Constraints on the curvature power spectrum from primordial black hole evaporation
CConstraints on the curvature power spectrum from primordialblack hole evaporation
Ioannis Dalianis aa Physics Division, National Technical University of Athens15780 Zografou Campus, Athens, Greece
Abstract
We estimate the maximum allowed amplitude for the power spectrum of the primordialcurvature perturbations, P R ( k ), on all scales from the absence of any detection signals of sub-solar mass black holes. In particular we analyze the constraints on the PBHs and we focus onthe low mass limit where the Hawking radiation is expected to significantly influence the bigbang observables, considering also different early cosmic histories. We derive the upper boundsfor the variance of density perturbations, σ ( M ), for any possible reheating temperature as wellas for the cosmological scenario of a scalar condensate domination. We expect our results tohave considerable implications for models designed to generate PBHs, especially in the lowmass range, and provide additional constraints to a large class of inflationary models. The primordial power spectrum of the comoving curvature perturbations P R ( k ) has been pre-cisely measured by the CMB probes in the range of scales between k ∼ − Mpc − and 1Mpc − . In smaller scales the P R ( k ) is poorly and indirectly constrained by observations ofnonlinear structures. The relevant limits are very weak coming mainly from the mass fractionof the universe, β ( M ), that collapsed and formed primordial black holes (PBH) of mass M .Black holes affect dynamical systems and cause microlensing events and so a bound on the β ( M ) value is obtained. The Hawking prediction [1, 2] that black holes radiate thermally withtemperature T BH = 1 . M M (cid:39) (cid:18) g M (cid:19) GeV , (1)and evaporate after a timescale τ ( M ) (cid:39) × (cid:18) M g (cid:19) s , (2)provides us with additional bounds on β ( M ) for small mass PBHs from the absence of anyevidence for black hole evaporation. Consequently, P R ( k ) bounds in the smallest range of scalescan be obtained. In the inflationary framework the measurement of the P R ( k ) can be regardedas an insight into the microscopic dynamics of the field(s) that dominated the energy densityof the early universe and generated the primordial perturbations. The purpose of this paper isto make use of the limits on β ( M ) coming primarily from the CMB and BBN observables toconstrain the variance of the density perturbations and therefore, the cosmological scenarios,such as inflation, designed to trigger PBH formation.1 a r X i v : . [ a s t r o - ph . C O ] D ec BHs form from the collapse of large-amplitude inhomogeneities [3–7]. In order to decouplefrom the background expansion it has to be
GM/R ∼
1, for a region of mass M over a scale R . This can be achieved if the power spectrum P R ( k ) is enhanced at a scale R − ∼ k ,characteristic of the PBH mass, by many orders of magnitude. Large wavenumbers yield lightPBH which if they have mass M (cid:46) g evaporate at timescales less than the age of theuniverse. PBHs with M > g would still survive today and would be dynamically coldcomponent of the dark matter in galactic structures. To distinguish between the nonevaporatedand the evaporated PBH we label the mass and the characteristic wavenumber of the formerwith a dark dot subscript, i.e. M • , k • respectively.The formation of a primordial black hole of mass similar to the black holes detected byLIGO [8], M • ∼ M (cid:12) requires P R (10 Mpc − ) ∼ − . Similar values for the P R ( k ) arerequired for the formation of lighter primordial black holes that, although lack observationalsupport, are well motivated dark matter candidates. Actually, it is the low mass window, M • (cid:28) M (cid:12) , the most promising one for explaining the dark matter in the galaxies, accordingto the current observational constraints. Several inflation models that achieve the required P R ( k ) enhancement have been proposed the last years [9–27] putting forward new ideas andelaborating further earlier works [28–33]. The shape of the P R ( k ) at small scales is mainlyconstrained at a scale k • where the abundance of relic PBHs maximizes. The aim of this workis to stress that, depending on the postinflationary expansion history, the shape of the P R ( k )at smaller scales k (cid:29) k • is crucial to affirm the viability of a model designed to generate darkmatter PBHs.Large values for the P R ( k ) at small scales may generate short-lived PBH that, althoughabsent from the cosmic structures today, evaporate in the early universe leaving potentiallyobservable signatures. The thermal emission of black holes affects the BBN [34–41] in themass range 10 − g and bounds on the fraction of the universe mass that collapses intoblack holes, β ( M ) are induced. In addition, the diffuse extragalactic γ -ray background putconstraints on the mass range 10 − g [42–46]. The most stringent constraint on themean number density of the short-lived PBHs comes from the CMB anisotropy damping [35]which limits the β ( M ) (cid:46) − [47] in the mass range 10 − g. In Ref. [41] these constraintsare outlined and further references can be found therein.Our investigation is focused on the β ( M ) bounds for M (cid:28) g it will be shown thatextra important constraints can be put on the P R ( k ). From a different perspective theseconstraints can be viewed also as an insight into the unknown cosmic history of the earlyuniverse if a measurement of the P R ( k ) on small scales is made possible. Apparently, the keyrelation is the one that connects the power spectrum P R ( k ) and the β ( M ). The knowledgeof the β ( M ) can constrain the P R ( k ) only if one assumes a model for the PBH formation. Inthe following analysis we assume spherical symmetric Gaussian primordial perturbations andthat the PBHs form on the high σ -tail according to the PressSchechter formalism [48]. Weconsider gravitational collapse during radiation era as well as during presureless matter erataking into account spin effects. This is actually a distinct ingredient of this work. We followthe monochromatic mass spectrum approximation and assume a one-to-one correspondencebetween the scale of perturbation and the mass of PBHs. We do not consider possible impactson the power spectrum from non-Gaussianities [49, 50] and quantum diffusion effects [51–54].The main result of this work is the derivation of upper bounds for the variance of comovingdensity contrast at horizon entry σ ( M ) on all scales for different reheating temperatures andcosmological scenarios and translate these bounds onto P R ( k ) bounds. In particular, we derivethe upper bound for the σ ( M ) in order that the CMB and BBN observables remain intact forany possible reheating temperature. These bounds are the most stringent ones for promptlyevaporating PBHs. Most of our analysis is general regardless the mechanism that generates theperturbation spectrum. We implicitly assume that it is inflation behind the P R ( k ) generation,however, we do not specify the inflaton dynamics apart from the energy scale that inflationends.Large-amplitude inhomogeneities are necessary for the PBH formation however, the forma-2ion rate may significantly increase [55, 56] if the equation of state of the background energydensity, w , becomes soft or zero. This is a rather plausible scenario for PBHs that form notlong after the end of the inflationary phase where the inflaton coherent oscillations result in anearly matter domination era. Other scenarios, such a modulus domination that is natural inseveral extensions of the Standard Model of particle physics, also result in an early non-thermalphase. The most important implications of such a pressureles phase for the models of PBHformation is that the variance of the density perturbations, which determines the β ( M ), canbe smaller for a fixed PBH abundance.The recent works [57,58] have also examined the P R ( k ) constraints on all scales in a similarcontext. In this work we present new results and elucidate different questions. In particular,we complement part of their analysis by including the spin effects for gravitational collapse,that are crucial and change considerably the corresponding bounds on the variance σ ( M ) andthe power spectrum P R ( k ). We also derive the constraints on the PBH production scenariosfor any reheating temperature and in addition we examine the scenario of a non-thermal phasedue to a modulus field. Finally, we estimate constraints for the spectral index value of the tailof the power spectrum with respect to the reheating temperature and estimate the maximumallowed value for the power spectrum taking into account the BBN and CMB constraints alongwith the f PBH bounds.The structure of the paper is the following. In section 2 we discuss the observationalbounds on the mass fraction of the universe that collapses into PBH, β ( M ), introducing the β ( M ) constraints for the matter domination (MD) era in addition to those for the radiationdomination (RD) era. In section 3 we derive the expressions that relate the PBH mass and thecomoving horizon scale for different cosmic histories, that we generalize in section 7. In section4 we derive the principle results of this work, that is the upper bound for the variance of thedensity perturbations for any reheating temperature. In section 5 we estimate the maximumpossible amplitude for the power spectrum, A max , with respect to the reheating temperatureconsidering constraints both on the relic PBH abundance and the Hawking radiation. In section6 we examine the cosmological scenario of an intermediate non-thermal phase due to moduluscondensate domination. In section 7 we present the full power spectrum constraints and brieflydiscuss additional constraints that apply at larger scales. We conclude in section 8 where weoutline and discuss the implications of the constraints derived in this work for particular classesof inflationary models. In the Appendix we assume a particular morphology for the P R ( k ) forlarge wavenumbers and illustrate the tension with the big bang observables that a wide powerspectrum peak may generate. β bound for PBH formation during radiation and matterdomination eras A PBH with mass M forms during a radiation dominated (RD) era if a preexisting overdensitywith wavelength k − enters the horizon after the reheating of the universe. The PBH mass isequal to γM hor where M hor is the horizon mass and γ a numerical factor which depends on thedetails of gravitational collapse. This consideration is regarded as the conventional one for PBHformation, for a different recent suggestion see Ref. [59]. The present ratio of the abundance ofPBHs with mass M over the total dark matter (DM) abundance, f PBH ( M ) ≡ Ω PBH ( M ) / Ω DM ,can be expressed as f PBH ( M ) = (cid:18) β ( M )7 . × − (cid:19) (cid:18) Ω DM h . (cid:19) − (cid:16) γ R . (cid:17) (cid:18) g ( T k )106 . (cid:19) − (cid:18) M g (cid:19) − / , (3)where we took again the effective degrees of freedom g ∗ and g s approximately equal. We alsowrote the numerical factor γ during RD as γ R in order to distinguish it from that during MD,labeled γ M , since their values are expected to be different. The observational constraints put3 rh > GeV E n t r op y D M BBNCMB E G γ Ω p b h WD S u b a r u H S C M A C H O / E R O S O G L E Dwarfs CMBGW P l a n ck - - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' τ [ s ] Femto E G γ WD SubaruHSC MACHOS / EROS / OGLEDwarfsCMBGW �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� - � �� - � �� - � ���������������� �� - �� �� - �� �� - �� �� - �� �� - � �� - � �� - � �� - � � �� � �� � � [ � ] � � � � = Ω � � � / Ω �� � / � ⊙ Figure 1:
Left panel:
The figure depicts updated upper bounds on β (cid:48) ( M ) (and β (cid:48) ( τ ) on theupper axis with t indicating the age of the universe) from Planck and LSP relics, BBN, CMBanisotropy, extragalactic gamma rays, due to evaporating black holes and density bounds onnonevaporated black holes for arbitrarily large reheating temperature, from Ref. [41]. The DMconstraint is depicted with dotted-dashed line because it is model dependent and subject tomuch uncertainty; the constraint associated with GWs is partial and included only as referencefollowing Ref. [41]. The focus of the present work is mostly on the red line, i.e. the BBN andCMB constraints. Right panel:
The fractional abundance of the nonevaporated PBH.upon the PBH dark matter fraction f PBH ( M ) an upper bound f max . However for evaporatingPBH it is more meaningful to use the β ( M ) instead, β (cid:48) ( M ) = 6 . × − f PBH (cid:18) M g (cid:19) / (4)where we defined β (cid:48) ( M ) ≡ γ / (cid:16) g ∗ . (cid:17) − / β ( M ) (5)and took Ω DM h = 0 .
