Quantum effects on the evaporation of PBHs: contributions to dark matter
Md Riajul Haque Suvashis Maity Debaprasad Maity Yann Mambrini





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            ^ariaj.0009@gmail.com^bsuvashis@physics.iitm.ac.in^cdebu@iitg.ac.in^dyann.mambrini@ijclab.in2p3.fr^1Centre for Strings, Gravitation and Cosmology,
            Department of Physics, Indian Institute of Technology Madras, 
            Chennai 600036, India^2Department of Physics, Indian Institute of Technology Guwahati, Guwahati, Assam, India^3 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
            We compute the relic abundance of dark matter in the presence of Primordial Black Holes (PBHs) beyond the semiclassical approximation. We take into account the quantum corrections 
            due to the memory burden effect, which is assumed to suppress the 
            black hole evaporation rate by the inverse power of its own entropy. Such 
            quantum effect significantly enhances the lifetime, rendering the 
            possibility of PBH mass ≲ 10^9 g being the sole dark matter (DM) candidate. 
            However, Nature can not rule out the existence of fundamental 
            particles such as DM. We, therefore, include the possibility of 
            populating the dark sector by the decay of PBHs to those 
            fundamental particles, adding the contribution to stable PBH whose lifetime is extended due to the quantum corrections. Depending on 
            the strength of the burden effect, we show that a wide range of 
            parameter space opens up in the initial PBH mass and fundamental dark matter mass plane that respects the correct relic abundance. 
             Quantum effects on the evaporation of PBHs: contributions to dark matter
                Yann Mambrini^3 d,
                April 29, 2024
            ========================================================================
            
            
            
            
            § INTRODUCTION 
            
            Primordial Black holes (PBHs) are fascinating objects 
            proposed more than 50 years ago <cit.>, 
            but they have recently gained widespread interest, 
            particularly in the context of dark matter (DM) <cit.> and searches
            for gravitational waves <cit.>. Indeed, 
            contrary to the conventional DM matter candidates, the proposal of PBHs as a possible relic could be considered a huge conceptual 
            jump, which does not require any new physics beyond the Standard Model and gravity. 
            However, black holes are metastable candidates and emit
            fundamental particles through the well-known phenomena 
            of Hawking radiation <cit.>. 
            In vanilla scenarios, PBHs weighing less than 10^15 grams have already evaporated, so they cannot be considered as potential dark matter candidates, even if they can still play an important role in reheating <cit.>, leptogenesis <cit.>, gravitational wave 
            <cit.> or dark matter production
            <cit.>.
            In fact, even PBHs of higher masses, in spite of having survived, 
            enter in conflict with observations of γ-ray <cit.>, light elements abundance during big bang nucleosynthesis (BBN) <cit.>
            or other indirect observations <cit.>. Therefore, the idea of PBHs being a real DM candidate is gradually fading away except within a very narrow mass window 10^16 - 10^23 grams (see the review <cit.> and references therein for a complete analysis). 
            
            However, this semiclassical phenomenon of black hole radiation heavily relies on the assumption that during the 
            evaporation process, the black hole remains classical till the end of its lifetime. Over the years, it has been realized that such a semiclassical approach 
            should not remain self-similar <cit.> throughout its entire lifetime, and new physics should come into play, 
            particularly in the parlance of the information loss paradox. It has recently been argued <cit.> that during its 
            evaporation, when the mass of a BH reaches a certain fraction of its initial value, the backreaction can not 
            be ignored, and it can potentially reduce the evaporation rate by the inverse power law of its entropy S^-k. This, in turn, 
            significantly enhances the lifetime of black holes, reopening the possibility of the BHs of mass <10^15 grams being the sole DM 
            candidate as recently discussed in <cit.>. 
            In this paper, we will be inclusive, considering also the possibility of a multi-component scenario, where part of the total DM could be fundamental particles produced through the evaporation process. Such inclusion and its implicit dependence on the enhanced BHs lifetime are, therefore, expected to alter the existing constraints <cit.> on particle DM and PBH parameter space significantly, which is the subject of our study.
            
            
            At this point, we shall mention that there are several potential mechanisms through which PBHs can form. The mechanism includes the collapse of the enhanced density perturbations that originated during  inflation <cit.>,
            bubble collisions <cit.>, 
            collapse 
            of domain walls <cit.>, 
            collapse of cosmic strings <cit.>, 
            electroweak phase transition <cit.>, 
            first-order phase transitions <cit.>, 
            and formation of other topological defects <cit.>.
            In this work, we will not focus on the details of how the PBHs are formed. Rather, our discussion spans around the evolution, especially the evaporation of the PBHs and the effects of memory burden on the process. 
            
