- Email: [email protected]
Formation of primordial black holes in the inflationary universe
Physics Reports 307 (1998) 133—139
Formation of primordial black holes in the inflationary universe Jun’ichi Yokoyama* Department of Physics, Stanford University, Stanford, CA 94305, USA and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract We propose a new double inflation model containing only one inflaton scalar field, in which new inflation follows chaotic inflation. It is shown that the large density fluctuation generated in the beginning of new inflation can result in significant formation of primordial black holes on astrophysically interesting mass scales. 1998 Elsevier Science B.V. All rights reserved. PACS: 98.80.!k
If overdensity of order of unity exists in the hot early universe, a black hole can be formed when the perturbed region enters the Hubble radius [1]. While the properties of the primordial black holes (hereafter PBHs) thus produced were a subject of extensive study decades ago, there were no observational evidence of their existence and only observational constraints were obtained against their mass spectrum [2,3]. Recently, however, possibilities of their existence have been raised from a number of astrophysical and cosmological considerations, for example, in an attempt to explain the origin of MACHOs [4] or a class of gamma-ray bursts [5]. It is difficult, unfortunately, to realize a desired spectrum of density fluctuations for PBH formation in inflationary cosmology [6—8] because usual inflation models predict a scale-invariant spectrum [9] whose amplitude has been normalized to O(10\) by the observed anisotropy of the cosmic microwave background radiation (CMB) [10]. But in order to produce PBHs on some specific scale, we must prepare density perturbation whose amplitude has a high peak of O(10\) on the corresponding scale. In the present paper, we propose a new scenario of multiple inflation as a model to produce the desired spectrum of density fluctuation for PBH formation. Unlike previous double inflation
* E-mail: [email protected]. 0370-1573/98/$ — see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 4 4 - 1
134
J. Yokoyama / Physics Reports 307 (1998) 133—139
models [11], our model contains only one source of inflation. We assume the Einstein gravity and employ an inflaton scalar field, , with a simple potential which has an unstable local maximum at the origin. This is the same setup as the new inflation scenario [7], but chaotic inflation [8] is also possible if has a sufficiently large amplitude initially. In fact, Kung and Brandenberger [12] studied the initial distribution of a scalar field with such a potential and concluded that chaotic inflation would be much more likely than new inflation. Thus, we also start with chaotic inflation, but we show that new inflation is also possible when evolves towards the origin after chaotic inflation. In fact, if the parameters of the potential are appropriately chosen, the scalar field acquires the right amount of kinetic energy after chaotic inflation and climbs up the potential hill to near the origin to start slow rollover there. Hence, in this model the initial condition for new inflation is realized not due to the high-temperature symmetry restoration nor for a topological reason [13], but by dynamical evolution of the field which has already become sufficiently homogeneous because of the first stage of chaotic inflation. We shall refer this succession of inflation simply to chaotic new inflation. With an appropriate shape of the potential, density fluctuations generated during new inflation can have larger amplitude than those during chaotic inflation. Furthermore, since the power spectrum of fluctuation generated during new inflation can be tilted, it can have a peak on the comoving Hubble scale when the inflaton enters the slow-rollover phase during new inflation. If the peak amplitude is sufficiently large, it results in formation of PBHs on the horizon mass scale when the corresponding comoving scale reenters the Hubble radius during radiation domination. We shall show below that such a scenario is indeed possible with a simple potential of the inflaton field. We adopt the following potential
j 1 1 »[ ]"! m # ln ! #» , v 4 4 2
(1)
that is, typical one-loop effective potential with nonvanishing mass term at the origin. A potential of this type but with a positive mass-squared at the origin was employed in the original inflation scenario [6], but for our purpose we adopt a negative mass term. The potential (1) has four parameters, but one of them, » , is fixed from the requirement that the vacuum energy density vanishes at the potential minima ,$ . Another parameter, say j, can K be fixed from the amplitude of large-scale CMB fluctuations using the COBE data as in chaotic inflation. Hence, we are essentially left with two free parameters, v and m. While v mainly controls the speed of around the origin and its fate, i.e., to which minimum it falls, and m mainly governs duration of new inflation, the entire dynamics is determined by a complicated interplay of the three parameters. For example, we cannot determine j until we calculate the duration of new inflation which also depends on j itself for fixed values of m and v. Hence, we must numerically solve the equations of motion iteratively to find out appropriate values of parameters to produce PBHs at the right scale with the right amount. In the new inflation regime the slow roll-over classical solution, (t), and the scale factor, a(t), read
m 8p» , H (t!t ) , a(t)Je&R, H"
(t)" exp 3M 3H .