12. The observational upper bound on f PBH ( M ) which from Eq. (4) istranslated into an upper bound on β ( M ) that we call C M , β (cid:48) ( M ) < C M ≡ β (cid:48) RD, max ( M ) . (6)The evaporation of the PBH formed will not affect the cosmological observables if the constraint(6) is satisfied. The constraint line C M is depicted in Fig. 1, following the results of Ref. [41].The inequality (4) is written in terms of the black holes lifetime and yield as β (cid:48) ( τ ) < . × ( τ /s ) / Y PBH . For lifetime τ < seconds there are limits on the amount of thethermal radiation from the PBHs evaporation due to the production of entropy, that may bein conflict with the cosmological photon-to-baryon ratio, [60], dark matter, e.g. the lightestsupersymmetric particle, or Planck-mass relics [61–63], labeled entropy, DM (with dotted-dashed line due to model dependence of this constraint) and Planck respectively in our figures.These ultra light PBHs give interesting constraints, however, in this work we will focus onPBHs with larger lifetimes since the presence of such PBHs might be in conflict with the
BBN and
CMB observables implying that the β ( M ) has to be particularly suppressed. For τ = 10 − s , that corresponds to M = 10 − g hadrodissociation processes becomeimportant and the debris deuterons and nonthermally produced Li; for τ = 10 − s, thatcorresponds to M = 10 − g, photodissociation processes overproduce He and D and putstrong constraints on β ( M ) [42–46]. In addition, the heat produced by PBHs evaporation afterthe time of recombination may damp small-scale CMB anisotropies contrary to observations.The electrons and positron scatter off the the CMB photons and heat the surrounding matter.The small scale CMB anisotropies will remain intact by the PBHs evaporation if β (cid:48) ( M ) (cid:46) × − ( M/ g) . for 2 . × (cid:46) M (cid:46) . × [41, 47, 64]. This is stronger than all4he other available limits on the β (cid:48) ( M ). In the next sections we will utilize the BBN boundand CMB (monochromatic) bounds β (cid:48) (5 × g) < − and β (cid:48) (2 . × g) < × − respectively, to derive the stringent constraints on the variance of the density perturbations onsmall scales.Apart from the constraints on evaporated PBHs, that is the primary interest of this work,there are numerous constraints on PBHs present in the late universe, which are the most com-monly applied. In the late universe, the PBH evaporation rate is constrained from the extragalactic gamma-ray background [42–46]. Black holes of mass above 10 g are subject to gravi-tational lensing constraints [65–67], labeled with Subaru HSC, MACHOS, EROS OGLE in theplots. The recent results of Ref. [68] remove the femtolensing constraints and we accordinglyupdated the plots. Also, black holes influence the trajectory and the dynamics of other astro-physial objects such as neutron stars and white dwarfs [69–73] that constrain the abundance oflight black holes, labeled WD . The CMB constrains the PBH with mass above 10 g becausethe accretion of gas and the associated emission of radiation during the recombination epochcould affect the CMB anisotropies [74]. Recently it has been claimed that the CMB boundson massive PBH may be relaxed due to uncertainties in the modeling of the relevant physicalprocesses [75–77]. Finally, there are indirect constraints from the pulsar timing array exper-iments on the gravitational waves ( GW ) associated with the formation of relatively massivePBH at the epoch of horizon entry. Notably, a very severe constraint, β ( M ) (cid:46) − , on themass band 10 − M (cid:12) comes from pulsar timing data since the large scalar perturbationswhich are necessary to produce the PBHs also generate second-order tensor perturbations [78].GW can constrain a larger window of relic PBHs mass. Bounds from GW are very interestingbut we do not examine them further since they apply on the large mass window of PBHs,beyond the scope of this work. Only for comparison with the other observational constraintswe include only the GW bound of Ref. [78] (depicted with dotted-dashed line in the figures)and add a brief discussion on secondary GW in section 8. We note finally that 21 cm obser-vations [79] could potentially provide a stronger constraint in the mass range around 10 g,with β (cid:48) ( M ) < × − ( M/ g) / for M > g but such limits do not exist at present.The combined upper bounds on β (cid:48) ( M ) and f PBH are collectively depicted in Fig. 1.
If the PBH form during pressureless reheating stage, i.e. matter domination (MD) era, thecorresponding wavelength k − enters the horizon before the complete decay of the inflaton andit is f (MD)PBH ( M ) = (cid:18) β MD ( M )2 . × − (cid:19) (cid:18) Ω DM h . (cid:19) − (cid:16) γ M . (cid:17) (cid:18) g ( T rh )106 . (cid:19) − (cid:18) M g (cid:19) − / (cid:18) kk rh (cid:19) − / (7)The extra factor ( k/k rh ) / accounts for the different redshift of the energy density of the thematter dominated universe compared to the radiation dominated universe. This scenario inthe realistic framework of the inflationary attractors has been examined in Ref. [25]. In thiscase we find that γ / (cid:16) g ∗ . (cid:17) − / β MD ( M ) = 6 . × − f (MD)PBH (cid:18) M g (cid:19) / (cid:18) kk rh (cid:19) − / . (8)As we will show in the Section 3, it is k/k rh ∝ γ / g − / ∗ T − / M − / , and the PBH dark matterfraction is written as, f (MD)PBH = 1 . × γ M β MD ( M ) (cid:18) T rh GeV (cid:19) . (9)5he f (MD)PBH value has to be below the observational bound f max and, on the same footing withthe definition (5), we define for the matter domination era, β (cid:48) MD ( M ) ≡ γ M β MD ( M ) (10)which is independent of thermal degrees of freedom g ∗ as it should. The corresponding obser-vational bound on β (cid:48) ( M ) in terms of the C M reads, β (cid:48) MD ( M ) < . (cid:18) T rh GeV (cid:19) − (cid:18) M g (cid:19) − / C M ≡ β (cid:48) MD, max ( M, T rh ) . (11)Hence, for k rh < k , or equivalently M/γ R < M rh , where M rh the horizon mass at the endof reheating, the maximum mass fraction of the universe allowed to collapsed into PBH istemperature dependent, see Fig. 1.In the inflationary framework the upper bound on β (cid:48) ( M ) is effective only if the formationof PBH with mass M is possible. The horizon mass right after inflation is M end = 4 πM /H end and the bounds are meaningful for PBH with masses M > γ M end , that is for H end > . × GeV (cid:18)
M/γ R g (cid:19) − (12)The above inequality yields a lower bound for the inflation energy scale. A PBH with mass M will form due to superhorizon perturbation if the corresponding wavelength k − is larger thanthe horizon distance at the end of inflation. Thus a different way to write the condition (12) is M > M end or k end > k . (13)In the following we will find the expression k ( M ) in order to recast the β (cid:48) ( M ) bound into σ ( k ) and put the constraints onto the power spectrum. This requires to determine the evolutionof the cosmic horizon during RD and MD eras. The relation between the mass M contained in the comoving horizon of size k − is requiredin order to specify the PBH mass generated at that scale. This relation is also necessary inorder to make contact of the β ( M ) bounds with the power spectrum of the comoving curvatureperturbation P R ( k ). In a radiation dominated universe the mass contained in the horizon ofsize k − is M hor ( k ) = M rh ( k/k rh ) − , where M rh the horizon mass at the time of the reheatingof the universe. In the following we will determine the scales k end and k rh . It is k end = k rh e ˜ N rh / , k rh = k e ∆ ˜ N (RD)0 . (14)where ˜ N rh are the efolds that take place after inflation until the onset of the RD phase and∆ ˜ N RD are the efolds that take place from the onset of the continuous RD phase until thereentry epoch of the scale k − . . The k . is the Planck pivot scale k = 0 .
05 Mpc − ≡ k . where the spectrum is tightly constrained [80]. In the case that an additional pressurelessnon-thermal phase that last for ˜ N X efolds follows the reheating of the universe then it is k rh = k . e ∆ ˜ N RD + ˜ N X / .The ˜ N rh and ∆ ˜ N (RD)0 . are related with the inflationary dynamics by the expression N . + ln (cid:18) H end H . (cid:19) INF = 12 ˜ N rh + 12 ˜ N X + ∆ ˜ N (RD)0 . ≡ ˜ N . . (15)where N X are the e-folds of a possible postinflationary non-thermal phase. In following sectionswe will consider the scenario of an early cosmic era dominated by a scalar-condensate with zeroeffective equation of state. 6 rh = GeV D M P l a n ck BBNCMB E G γ Ω p b h WD S u b a r u H S C M A C H O / E R O S O G L E DwarfsCMBGW - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' τ [ s ] T rh = GeV P l a n ck D M BBNCMB E G γ Ω p b h WD S u b a r u H S C M A C H O / E R O S O G L E DwarfsCMBGW - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' τ [ s ] T rh = GeV P l a n ck D M BBNCMB E G γ Ω pbh WD S u b a r u H S C M A C H O / E R O S O G L E DwarfsCMBGW - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' τ [ s ] T rh = GeV P l a n ck D M BBNCMB E G γ Ω pbh WD S ub a r u H S C M A C H O / E R O S O G L E DwarfsCMBGW - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' τ [ s ] Figure 1: The plots depict combined upper bounds on β (cid:48) ( M ) for reheating temperatures T rh = 10 , , , GeV. It is β (cid:48) ( M ) = γ / ( g ∗ / . − / β ( M ) for M > γ R M rh and β (cid:48) ( M ) = γ M β ( M ) for M < γ R M rh . The vertical line indicates the PBH mass forming at theepoch of reheating. The dotted lines depict the β (cid:48) ( M ) constraints for arbitrarily large reheatingtemperature, as in Fig. 1.The number of e-folds during inflation N . are analyzed as [81, 82], N . = 67 − ln (cid:18) k . a H (cid:19) + 14 ln (cid:18) V . M ρ end (cid:19) + 1 − w rh w rh ) ln ρ rh ρ end −
14 ˜ N X , (16)where we considered ¯ w X = 0 for the scalar-condensate domination. The measured value P R = V . / (24 π (cid:15) . M ) = 2 . × − gives that ln( V / . M − / √
3) = − . / (cid:15) . ) andln( k . / ( a H )) (cid:39) .