            
            
            The paper is organized as follows. In section <ref>, we discuss the evolution of PBH mass when the memory effect is active after a certain fraction of the initial mass of the PBH remains due to Hawking evaporation.
            In section <ref>, we analyze the effects of the memory burden on the DM that is produced due to the evaporation of PBHs.
            We shall distinguish two different scenarios in this context. In section <ref>, we consider the mass range of PBHs that evaporates  before BBN completely, whereas in section <ref>, we look at the case where the initial phase of evaporation happens before BBN while the PBHs are 
             stable till present day. We then conclude in section <ref>.
            
            Before we begin, let us list out the main notations that we will be using throughout our analysis:
            
            [colback=gray!5!white,colframe=gray!50!black,colbacktitle=gray!75!black,title=Notations about scale factor and time]
            
              ,: Scale factor and time  at the PBH formation
            
              ,: Scale factor and time at the onset of memory burden 
            
              ,: Scale factor and time at PBH domination
            
              ,: Scale factor and time at the PBH evaporation 
            
              , : Scale factor and time at BBN
             
            
            
            § MASS EVOLUTION OF PBHS DUE TO EVAPORATION AN THE EFFECTS OF MEMORY BURDEN
            
            As a first step, we need to study the evaporation process of the black hole in detail, taking into account the memory burden effects (for a detailed discussion see Refs. <cit.>).
            
            After the formation, PBH energy density = is mostly governed by two quantities: the PBH number density,  which decreases as the universe expands 
            
            as ∼ a^-3 (a be the scale factor),
            and the mass of PBHs , which changes due to the mechanisms of evaporation and accretion.
            We shall first present a brief summary of the evaporation process
            in the semiclassical approximation before considering the quantum effect. 
            The Hawking temperature,  and entropy, S associated with a PBH with mass  are given by
            =^2,     S=12ł()̊^2=12ł()̊^2,
             
            where = 1/√(8 π G_N)≃ 2.4× 10^18 GeV
            is the reduced Planck mass.
            If we consider that the evaporation happens by purely Hawking radiation then the rate of change of mass at any point is <cit.>
            ṭ=-ϵ^4^2,
             
            where ϵ =274π g_∗()480, 
            with g_∗() being the number of degrees of freedom 
            associated with the PBH temperature. The factor 27/4 accounts for the graybody factor[For instance, a more thorough formulation for the greybody factor can be found in Refs. <cit.>.]<cit.>, and the negative sign on the right-hand side is since the PBH mass decreases with time due to the evaporation. We shall note that the accretion effect is negligible here.
            
            After integrating Eq. (<ref>) from the time of the formation of PBH,  to the time t, we find the PBH mass at t is given by
            (t)=ł[1-^0(t-)]̊^1/3,
             
            where  is the initial mass of the PBH, which is related to
            the horizon size at the time of formation, 
            =4/3πγ^-3
            = 4πγ^2/H_ in ,
            
            where we assume the formation of PBHs in a radiation-dominated universe, so ^2=ρ_R()/(3 ^2), and ρ_R() is the background radiation energy density and can be connected with the formation temperature T_ in, ρ_R()=α T_ in^4 with α=π^2 g_*/30. Here, g_* is the relativistic degrees of freedom associated with the thermal bath, which we assume g_*=106.75. 
            In Eq. (<ref>),
             γ represents the efficiency factor for collapse, which defines what fraction of the total mass inside the Hubble radius collapses to form PBHs. For standard radiation domination, γ∼ 0.2<cit.>. 
            The time of the PBH formation can be related to the formation mass of the PBH as 
            =/(8πγ^2), 
            where we have considered a radiation-dominated Universe, H(t)=1/(2t). 
            The quantity ^0=3ϵ^4/^3 is the decay width associated with the evaporation of PBH. 
            Note that the evolution of the PBH mass
            described by Eq. (<ref>) is really abrupt, hence Stephen Hawking's own description of it as an `explosion'. The mass of the PBH (t) remains almost constant 
            ∼ during the whole process of evaporation, and then the PBH
            rapidly decays at t=.
            
            We find the lifetime, t_ ev
            of the PBH by solving (t_ ev)=0 in Eq. (<ref>)=1^0=^33ϵ^4≃ 2.4× 10^-28(/1 g)^3 s ,
             
            where we supposed ≫.
            During radiation domination, the Hubble parameter 
            is related to the radiation temperature as H=√(α/3) T^2/.
            One can then connect time and temperature during the radiation era as 
             
            t=12√(3α)T^2 .
             
