Primordial black holes from ultra-slow-roll inflation Yungui Gong1, ∗ 1
School of Physics, Huazhong University of Science
arXiv:1707.09578v2 [astro-ph.CO] 28 Aug 2017
and Technology, Wuhan, Hubei 430074, China
Abstract The next generation of space-borne gravitational wave detectors may detect gravitational waves from extreme mass-ratio inspirals with primordial black holes. To produce primordial black holes which contribute a non-negligible abundance of dark matter and are consistent with the observations, a large enhancement in the primordial curvature power spectrum is needed. For a single field slow-roll inflation, the enhancement requires a very flat potential for the inflaton, and this will increase the number of e-folds. To avoid the problem, an ultra-slow-roll inflation at the near inflection point is required. We elaborate the conditions to successfully produce primordial black hole dark matter from inflation and propose a toy model with polynomial potential to show the result.
∗
[email protected]
1
I.
INTRODUCTION
The detection of the gravitational waves (GWs) from black hole mergers by the LIGO Collaboration and Virgo Collaboration opens a new window to probe black hole physics [1–3]. The next generation of space-borne GW detectors will operate in the 0.1−100mHz frequency band and detect GWs from supermassive black hole binaries, Galactic white-dwarf binaries and extreme mass-ratio inspirals with primordial black holes (PBHs) [4, 5]. PBH can be taken as dark matter candidate. Due to the failure of direct detection of particle dark matter such as weakly interacting massive particles and axions, the interest in PBH as dark matter has grown recently [6–24]. PBHs with mass smaller than 1015 g would have evaporated by now through Hawking radiation [5]. Observations from femtolensing of gamma-ray bursts [25, 26], millilensing of compact radio sources [27], microlensing of quasars [28], the Milky way and Magellanic Cloud stars [29–32] constrained the mass range of PBHs to be from 1015 g to 2 × 1017 g and from 2 × 1020 g to 4 × 1024 g [10]. However, it was shown that the mass window for PBH as all dark matter is around 1020 g [17], or around 5×10−16 M , 2×10−14 M and 25 − 100M [24], so the mass window for PBH is narrow and needs further study. PBH forms in the radiation era as a result of gravitational collapse of density perturbations generated during inflation. To produce appreciable abundance of PBH dark matter, the curvature power spectrum needs to be amplified to the order of 0.01 near the end of inflation, so the slow-roll parameter is required to decrease by at least 7 orders of magnitude. This can be realized at a near inflection point where the potential becomes almost a constant [18, 33]. However, inflation either ends or the slow-roll condition is violated before reaching the near inflection point[21, 23, 34]. On the other hand, the deep decrease in will increase the number of e-folds N . Furthermore, it is even possible that the sudden change in lasts for as long as 60 e-folds, then the power spectrum is featureless only in a narrow range of scales around k∗ , and the µ distortion of the power spectrum becomes large. In [18, 33], the models avoid the violation of slow-roll and the jump in N at the near inflection point is around ∆N ∼ 30, but the enhancement of the power spectrum is less than 5 orders of magnitude. The reason for a small number of e-folds spent at the near inflection point is a period of ultra-slow-roll inflation during which the inflton rolls down the potential faster because the acceleration of the inflaton is locked by the friction term. In this letter, we give the conditions for the reach of the ultra-slow-roll inflation and use a toy model with the 2
polynomial potential to show that the successful production of the primordial dark matter can be achieved by ultra-slow-roll inflation and the problems for the slow-roll inflation do not exist for the ultra-slow-roll inflation.
II.
PRIMORDIAL BLACK HOLES FROM INFLATION
By using the Press-Schechter theory and smoothing on the scale M with a Gaussian, the fractional energy density of PBH in the Universe is δc ρPBH ≈ erfc √ = erfc β= ρtot 2Pδ
9δ pc 4 2Pζ
! ,
(1)
where the critical density perturbation is assumed to be in the range δc = 0.07 − 0.7 [5, 12– 14, 35–38], the density contrast δ is related with the primordial curvature perturbation ζ as δ = 4ζ/9 during radiation domination, and the power spectrum for the primordial scalar perturbation is Pζ =
H2 V (φ) (φ∗ ) ≈ Pζ (k∗ ) , 2 8π V (φ∗ ) (φ)
(2)
−2 where we choose MPl = 8πG = 1, the slow-roll parameter = 21 (Vφ /V )2 and Vφ = dV /dφ.