4. For ¯ w rh = 0 we attain the e-folds ∆ ˜ N (RD)0 . when the expression (15) isplugged in ∆ ˜ N (RD)0 . = 57 . −
12 ln (cid:18) H . H end (cid:19) INF + 14 ln r . − (cid:16) ˜ N rh + ˜ N X (cid:17) (17)The H . is written in terms of the inflationary observable r . as H . ( r ) = 8 . × GeV × (cid:112) r . / . N (RD)0 . (cid:39) (57 . − .
3) + 12 ln (cid:18) H end GeV (cid:19) − (cid:16) ˜ N rh + ˜ N X (cid:17) . (18)Now, the ˜ N rh is related to the reheating temperature as ρ rh = ρ end e − N rh = π g ∗ T /
30 andwe attain ˜ N rh ( T rh , H end , g ∗ ) = −
43 ln (cid:34)(cid:18) π g ∗ (cid:19) / T rh ( H end M Pl ) / (cid:35) . (19)7ne sees that the T rh maximizes when ˜ N rh = 0 and H end is maximum. The ∆ ˜ N (RD)0 . is recastinto ∆ ˜ N (RD)0 . ( T rh , g ∗ ) (cid:39) . (cid:34)(cid:18) π g ∗ (cid:19) / T rh (GeV M Pl ) / (cid:35) −
34 ˜ N X , (20)where the H end has canceled out, as expected. From Eq. (14) it is k end = k . e ( ˜ N rh + ˜ N X ) / N (0 . k end ( T rh , H end , g ∗ ) (cid:39) k . e . (cid:18) H end GeV (cid:19) / (cid:34)(cid:18) π g ∗ (cid:19) / T rh (GeV M Pl ) / (cid:35) / e − ˜ N X / (21)This is the wavenumber that corresponds to the horizon mass at the end of inflation, M end .Accordingly we obtain the horizon wavenumber, k rh = k . e ∆ ˜ N RD + ˜ N X / , at the moment ofreheating, Γ inf = H . It is k rh ( T rh , g ∗ ) (cid:39) k . e . (cid:34)(cid:18) π g ∗ (cid:19) / T rh (GeV M Pl ) / (cid:35) e − ˜ N X / (22)During pressureless reheating (MD), the relation between the the scale k − and the horizonmass M/γ M is k MD = k end (cid:18) πM H end (cid:19) / (cid:18) Mγ M (cid:19) − / , for k MD > k rh . (23)Utilizing the relation (21) and after normalizing the PBH mass, the reheating temperature andthe relativistic degrees of freedom we obtain for N X = 0, k MD ( M, T rh , g ∗ ) = 8 . × Mpc − γ / (cid:18) M g (cid:19) − / (cid:18) T rh GeV (cid:19) / (cid:16) g ∗ . (cid:17) / . (24)After the completion of reheating the universe is in a thermal equilibrium state with temper-ature T rh and the radiation domination phase commends. The horizon mass at that stage is M rh = M hor ( T rh , g ∗ ), M rh = 4 π (cid:18) π g ∗ (cid:19) − / M T = 9 . × g (cid:18) T rh GeV (cid:19) − (cid:16) g ∗ . (cid:17) − / . (25)During the RD era the relation between the scale k − and the horizon mass M/γ R is k RD = k rh (cid:18) M/γ R M rh (cid:19) − / , for k RD < k rh , (26)where we substituted the expression for k rh . Plugging in numbers we obtain k RD ( M ) = 1 . × Mpc − γ / (cid:18) M g (cid:19) − / . (27)Collectively we write the PBH mass M and the horizon scale relation, k ( M, T rh , g ∗ ) = k MD ( M, T rh , g ∗ ) , for k > k rh k RD ( M ) for k < k rh (28)For a range of reheating temperatures the k = k ( M ) relation is depicted in Fig. 2. Thisrelation is necessary in order that someone to apply the β ( M ) constraints onto an inflationarymodel that yields a particular P R ( k ). Throughout this paper we assume a one-to-one cor-respondence between the scale of perturbation and the mass of the PBHs. Our analysis issupported by the findings of the Refs. [83, 84] that the typical mass of the PBHs is about thehorizon mass at the moment of formation. Nevertheless, the work of [84] points out that atiny amount of black holes are created at the low-mass tail of the near-critical collapse. Thisfinding is rather interesting, nevertheless in the current analysis we omit possible effects fromPBHs in the low-mass tail. 8 end k rh H end = - M Pl - M Pl - M Pl - M Pl T rh ( GeV ) k end - r h ( M p c - ) T rh = GeV T rh = GeV T rh = GeV T rh = GeV M k ( g ) k ( M p c - ) Figure 2:
Left panel : The dependence of the wavenumber at the end of inflation, k end , (solidlines) and at the beginning of the radiation dominated era, k rh (dashed line) on the reheatingtemperature are depicted. Four different values for the H end were considered. The black dotsshow the k end , k rh when the reheating temperature is maximum, i.e. N rh = 0. Right panel :The k = k ( M ) relation (28) for four different reheating temperatures T rh . The change of theslope happens at k = k rh . For smaller T rh the smaller the k end at the end of inflation is. Thesolid (dotted) line is for γ R = γ M = 1 (0.2). The transition from matter to radiation
A key quantity is the moment that the transition from the matter domination to radiationtakes place. If it happens instantaneously then the horizon mass at the transition epoch isequal to M rh . But, the decay of the inflaton condensate is not an instantaneous process. Ithappens with a decay rate Γ inf and the completion is usually defined at the moment that H = Γ inf . However radiation is gradually generated by the partial inflaton decay implies thatthe transition from MD to RD may take place either before or after the moment H = Γ inf .Let us define the moment of the transition as H tr = Γ inf /α . Then the efolds that takeplace from the end of inflation until the transition epoch are N tr = N rh + ln α . Also, thewavenumber at the transition is k tr = α − / k rh and the horizon mass, M tr = α / M rh . Themoment of the transition might be when the energy density of the universe is equally partitionedbetween the inflaton condensate and the entropy produced by the inflaton decay, or when theprobability for PBH formation coincides for the two production mechanisms [85]. In the firstcase it is α > N tr > N rh . In the second case it is α < N tr (cid:39) N rh < N rh [85]. For clarity and simplicity, in the following analysis we approximate N tr = N rh , k tr = k rh and M tr = M rh . PBH formation is possible during the early stages of the Universe when superhorizon fluc-tuations in the curvature of spacetime cross into the horizon and collapse under their ownself-gravitation. We will assume the approximation that the mass distribution of the PBHformed is contracted about the horizon mass. In the heated universe the PBHs are expectedto form with mass M = γ R M hor when the cosmic temperature is T ( M ) = 9 . × GeV γ / (cid:18) M g (cid:19) − / (cid:16) g ∗ . (cid:17) − / . (29)For M = 5 × g we define the BBN critical temperature where PBHs of that mass form,since the flux of the Hawking thermal radiation in a timescale τ ( M ) might alter the BBN9bservables, T bbn ≡ . × GeV γ / (cid:16) g ∗ . (cid:17) − / . (30)Respectively, for M = 2 . × g we define the CMB critical temperature, that PBHs withlifetime τ ( M ) relative to the timescale of the CMB physics form, T cmb ≡ . × GeV γ / (cid:16) g ∗ . (cid:17) − / . (31) The β (cid:48) ( M ) < C M bound applies for PBH with mass M > γ R M hor ( T rh ), or equivalently forscales k − that enter after the completion of the reheating phase, i.e. k < k rh . From Eq. (22)this is recast into the condition for the reheating temperature, T rh > . × GeV γ / (cid:18) M g (cid:19) − / (cid:16) g ∗ . (cid:17) − / (32)For that large reheating temperatures the mass fraction β (cid:48) of the universe that collapsesinto PBH has to be smaller than C M , β RD ( M ) (cid:39) √ π σ ( M ) δ c e − δ c / σ ( M ) < γ − / (cid:16) g ∗ . (cid:17) / C M ≡ β RD, max ( M, γ R , g ∗ ) . (33)We have assumed that the fluctuations at horizon crossing are Gaussian with variance σ ( M ).A black hole forms if the density contrast at horizon crossing k = aH exceeds a critical value δ c . The value of δ c varies in the literature, e.g. δ c = 1 / δ c = 3(1 + w )5 + 3 w sin π √ w w . (34)For w = 1 / δ c = 0 .
41. We note that different values for δ c are cited in the literature,see also [83, 88–90]. The numerical value of the γ M is unknown. It depends on the details ofgravitational collapse. Simple analytical calculation suggests that it is γ R ∼ . γ M value [92,93]. In our analysiswe leave the γ numerical values unspecified and for clarity in some expressions we will normalizethe γ R with 0 . γ M with 0.1. BBN and CMB constraints
The β ( M ) depends very mildly on the degrees of freedom thus for simplicity and without costin the accuracy we assume below that g ∗ = 106 .
75. For | ln( C M / √ γ R ) / | (cid:29) | ln( σ/δ c ) | the (33)rewrites σ ( M ) < δ c √ (cid:20) ln (cid:18) √ γ R √ π C M (cid:19)(cid:21) − / ≡ σ RD, max ( M, δ c , γ R ) . (35)For M (cid:39) × g the BBN constraint is C M (cid:39) − and for M (cid:39) × g the CMBconstraint is C M (cid:39) × − , hence we can obtain the σ (5 × g) and the σ (2 . × g)bound respectively. Using the expression (27) the constraints read in the momentum space, σ (7 . × k . ) (cid:46) . (cid:18) δ c . (cid:19) (cid:104) .
