Mass bound for primordial black hole from trans-Planckian censorship conjecture
MMass bound for primordial black hole from trans-Planckian censorship conjecture
Rong-Gen Cai ∗ CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China
Shao-Jiang Wang † Tufts Institute of Cosmology, Department of Physics and Astronomy,Tufts University, 574 Boston Avenue, Medford, Massachusetts 02155, USA
The recently proposed trans-Planckian censorship conjecture (TCC) imposes a strong constrainton the inflationary Hubble scale, of which the upper bound could be largely relaxed by consideringa noninstantaneous reheating history. In this paper we will show that, if the primordial blackholes (PBHs) are formed at reentry in the radiation-dominated era from the enhanced curvatureperturbations at small scales, the TCC would impose a lower bound on the PBH mass M PBH >γ ( H end / GeV) M (cid:12) regardless of the details for reheating history, where γ is the collapse efficiencyfactor and H end is the Hubble scale at the end of inflation. In particular, the current open windowfor PBHs to make up all the cold dark matter could be totally ruled out if the inflationary Hubblescale is larger than 10 TeV. For the case of PBHs formed in an early matter-dominated era, anupper mass bound is obtained. I. INTRODUCTION
Perhaps the most astonishing insight on cosmology isthat the phenomena at the largest scale, like the cos-mic microwave background (CMB) [1–3] and large-scalestructure (LSS), could emerge from the phenomena ofquantum fluctuations at the smallest scale, which canbe traced back to an accelerating expansion phase inthe early Universe with shrinking comoving Hubble hori-zon 1 / ( aH ) in the inflationary cosmology [4–8], as wellas some alternative scenarios (see, e.g., Ref. [9] for areview). Such an early inflationary phase, if it lastedlong enough, would eventually stretch even the smallestquantum fluctuations of Planck size out of the Hubblehorizon, after which they would become classical andfrozen until reentry to be observed today. This raisesthe well-known inflationary trans-Planckian problem [10–14], since, in a consistent theory of quantum gravity,these trans-Planckian quantum fluctuations should re-main quantum so as not to jeopardize the effective fieldtheory (EFT) treatment on inflation. This leads to therecent claim [15] of the trans-Planckian censorship con-jecture (TCC) that no trans-Planckian mode should everexit the Hubble horizon that would otherwise belong tothe swampland.The TCC puts a strong constraint on the duration ofan early inflationary phase: a f a i < M Pl H f , (1)where a i,f are the scale factors at the beginning and endof that inflating phase, and H i,f are the corresponding ∗ [email protected] † [email protected] Hubble scales. Working with the approximation of a con-stant inflationary Hubble scale H i ≈ H f ≈ H inf , Eq. (1)could be translated into an upper bound on the inflation-ary e-folding number N inf (the e-folding number at theend of inflation is fixed to be zero throughout the paper):e N inf < M Pl H inf , (2)which serves as a stronger bound compared to an earlyestimation N inf < M /H from quantum gravity [16].If such an early inflationary phase is directly connectedto the phase of standard big bang expansion with instan-taneous reheating history, it could be quickly observed[17] from Eq. (2) that N inf = 46 .
