Scalar spectral index in the presence of Primordial Black Holes
We study the possibility of reheating the universe in its early stages through the evaporation of Primordial Black Holes (PBHs) that are formed due to the collapse of the inhomogeneities that were generated during inflation. By using the current results of the baryon-photon ratio obtained from BBN and CMB observations, we impose constraints on the spectral index of perturbations on those small scales that cannot be estimated through CMB anisotropy and CMB distortions. The masses of the PBHs constrained in this study lie in the range of and g, which corresponds to those PBHs whose maximal evaporation took place during the redshifts . It is shown that the upper bound on the scalar spectral index, can be constrained for a given threshold value, , of the curvature perturbations for PBHs formation. Using Planck results for cosmological parameters we obtained for and for respectively. The density fraction that has contributed to the formation of Primordial Black Holes has also been estimated.
pacs:98.80-k, 98.80.Cq, 04.70.Dy,
The universe was to a high degree homogeneous and isotropic in its early phase, in which tiny fluctuations are believed to have been generated during the inflation (Carr1994, ; kholpov-inf, ; Naselsky-inf, ) due to quantum processes. The fluctuations zeldovich (); Carr1975 () of cosmological relevance were smaller than the Hubble radius () at the onset of inflation, exited to grow to super horizon scales during inflation and re-entered during the post inflationary phase. If these density perturbations are of size of the order of horizon scale and amplitude above a certain critical threshold a few regions become sufficiently compressed for gravity to overpower pressure forces and the rate of expansion and in turn cause collapse to a black holeHawking1971 (). Such black holes are being referred as Primordial Black Holes (PBHs). Carr and Hawking(1974)carr-hawk () also showed that the PBHs would not have grown much due to accretion which implies that their masses today should be same as it was during its formation. In general, the collapsing region during the radiation dominated epoch is assumed to be spherically symmetric dorosh (); bardeen (). The critical density required for a region to collapse and form a PBH strongly depends on the density profile of the overdense region.
The PBH mass at the time of its formation can be estimated to be equal to or smaller than the cosmic horizon mass.
Unlike stellar mass black holes, which have a minimum mass criterion a range of different masses are possible in the case of PBHs. Those for which the mass was less than g at the time of their formation must have evaporated away by the present epoch by virtue of Hawking radiationhawking-nature () i.e, black body radiation being emitted from a black hole at a temperature which is inversely proportional to its mass. On the other hand, PBHs whose mass g could survive in the present universe.
In principle, the formation of PBHs could also take place due to several other mechanismsCarr2009 (); Hawking-1989 (); Garriga (); Caldwell (); Gibbon-1998 () like topological defects and bubble collisionscrawford-bubble (); Hawking-bubble (); kodama (); La (). In the observable universe, the PBHs abundance can constrain the primordial power spectrum, and hence models of inflation, on scales much smaller than those that are accessible via observations of the CMB and LSS (which provide strong constraints on the primordial power spectrum for scales between k Mpc). Thus, PBHs can act as a significant probe to study the earliest phase of the universe complimenting the information from CMB and LSS analysis.
Our work in this article is organised as follows. In section II we describe how the evaporating PBHs can inject energy into the photon-baryon fluid in the early stages of our universe. In section III we describe how we have estimated the energy injected into the background fluid via Hawking radiation. In section IV we describe the constraints on which reflects the abundance of regions of mass M that collapses to form PBHs. In section V we try to obtain the upper limit on the spectral index of perturbations with a power law type power spectrum and contrast it in light of WMAP data and Planck data. We end with our conclusions in section VI.
Ii Reheating by PBHs
In the early universe, the PBHs inject energy (produced by the process of Hawking radiation) into photon-baryon fluid. If this energy injection happens before the CMB distortion era () the fluid can achieve black-body spectrum by photon-electron interaction via double Compton scatterings and Bremsstrahlung hecklerprl (); hecklerprd (); Page:2007yr () by the epoch of recombination. This is so because the time scales of the interaction for these processes is smaller than the cosmological time scale. However, these processes (double Compton process and Bremsstrahlung) result in the increase of photon number, while the total number of baryons remains unchanged. Consequently, a baryon-photon ratio decreases due to this energy pumped due to evaporating PBHs.
