a view that determines fate of and that controls evolution of but not vice versa. In order that does not aect evolution of the inequality must be satised, while we must have ; (5) ; (6) so that does not aect the motion of. We assume these inequalities hold below. When < c, both and have nontrivial minima which we denote by m and m, respectively. From V = + =0; (7) and V = ( c) + =0; (8) with V j,we nd m(t) = m(t) (9) m (t) = c (t) + m (t) = c (t) ; (0) 9

10 where (6) was used in the last expression. In the early inationary stage when c, the eective mass-squared of j, V jj, at the potential minimum is given by V [ ; m ; m ]= m = ( c ) ' c = c H I ; () V [ ; m ; m ]= c H I ; () where c is the epoch when = c.wechoose parameters such that > c ; and > c : () Then V and V are larger than HI initially at the potential minimum, so that both and settle down to m (t) and m (t), respectively. m (t) decreases down to zero at = c when V [ m ] also vanishes. Then V [ m = 0] starts to increase according to and soon acquires a large positive value, which implies that practically traces the evolution of m (t) down to zero without delay. On the other hand, evolves somewhat dierently because it does not acquire a positive eective mass from at the origin. Although (t) traces m (t) initially, asv [ m ] becomes smaller it can no longer catch up with m (t). From a generic property of a scaler eld with a small mass in the De Sitter background, one can show that this happens when the inequality gets satised, or at with = m (t) 0 d m (t) + s > V [ m ] H ; (4) A c l = c ; (5) ( l) l = s H I l : (6) Thus slows down its evolution. In the meantime vanishes. Then is governed by the quartic potential. We can therefore summarize its evolution during ination as () = 8 >< >: m() = H I c ; > l ; l + l H I ( l ) ; 0< < l : 0 (7)

11 Since l is adequately smaller than H I, remains practically constant in the latter regime. Let us also write down time dependence of its eective mass-squared for later use. V [ ] = 8 H >< I c ; > l ; >: l + l H I 5 Generation of uctuations ( l ) ; 0< < l : We now consider uctuations in both the scalar elds and metric variables in a consistent manner. We adopt Bardeen's (980) gauge-invariant variables A and H, with which the perturbed metric can be written as (8) ds = (+ A )dt + a(t) (+ H )dx ; (9) in the longitudinal gauge. In this gauge scalar eld uctuation j coincides with the corresponding gauge-invariant variable by itself. Assuming an exp(ikx) spatial dependence and working in the Fourier space, the perturbed Einstein and scaler eld equations are given by j +H _ j + A + H = 0 (40) _ H + H H = 4G( _ + _ + _ ) (4) k a (t) + V jj! j =V j A + _ A _ j _ H _ j X i6=j V ji i ; (4) Note that all the uctuation variables are functions of k and t. The above system is quite complicated at a glance. However, using constraints on various model parameters we have obtained so far, i.e. (5), (6), and (), it can somewhat be simplied. First, since has an eective mass larger than H I during ination except in the vicinity of= c, quantum uctuation on is suppressed and moreover its energy density practically vanishes by the end of ination. We can therefore neglect uctuations in. On the other hand, we can show that j _ j s j _ jj _ j; (4) with the help of (6) and (). Hence (4) and (4) with j = reduce to _ H + H H = 4G _ (44)

12 +H _ + k a (t) + V! =V A 4 _ H _ : (45) Thus only contributes to adiabatic uctuations and it can be calculated in the same manner as in the new ination model with a single scalar eld. This is as expected because dominates the energy density during ination. In fact since we are only interested in the growing mode on the super-horizon regime which turns out to be weakly time-dependent as can be seen from the nal result, we can consistently neglect time derivatives of metric perturbations and terms with two time derivatives in (44) and (45) during ination. We thus nd H = A = 4G H I _ : (46) The resultant amplitude of scale-invariant adiabatic uctuations depends on and one can normalize its value using the COBE observation (Smoot et al. 99) as =:0 : (47) On the other hand, satises (4) with j =. From quantum eld theory in De Sitter spacetime, it has a root-mean-square amplitude = (H =k ) = when the k-mode leaves the Hubble radius if V is not too large. Since V vanishes when = m and V A remains small even for l,we can neglect all the terms in the right-hand side, to yield when k a(t)h I. condition, +H I _ + V = 0; (48) We can nd a WKB solution with the appropriate initial (k; t) = s HI! S(t k ) Z t exp S(t 0 )H k I S(t) t k H I dt 0 ; (49) s S(t) 4V 9H ; where t k is the time when k-mode leaves the Hubble radius: k = a(t k )H I. The above expression is valid when j _ SjS. In terms of, (49) can be expressed as (k; ) = s HI! Z S( k ) k exp S( 0 )d 0 k S() ( k ) ; (50)