(2)
J. Yokoyama / Physics Reports 307 (1998) 133—139
135
where the subscript s stands for the onset of slow-roll new inflation. The linear curvature perturbation on the comoving scale l"2p/k is given by H H H , ,f D( (t )) 3f (3) I m 2p (t ) 2p" Q (t )" I I where t is the epoch when k-mode left the Hubble radius during inflation. The above expression of I U(l) is valid until the scale l reenters the Hubble radius with f" () in the matter (radiation) domination. We find the spectral index of power-law density fluctuation is given by n"1!2m/(3H). Thus, the power spectrum can be significantly tilted. The linear perturbation has a peak amplitude U(l)"f
H H (4) max D"3 ,D , m 2p
on the comoving scale l ,2p/k where k ,a(t )H . It can be large if turns out to be small. We apply the stochastic inflation method [14] to estimate the probability distribution function (PDF) of curvature fluctuations. Since our potential can be approximated as »[ ]"» !m in the new inflation regime, the PDF of the coarse-grained inflaton field K (x, t), P[ K " , t], can be analytically calculated by solving the Fokker—Planck equation jP 1 j H jP " » [ ]P# . (5) jt 3H j
8p j Taking the initial condition P[ , t ]"d( ! ), its solution is given by the Gaussian: 2m ( ! (t)) 3H 1 exp exp ! , p(t)" H (t!t ) !1 . (6) P[ , q]" 3H 2p(t) 8pm (2pp(t) From this distribution function, we estimate the PDF of metric perturbation. In order to incorporate nonlinear effects at least partially, we write the coarse-grained metric in the quasiisotropic form,
ds"!dt#aL ( K (t, x)) dx ,
(7)
following Ivanov [15], where the scale factor aL now depends on the coarse-grained spatial coordinate through K (x, t). We quantify the metric perturbation in terms of hK ,aL (t, x)/a(t)!1, with a(t) being the average scale factor. In order to estimate the abundance of PBHs produced we calculate the PDF of hK , P[hK "h], from Eq. (6). We are interested in the metric perturbations on the comoving scales that leave the Hubble radius in the period between t"t and tKt #H\ when the classical solution has rolled down to (t #H\)" exp(m/3H), , because these scales correspond to the peak of the power spectrum and dominates the formation of PBHs when they reenter the Hubble radius during radiation domination. The desired PDF is approximately given by
1 m
P , (1#ln(1#h)) . P[hª "h]" H 3H 1#h
(8)
136
J. Yokoyama / Physics Reports 307 (1998) 133—139
Explicitly, we find the probability that h exceeds a threshold value h as P[h'h ] c [(1#h )A!1] [(1#h )A!e\A] K exp ! , 2c[(1#h )A!e\A] (2p (1#h)A[(1#h)A!1][(1#h)A!e\A] where we have defined c"m/(3H) and c"cD/2. The criterion for black hole formation has been numerically investigated by Nadezhin et al. [16] and by Biknell and Henriksen [17]. Although it depends on the shape of the perturbed region, the generic value of the threshold reads h '0.75—0.9. Here we take the black hole threshold as h "0.75. Then we can express the initial volume fraction of the region collapsing into PBHs as b(M )"P[h'0.75], where the typical black hole mass, M , is equal to the horizon mass when the comoving scale l reenters the Hubble radius. Note that since our model predicts density fluctuation which is highly peaked on the comoving scale l the resultant mass function of PBHs is also sharply peaked at the mass around M . Let us now consider a specific example of formation of MACHO-PBHs [18]. For this purpose we must realize a peak with b&10\ on the comoving scale leaving the Hubble radius 35 e-folds expansion before the end of new inflation, that is, we must have N,H (t !t )"35, where t is the time new inflation is terminated. After some iterative trials we have chosen j"3;10\ and m"6;10\M , and then solved the equation of motion for various values of v. In this choice of . j and m we find new inflation lasts for more-than ten e-folds expansion if we take v in the range v"0.2131M !0.2147M . Hence, we do not need much fine-tuning of the model parameters to . . realize a new inflationary stage itself. We also find that settles down to if v50.21384364M K . and to ! if v40.21384363M . K . We can obtain the appropriate spectrum of fluctuation for MACHO-PBHs if we take v"0.21384360M . Figs. 1 and 2 depict evolution of the scale factor and the inflaton , respective. ly, with the initial condition a "1 at "3.5M . The chaotic inflation ends at K0.89M and G G . . the slow roll-over new inflation starts at "!4.03;10\M , . In this case we find . + c"0.300590 and the abundance of the PBHs at formation reads