2, leading to a strongconstraint on the inflationary Hubble scale and tensor-to-scalar ratio, H inf M Pl < e − N inf = 8 . × − , (3) r ≡ π P R (cid:18) H inf M Pl (cid:19) < . × − , (4)where P R ≈ . × − is used from Planck 2018 [3].However, the upper bound in Eq. (4) is so strong forslow-roll inflation models that it would cause a severefine-tuning of initial conditions.Fortunately, the upper bound [Eq. (4)] could belargely relaxed by considering a noninstantaneous re-heating history [18] (See also Refs. [19, 20] for non-thermal/nonstandard post-inflationary history). Start-ing with the observation that the current comoving Hub-ble horizon should be originated from the comoving Hub-ble horizon at the beginning of the inflation, 1 / ( a H ) (cid:46) / ( a i H i ), one arrives at H i M Pl < a H a f H f (5) a r X i v : . [ a s t r o - ph . C O ] F e b after appreciating the TCC bound [Eq. (1)]. To furtherevaluate the denominator a f H f at the end of inflation interms of a general reheating history characterized by ane-folding number N reh and an equation-of-state (EOS)parameter w reh , one could use the following relations: a f a reh = e − N reh , a reh a = (cid:18) g reh (cid:19) T T reh , (6)3 M H f e − N reh (1+ w reh ) = π g reh T , (7)where T reh , g reh , and a reh are the reheating temperature,the degrees of freedom of relativistic species, and the scalefactor at the end of reheating, respectively. The inflation-ary Hubble scale is therefore bounded from above by H i M Pl < e − N reh (1+3 w reh ) (11 / / ( π / / g − / H M Pl T T reh (8) (cid:46) H M Pl T T reh ≈ T T reh = 1 . × − (cid:18) T reh (cid:19) (9)where the first inequality in the second line is taken for anear-critical expansion after inflation with the EOS pa-rameter w reh (cid:38) − / r (cid:46) . × − (cid:18) T reh (cid:19) . (10)Now the upper bound in Eq. (10) with the reheating tem-perature at the lowest possible temperature required bybig bang nucleosynthesis (BBN) could be realized in somesupergravity- or string-inspired inflation models. On theother hand, the reheating temperature cannot be toolarge; otherwise, the inflationary energy density boundedby Eq. (8) could be smaller than the reheating energydensity. Therefore, by requiring 3 M H i (cid:38) π g reh T for Eq. (8), one obtains (cid:18) T reh M Pl (cid:19) (cid:46) π H T ; (11)namely, T reh (cid:46) . × GeV. See Ref. [21–24] for otherdiscussions on the TCC from the viewpoints of initialstate, dark matter, and warm inflation.In this paper, we will discuss the TCC implication onthe mass bound for the primordial black holes (PBHs)formed in the radiation-dominated era (Sec. II A) and anearly matter-dominated era (Sec. II B), assuming thatthe PBH formation at reentry comes from large curva-ture perturbations at small scales. We conclude in thelast section. It is worth noting that the derived PBHmass bounds would not be applicable to other scenariosof PBH production from curvatons [25, 26], scalar lumps[27, 28], cosmic strings [29, 30], domain walls [31, 32],primordial bubbles [33], bubble collisions [34, 35], andpreheating instability [36], to name just a few.
II. MASS BOUND FOR PBH FROM TCC
The only existing lower bound on the PBH mass, M PBH (cid:38) g, comes from the observation of theabsence of extragalactic photons [37, 38] during PBHevaporation[39, 40]. Here we will derive the mass boundfor PBH from a theoretical perspective of the TCC, irre-spective of details of the reheating history. A. PBHs formed in the radiation-dominated era
For PBHs collapsed from the horizon mass with effi-ciency factor γ ≈ . M PBH = γ πH − (3 M H ) = 4 πγM H form . (12)For PBHs formed in the radiation-dominated era (asshown in the left panel of Fig. 1), the Hubble scale atPBH formation, H form , should be at least smaller thanthe Hubble scale H end at the end of inflation, which isfurther constrained by the TCC bound from the total e-folding number of inflation set by the current observablescale, namely M PBH = 4 πγM H form ≥ πγM H end ≥ πγM Pl e N inf . (13)Therefore, it is expected to see a lower bound on the PBHmass formed in the radiation era.To be specific, H form could be related to the Hubblescale H PBH at the exit of the corresponding curvatureperturbations via the comoving relation a form H form = a PBH H PBH and the scaling relation H ∝ a − w ) / ,namely H form H PBH = a a a a a a H PBH H form = e − N PBH e − N reh H form H reh H PBH H form = e − N PBH + N reh ) H end H reh H PBH H end = e − N PBH + N reh ) e N reh (1+ w reh ) H PBH H end = e − N PBH + N reh (1 − w reh ) − ln H PBH H end ] . (14)The exponential factor in the above equation could berearranged in such a way that all dependence on the re-heating history could be totally removed by noting that[50] k CMB a H = a CMB a end a end a reh a reh a H CMB H = e − N CMB − N reh (cid:18) (cid:19) / g − / T T reh H CMB H (15)= e − N CMB − N reh (1 − w reh ) ( π / / (11 / / g − / T H H CMB M / H / ; a i a CMB a PBH a f a reh a form a M Pl - H inf - H form - H - H CMB - H PBH - H end - ⇤ - Δ N PBH - ⇥ N PBH ⇤ - N reh -- ⇥⇤ ---- N CMB ----- ⇥⇤ ------- N inf ------ ⇥ - - - - - - - - -
10 10987654321 Log [ M PBH / M ☉ ] Log f D M PB H Log [ γ H end / GeV ] E xc l uded i f γ H end > G e VE xc l uded i f H end > T e V EG γ Femto WD HSC EROS / MACHO U F D CMB
FIG. 1.