            From Eq. (<ref>), we recover that PBHs of mass ≳ 10^15 grams
            should not have decayed yet, whereas PBHs of mass below ≲ 10^9 grams should have decayed before the BBN epoch, or equivalently, before 1 second[Note that all the above conclusions are under the assumption that the PBHs radiate particles in a self-similar semiclassical process until their end of life.]. 
            
            
            We now consider a special type of correction to the evaporation process proposed in <cit.>. Such correction is called the memory burden's effect, which essentially suggests that the quantum modes associated with the entropic degrees of freedom of a black hole necessarily have a strong backreaction effect on its own evaporation process. Therefore, the initial semiclassical Hawking evaporation will no longer be valid after a certain time scale. For the purpose of our analysis, we can suppose that the semiclassical regime is valid
            until the mass of the PBH reaches a certain value, 
            i.e. =q, with 0<q<1. 
            The authors of <cit.> 
            proposed that the quantum
            effects begin to be important when =(1/2),
            or q=1/2. However, such value is subjected to the detailed quantum mechanical modeling of a black hole. 
            To keep our study as general as possible, let  be the time at the end of the semiclassical phase.
            Then, from Eq. (<ref>) we obtain 
            = 1-q^3^0,
             
            where ^0 is defined before.
            In the above equation, with the substitution of q=0, one shall recover the full evaporation time, .
            Once the mass of the PBH reaches q, at ,
            the quantum memory effect starts dominating. Upon parameterizing 
            the memory burden effect in the second phase, the evolution of the mass which is given in Eq. (<ref>) modifies to[Note that  being almost constant during the evaporation process, S()∼ S(), which corresponds to the approximation made in <cit.>, and which we will apply for our numerical analysis.]ṭ=-ϵł[S()]̊^k^4^2,
             
            where S() is the BH entropy defined in Eq. (<ref>). The parameter k characterizes the efficiency of the backreaction effect. So far, we do not have any theoretical constraints on the value of the power k except for its probably being a positive number. For our present purpose, we would take it to be an integer.
            We can understand the quantum effect as a slowdown of the decay due to an 
            excess of entropy, produced by its evaporation, surrounding the PBH. 
            After integrating Eq. (<ref>) we obtain
            =qł[1-^k(t-)]̊^1/3+2k,  with  ^k= 2^k(3+2k) ϵł(q)̊^3+2k.
            
            
            
            
            
            From Eq. (<ref>), we obtain that the second phase of evaporation will occur for a time ∼ 1/^k. Hence, the total evaporation
            time is given by ^k=+1/^k≃ 1/^k. 
            It is straightforward to see that for k=0 and q=1, ^k will be equal to as given in Eq. (<ref>). Also for k>0, we have ^k≫ (for a detailed discussion see Ref. <cit.>) and
            we obtain
            
            
            
            ^k ≃q^3+2k/2^k(3+2k)(/4.3×10^-6 g)^3+2k 5.7 × 10^-44 s ,
            which gives, for instance, for k=1 and q=1/2
            t_ ev^k=1≃ 1.2× 10^-19(/1 g)^5  s≃ 4× 10^17(/2×10^7 g)^5 s .
            
            We then conclude that even in the minimal k=1 case, 
            a PBH of only 2× 10^7 grams can survive until the present time and contribute to the DM relic abundance.
            To illustrate our result, we show in the left panel of Fig. <ref>  the evolution of PBH mass as 
            a function of time for q=1/2 and different values of 
            k=0, 1, 3 and 5, which are plotted in dotted, dot-dashed, dashed and solid lines respectively. Obviously, we see that for k=0, there is no second phase of evolution because the treatment is completely semiclassical.
            As the value of k increases, the greater is the quantum effect, slowing down the process of decay even further. We also presents some numerical results of
            the evaporation time  in Table <ref> for 3 PBHs masses (1, 10^4 
            and 10^8 g) with and without taking into account the memory burden effect for different values of k, and with the 
            formation time  for information.
            
            
            
            We focus on the case where the semiclassical phase of the evaporation ends before BBN. The maximum initial 
            mass of a PBH for which it will half evaporate before BBN can be obtained 
            from Eq. (<ref>) with q=1/2. This is independent of the value of k 
            and is represented by the red line on top of the right panel in  Fig. <ref>. The red-shaded region above the line indicates the region where the Hawking evaporation of PBHs happens after BBN, i.e., >. Note that we assume =1 second throughout our analysis. On the other hand,
            the magenta line corresponds to the maximum value of  for which the complete evaporation happens before BBN, i.e., ^tot≤ and the value is given by
            =qł((3+2k) ϵ  2^k )̊^1/(2k+3) .
             