b"0.888D exp(!0.131072D\) . (9) One can also obtain an approximate shape of the mass spectrum of the PBHs using Eq. (9) with D replaced by D"H/(2p" Q ") at different epoch corresponding to different black hole mass. More specifically the mass of black holes, M, and their initial fraction, b(M), can be written by an implicit function of t as I M"exp(2[H (t !t )!35])M , b(M)K0.888D( (t ))exp(!0.131072D\( (t ))) . (10) D I > I I Fig. 3 depicts the mass spectrum of black holes obtained from Eq. (10). Thus, the PBH abundance is sharply peaked. Note, however, that the shape of the large-mass tail is not exactly correct which corresponds to the regime where slow-roll solution is invalid. Nonetheless, this figure correctly describes the location of the peak up to a factor of order of unity. A more proper analysis of the mass function [21] based on a newer numerical calculation of PBH formation [22] would also change its shape slightly.
J. Yokoyama / Physics Reports 307 (1998) 133—139
137
Fig. 1. Evolution of the inflaton in chaotic new inflation. Time and are displayed in units of the Planck time and M , . respectively.
Fig. 2. Evolution of the scale factor in chaotic new inflation.
We can also apply our model for the formation of PBHs with different masses and abundance. For example, we may produce PBHs with MK10M which may act as a central engine of AGNs > with the current density, say, n&10\ Mpc\ corresponding to b&10\ at formation [19]. From the first equation of Eq. (10) we find M"10M corresponds to N"44 and the desired > spectrum is realized for j"3.7;10\, m"6.1;10\M , and v"0.21532324M under the . . COBE normalization [10]. Another interesting possibility is to produce a tiny amount of PBHs
138
J. Yokoyama / Physics Reports 307 (1998) 133—139
Fig. 3. Expected mass spectrum of PBHs in chaotic new inflation (Eq. (10)). Mass is displayed in units of the solar mass.
which are evaporating right now [20], with the initial mass MK10 g. With the current abundance XK10\ or bK10\ at formation, they may explain a class of gamma-ray bursts [5]. In this case we should have only a short period of slow-roll new inflation, N"14. We find b"2;10\ at the right mass scale if we choose j"3;10\, m"5;10\M , and . v"0.16557604828M . . In summary, we have proposed a new double inflation model in which chaotic inflation is followed by new inflation and large density fluctuation is generated in the beginning of the latter regime. We have applied it for formation of PBHs and shown that we can choose values of model parameters so that significant numbers of PBHs are produced in the mass scales of astrophysical interest [23]. The author is grateful to Professor Andrei Linde for his hospitality at Stanford University, where this work was done. This work was partially supported by the Monbusho.
References [1] Ya.B. Zel’dovich, I.D. Novikov, Sov. Astronomy 10 (1967) 602; S.W. Hawking, Mon. Not. R. Astron. Soc. 152 (1971) 75. [2] B.J. Carr, Astrophys. J. 201 (1975) 1. [3] B.J. Carr, Astrophys. J. 206 (1976) 8; S. Miyama, K. Sato, Prog. Theor. Phys. 59 (1978) 1012; I.D. Novikov, A.G. Polnarev, A.A. Starobinsky, Ya.B. Zel’dovich, Astron. Astrophys. 80 (1979) 104. [4] C. Alcock et al., Nature 365 (1990) 623; Phys. Rev. Lett. 74 (1995) 2867; Astrophys. J. 486 (1997) 697; E. Aubourg et al., Nature 365 (1993) 623; Astron. Astrophys. 301 (1995) 1. [5] D. Cline, D.A. Sanders, W. Hong, Astrophys. J. 486 (1997) 169. [6] A.H. Guth, Phys. Rev. D 23 (1981) 347; K. Sato, Mon. Not. R. Astron. Soc. 195 (1981) 467.