Left : Demonstration for PBH formation in the radiation-dominated era. The late matter- and dark-energy-dominatederas are omitted for clarity.
Right : The implication of our lower bound on PBH mass. The colored regions are excluded by theobservations ( EG γ : Extragalactic gamma ray [38], Femto: Femtolensing [41], WD: White dwarfs explosions [42], HSC: SubaruHyper Suprime-Cam microlensing [43], EROS/MACHO: EROS [44] and MACHO [45], UFD: Ultrafaint dwarfs [46], and CMB[47, 48]) on the PBH abundance in DM for a given PBH mass. The gray region could be excluded by our lower bound on PBHmass from the TCC if the inflationary Hubble scale is larger than the certain value indicated by the top frame ticks. namely,14 N reh (1 − w reh ) = − N CMB + 12 ln H M Pl H end + ln (cid:20) T k CMB ( π / / (11 / / g − / (cid:21) , (16)where Eqs. (6) and (7) are used to replace T reh in Eq.(15). Therefore, one finally arrives at H form H PBH = e − N tot − ∆ N PBH + ln H CMB H PBH ] , (17)where ∆ N PBH ≡ N CMB − N PBH is the difference in thee-folding number at the exit of the CMB pivot scale k CMB = 0 .
002 Mpc − with respect to the exit of cur-vature perturbations that collapse into PBHs at reentry,and N tot is an abbreviation for the combination N tot ≡ ln (cid:20) T k CMB ( π / / (11 / / g − / (cid:21) + 14 ln (cid:18) π r P R (cid:19) ≈ .
99 + 14 ln( r P R ) ≈
65 + 14 ln (cid:18) π H M (cid:19) . (18)Now the PBH mass can be expressed as M PBH = 4 √ γ (cid:18) H CMB H PBH (cid:19) e − ∆ N PBH ) M Pl . (19) For PBH fluctuations reentering the horizon after thereheating era, N PBH + N reh > N reh (1 + w reh ), hence∆ N PBH ≤ N inf − N reh (1 + 3 w reh ) ≤ ln M Pl H end − N reh (1 + 3 w reh ) , (20)where N CMB < N inf and the TCC bound are used. Afterusing H CMB (cid:38) H PBH , a lower bound is expected, M PBH > γ (cid:18) H end . × GeV (cid:19) e (1+3 w reh ) N reh M (cid:12) > γ (cid:18) H end . × GeV (cid:19) M (cid:12) , (21)which is independent of specific configurations of reheat-ing history. It is easy to see that if there is a lower boundon the inflationary Hubble scale, there would be a cor-responding lower bound on the PBH mass, below whichthere are no PBHs, as shown by the excluded gray re-gions in the right panel of Fig. 1. On the other hand,the PBH abundance in cold dark matter (DM) cannot beconstrained by the TCC, since it is exponentially sensi-tive to the small-scale enhancement in curvature pertur-bations.Several implications from our lower bound on PBHmass [Eq. (21)] are as follows: First, the observationallower bound M PBH (cid:38) − M (cid:12) can always be fulfilled aslong as the inflationary scale H end (cid:38) γ − / GeV. Second,no PBH with mass smaller than 10 M (cid:12) is allowed if theinflationary Hubble scale is larger than 10 γ − / GeV.Fortunately, such LIGO-type PBHs could be allowed bythe TCC, because the reheating-assisted TCC bound onthe inflationary Hubble scale [Eq. (8)] forbids any infla-tionary scale larger than 10 GeV unless the reheatingtemperature is lower than 1 MeV that would otherwiseviolate the BBN constraint on reheating temperature.Third, currently there is an open window [51, 52] be-low sublunar mass for PBHs making up all the cold DM;however, such a window could be totally closed if theinflationary Hubble scale is larger than 10 TeV scale.