13 S() = 8 >< >: c + l H I ; > l ; + l H I! ( l ) ; 0< < l ; where k (t k ) and 4 ination at t = t f or =0.. The above equality isvalid until the end of 6 Evolution of the universe after ination Let us assume the universe is rapidly and eciently reheated at t = t f for simplicity to avoid further complexity. (see, e.g. Kofman et al. (994), Shtanov et al. (995), and Boyanovsky et al. (995) for recent discussion on ecient reheating.) Then the reheat temperature is given by T R = 0: qh I M Pl : (5) If there is no further signicant entropy production later, one can calculate the epoch, (L), when the comoving length scale corresponding to L pc today left the Hubble horizon during ination as (L) =7+ln L!+ pc ln H I 0 0 GeV : (5) Then the comoving horizon scale at t =0 6 sec, or L =0:0 pc corresponds to = 4 m and the present horizon scale ' 000Mpc to = 59. On the other hand, (t) and (k;t)evolve according to +H _ + +m = 0; (5) +H _ +( +m) = 0; (54) where the Hubble parameter is now time-dependent: H = =t, and the latter equation is valid for k ah. When H becomes smaller than ( m ), both and start rapid oscillation around the origin. Using (50) one can express the amplitude of the gauge-invariant comoving fractional density perturbation of,,as = _ A ' 4 =0 4 f f ; (55)

14 in the beginning of oscillation. Here _ m (56) is the energy density of. Using the virial theorem one can easily show that it decreases in proportion to a 4 (t) as long as > m.thus the amplitude of decreases with a (t). On the other hand, has a rapidly oscillating mass term when > m, which causes parametric amplication. We have numerically solved equations (5) and (54) with various initial conditions with j jj j initially. Wehave found that in all cases the amplitude of remains constant as long as m is negligible. Thus increases in proportion to t = in this regime and becomes as large as = 4 f f f m! ; (57) while the ratio of to the total energy density, tot, which isnow dominated by radiation, remains constant: tot = f M Pl! : (58) As becomes smaller than m, and come to satisfy the same equation of motion, see (5) and (54), and saturates to the constant value (57). At the same time starts to decrease less rapidly than radiation, in proportion to a (t). Since contributes to the total comoving density uctuation by the amplitude = tot ; (59) it increases in proportion to a(t) / t =. In the beginning of this stage, we nd H = m 4 = f, to yield = 8 f f f m!!! f m t p ; (60) M Pl f at a later time t. In order to relate it with the initial condition required for PBH formation, we must estimate it at the time k-mode reenters the Hubble radius, t k, dened by k =a(t k )H(t k )= a f : (6) (t k t f) 4

15 Since k can also be expressed as k = a f e k H I, the amplitude of comoving density uctuation at t = t k is given by (k; t k ) = 8 f f p!! f f e k : (6) H I M Pl 7 Constraints on model parameters In x we discussed the necessary condition on the amplitude of uctuations for PBH formation using the uniform Hubble constant gauge. Hence we should calculate the predicted amplitude in this gauge, which is a linear combination of and the gauge-invariant velocity perturbation. However, in the present case in which grows in proportion to a(t) in the radiation-dominant universe, one nds that the latter quantity vanishes and that density uctuation in the uniform Hubble constant gauge coincides with (Kodama & Sasaki 984). Thus we nally obtain the quantity to be compared with BH in (6), namely, the root-mean-square amplitude of density uctuation on scale r = =k at the horizon crossing, (r), as with (r) = " # 4k () j(k; t k)j = 4 p H I f M 4 Pl! e k C f ( k ; c ; ); (6)! C f ( k ; c ; ) S( Z k) k exp S( 0 )d 0 S(0) 0 k ; (64) where we have used (50). We also nd f = s 8 + s! 0 + A 8 H I c ; (65) from (6) and (7). The remaining task is to choose values of parameters so that (r) has a peak on the comoving horizon scale at t =0 6 sec, which we denote by r m, corresponding to k = m = 4, with its amplitude (rm ) = 0:05. We thus require d ln (r) d k = S0 ( k ) S( k ) + S( k) = c 4 k [ c k + ( c k )] + + c k (66) 5