028 ln (cid:16) γ R . (cid:17)(cid:105) − / , (BBN) (36) σ (3 . × k . ) (cid:46) . (cid:18) δ c . (cid:19) (cid:104) .
023 ln (cid:16) γ R . (cid:17)(cid:105) − / , (CMB) (37)10or T rh > . × γ / GeV and T rh > . × γ / GeV respectively. Numerically we findthat the approximation (35) differs from the exact only about 0.1%. In the next section, wherethe gravitational collapse during MD era will be examined, we will see that the constraints(36) and (37) apply also for smaller reheating temperatures -about two orders of magnitudesmaller- due to the finite time required for the gravitational collapse.The above constraints on the variance of perturbations can be applied on the power spec-trum of the primordial comoving curvature perturbations. An explicit constraint on P R canbe found only if the P R is known in a range of momenta k . Also, one has to consider a windowfunction to smooth the density contrast. During RD the relation between the variance of thecomoving density contrast and the P R reads σ ( k ) = (cid:18) (cid:19) (cid:90) dqq W (cid:16) qk (cid:17) (cid:16) qk (cid:17) P R ( q ) , (38)where W ( z ) represents the Fourier transformed function of the Gaussian window, W ( z ) = e − z / . For an order of magnitude estimation we can approximate σ ∼ (4 / P R / and theconstraints on the power spectrum read in momentum space P R ( k ) (cid:46) O (10 − ). Increasing the δ c value the bounds become weaker, for example for δ c = 0 . P R are relaxed 1.5times. T rh > GeV
DwarfsMachos / Eros / OgleSubaru
HSC Ω pbh WDEG γ CMBBBNDMPlanck GW 10 k [ Mpc - ] σ m ax M [ g ] T rh = Ge V Planck BBNCMB EG γΩ pbh WD Subaru
HSC Machos / Eros / OgleDwarfsGWDM 10 - k [ Mpc - ] σ m ax M [ g ] T rh = Ge V PlanckDM BBNCMB EG γ WD Ω pbh Subaru
HSC Machos / Eros / OgleDwarfsGW10 - - k [ Mpc - ] σ m ax M [ g ] T rh = GeV
PlanckDM BBNCMB EG γ WD Ω pbh Subaru
HSC Machos / Eros / OgleDwarfsGW - k [ Mpc - ] σ m ax M [ g ] Figure 3: The plots depict combined upper bounds on σ ( k ) and σ ( M ) for reheating tempera-tures arbitrary large (upper left panel) and T rh = 10 , , GeV. The vertical line indicatesthe PBH mass forming at the epoch of reheating and the horizontal line the threshold value σ thr = 0 . σ thr depict the σ ( M ) upper bound for vanishing spin.The dashed lines next to the reheating scale show the maximum σ if one neglects the finite timefor the gravitational collapse. In the plots benchmark values, γ R = 0 . γ M = 0 . δ c = 0 . .2 Matter domination era A presureless matter domination era is naturally realized in the early universe due to thecoherent oscillations of the inflaton or other scalar fields. During matter era the Jeans pressureis negligible and scalar perturbations, that would be minor in the radiation domination era, cangrow linearly with the scalar factor and lead to PBH formation. If the reheating temperatureis T rh < . × GeV γ / (cid:18) M g (cid:19) − / (cid:16) g ∗ . (cid:17) − / , (39)then the formation rate of PBH with mass less than γ R M rh might change drastically. In additionthe PBH abundance formed during MD era scales differently with time. For scales k that enterduring pressureless reheating the bound is β (cid:48) MD ( M ) < γ − / ( g ∗ / . / ( k/k rh ) / C M , andin terms of M and T rh is recast into β MD ( M ) < C M (4 . / γ − (cid:18) T rh GeV (cid:19) − (cid:18) M g (cid:19) − / ≡ β MD, max ( C M , M, T rh , γ M ) (40)The β (cid:48) MD ( M ) constraint is depicted in Fig. 1.PBHs formed during a pressureless matter dominated (MD) era has been considered inRef. [92–96]. Employing the results of Ref. [55] for spinless gravitational collapse during MDera the formation rate, which depends on the fraction of the regions which are sufficientlyspherically symmetric, is given by β MD ( M ) (cid:39) . σ ( M ) . (41)We comment that in Ref. [92, 93] an additional suppression factor σ / is included to take intoaccount inhomogeneity effects that we do not consider here, following Ref. [55]. The PBHproduction rate in the MD era is larger than that in the RD era for σ (cid:46) .
05 whereas for largervariance, due to the absence of relativistic pressure, the nonspherical effects suppress the PBHformation rate in MD eras [55]. From Eq. (41) we attain a relation that relates the varianceof the comoving density contrast σ ( M ) to the observational bound C M , σ MD ( M ) | spinless < . γ − / (cid:18) T rh GeV (cid:19) − / (cid:18) M g (cid:19) − / C M / (42)The PBH production rate is modified when the collapsing region has spin. The angularmomentum suppresses the formation rate which now reads [56], β MD ( M ) = 2 × − f q ( q c ) I σ ( M ) e − . I / σ ( M )2 / . (43)Benchmark values are q c = √ f q ∼ I is a parameter of order unity [56]. Accordingto [56] this expression applies for σ ( M ) (cid:46) . ≡ σ thr , whereas the equation β MD ( M ) (cid:39) . σ ( M ) applies for 0 . (cid:46) σ ( M ) (cid:46) . The finite duration of the PBH formation
An additional critical parameter is the duration of the gravitational collapse. PBH formationis strongly suppressed by a centrifugal force and it completes, that is to enter into the nonlinearregime, only if the MD era lasts sufficiently long. According to [56] the finite duration of thePBH formation can be neglected if the reheating time t rh satisfies t rh > (cid:18) I σ (cid:19) − t , (44)12here t is the time of the horizon entry of the scale k − . In terms of wavenumbers andtemperatures the above condition rewrites respectively, k rh < (cid:18) I σ (cid:19) / k or T rh < (cid:18) I σ (cid:19) / T , (45)where T the temperature that the scale k − enters the horizon. If these conditions are notfulfilled then the time duration for the overdensity to grow and enter the nonlinear regime isnot adequate. Due to the fact that the collapse does not happen instantaneously after thehorizon crossing the formation rate (43) applies only for the scales k that experience a varianceof the comoving density contrast at horizon entry that is larger than σ > σ cr ≡ I − (cid:18) k rh k (cid:19) . (46)In terms of temperature this translates into σ > / I − ( T rh /T ) . If σ < σ cr we will considerthat the formation rate is that of the radiation era and in our numerics we will choose I = 1. BBN and CMB constraints
The constraints on the variance during MD apply only for those perturbations that have enoughtime to gravitationally collapse during the reheating era, since the collapsing process is not in-stantaneous. For a given scale k − and variance σ ( k ) there is a maximum reheating temperaturethat the collapse is realized during the MD era. For PBHs to form during MD with masses, M bbn , M cmb , associated with the BBN and CMB constraints it has to be T rh < T (MD)bbn ≡ (cid:18) I (cid:19) / T bbn σ / ( M bbn ) , T rh < T (MD)cmb ≡ (cid:18) I (cid:19) / T cmb σ / ( M cmb )(47)Otherwise, the upper bound on the variance should be determined by the RD era dynamics sincewe expect the relativistic pressure at times Γ − to cause a bounce on the ongoing collapsingprocess. Thus, the constraints on the variance of the density perturbations given by Eq. (36)and (37) apply respectively for T rh > T (MD)bbn and T rh > T (MD)cmb . The CMB constraint, as willdiscuss below, is the stringent one except if the reheating temperature is in the window T (MD)cmb (cid:46) T rh (cid:46) T (MD)bbn , (48)For such reheating temperatures the PBH that form during MD influence the BBN but notthe CMB observables.The MD variance of the comoving density contrast at horizon entry is constrained by theBBN and CMB observables for T rh < T (MD)bbn and T rh < T (MD)cmb respectively at the scales, σ (5 × g) = σ (cid:32) γ / (cid:18) T rh GeV (cid:19) / k . (cid:33) (49) σ (2 . × g) = σ (cid:32) × γ / (cid:18) T rh GeV (cid:19) / k . (cid:33) . (50)In order to recast the mass fraction β ( M ) constraints into constraints on the variance σ ( M )we have to solve the inequality β MD ( σ ( M )) < β MD, max ( C M , M, T rh , γ M ) . (51)However, an analytic solution can be found only for the case of spinless gravitational collapse.For the spinning case any analytic approximation is not accurate enough and numerical solu-tions have to be pursued. It is actually the spin effects that determine the maximum value forthe variance σ and cannot be ignored. 13et us first calculate the constraints on the variance of the comoving density contrast forthe spinless collapse approximation, i.e. for σ > . σ (5 × g) (cid:12)(cid:12) spinless (cid:46) . × − γ − / (cid:18) T rh GeV (cid:19) − / (BBN) (52) σ (2 . × g) (cid:12)(cid:12) spinless (cid:46) . × − γ − / (cid:18) T rh GeV (cid:19) − / (CMB) (53)One sees that for the BBN constraint it is σ > .
005 for T rh (cid:46) T rh < T (MD)cmb the CMB constraint is always the stringent one and there the spineffects (that we discuss right after) determine the maximum allowed variance, σ max , see theright panel of Fig. 4. Hence, the bounds (52) and (53) should be seen only as indicative ones.Turning now to the PBH formation rate considering spin effects, dictated by Eq. (43), onehas to solve numerically the inequality (40) in order to derive upper bounds for the varianceof the density perturbations. After fitting the numerical solution we find the BBN and CMBconstraints for the variance, σ (5 × g) (cid:12)(cid:12) +spin (cid:46) Exp [ − .
22 + 0 .
196 ln T rh GeV + 6 . × − (cid:18) ln T rh GeV (cid:19) (54) − . × − (cid:18) ln T rh GeV (cid:19) (cid:35) (BBN)for T rh < T (MD)bbn and, σ (2 . × g) (cid:12)(cid:12) +spin (cid:46) Exp [ − . − .