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[7] A.D. Linde, Phys. Lett. 108B (1982) 389; A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [8] A.D. Linde, Phys. Lett. 129B (1983) 177. [9] S.W. Hawking, Phys. Lett. 115B (1982) 295; A.A. Starobinsky, Phys. Lett. 117B (1982) 175; A.H. Guth, S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110. [10] C.L. Bennet et al., Astrophys. J. Lett. 464 (1996) 1. [11] L.F. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Lett. B 157 (1985) 361; J. Silk, M.S. Turner, Phys. Rev. D 35 (1987) 419; D. Polarski, A.A. Starobinsky, Nucl. Phys. B 385 (1992) 623. [12] J.H. Kung, R.H. Brandenberger, Phys. Rev. D 42 (1990) 1008. [13] A.D. Linde, Phys. Lett. B 327 (1994) 208; A. Vilenkin, Phys. Rev. Lett. 72 (1994) 3137. [14] A.A. Starobinsky, in: H.J. de Vega, N. Sanchez (Eds.), Field Theory, Quantum Gravity, and Strings, Lecture Notes in Physics, vol. 246, Springer, Berlin, 1986, 107. [15] P. Ivanov, Phys. Rev. D 57 (1998) 7145. [16] D.K. Nadezhin, I.D. Novikov, A.G. Polnarev, Sov. Astron 22 (1978) 129. [17] G.V. Bicknell, R.N. Henriksen, Astrophys. J 232 (1978) 670. [18] J. Yokoyama, Astron. Astrophys. 318 (1997) 673. [19] E.L. Turner, Astron. J 101 (1991) 5. [20] S.W. Hawking, Nature 248 (1974) 30; Comm. Math. Phys. 43 (1975) 199. [21] J. Yokoyama, Phys. Rev. D 58 (1998) 107502. [22] J.C. Niemeyer, K. Jedamzik, Phys. Rev. Lett. 80 (1998) 5481. [23] J. Yokoyama, Phys. Rev. D 58 (1998) 083510.
Formation of primordial black holes in the inflationary universe Jun’ichi Yokoyama* Department of Physics, Stanford University, Stanford, CA 94305, USA and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract We propose a new double inflation model containing only one inflaton scalar field, in which new inflation follows chaotic inflation. It is shown that the large density fluctuation generated in the beginning of new inflation can result in significant formation of primordial black holes on astrophysically interesting mass scales. 1998 Elsevier Science B.V. All rights reserved. PACS: 98.80.!k
If overdensity of order of unity exists in the hot early universe, a black hole can be formed when the perturbed region enters the Hubble radius [1]. While the properties of the primordial black holes (hereafter PBHs) thus produced were a subject of extensive study decades ago, there were no observational evidence of their existence and only observational constraints were obtained against their mass spectrum [2,3]. Recently, however, possibilities of their existence have been raised from a number of astrophysical and cosmological considerations, for example, in an attempt to explain the origin of MACHOs [4] or a class of gamma-ray bursts [5]. It is difficult, unfortunately, to realize a desired spectrum of density fluctuations for PBH formation in inflationary cosmology [6—8] because usual inflation models predict a scale-invariant spectrum [9] whose amplitude has been normalized to O(10\) by the observed anisotropy of the cosmic microwave background radiation (CMB) [10]. But in order to produce PBHs on some specific scale, we must prepare density perturbation whose amplitude has a high peak of O(10\) on the corresponding scale. In the present paper, we propose a new scenario of multiple inflation as a model to produce the desired spectrum of density fluctuation for PBH formation. Unlike previous double inflation
* E-mail: [email protected]. 0370-1573/98/$ — see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 4 4 - 1
134
J. Yokoyama / Physics Reports 307 (1998) 133—139
models [11], our model contains only one source of inflation. We assume the Einstein gravity and employ an inflaton scalar field, , with a simple potential which has an unstable local maximum at the origin. This is the same setup as the new inflation scenario [7], but chaotic inflation [8] is also possible if has a sufficiently large amplitude initially. In fact, Kung and Brandenberger [12] studied the initial distribution of a scalar field with such a potential and concluded that chaotic inflation would be much more likely than new inflation. Thus, we also start with chaotic inflation, but we show that new inflation is also possible when evolves towards