In the era, our focus, in particular, lies in exploring the thermal history of the universe between and . This redshift range corresponds to energy injections of those PBHs whose mass range lies between and g at the time of their formation.
From an observational point of view, both BBN and CMB provide constraints on at their respective epochs. If a process pumps energy in a redshift interval , we can define the fractional density excess as
where is the background energy density of photons which scales as .
The ratio of the number of baryons per photon at recombination () to that at BBN () is related to the fractional density excess of the energy pumped between the epoch of BBN and recombination as Nakama:2014vla ()
Nakama et al. have used the observationally inferred values of and as and respectively Ade:2015xua (); Nollett:2013pwa (). Using these values, they obtained an upper bound on the density fraction of the injected energy as
In our study, we calculate the fractional density excess for PBHs evaporation. This fractional density excess depends on . Further, using Eq.(4) we place a bound on .
For a certain region to collapse into a black hole for a flat Friedmann background, it was found that , the fractional mass difference within the initial horizon) should be of the order of Carr1975 ().
Shibata and Sasaki Shibata:1999zs () had obtained, using numerical simulations, the threshold value of density fluctuations required for the overdense regions to collapse into PBHs in the horizon crossing epoch. We use these results in our work.
The gauge-invariant curvature perturbation on the uniform-density hyper surface is defined as Wands:2000dp ()
where and are the background density and the perturbed density, respectively, is the curvature perturbation and dot (.) represents the time derivative. Green et al.Green:2004wb () derived the relation between the threshold of density perturbations and the threshold value of (). Corresponding to a threshold density perturbation of 0.5 and 0.3 the turns out to be 1.2 and 0.7 respectively Green:2004wb ().
In our analysis, we assume a power-law primordial power spectrum, . We have also used at the scale from WMAP Spergel:2006hy () as well as at the scale from Planck Ade:2015xua () to calculate in each case. When the density fields are smoothed by a Gaussian window function with comoving size , the peak theory gives the comoving number density of the peaks which are higher than and can be expressed in the high peak limit() as Green:2004wb ()
Here can be expressed in terms of the threshold value of the curvature perturbation for the PBH formation by
We consider Eq. (6) to be the comoving number density of PBHs formed from the collapse of the overdense regions with scale .
The smoothing scale can be related to the PBH mass which further depends on the initial environment around the peak as well as the threshold value. For the sake of simplicity, we assume that PBHs with the horizon mass are produced when the regions with overdensity above the threshold value enter the horizon. The scale at this epoch is given by
The comoving entropy conservation gives
where and denote the number of relativistic degrees of freedom and the redshift of the matter-radiation equality, respectively.
Tashiro et al.Tashiro:2008sf () have calculated the number density of PBHs in the mass interval and at any epoch, ,
We have considered the high peak limit, in obtaining Eq.(II).
Iii Estimation of injected energy via PBHs radiation
In this section, we summarize the calculations for the estimation of the energy pumped into the background fluid due to the Hawking radiation of PBHs in the redshift range and . Consider a Schwarzschild black hole with mass which emits particles near the horizon with spin Page-1976 (). The rate of total energy emitted between and per degree of freedom is given by Hawking-1975 (); Gibbon-1991 (),
where the temperature of a black hole is related to its mass as
Here signifies the dimensionless absorption probability of emitted species. In case of photons, can be expressed as Gibbon-Webber ()
The PBH loses its mass by virtue of Hawking evaporation. The rate of mass loss of a PBH can be given as
We can approximately integrate Eq.(16) and evaluate the time evolution of the PBH mass which was formed at some earlier epoch with an initial mass by
where being a weak function of takes typical values viz. 1.0, 1.6, 9.8 or 13.6 for g, g, g, g respectively MacGibbon:1991tj (). We can express the energy injection rate due to evaporating PBHs as
where is the horizon mass at a time and is the minimum initial mass of the PBHs. Using Eq.(2) and Eq.(18), we estimate the total energy dissipated by evaporating PBHs into the background fluid between the redshifts and .