087 ln T rh GeV + 2 × − (cid:18) ln T rh GeV (cid:19) (55) − × − (cid:18) ln T rh GeV (cid:19) (cid:35) (CMB)for T rh < T (MD)cmb . The Eq. (49) translates them into the k -space. Compared to the spinlesscase, these bounds are weaker but they are the effective ones for the matter domination eraregardless the reheating temperature. They are depicted in Fig. 4. Collectively, the upperbounds on σ ( M ) for any PBH mass M , written also in the momentum space, consideringcollapse during RD era and during MD era with and without spin effects, are presented in Fig.3. Now that we derived the expressions (54) and (55) for the temperature dependent variancewe can estimate the T (MD)bbn and T (MD)cmb from Eq. (47). Plugging in the the upper bound valuefor the σ MD ( M ) for spinning collapse we find the values, T (MD)bbn = 3 . × GeV , T (MD)cmb = 1 . × GeV . (56)After rewritting the Eq. (47), we can also find the maximum reheating temperature valuethat a PBH with arbitrary mass M forms during matter domination era. This is found aftersolving the equation T rh-max = (cid:18) I (cid:19) / T ( M ) σ / ( M, T rh-max ) , (57)where T ( M ) is given by Eq. (29). We numerically solve this equation to find the σ and themass of the transition from MD collapse to RD collapse for particular reheating temperaturesand make the plots in Fig. 3, as well as in Fig. 8 and 9 presented in the following sections. Inthese figures the dashed lines in the region of transition from MD to RD give the upper boundfor σ ( M ) if the collapse had been instantaneous.14lugging in benchmark values, e.g, γ M = 0 .
1, and assuming T rh = T (MD)cmb (cid:39) GeVwhere the bounds become stringent, we get σ (4 . × k . ) < × − for the BBN and σ (1 . × k . ) < × − for the CMB. The constraints on the variance of the comovingdensity contrast can be applied on the power spectrum if there is an explicit form of the P R at hand. In a matter domination era the relation between the variance and the P R reads σ ( k ) = (cid:18) (cid:19) (cid:90) dqq W (cid:16) qk (cid:17) (cid:16) qk (cid:17) P R ( q ) . (58)For an order of magnitude estimation we can approximate σ ∼ (2 / P R / and for γ M =0 . T rh = 10 GeV the constraints on the power spectrum read in momentum space, P R (4 . × k . ) (cid:46) O (4 × − ) and P R (1 . × k . ) (cid:46) O (9 × − ) for the BBN andCMB respectively. In the next section we are going to derive constraints on the P R and thereheating temperature assuming a particular but representative enough form for the powerspectrum. Spinless + Spin - - T rh ( GeV ) σ m ax ( × g ) BBN constraint
Spinless + Spin - - T rh ( GeV ) σ m ax ( . × g ) CMB constraint
Figure 4:
Left panel : The maximum sigma for M = 5 × g. The blue line depicts the σ forspinless collapse and the orange when spin effects are included. The solid line gives the correctupper bound if spin is considered. Right panel : The maximum sigma for M = 2 . × g.Benchmark values γ M = 0 . γ R = 0 . σ thr = 0 .
005 threshold. The change of the formation probability is determined respectivelyby the temperatures T (MD)bbn and T (MD)cmb , see Eq. (56). P R ( k ) If the universe is reheated right after the formation of a PBH with mass M • then we call thisreheating temperature M • -critical and we label it T (MD) • . In a matter domination universe,contrary to the radiation dominated case, the black hole formation is not instantaneous. Afterthe reentry of the overdensity with wavenumber k a finite time for the collapse is required. Itis T (MD) • = T • (cid:18) I σ ( M • ) (cid:19) / , (59)15here T • the temperature that the scale k − • enters the Hubble horizon. For the four massexamples of PBH considered in this paper the critical temperatures read T (MD) • ( M • = 10 g) = 9 . × γ / (cid:16) g ∗ . (cid:17) − / (cid:18) I σ ( M • ) (cid:19) / GeV T (MD) • ( M • = 10 g) = 9 . × γ / (cid:16) g ∗ . (cid:17) − / (cid:18) I σ ( M • ) (cid:19) / GeV T (MD) • ( M • = 10 g) = 31 γ / (cid:16) g ∗ . (cid:17) − / (cid:18) I σ ( M • ) (cid:19) / GeV T (MD) • ( M • = 10 g) = 3 . × − γ / (cid:16) g ∗ . (cid:17) − / (cid:18) I σ ( M • ) (cid:19) / GeV . The numerical value of the T (MD) • depends on the variance of the density perturbation atthe scale k • ≡ k ( M • ). Assuming the maximum allowed σ MD ( M, T rh ), we find after solvingthe algebraic equation (59) for σ MD given by Eq. (42) -spin effects can be ignored here- that, T (MD) • = 2 . × GeV, 3 . × GeV, 2 . × − GeV for PBH masses M • = 10 g,10 g, 10 g and 10 g respectively. P R ( k ) amplitude For the general definition P R ( k ≥ k peak ) ≡ A max f ( k ), the maximum amplitude of the powerspectrum of the comoving curvature perturbation is A max = P R ( k ) [ f ( k )] − . For the varianceof the comoving density contrast σ ( k ) = θ P R ( k ) the general constraints on σ max ( k ) obtainedin the previous sections can be applied on the power spectrum maximum amplitude, A max ≤ σ ( k ) θ [ f ( k )] − , (60)where we assumed that the PBHs mass distribution is contracted about the horizon mass. Fora power spectrum P R ( k ) designed to trigger a sizable PBH formation the A max is bounded bythe dynamical constraints on the nonevaporated PBH relics, that is the fractional abundance f PBH must not violate the bounds depicted in the right panel of Fig. 1. Nevertheless, theevaporating PBH put additional bounds on the A max for a fixed form of the power spectrumtail, that is a fixed f ( k ). In particular the maximum amplitude of the power spectrum has tosatisfy the constraints A max ≤ Min (cid:26) σ ( k • ( T rh )) θ [ f ( k • ( T rh ))] − , σ ( k bbn ( T rh )) θ [ f ( k bbn ( T rh ))] − (cid:27) , (61)for T (MD)cmb (cid:46) T rh (cid:46) T (MD)bbn and A max ≤ Min (cid:26) σ ( k • ( T rh )) θ [ f ( k • ( T rh ))] − , σ ( k cmb ( T rh )) θ [ f ( k cmb ( T rh ))] − (cid:27) , (62)for T rh (cid:46) T (MD)cmb . For f ( k ) = ( k/k • ) − p , that we exemplify in the Appendix, it is θ =1 / / Γ (cid:0) − p (cid:1) . The ratio k/k • depends on the reheating temperature. According tothe expressions (24) and (27) it is kk • = k RD ( M ) k RD ( M • ) = (cid:16) MM • (cid:17) − / , for T rh > T kk MD ( M ) k RD ( M • ) = η (cid:16) M g (cid:17) − / (cid:16) M • g (cid:17) / (cid:16) T rh GeV (cid:17) / for T • < T rh < T kk MD ( M ) k MD ( M • ) = (cid:16) MM • (cid:17) − / for T rh < T • (63)16here η ≡ . × γ − / γ / ( g ∗ / . / and k − identified either as the CMB or theBBN scale. T k is the temperature that the k − scales enters the horizon. In the Fig. 5the σ ( k • ( T rh )) /θ [ f ( k • ( T rh ))] − bound is depicted with dot-dashed lines and the combinedCMB and BBN bound, σ ( k ( T rh )) /θ [ f ( k ( T rh ))] − , with solid lines. The CMB+BBN bounddepends both on the maximum value of the power spectrum and on the form of the powerspectrum tail. Following the analysis of the previous section we depict in Fig. 5 three different P R ( k ) slopes with green, blue and black color respectively. p = = = f PBH - - - T rh ( GeV ) M ax i m u m P ℛ ( k ● ) f PBH + CMB + BBN constraints for M ● = g p = = = f PBH - - - T rh ( GeV ) M ax i m u m P ℛ ( k ● ) f PBH + CMB + BBN constraints for M ● = g p = = = f PBH - - - T rh ( GeV ) M ax i m u m P ℛ ( k ● ) f PBH + CMB + BBN constraints for M ● = g p = = = f PBH - - - T rh ( GeV ) M ax i m u m P ℛ ( k ● ) f PBH + CMB + BBN constraints for M ● = g Figure 5: The plots depict upper bounds on the power spectrum amplitude, A max = P R ( k • ),from the relic PBH abundance (dot-dashed line) and from the CMB and BBN constraintson the evaporating PBH (solid lines) for three different slopes for the power spectrum tailas described in the Appendix: green (p=0.1), blue (p=0.5) and black (p=1). The step-likechanges take place at T (MD) • , T • , T cmb and T bbn from left to right. Benchmark values δ c = 0 . γ M = 0 . γ R = 0 . A generic prediction of beyond the Standard Model physics is the existence of additionalscalar fields. These scalars under general initial conditions predict an epoch of early mat-ter domination following inflation. The mass and decay rate of these scalars vary. For ex-ample, in the stringy and supersymmetric frameworks there are scalars, collectively calledmoduli, that decay gravitationally and their mass is determined by the scale of the symme-try breaking. A gravitationally decaying scalar X with mass m X reheats the universe at T dec X ∼ m X / GeV) / . The production of entropy by the modulus dilutes the thermalplasma ∆ X times, ∆ X (cid:39) T dom X T dec X (64)17here T dom X the temperature that the scalars dominated the energy density of the universe and T dec X the late reheating temperature. T rh ⩾ GeV X = M Pl T X dec =
10 MeV
Modulus
Domination10 - M ( g ) k ( M p c - ) T rh ⩾ GeV X = - M Pl T X dec =
10 MeV
Modulus