PBH with mass greater than M forms when the density contrast δ exceeds the critical value δc . The mass M of the PBH that forms during radiation domination is of the same order of the horizon mass MH , M = γMH , where MH =
4πρ 4π = . 3 H H
(3)
Since its formation, the energy density ρPBH of PBH scales slower than the total energy density ρtot of the Universe during radiation domination, so the relative contribution to the total energy density grows. Ignoring the mass accretion and evaporation and assuming that β(M ) is a constant, we have [17, 21, 39] −9 −1/2
β = 3.09 × 10 γ
ΩPBH h2 0.12
M M
1/2 .
(4)
If all the dark matters are PBHs with M = (10−13 M , 2 × 10−9 M ) [10, 22] and γ = 1, then we get β = (9.76 × 10−16 , 1.38 × 10−13 ). If M = (5 × 10−19 M , 10−16 M ) [10, 22], here the lower bound is set by Hawking radiation [5], then β = (2.2 × 10−18 , 3.1 × 10−17 ). Since Pζ (k∗ ) = 2.2 × 10−9 [40], for appreciable dark matter to be in the form of PBHs, we need a large enhancement on Pζ , for example, at least 7 orders of enhancement from 3
the slow-roll parameter . This requires the potential to be very flat at the enhancement point. To keep slow-roll inflation, the enhancement point should be a near inflection point. However, this enhancement not only decreases the field excursion [41], but also increases the number of e-folds because φe
Z N= φ
dφ p . 2(φ)
(5)
At the near inflection point φinfl , the slow-roll parameter decreases by 7 orders of magnitude, unless this change happens around ∆φ = 10−6 or less, it will contribute a lot to N . In fact, almost all the number of e-folds comes from the contribution around φinfl , then the power spectrum gets big boost in the small scales, this will cause big µ distortion [39, 42] because the µ distortion is [42, 43] Z
∞
µac ≈ kmin
dk Pζ (k)Wµ (k), k
(6)
where " Wµ (k) =2.8A2 exp − "
kˆ − exp − 32
2 ˆ [k/1360] 0.3 + k/340 ˆ ˆ 1 + [k/260] #2
! (7)
,
kmin ≈ 1Mpc−1 , A ≈ 0.9 and kˆ = k/[1 Mpc−1 ]. If changes too quickly at the near inflection point, then the slow-roll condition will be violated [21] because d √ = 2|η − 2|, dφ
(8)
where η = Vφφ /V . Therefore, we expect either of the following problems for single field inflation: (1) the slow-roll approximation breaks down around the near inflection point φinfl ; (2) N is much larger than 60; (3) the power spectrum is enhanced over almost the whole range of the scales k > k∗ and the µ distortion becomes large. However, if an ultra-slow-roll inflation [44–46] in reached, then the slow-roll formula (5) may overestimate the number of e-folds because the acceleration of the inflaton is locked by the friction term and the inflaton rolls down the potential faster [47, 48]. For example, in the critical Higgs inflation with the potential V (φ) = λ(φ)φ4 /4 and the nonminimal coupling 4
ξ(φ)φ2 R [33], where λ(φ) = λ0 + bλ ln2 (φ/µ),
(9)
ξ(φ) = ξ0 + bξ ln(φ/µ),
(10)
an ultra-slow-roll inflation is reached around the near inflection point φ/µ = 0.785 and we get the enhancement on the power spectrum by about three orders of magnitude if we take λ0 = 2.69 × 10−7 , ξ0 = 9.22, µ2 = 0.118, bλ = 1.1 × 10−6 and bξ = 10.9 [33]. In this model, the number of e-folds spent on the near inflection point is only ∆N = 35. However, the enhancement is not big enough to produce significant PBH dark matter. We propose a toy model with the polynomial potential to produce a significant amount of PBH dark matter without the problems for single slow-roll inflation mentioned above. We consider the polynomial potential " m # m=5 X φ , φ ≥ 0, λm V0 1 + MPl m=1 " V (φ) = m # m=3 X φ λm , φ < 0, V0 1 + MPl m=1
(11)
where the first three coefficients λ1 , λ2 and λ3 are determined from ns , r and n0s = dns /d ln k, the coefficients λ4 and λ5 are determined from the near-inflection condition Vφ ≈ 0 and Vφφ ≈ 0 1 . Choosing φ∗ = −0.5 and φinfl = 0.0376, and using the Planck 2015 results, k∗ = 0.05Mpc−1 , ns = 0.9674, r = 0.005, n0s = −0.0008 and Pζ = 2.2 × 10−9 [40], we get λ1 = −0.0353553, λ2 = −0.0115783, λ3 = −0.00235702, λ4 = 728.239, λ5 = −11882.9 and V0 = 1.63 × 10−10 . The potential along with its slow-roll parameters and η are shown in Fig. 1. From Fig. 1, we see that the slow-roll parameter η > 1 before the inflaton reaches the near inflection point φinfl . Even though η becomes larger than 1 briefly, the slow-roll condition is still satisfied. Furthermore, this property is important for the reach of the ultra-slow-roll inflation. By solving the equation of motion numerically, we find that inflation ends at φe = 0.14, the inflaton excursion is ∆φ = φe − φ∗ = 0.64MPl , and the number of e-folds N before the end of inflation when the scale k∗ = 0.05Mpc−1 exits the horizon is N = 60.99. We show 2 ˙ the evolution of the scale factor a(t), the Hubble flow slow-roll parameter H = −H/H and 1
This is the reason why we need to consider a polynomial with at least fifth degree.