the origin after chaotic inflation. In fact, if the parameters of the potential are appropriately chosen, the scalar field acquires the right amount of kinetic energy after chaotic inflation and climbs up the potential hill to near the origin to start slow rollover there. Hence, in this model the initial condition for new inflation is realized not due to the high-temperature symmetry restoration nor for a topological reason [13], but by dynamical evolution of the field which has already become sufficiently homogeneous because of the first stage of chaotic inflation. We shall refer this succession of inflation simply to chaotic new inflation. With an appropriate shape of the potential, density fluctuations generated during new inflation can have larger amplitude than those during chaotic inflation. Furthermore, since the power spectrum of fluctuation generated during new inflation can be tilted, it can have a peak on the comoving Hubble scale when the inflaton enters the slow-rollover phase during new inflation. If the peak amplitude is sufficiently large, it results in formation of PBHs on the horizon mass scale when the corresponding comoving scale reenters the Hubble radius during radiation domination. We shall show below that such a scenario is indeed possible with a simple potential of the inflaton field. We adopt the following potential
j 1 1 »[ ]"! m # ln ! #» , v 4 4 2
(1)
that is, typical one-loop effective potential with nonvanishing mass term at the origin. A potential of this type but with a positive mass-squared at the origin was employed in the original inflation scenario [6], but for our purpose we adopt a negative mass term. The potential (1) has four parameters, but one of them, » , is fixed from the requirement that the vacuum energy density vanishes at the potential minima ,$ . Another parameter, say j, can K be fixed from the amplitude of large-scale CMB fluctuations using the COBE data as in chaotic inflation. Hence, we are essentially left with two free parameters, v and m. While v mainly controls the speed of around the origin and its fate, i.e., to which minimum it falls, and m mainly governs duration of new inflation, the entire dynamics is determined by a complicated interplay of the three parameters. For example, we cannot determine j until we calculate the duration of new inflation which also depends on j itself for fixed values of m and v. Hence, we must numerically solve the equations of motion iteratively to find out appropriate values of parameters to produce PBHs at the right scale with the right amount. In the new inflation regime the slow roll-over classical solution, (t), and the scale factor, a(t), read
m 8p» , H (t!t ) , a(t)Je&R, H"
(t)" exp 3M 3H .
(2)
J. Yokoyama / Physics Reports 307 (1998) 133—139
135
where the subscript s stands for the onset of slow-roll new inflation. The linear curvature perturbation on the comoving scale l"2p/k is given by H H H , ,f D( (t )) 3f (3) I m 2p (t ) 2p" Q (t )" I I where t is the epoch when k-mode left the Hubble radius during inflation. The above expression of I U(l) is valid until the scale l reenters the Hubble radius with f" () in the matter (radiation) domination. We find the spectral index of power-law density fluctuation is given by n"1!2m/(3H). Thus, the power spectrum can be significantly tilted. The linear perturbation has a peak amplitude U(l)"f
H H (4) max D"3 ,D , m 2p
on the comoving scale l ,2p/k where k ,a(t )H . It can be large if turns out to be small. We apply the stochastic inflation method [14] to estimate the probability distribution function (PDF) of curvature fluctuations. Since our potential can be approximated as »[ ]"» !m in the new inflation regime, the PDF of the coarse-grained inflaton field K (x, t), P[ K " , t], can be analytically calculated by solving the Fokker—Planck equation jP 1 j H jP " » [ ]P# . (5) jt 3H j
8p j Taking the initial condition P[ , t ]"d( ! ), its solution is given by the Gaussian: 2m ( ! (t)) 3H 1 exp exp ! , p(t)" H (t!t ) !1 . (6) P[ , q]" 3H 2p(t) 8pm (2pp(t) From this distribution function, we estimate the PDF of metric perturbation. In order to incorporate nonlinear effects at least partially, we write the coarse-grained metric in the quasiisotropic form,