Domination10 - M ( g ) k ( M p c - ) Figure 6: The plots depict the k = k ( M ) relation for the two modulus domination scenariosconsidered in the text for γ R = γ M = 1. The gridlines indicate, from left to right, the transitionfrom the reheating to the thermal phase, the modulus domination and the final thermal phase.The fractional PBH abundance formed during a modulus domination era reads f (MD)PBH ( M ) = (cid:18) β MD ( M )1 . × − (cid:19) (cid:16) γ M . (cid:17) (cid:18) g ( T dec X )10 . (cid:19) − (cid:18) M g (cid:19) − / (cid:18) kk dec X (cid:19) − / , (65)where k dec X ( T dec X , g ∗ ) = 1 . × Mpc − (cid:18) T rh MeV (cid:19) (cid:18) g ∗ ( T dec X )10 . (cid:19) / , (66)and k X ( M, T dec X , g ∗ ) = 7 . × Mpc − γ / (cid:18) M g (cid:19) − / (cid:18) T dec X MeV (cid:19) / (cid:18) g ∗ ( T dec X )10 . (cid:19) / (67)for k dec X < k X < k dom X . The ˜ N X are the efolds of modulus-condensate domination, ˜ N X = ln(∆ X g / /g / ), and it is k dom X = e ˜ N X / k dec X hence k dom X (cid:39) ∆ / X k dec X .Also, we can define the k -dependent ”dilution” size , ∆ k ≡ (cid:0) k/k dec X (cid:1) / . Due to themodulus domination the abundance of PBH formed at scales k > k dom X are diluted ∆ X timeswhile the abundance of PBH formed at scales k during the modulus domination era are partially”diluted” ∆ k times. The horizon masses at the scales are k dec X and k dom X are respectively, M hor ( T dec X , g ∗ ) (cid:39) × g (cid:18) T dec X MeV (cid:19) − (cid:16) g ∗ . (cid:17) − / (68) M hor ( T dom X , g ∗ ) = M hor ( T dec X , g ∗ ) (cid:18) k dom X k dec X (cid:19) − = M hor ( T dec X , g ∗ ) ∆ − X (69)For the entire postinflationary phase the PBH mass M and the horizon scale k − are related Strictly speaking this is not a dilution. It accounts for the absence of expansion effects on the PBH abundanceduring the modulus domination. k ( M ) = k MD ( M, T rh , g ∗ ) = (Eq.(24)) × e − ˜ N X / , for k rh < k < k end k RD ( M, T rh , g ∗ ) = k rh γ / M / ( T rh , g ∗ ) M − / for k dom X < k < k rh k X ( M, T dec X , g ∗ ) = (Eq.(67)) for k dec X < k < k dom X k RD ( M, T dec X , g ∗ ) = k dec X γ / M / ( T dec X , g ∗ ) M − / for k < k dec X (70)We assume that the scalar X decays just before the BBN nucleosynthesis, thus we assume M X ∼ GeV and gravitational interactions. We let the free parameter to be the ∆ X , orequivalently, the T dom X . To make the distinction clear, we also assume that the inflaton fielddecays fast with T rh ∼ GeV, though a late decaying inflaton together with a modulusdomination era is also an interesting possibility. The corresponding bounds on the β (cid:48) ( M ) read β (cid:48) MD ( M ) ≡ γ M β MD ( M ) < × (cid:18) T dec X MeV (cid:19) − (cid:18) M g (cid:19) − / C M . (71)The bound on β ( M ) can be translated into a bound on the variance σ . For M hor ( T dom X ) HSC Machos / Eros / Ogle DwarfsCMBGW T rh = GeV X = M Pl T X dec = 10 MeV - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' τ [ s ] P l a n ck DM BBNCMB E G γ Ω pbh WD Subaru HSC Machos / Eros / Ogle T rh = GeV X = - M Pl T X dec = 10 MeV DwarfsCMBGW - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' τ [ s ] Figure 7: Left panel : The combined upper bounds on β (cid:48) ( M ) for the cosmological scenariowith reheating temperatures T rh = 10 GeV and a modulus field that dominates the energydensity at T dom X = 5 × GeV (left gridline) and decays at T dec X = 10 MeV (right gridline).The dotted lines depict the β (cid:48) ( M ) constraints for arbitrarily large reheating temperature. Rightpanel : As in the right panel with the difference that the modulus field dominates the energydensity at T dom X = 10 GeV.Benchmark values for the T dom X are determined by the initial value of the modulus potential.Assuming the simple quadratic potential for the modulus field V ( X ) = m X X / lanck DM BBNCMB EG γ WD Subaru HSC Machos / Eros / OgleDwarfsGW Ω pbh T rh = GeV X = M Pl T X dec = 10 MeV - k [ Mpc - ] σ m ax M [ g ] Subaru HSC Machos / Eros / OgleDwarfs Ω pbh WDEG γ CMBBBNDMPlanck GW T rh = GeV X = - M Pl T X dec = 10 MeV k [ Mpc - ] σ m ax M [ g ] Figure 8: Left panel : The combined upper bounds on σ ( M ) for the cosmological scenariowith reheating temperatures T rh = 10 GeV and a modulus field that dominates the energydensity at T dom X = 5 × GeV (left gridline) and decays at T dec X = 10 MeV (right gridline).The dotted lines, for the CMB and extra-galactic gamma rays constraints, give the σ boundfor spinless gravitational collapse. The dashed lines next to k rh give the maximum variance forinstantaneous gravitational collapse. Right panel : As in the left panel with the modulus fielddominating the energy density at T dom X = 10 GeV. m X ∼ GeV, it is the initial displacement X from the zero temperature minimum thatdetermines the T dom X . The initial displacement of the modulus field is model dependent. For ouranalysis we assume two distinct cases, a fist with the maximum possible initial displacement(to avoid a late inflationary phase) X ∼ M Pl , and a second with an intermediate initialdisplacement, X ∼ − M Pl . In both cases we assume that the effective mass of the modulusduring inflation is larger than the Hubble scale so that the spectrum of the de-Sitter fluctuationsis not transferred to the modulus field, that could otherwise act as a curvaton field. For these X values we get respectively,1. T dom X (cid:39) × GeV and ∆ X (cid:39) × for X ∼ M Pl T dom X (cid:39) GeV and ∆ X = 10 for X ∼ − M Pl .For these two benchmark cases the horizon mass at the time of the modulus decay is M hor ( T dec X = 10MeV) (cid:39) × g, and at the time the modulus dominates the energy densityis M hor ( T dom X = 5 × GeV) (cid:39) × g and M hor ( T dom X = 10 GeV) (cid:39) g respectively. The bounds obtained from evaporating and unevaporating PBH constrain the power spectrumover 45 decades of mass, whereas the CMB direct measurements span only 5 decades. Theanalysis method followed here to obtain the bounds is illustrated in Fig. 10. In Fig. 9 theupper bounds for the power spectrum of the comoving curvature perturbation are depicted afterthe assumption that P R ( k ) (cid:39) (9 / σ ( k ) for a RD era and P R ( k ) (cid:39) (5 / σ ( k ) for a MD era.Each panel corresponds to a different cosmic history. Scenarios with reheating temperatures, T rh > and T rh = 10 , , − GeV as well as scenarios with an intermediate non-thermalphase due to a scalar condensate domination have been examined. For T rh (cid:39) T (MD)cmb (cid:39) GeVthe constraint on the power spectrum is the stringent one after the direct ∆ T /T observationalconstraint at k ∼ . 05 Mpc − [80]. At that scale, that corresponds to the horizon mass2 . × g /γ M , the power spectrum of the comoving curvature perturbation has to be P R ( k ) (cid:46) × − .In Fig. 9 the observational constraints, e.g. the CMB constraint, is located in differentposition in the k -space, albeit the position of the constraints on the axis of mass remains thesame. This is either due to the different reheating temperatures or due to a postinflationary20on-thermal phase. In the Fig. 9 we also included the Planck 2018 constraints on the powerspectrum [80]. As in the previous figures, the constraint associated with GWs is depicted withdotted-dashed lines because it is partial and included only as reference. Moreover, in the rightupper panel of Fig. 9 we have added the k -range where future observational probes, that webriefly outline below, can constrain further and significantly the power spectrum amplitude. Apart from the direct constraints coming from nonevaporated PBH the power spectrum at largescales can be constrained by other effects. PBH generation requires large density perturbationsthat in turn source the generation of gravitational waves, see Ref. [97] for a review. Scalar per-turbations and tensor perturbations are coupled beyond the linear order [98, 99] and hence theinduced gravitational waves are also stochastic. These gravitational waves are produced at thehorizon crossing of the scalar perturbations, hence simultaneously with the potential PBH gen-eration, and their frequency is related with the PBH mass as f GW (cid:39) × − γ / ( M • /M (cid:12) ) − / Hz. The amount of the gravitational waves depends on the type and the amplitude of the cur-vature power spectrum. Low frequency gravitational waves are severely constrained by thepulsar timing experiments [78, 100] whereas, higher frequencies will be subject to future obser-vational probes, see e.g. [101, 102]. An interesting scenario is that PBHs with mass 10 − M (cid:12) can comprise most of the dark matter in the universe and in such a case their production isassociated with a mHz gravitational wave signal that can be tested by LISA [103, 104].In addition due to the Silk damping, that is the erase of acoustic oscillations of k − thatfalls within the photon diffusion scale, energy is transferred to the background homogeneousplasma [105]. Depending on the redshift z > that the damping occurs there exist two typesof CMB distortions, the µ -distortion at scales 50 Mpc − (cid:46) k (cid:46) Mpc − and y -distortion onlarger scales. So far µ -type spectral distortion of the CMB has not be detected and for Gaussianprimordial density perturbations PBHs in the mass range 2 × M (cid:12) (cid:46) M • (cid:46) × M (cid:12) areexcluded. Smaller scales, that correspond to PBH masses M • (cid:46) × M (cid:12) are still possible tobe probed by the measurement of the baryon-to-photon ratio and put constraints on the powerspectrum amplitude [106, 107]. We comment that if the scalar perturbations are non-Gaussianthen the constraints on the P R ( k ) change [108]. In such a case, the µ -distortion as well thebounds from the stochastic gravitational wave background can weaken depending on the degreeof the non-Gaussianity of the primordial perturbations.The µ and y type distortions of the CMB black body spectrum as well as the secondarystochastic gravitational waves usually correspond to very large PBH masses and the interestof this work is mainly on PBH with short lifetime, M • (cid:28) M (cid:12) , and the P R ( k ) features at largewavenumbers. We neither examine here the implications for the P R ( k ) due to the µ and y typedistortions nor due to stochastic gravitational waves, see e.g. [109] for a recent work. We do notexamine implications on the power spectrum of ultracompact minihalos of dark matter [110],or from the decay of metastable vacua [111–113] either. Following Ref. [41] we included onlythe GW constraint, β ( M ) (cid:46) − , on the mass band 10 − M (cid:12) that comes from pulsartiming data due to the generation second-order tensor perturbations [78]. Also for comparison,in