5
2.0
V/V0 ϵ
1.5
η 1.0
0.5
0.0
-0.5
-1.0 -0.6
-0.4
-0.2
0.0
0.2
ϕ/Mpl
FIG. 1. The potential along with its slow-roll parameters for the polynomial model.
¨ ˙ in Fig. 2. From Fig. 2, we see that we have the ultra-slow-roll inflation ηH = −φ/(H φ) with ηH ≈ 3 around φinfl , the number of e-folds spent at φinfl is ∆N = 42 and H decreases by 7 orders of magnitude at φinfl . Note that slow-roll inflation is kept when η > 1. The condition η > 1 before φinfl guarantees the reach of ultra-slow-roll inflation at φinfl because the condition causes V 0 to decrease faster so that it becomes much smaller than the friction term 3H φ˙ at φinfl . To ensure enough number of e-folds and keep the power spectrum to be smooth and featureless over a wide range of scales, we also require a long period of slow-roll inflation, this can be achieved by choosing the potential to be a cubic polynomial. This is the reason we take the cubic polynomial for φ < 0. The power spectrum from the numerical solution is shown in Fig. 3. By using the result for the power spectrum, we find that the µ distortion is µac = 1.95 × 10−8 which is consistent with the observations [42]. From Fig. 3, we find that the maximum amplitude of the power spectrum is 0.0157. If we choose δc = 0.12, we get β = 0.031. If we choose δc = 0.45 or ζc = 1.01 [23], we get β = 7.59 × 10−16 . Therefore, the toy model can produce a significant amount of PBH dark matter and avoids the problems of large number of e-folds and large µ distortion.
III.
CONCLUSIONS
Dependent on the value of δc , the maximum value and the enhancement of the power spectrum needed for the production of appreciable amount of PBH dark matter varies significantly. To keep the slow-roll condition and inflation, the decrease in cannot happen √ instantly. Since N is inversely proportional to , the not-so-fast change of will cause the 6
3 2
ηH
1 0 -1 -2 -3 0
Log10 (ϵH )
-2 -4 -6 -8 -10 50 40
ΔN
30 20 10 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
ϕ/Mpl
FIG. 2. The evolutions of the number of e-folds ∆N = ln[a(φ)/a(0)], the Hubble flow slow-roll parameter H and ηH . 10-2
Pζ
10-5
10-8
10-11 0
10
20
30
40
50
ln(k/k* )
FIG. 3. The power spectrum for the polynomial model. We choose k∗ = 0.05Mpc−1 .
number of e-folds spent at φinfl to be larger than 60. This will lead to the enhancement of the power spectrum even at the scale k = 0.1Mpc−1 , and cause the µ distortion to be large. To overcome these problems for slow-roll inflation, an ultra-slow-roll inflation should be reached at φinfl because this will cause the inflaton to roll down the potential faster and the number of e-folds spent at φinfl will decrease while the enhancement of the power spectrum remains. To reach the ultra-slow-roll inflation, V 0 needs to decrease fast before φ reaches φinfl so that it becomes much smaller than the friction term 3H φ˙ at φinfl , this condition requires 7
|η| > 1 for a short period of time before φ reaches φinfl . Since |η| > 1 happens for a short period of time, inflation will continue. Because the potential decreases, the inflaton will not be trapped in the ultra-slow-roll inflation. In this letter, we consider the enhancement of the power spectrum by 7 orders of magnitude with the polynomial potential. The number of e-folds spent at φinfl is ∆N = 42 and the maximum amplitude of the power spectrum is Pζ = 0.0157, so β = 0.031 if we choose δc = 0.12 and β = 7.59 × 10−16 if we choose δc = 0.45.
ACKNOWLEDGMENTS
This research was supported in part by the Major Program of the National Natural Science Foundation of China under Grant No. 11690021, and the National Natural Science Foundation of China under Grant No. 11475065.
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