ds"!dt#aL ( K (t, x)) dx ,
(7)
following Ivanov [15], where the scale factor aL now depends on the coarse-grained spatial coordinate through K (x, t). We quantify the metric perturbation in terms of hK ,aL (t, x)/a(t)!1, with a(t) being the average scale factor. In order to estimate the abundance of PBHs produced we calculate the PDF of hK , P[hK "h], from Eq. (6). We are interested in the metric perturbations on the comoving scales that leave the Hubble radius in the period between t"t and tKt #H\ when the classical solution has rolled down to (t #H\)" exp(m/3H), , because these scales correspond to the peak of the power spectrum and dominates the formation of PBHs when they reenter the Hubble radius during radiation domination. The desired PDF is approximately given by
1 m
P , (1#ln(1#h)) . P[hª "h]" H 3H 1#h
(8)
136
J. Yokoyama / Physics Reports 307 (1998) 133—139
Explicitly, we find the probability that h exceeds a threshold value h as P[h'h ] c [(1#h )A!1] [(1#h )A!e\A] K exp ! , 2c[(1#h )A!e\A] (2p (1#h)A[(1#h)A!1][(1#h)A!e\A] where we have defined c"m/(3H) and c"cD/2. The criterion for black hole formation has been numerically investigated by Nadezhin et al. [16] and by Biknell and Henriksen [17]. Although it depends on the shape of the perturbed region, the generic value of the threshold reads h '0.75—0.9. Here we take the black hole threshold as h "0.75. Then we can express the initial volume fraction of the region collapsing into PBHs as b(M )"P[h'0.75], where the typical black hole mass, M , is equal to the horizon mass when the comoving scale l reenters the Hubble radius. Note that since our model predicts density fluctuation which is highly peaked on the comoving scale l the resultant mass function of PBHs is also sharply peaked at the mass around M . Let us now consider a specific example of formation of MACHO-PBHs [18]. For this purpose we must realize a peak with b&10\ on the comoving scale leaving the Hubble radius 35 e-folds expansion before the end of new inflation, that is, we must have N,H (t !t )"35, where t is the time new inflation is terminated. After some iterative trials we have chosen j"3;10\ and m"6;10\M , and then solved the equation of motion for various values of v. In this choice of . j and m we find new inflation lasts for more-than ten e-folds expansion if we take v in the range v"0.2131M !0.2147M . Hence, we do not need much fine-tuning of the model parameters to . . realize a new inflationary stage itself. We also find that settles down to if v50.21384364M K . and to ! if v40.21384363M . K . We can obtain the appropriate spectrum of fluctuation for MACHO-PBHs if we take v"0.21384360M . Figs. 1 and 2 depict evolution of the scale factor and the inflaton , respective. ly, with the initial condition a "1 at "3.5M . The chaotic inflation ends at K0.89M and G G . . the slow roll-over new inflation starts at "!4.03;10\M , . In this case we find . + c"0.300590 and the abundance of the PBHs at formation reads
b"0.888D exp(!0.131072D\) . (9) One can also obtain an approximate shape of the mass spectrum of the PBHs using Eq. (9) with D replaced by D"H/(2p" Q ") at different epoch corresponding to different black hole mass. More specifically the mass of black holes, M, and their initial fraction, b(M), can be written by an implicit function of t as I M"exp(2[H (t !t )!35])M , b(M)K0.888D( (t ))exp(!0.131072D\( (t ))) . (10) D I > I I Fig. 3 depicts the mass spectrum of black holes obtained from Eq. (10). Thus, the PBH abundance is sharply peaked. Note, however, that the shape of the large-mass tail is not exactly correct which corresponds to the regime where slow-roll solution is invalid. Nonetheless, this figure correctly describes the location of the peak up to a factor of order of unity. A more proper analysis of the mass function [21] based on a newer numerical calculation of PBH formation [22] would also change its shape slightly.
J. Yokoyama / Physics Reports 307 (1998) 133—139
137
Fig. 1. Evolution of the inflaton in chaotic new inflation. Time and are displayed in units of the Planck time and M , . respectively.
Fig. 2. Evolution of the scale factor in chaotic new inflation.