the upper right panel of Fig. 9, where a net radiation dominated phase is presented, we addwith dotted lines the k -range where the µ -distortion constraints apply and the range that willbe probed by gravitational wave antennas LISA and Square Kilometre Array (SKA), as wellas pulsar time arrays (PTA) that can search for secondary gravitational waves. The gravitational observation of black hole mergers by LIGO offers us an unprecedented pieceof information about the dark sector of the universe. This direct observation of black holesmotivated the cosmologists to explain the LIGO events by PBHs [77] as well as to investigatethe scenario that lighter PBHs may comprise a significant fraction of the dark matter in the21 rh > GeV PlanckEntropyBBN DMCMBEG γ WD Ω pbh Subaru HSCMachos / E / ODwarfsCMBCMB LISAPTA, SKA μ - d i s t o r t i on s - - - - - - - - - - k [ Mpc - ] M ax i m u m P ℛ ( k ) M [ g ] T rh = 10 MeV PlanckBBNCMBEG γ Subaru HSCWDMachos / E / O Ω pbh DwarfsCMBCMB 1 10 - - - - - - - - - - k [ Mpc - ] M ax i m u m P ℛ ( k ) M [ g ] T rh = GeV PlanckDMBBNCMBEG γ WD Ω pbh Subaru HSCMachos / E / ODwarfsCMBCMB 1 10 - - - - - - - - - - k [ Mpc - ] M ax i m u m P ℛ ( k ) M [ g ] CMB CMB Dwarfs Subaru HSCMachos / E / O Ω pbh WDEG γ CMBBBN Planck T rh = GeV X = M Pl T X dec = 10 MeV - - - - - - - - - - k [ Mpc - ] M ax i m u m P ℛ ( k ) M [ g ] T rh = GeV CMBCMB Dwarfs Machos / E / OSubaru HSC Ω pbh WDEG γ CMBBBN Planck1 10 - - - - - - - - - - k [ Mpc - ] M ax i m u m P ℛ ( k ) M [ g ] CMB CMB DwarfsMachos / E / OSubaru HSC Ω pbh WD EG γ CMBBBN PlanckDM T rh = GeV X = - M Pl T X dec = 10 MeV - - - - - - - - - - k [ Mpc - ] M ax i m u m P ℛ ( k ) M [ g ] Figure 9: The plots depict upper bounds on the (full) power spectrum of the comovingcurvature perturbation coming from constraints on evaporated and nonevaporated PBHs fordifferent early universe cosmic histories after the assumption that P R ( k ) = θ − σ ( k ), where θ = 2 / / σ = 0 . 005 theoretical threshold. For reference, in the right upperpanel the probing k -range of GW antennas and µ -distortions is also depicted.22niverse. Since this sort of dark matter candidates originate from the primordial densityperturbations, the presence or the absence of PBHs provides us with an indirect insight intothe spectrum of the primordial density perturbation far beyond the scales directly accessiblein the CMB.In this work we focused on PBH scenarios with masses smaller than 10 g which, if evergenerated, will have evaporated by now. These PBHs, although absent from our galaxies, areexpected to have interesting cosmological implications for the mechanisms that generate the P R ( k ) as well for the details of the early cosmic history. We have shown that models designedto produce relic PBHs with mass M • > g have to pass strict constraints in scales significantsmaller where ephemeral PBHs form with mass M (cid:28) M • . In addition, these constraints aremuch sensitive to the reheating temperature of the universe. In particular, we investigated theimplication of the evaporating PBHs on the variance of the density perturbations for differentreheating temperatures and in scenarios where the early universe has been dominated by amodulus scalar field, see Fig. 3 and Fig. 8. We explicitly examined the reheating temperatureconstraints in scenarios with relic-PBHs in four different mass scales: the asteroid mass range, M • ∼ g, the lunar mass range, M • ∼ g, the planet mass range M • ∼ g, and theLIGO mass range (also called intermediate black hole mass range ) M • ∼ g, see Fig. 11,12 and 5.The main result of this work is that the variance of the density perturbations generatedby any inflationary model has to satisfy strict constraints in the large k limit of the spectrum.Additionally to the dynamical constraints for the nonevaporated PBH relics, we found the σ ( k )constraint given by the Eq. (37) for reheating temperatures T rh (cid:38) T (MD)bbn , the constraint givenby Eq. (54) for T (MD)cmb (cid:46) T rh < T (MD)bbn , and the constraint given by Eq. (55) for T rh (cid:46) T (MD)cmb ,where T (MD)bbn (cid:39) × GeV and T (MD)cmb (cid:39) GeV. The combined constraints with respect tothe reheating temperature are depicted in Fig. 4.We conclude that significant power in the large k -limit might be in conflict with the obser-vations. Mechanisms that generate PBH relics with asteroid or lunar mass scale are required,by the derived bounds on evaporating PBHs, to have a very narrow P R ( k ) peak, see Appendixfor details. Remarkably, these mass windows are rather interesting because the relic PBHs canexplain the entire dark matter found in the galaxies. For heavier PBH relics there is more free-dom however, since the P R ( k ) amplitude maximizes closer to the Planck pivot scale k ∼ . − ≡ k . where the spectrum is tightly constrained [80], these scenarios are often inconflict by the spectral index value and running. As expected, the most strict observationalconstraints on the power spectrum come actually from the CMB anisotropies at the scale k . .Next to this constraint it is usually found to be the CMB bound from PBH evaporation at the k = k (2 . × g , T ). It stringent for T rh = T (MD)cmb ∼ GeV, where the power spectrum ofthe comoving curvature perturbation has to be P R ( k ) (cid:46) × − .The derived constraints on σ ( k (cid:29) k . ), translated into constraints on the spectral indexvalue of the power spectrum tail, see Fig. 12 of the Appendix, constrain numerous inflationmodels and have considerable implications for the inflationary model building. A brief reviewof relevant to our discussion inflation models can be found in Ref. [97]. Models that generatea broad peak such as running mass inflation models or inflaton-curvaton models are severelyconstrained. Running mass inflation models can realize a blue-tilted spectrum and are requiredto have an appropriately balanced running and running of the running of the spectral index.Curvaton models have a scale invariant spectrum in smaller scales and the mass spectrum ofthe PBHs formed will be broad. In these models the large scale perturbations are generatedby the inflaton field and the small scale by the curvaton. According to our analysis, elaboratedfurther and illustrated in detail in the Appendix, this sort of models are in conflict with the theBBN observables if the maximum amplitude is A max (cid:38) − for large reheating temperatures,whereas for T rh (cid:46) × GeV it must be A max < − , implying practically a zero relicPBH abundance, see Fig. 5. Double inflation or inflection point models are also subject toconstraints, if the decrease of the power spectrum at large k is not as fast enough as the Eq.(37), (54) and (55) dictate. 23he results of this work show that the PBHs can be regarded as a powerful tool to probethe primordial fluctuations on much smaller scales and give us insights into the dynamics thatgenerated the seeds of the cosmic structure, even if PBHs do not comprise the observed darkmatter in the universe. β max ( M ) Y PBH , max β MD , max ( M, T rh ) β RD , max ( M ) σ MD , max ( M, T rh ) σ RD , max ( M ) σ +spinMD , max ( M, T rh ) σ spinlessMD , max ( M, T rh ) P R , max ( k ) M > γM rh M < γM rh σ max > σ cr ( M, T rh ) σ max < σ cr ( M, T rh ) σ max < . σ max > . k ( M, T rh ) k ( M ) Figure 10: The graph illustrates the steps followed to derive the upper bounds for the powerspectrum of the comoving curvature perturbations. Acknowledgments I would like to thank Antonio Riotto and Jun’ichi Yokoyama for their comments. This workis supported by the IKY Scholarship Programs for Strengthening Post Doctoral Research,co-financed by the European Social Fund ESF and the Greek government. A Parametrizing the morphology of the P R ( k ) In the main text of this paper, we computed upper bounds for the variance of the comovingdensity contrast without specifying the power spectrum -with the exception of the Fig. 5. Inthis Appendix we assume a particular morphology for the power spectrum of the comoving24urvature perturbation in order to illustrate the sort of the power spectra that are compatiblewith the constraints derived in this work. For simplicity, we omit possible critical collapse effectsand assume the horizon-mass approximation for the PBHs mass function. In this regard, thebounds that we derive are conservative since critical effects are expected to increase the PBHsabundance in the low mass tail [84, 114] that we focus on.We recall that the comoving curvature perturbation, R k , is related to the metric perturba-tions Φ( k ) for modes k outside the horizon via the relation R k = − (5 + 3 w )Φ( k )3 + 3 w (73)where the metric in the longitudinal gauge reads, ds = − (1 + 2Φ) dt + a ( t )(1 − dx .Typical inflationary scalar perturbations are well approximated by the expansionln P R ( k ) = ln A . + ( n s − 1) ln( k/k . ) + 12 d ln n s d ln k ln( k/k . ) + 16 d ln n s d ln k ln( k/k . ) + ... where k . = 0 . 