B. PBHs formed in an early matter-dominated era
To minimally extend the previous discussion on massbound for PBHs to other production channels, onecould also consider PBH formation in an early matter-dominated era right after inflation but before the reheat-ing era [53–55], which is shown by the gray region in theleft panel of Fig. 2. After simple manipulations withthe comoving relation a form H form = a PBH H PBH and thescaling relation H ∝ a − w ) / , one could express theHubble scale at PBH formation in terms of the Hubblescale at the exit of the enhanced small-scale fluctuationsdirectly by H form H PBH = a PBH a form = a a a a H H = e − N PBH H H (22)without referring to the reheating era. Hence the PBHmass becomes M PBH = 4 πγ M H H e N PBH < πγ M H H e N inf < πγ M H H end (cid:46) πγ M H , (23)where we have used N PBH < N inf < ln( M Pl /H end ) and H PBH (cid:38) H end . Now the PBH mass formed in an earlymatter-dominated era has an upper bound as M PBH < γ (cid:18) . × GeV H end (cid:19) M (cid:12) , (24) above which is excluded, as shown in gray in the rightpanel of Fig. 2 if the inflationary scale is larger thana certain value. Several implications from the abovebound are in order: First, no PBH with mass larger than10 − M (cid:12) is allowed if the inflationary Hubble scale islarger than 10 GeV. Second, LIGO-type PBHs cannotfit the upper bound [Eq. (24)] if the inflationary Hub-ble scale is larger than 10 GeV. Third, PBHs formedin an early matter-dominated era with mass larger than10 M (cid:12) are not allowed if the inflationary Hubble scaleis larger than 10 GeV.
III. CONCLUSION
In this paper, using only the recently proposed TCCbound on the inflationary e-folding number, we derivethe mass bounds for PBHs formed in the radiation-dominated and early matter-dominated eras from theenhanced curvature perturbations at small scales in theinflationary cosmology. The explicit dependences on thedetail configurations of reheating history are carefully re-moved. The resulting mass bounds for PBH thereforeonly rely on the inflationary Hubble scale. In particu-lar, for PBHs formed in the radiation-dominated era, theasteroid-mass PBHs observationally allowed to make upall the cold DM cannot exist if the inflationary Hubblescale is higher than 10 TeV scale.
ACKNOWLEDGMENTS
We thank Shi Pi for helpful discussion. R.-G. C.was supported by the National Natural Science Foun-dation of China Grants No. 11690022, No. 11821505and No. 11851302, and by the Strategic Priority Re-search Program of Chinese Academy of Sciences GrantNo. XDB23030100, and by the Key Research Programof Frontier Sciences of the Chinese Academy of Sciences.S.-J. W. is supported by the postdoctoral scholarship ofTufts University from the National Science Foundation. [1] K. M. Gorski, A. J. Banday, C. L. Bennett, G. Hinshaw,A. Kogut, G. F. Smoot, and E. L. Wright, “Powerspectrum of primordial inhomogeneity determined fromthe four year COBE DMR sky maps,” Astrophys. J. (1996) L11, arXiv:astro-ph/9601063 [astro-ph] .[2]
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