Iv The PBH Abundance
To describe the PBH mass abundance, we define a function which represents the fraction of total energy density which collapses to form the black holes, as
Here denotes the energy density of black holes and represents the background energy density of the universe at the formation of black hole of mass .
V Results and Discussion
We assumed power-law primordial spectrum, . The mass range of the PBHs constrained in this study lies between and g, which corresponds to those PBHs whose maximal evaporation took place in the redshift range . We have obtained an upper bound on the spectral index of density fluctuations by using observational parameters from WMAP as well as Planck. In Fig.1 we have shown the results for the case of WMAP where is at the scale Spergel:2006hy (). In this case, we found the upper bound on the spectral index to be for and for . On the other hand, in Fig.2 using Planck results for cosmological parameters where is at the scale Ade:2015xua () we have found the upper bound on the spectral index to be for and for . In obtaining the above mentioned bounds on we have only considered the contribution of photons that are being directly emitted from the black holes. However, if the contribution of secondary photon emission is considered, our bounds on remains same upto the second decimal place.
In our work, we have considered a blue spectrum for the initial density fluctuations. To normalise the spectrum, we have used WMAP and Planck data. The pivot scale for normalisation considered in the Planck data ( Mpc) is greater than that of WMAP data ( Mpc) but the amplitudes on both the scales are almost same. Since we have considered a blue spectrum, it turns out that the spectrum amplitude for the scales of our interest for Planck data is smaller compared to the WMAP data. Due to this fact, the number of black holes formed for the scales of our interest are larger for WMAP data than the Planck data. This is the reason, we are getting a tighter constraint on in the case of WMAP data compared to the case of Planck data.
In estimating the fractional density excess for power law behaviour with different values of , we assume that mass black hole formed after reheating. Hence reheating takes place at a temperature larger than the temperature of the universe when the black holes of mass is formed ( is the mass of that black hole which evaporate away at the epoch of BBN). If the reheating takes place below this temperature then the masses of the black holes formed will be greater than . It means that as the temperature of reheating decreases, mass range of the black holes which evaporate away between the epoch of BBN and CMB- distortions also decreases. The maximum contribution to the excess density fraction comes from the minimum mass of the black hole in this range. This allows us to constrain the mass fraction of the black holes using the constraint on the spectral index. Further we take different reheating temperatures and calculate their corresponding mass scales and determine the bounds on the spectral index. Using this mass and the corresponding bound on spectral index, we calculate the beta function.
In Fig.(3) we show the upper bound obtained on the PBHs abundance as a function of their masses. The upper bound turns out to be for the investigated mass range ().
In the present work, we have considered evaporating PBHs that had injected energy into the background fluid. Since the processes like double Compton scattering and Bremsstrahlung are very significant during the early phases of the universe, the number density of background photons increases while the number density of baryons remain unchanged. As a result of this, the baryon-photon number ratio decreases due to additional reheating. Since modern cosmological observations such as BBN and CMB well constraints this baryon- photon number ratio we can use this to obtain an upper bound on the spectral index of density fluctuations at small scales which are difficult to be probed through the observations of CMB anisotropies and distortions. Using Planck results for cosmological parameters we obtained the upper bound on for and for respectively. In our study, the constraint on the PBH density fraction is found to be . This is a significant probe of the amplitude of density fluctuations on small scales that cannot be accessed by CMB anisotropies.
Acknowledgements.GG would like to thank University Grants Commission, Govt. of India to provide financial support via Dr. D. S. Kothari Postdoctoral Fellowship. RS acknowledges CSIR, India for the financial support through grant 09/045(1343)/2014-EMR-I. TRS acknowledges SERB for the project grant EMR/2016/002286. The authors also acknowledge the services of IRC, Delhi.
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