16 vanishes at k = m, which gives us a relation between c and. Since c roughly corresponds to the comoving scale where scale-invariance of primordial uctuation is broken, the peak at (r m ) becomes the sharper, the closer c approaches m. For example, if we take c =0we nd so that = In order to have (r m )=0:05, we nd 4 = 00; (67) C f =0: and f =0:045 H I : (68) p HI M Pl =:70 5 ; (69) which can easily be satised with some reasonable choices of and H I.However, it is not the nal constraint. Since we are assuming that the universe is dominated by radiation at this time, we require tot =! f M Pl! m p e m : (70) f H I Furthermore should decay some time after t =0 6 sec so as not to dominate the energy density of the universe which would hamper the primordial nucleosynthesis. Assuming that it decays only through gravitational interaction, its life time is given by M = Pl =0 5:5 m sec : (7) m 0 6:5 GeV Now wehave displayed all the necessary equalities and inequalities the model parameters should satisfy. Since there is a wide range of allowed region in the multidimensional space of parameters, we do not work out the details of the constraints but simply give one example of their values with which all the requirements are satised: H I = :7 0 0 GeV; m = : 0 6 GeV; = : 0 ; (7) = :4 0 6 ; = =6:70 8 ; = :0 0 ; 6

17 for which = tot =0:att=0 6 sec and inequalities (5) and (6) are maximally satised. In gure we have depicted the qualitative mass spectrum of produced PBHs for dierent values of c where! (M) =(r) exp (7) 8(r) has been shown as a function of the horizon mass when the scale r reenters the horizon. 8 Conclusion In the present paper wehave considered possibility to produce a signicant amount of PBHs on a specic mass scale by generating appropriate spectrum of density uctuations in inationary cosmology. We have reached a model with the desired feature making use of a simple polynomial potential () without introducing any break in the potential of the scalar elds. We have chosen values of the model parameters so that these PBHs can account for the observed MACHOs. In order to set the order of magnitude of the mass scale of the black holes and that of their abundance correctly, we had to tune some combinations of model parameters such as (67) and (69) with two digits' accuracy. However, there exists a wide range of allowed region in the parameter space to realize it. We also note that the precise values such as those quoted in (7) are not of much signicance, primarily because the formula for the fraction of PBHs (6) is only a qualitative one. In this respect we have restricted ourselves to the analytic treatment of evolution of uctuation, which is not exponentially accurate. In the event a more precise formula for PBH fraction is obtained, full numerical analysis of uctuations would be required. At the present stage, however, analytic treatment is more appropriate with which dependence of the results on physical parameters are understood more clearly. It is evident that our model can be applicable to produce PBHs with a dierent mass and abundance by slightly changing values of parameters. For example, we could produce black holes with mass 0 6 M which would act as a central engine of AGNs. These black holes are usually considered to have formed in the postrecombination universe (Loeb 99), but they might have formed in the early universe at t 0sec corresponding to the onset of primordial nucleosynthesis. Note that this would not hamper successful nucleosynthesis because the root-meansquare amplitude of density uctuation required for such black hole formation is 7

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19 Linde, A.D., 994, Phys. Lett. B7, 08 Loeb, A., 99, ApJ 40, 54 Nadezhin, D.K., Novikov, I.D. & Polnarev, A.G., 978, SvA, 9 Novikov, I.D., Polnarev, A.G., Starobinsky, A.A., Zel'dovich, Ya.B., 979, A&A 80, 04 Olive, K.A., 990, Phys. Rep. 90, 07 Richer, H.B. & Fahlman, G.G., 99, Nature 58, 8 Sato, K. 98, MNRAS, 95, 467 Smoot, G.F. et al., ApJ, 96, L Shtanov, Y., Traschen, J., & Brandenberger, R., 995, Phys. Rev. D5, 548. Starobinsky, A.A., 98, Phys. Lett. B7, 75 Vilenkin, A., 994, Phys. Rev. Lett. 7, 7 Zel'dovich, Ya.B. & Novikov, I.D., 967, SvA 0, 60 Figure captions Figure Fraction of primordial black holes as a function of root-mean-square amplitude of density uctuations (eq.[6]) Gaussian distribution of uctuations is assumed. Figure Expected mass spectrum of primordial black holes with dierent values of c. 9