05 Mpc − is the pivot scale used by the Planck collaboration [80] and n s thescalar spectral index. If the running of the spectral index d ln n s /d ln k and the running of therunning are nonzero the power spectrum amplitude may increase significantly at smaller scales.The enhancement of the power spectrum with amplitude A max can generate PBHs at the scale k − .We focus on the power spectrum tail since it is this part of the spectrum that is relevantto the PBH evaporation constraints. A general description if P R ( k ) = P R ( k peak ) f ( k ) for k >k peak . Let us parametrize the tail of the power spectrum by the function f ( k ) = ( k/k peak ) s ( k,p,α ) where s ( k, p, α ) ≡ − p (cid:16) log kk peak (cid:17) α − . For the values α = 1 or α = 2 this parametrizationdescribes well a large class of power spectra designed to generate PBH, e.g inflationary modelswith inflection point. In the following we will examine analytically the simplest α = 1 case, P R ( k ≥ k peak ) = A max (cid:18) kk peak (cid:19) − p (74)where A max ≡ P R ( k peak ). The power p can be viewed as the spectral index of the powerspectrum tail, p ≡ − n (tail) s . A.1 Radiation domination In the approximation that the inflation stage is instantaneously followed by a thermal radiationdominated epoch it is k rh = k end . The variance of the perturbation during RD era thatinstantaneously follows inflation is given by σ ( k ≥ k peak ) = (cid:18) (cid:19) P R ( k peak ) (cid:90) k end k peak dqq W (cid:16) qk (cid:17) (cid:16) qk (cid:17) (cid:18) qk peak (cid:19) − p . (75)The integration gives σ ( k ≥ k peak ) = (cid:18) (cid:19) A max (cid:18) kk peak (cid:19) − p (cid:34) Γ (cid:32) − p , k k (cid:33) − Γ (cid:18) − p , k k (cid:19)(cid:35) (76) (cid:39) (cid:18) (cid:19) A max (cid:18) kk peak (cid:19) − p Γ (cid:32) − p , k k (cid:33) , (77)where we took into account that Γ (cid:18) − p , k k (cid:19) (cid:29) Γ (cid:16) − p , k k (cid:17) . For k > k peak theincomplete Gamma function can be expanded and the variance is approximated as σ ( k ≥ k peak ) (cid:39) (cid:18) (cid:19) A max (cid:18) kk peak (cid:19) − p Γ (cid:16) − p (cid:17) + (cid:18) kk peak (cid:19) p p − (cid:32) k k (cid:33) + O (cid:32) k k (cid:33) σ ( k ) (cid:39) (cid:18) (cid:19) A max (cid:18) kk peak (cid:19) − p Γ (cid:16) − p (cid:17) (cid:39) . (cid:16) − p (cid:17) P R ( k ) ≡ θ P R ( k ) for k > k peak . (78) A.1.1 Constraints on the power spectrum tail The constraint (35) can be now applied on the power spectrum and reads for the BBN andCMB respectively P R (cid:0) k (5 × g) (cid:1) (cid:46) . (cid:0) − p (cid:1) (cid:18) δ c . (cid:19) (cid:104) . 028 ln (cid:16) γ R . (cid:17)(cid:105) − (BBN) (79) P R (cid:0) k (2 × g) (cid:1) (cid:46) . (cid:0) − p (cid:1) (cid:18) δ c . (cid:19) (cid:104) . 023 ln (cid:16) γ R . (cid:17)(cid:105) − (CMB) (80)These upper bounds, P R bound , constrain the power spectrum maximum amplitude A max andthe power p at a particular scale. It is P R ( M ) = A max ( M/M • ) p/ for M < M • , thus we get p (cid:38) A max − ln P R bound ( M )ln M • − ln M . (81)where M • is the characteristic relic PBH mass that an inflationary model predicts. Scenarioswith flat power spectra, p ∼ 0, that generate a significant amount of PBH relics are ruled out .The stringent constraint is for the CMB, which for benchmark values γ R = 0 . δ c = 0 . p (cid:38) A max / − ) − . M • / g) + 15 . . (82)For different PBH relic masses, M • , we plot in Fig. 12 the bound on the power spectrumamplitude, A max , against the spectral index p of the power spectrum tail. A.2 Matter domination Let us assume that after inflation reheating follows that lasts N rh e-folds of expansion, thenthe (74) rewrites, P R ( k ≥ k rh ) = P R ( k rh ) ( k/k rh ) − p . The variance of the comoving densitycontrast during pressureless reheating is σ ( k ≥ k rh ) = (cid:18) (cid:19) P R ( k rh ) (cid:90) k end k rh dqq W (cid:16) qk (cid:17) (cid:16) qk (cid:17) (cid:18) qk rh (cid:19) − p . (83)Also here the integration, after taking into account that Γ (cid:16) − p , k k (cid:17) (cid:29) Γ (cid:16) − p , k k (cid:17) andexpanding the incomplete Gamma function, gives the variance squared that at leading order, σ ( k ) (cid:39) (cid:18) (cid:19) Γ (cid:16) − p (cid:17) P R ( k ) ≡ θ P R ( k ) for k > k rh . (84)In the Fig. 11 power spectra that generate four representative relic PBH masses M • = 10 g,10 g, 10 g and 10 g are depicted for three different tails with steepness, p = 0 . p = 0 . p = 1 (in black). In red it is depicted the less steep allowed slope, thecritical slope, that separates the allowed from the disallowed power spectra. The reheatingtemperature is chosen to be T rh = 10 GeV where the CMB constraints are the stringent onesand the role of the evaporating PBHs on the inflationary model selection more manifest. Onthe plots, the total PBH fractional density f PBH, tot = (cid:82) M d ln M f PBH ( M ) is also computed.We have assumed a one-to-one correspondence between the wavenumber k of the perturbationsand the PBHs masses. 26 blue = A red = A black = p black = k peak = × Mpc - T rh = GeV10 - - - - - - - - - - [ Mpc - ] P ℛ ( k ) M = g T rh = GeV f blue = % , f red = % , f black = % - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' A green = A blue = A red = A black = = k peak = × Mpc - T rh = GeV10 - - - - - - - - [ Mpc - ] P ℛ ( k ) M = g T rh = GeV f green = % , f blue = % , f red = % , f black = % - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' A green = A blue = A red = A black = p black = k peak = × Mpc - T rh = GeV10 - - - - - - - - - - [ Mpc - ] P ℛ ( k ) M = g T rh = GeV f green = % , f blue = % , f red = f black = % , f black = % - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' A green = A blue = A red = , A black = p black = k peak = × Mpc - T rh = GeV10 - - - - - - - - - - - - - [ Mpc - ] P ℛ ( k ) M = g T rh = GeV f green = % , f blue = % , f red = % , f black = % - - - - - - - - - - - - - - - - - - - - t M [ g ] β ' Figure 11: Left panels : Typical P R ( k ) are depicted designed to produce PBH with M • = M peak = 10 , , , g for an inflaton that decays at T rh = 10 GeV. The differentslopes of the power spectra are fitted by the Eq. (74) for p = 0 . P R ( k ) peak, A max , differs so that the PBH abundancemaximizes. The black line corresponds to the critical slope and separates the P R ( k ) lines ruled out by PBH evaporation (above the black line) from those ruled in (below). The verticalgridlines from left to right show the scales k . , k rh , k end for T rh = 10 GeV. The dotted endsextend to k end if it was T rh = 10 GeV. Right panels : The mass fraction of the universethat collapses into PBH and the f PBH for each color against the observational constraints aredepicted. The dashed lines correspond to spinless collapse.27 ● = g M ● = g M ● = g M ● = g = - n s ( tail ) [ spectral index of the P ℛ ( k ) tail ] A m ax CMB CONSTRAINT ( for T rh ≫ T cmb ( MD ) ) M ● = g M ● = g M ● = g M ● = g p = - n s ( tail ) [ spectral index of the P ℛ ( k ) tail ] T r h ( G e V ) CMB CONSTRAINT ( for T rh ⩽ T cmb ( MD ) ; A max = - ) Figure 12: Left panel : The curves enclose the colored ( p, A max ) contour regions that satisfythe bound (81) imposed by the CMB observables, for cosmological scenarios with reheatingtemperature, T rh (cid:29) T (MD)cmb for the power spectrum (74) and for four representative mass scalesof PBH relics, M • . The solid lines correspond to δ c = 0 . 41 and the dotted to δ c = 0 . Rightpanel : The curves enclose the ( p, T rh ) contour regions that satisfy the bound (87) imposed byCMB observables, for fixed P R ( M • ) = A max = 10 − and cosmological scenarios with reheatingtemperature, T rh (cid:46) T (MD)cmb . The plot demonstrates that the power spectra with a large powerat small scales are much constrained, with the stringent constraint being for scenarios thatpredict light PBH relics. A.2.1 Constraints on the power spectrum tail and the reheating temperature The power spectrum value at the time of reheating, P R ( k rh ) depends on the reheating tem-perature, and we can pursue further the implications of the constraint (40), that we rewrite ithere as σ ( M ) < σ MD,max ( C M , M, T rh , γ M ) , (85)where σ ( M ) is given by Eq. (84) and σ max given by the observational constraints (independentof the P R ( k ) form). It is σ ( k ) = θ P R / ( k rh ) ( k/k rh ) − p/ and the power of the comovingcurvature perturbations that reentrer the horizon at the time of reheating is P R ( k rh ) = A max (cid:0) . × − (cid:1) p γ p/ (cid:18) T rh GeV (cid:19) − p (cid:18) M • g (cid:19) − p/ (cid:16) g ∗ . (cid:17) − p/ . (86)Substituting the ratio k/k rh from Eq. (22) and (28) and neglecting the finite time of thegravitational collapse we get the constraint on the reheating temperature, T p/ ≥ ξ ( A max , γ M , γ R , M • , M, p ) σ MD,max ( C M , M, T rh , γ M ) , (87)where ξ is given by the expression, ξ ≡ θ A / (2 . × − ) p/ γ p/ γ − p/ (cid:16) g ∗ . (cid:17) − p/ (cid:18) M • g (cid:19) − p/ (cid:18) M g (cid:19) p/ (88)In the approximation of the spinless collapse the σ MD,max ( M, T rh , γ M ) is explicitly calcu-lated, see Eq. (42), and the reheating temperature is constrained to be (cid:18) T rh GeV (cid:19) p − > ξ ( A max , γ M , γ R , M • , M, p ) 2 . γ / (cid:18) M g (cid:19) C M − / . (89)The above constraint has two branches. For p < . p > . σ max ( M ) ∼ β max ( M ) / bound becomesweaker as the reheating temperature decreases. But when the tail is steep the decrease in thereheating temperature implies also smaller values for the k that approaches the k peak where P R maximizes. For steep enough P R tail, p > . 2, the constraints are satisfied only for largereheating temperatures. We note that due to the finite time of the gravitational collapse theconstraint on T rh has to be translated into T (MD)rh .In fact spin effects cannot be neglected at the mass range relevant to the CMB observablesand one has to solve numerically the (87). Numerics show that, qualitatively, a similar behaviorto the condition (89) is found. There is a maximum T rh as the power p goes to zero and aminimum T rh for steeper P R slopes. In the left panel of Fig. 12 we see that, in inflationaryscenarios for PBH production, only ( p, T rh ) values from the colored contour regions leave intactthe CMB observables. Flat or not steep power spectrum tails are ruled out for any reheatingtemperature T rh (cid:46) T (MD)cmb ∼ GeV. 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