We can also apply our model for the formation of PBHs with different masses and abundance. For example, we may produce PBHs with MK10M which may act as a central engine of AGNs > with the current density, say, n&10\ Mpc\ corresponding to b&10\ at formation [19]. From the first equation of Eq. (10) we find M"10M corresponds to N"44 and the desired > spectrum is realized for j"3.7;10\, m"6.1;10\M , and v"0.21532324M under the . . COBE normalization [10]. Another interesting possibility is to produce a tiny amount of PBHs
138
J. Yokoyama / Physics Reports 307 (1998) 133—139
Fig. 3. Expected mass spectrum of PBHs in chaotic new inflation (Eq. (10)). Mass is displayed in units of the solar mass.
which are evaporating right now [20], with the initial mass MK10 g. With the current abundance XK10\ or bK10\ at formation, they may explain a class of gamma-ray bursts [5]. In this case we should have only a short period of slow-roll new inflation, N"14. We find b"2;10\ at the right mass scale if we choose j"3;10\, m"5;10\M , and . v"0.16557604828M . . In summary, we have proposed a new double inflation model in which chaotic inflation is followed by new inflation and large density fluctuation is generated in the beginning of the latter regime. We have applied it for formation of PBHs and shown that we can choose values of model parameters so that significant numbers of PBHs are produced in the mass scales of astrophysical interest [23]. The author is grateful to Professor Andrei Linde for his hospitality at Stanford University, where this work was done. This work was partially supported by the Monbusho.
References [1] Ya.B. Zel’dovich, I.D. Novikov, Sov. Astronomy 10 (1967) 602; S.W. Hawking, Mon. Not. R. Astron. Soc. 152 (1971) 75. [2] B.J. Carr, Astrophys. J. 201 (1975) 1. [3] B.J. Carr, Astrophys. J. 206 (1976) 8; S. Miyama, K. Sato, Prog. Theor. Phys. 59 (1978) 1012; I.D. Novikov, A.G. Polnarev, A.A. Starobinsky, Ya.B. Zel’dovich, Astron. Astrophys. 80 (1979) 104. [4] C. Alcock et al., Nature 365 (1990) 623; Phys. Rev. Lett. 74 (1995) 2867; Astrophys. J. 486 (1997) 697; E. Aubourg et al., Nature 365 (1993) 623; Astron. Astrophys. 301 (1995) 1. [5] D. Cline, D.A. Sanders, W. Hong, Astrophys. J. 486 (1997) 169. [6] A.H. Guth, Phys. Rev. D 23 (1981) 347; K. Sato, Mon. Not. R. Astron. Soc. 195 (1981) 467.
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[7] A.D. Linde, Phys. Lett. 108B (1982) 389; A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [8] A.D. Linde, Phys. Lett. 129B (1983) 177. [9] S.W. Hawking, Phys. Lett. 115B (1982) 295; A.A. Starobinsky, Phys. Lett. 117B (1982) 175; A.H. Guth, S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110. [10] C.L. Bennet et al., Astrophys. J. Lett. 464 (1996) 1. [11] L.F. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Lett. B 157 (1985) 361; J. Silk, M.S. Turner, Phys. Rev. D 35 (1987) 419; D. Polarski, A.A. Starobinsky, Nucl. Phys. B 385 (1992) 623. [12] J.H. Kung, R.H. Brandenberger, Phys. Rev. D 42 (1990) 1008. [13] A.D. Linde, Phys. Lett. B 327 (1994) 208; A. Vilenkin, Phys. Rev. Lett. 72 (1994) 3137. [14] A.A. Starobinsky, in: H.J. de Vega, N. Sanchez (Eds.), Field Theory, Quantum Gravity, and Strings, Lecture Notes in Physics, vol. 246, Springer, Berlin, 1986, 107. [15] P. Ivanov, Phys. Rev. D 57 (1998) 7145. [16] D.K. Nadezhin, I.D. Novikov, A.G. Polnarev, Sov. Astron 22 (1978) 129. [17] G.V. Bicknell, R.N. Henriksen, Astrophys. J 232 (1978) 670. [18] J. Yokoyama, Astron. Astrophys. 318 (1997) 673. [19] E.L. Turner, Astron. J 101 (1991) 5. [20] S.W. Hawking, Nature 248 (1974) 30; Comm. Math. Phys. 43 (1975) 199. [21] J. Yokoyama, Phys. Rev. D 58 (1998) 107502. [22] J.C. Niemeyer, K. Jedamzik, Phys. Rev. Lett. 80 (1998) 5481. [23] J. Yokoyama, Phys. Rev. D 58 (1998) 083510.