            We see that with the increase in the value of k, the PBHs become more stable hence the maximum value of  that evaporates before BBN decreases.
            
            
            However, the interesting situation is when PBHs' lifetime in the semiclassical approximation is  below the BBN limit (below the red line) while the quantum effect renders them stable at the scale of the age of the Universe. This region appears to be above the blue line in  Fig. <ref>.
            In other words, considering the quantum effect, some of the PBHs with mass ≤ 10^8 g will survive till today. For instance, for k=1, 2 and 3 the lower limit of the mass that should have survived are 
             ∼ 2× 10^7 g, 7× 10^3 g, and 80 g respectively.
            However, being stable at the scale of the age of the Universe is not sufficient. 
            Indeed, the metastability of PBHs implies the possibility to observe them through 
            their radiation in the CMB or extragalactic γ-ray. The non-
            observation of such signals exclude part of the region, shaded in 
            green and orange of Fig. <ref> respectively, to be the 
            sole DM candidate. Therefore, if at all PBHs form in those mass ranges, their fractional abundance must be very 
            small, and an additional component needs to be incorporated to match the present DM abundance, under the form of dark particles, for instance. We will calculate 
            in detail those contributions of additional particles like DM 
            components produced directly from the PBHs decay.  
            
            
            To be more precise, PBHs can act as DM due to their cold, non-interacting nature.
            The fraction of total DM that is given by the stable PBHs today is often referred to as .
            In Fig. <ref>, the region above the blue line has two additional constraints on  coming from extragalactic γ rays and from CMB anisotropies which are shaded in orange and green, respectively. PBHs can not contribute to the total dark matter in these regions, i.e., <1.  
            The contribution to the extragalactic γ rays comes from the evaporation of PBHs between the time of recombination and the present time.
            On the other hand, the evaporation of PBHs can inject energy into the neutral medium and ionize it after the recombination.
            This restricts the amount of  as the injected energy affects the angular power spectrum of temperature and polarization of CMB due to the rescattering of CMB photons. A detailed discussion about these constraints can be found in Ref. <cit.>.
            
            
            As emphasized earlier, in this paper, we consider the present-day DM to be composed of both PBHs and fundamental particles. Our work will then focus on these two interesting regions, the unshaded ones in Fig. <ref>, 
            where dark matter can be produced by PBH evaporation while not being excluded by other constraints.
            Indeed, whereas PBHs can be dark matter candidates
            within this parameter space, the fact that the semiclassical
            process takes place before BBN also creates the possibility 
            of producing a particle-like dark matter candidate through the PBH's early decay. In that case, the relic abundance should be the sum of the PBH dark decay product and the surviving PBH density.
            Note that the black line corresponds to the minimum value of  set by the maximum allowed value of the inflationary energy scale H_ inf∼ 5× 10^13 GeV, calculated considering the upper bound of the tensor-to scalar ratio r<0.036 from Planck together with latest BICEP/Keck data <cit.>.
            
            
            
            
            
            
            
            § PRODUCTION OF DARK MATTER AND THE EFFECTS OF MEMORY BURDEN 
            
            
            To study the effects of the memory burden on the production of dark matter from the evaporation of PBHs, we should consider two possible scenarios. 
            We shall first consider the case when the PBHs evaporate completely before BBN, and the dark matter generated from the evaporation satisfy the present dark matter relic. 
            This is represented by the white region below the magenta line in Fig. <ref>. 
            Another possibility is to consider that the first phase of Hawking evaporation (semiclassical approximation) 
            happens  before the BBN, whereas the PBH remains 
            stable till today due to the memory effect.  
            This is the white region above the blue line of Fig. <ref>. 
            First, we need to compute the number of particles that are emitted from the evaporation of one PBH. 
            Note that from now on, we will coin  phase-I  the semiclassical evaporation phase,  and  phase-II the second phase, where quantum correction is effective.
            
            
            
            
            
            
             §.§ Number of dark matter particles emitted from the evaporation of a PBH
            
            
            Let N_j be the number of particles that are emitted from the evaporation of one PBH. 
            We divide N_j into two parts N_1j and N_2j, which are the number of particles emitted from PBHs for  phase-I and  phase-II, respectively.
            We shall mention that BH mass and spin affect the production rate of any species that is generated from evaporation. 
            In this work, for simplicity, we restrict ourselves to the case of spin-zero
            Schwarzschild BH.
            The emission rate of particles j with mass m_j and internal degrees of freedom g_j that escape the horizon of radius R_S 
            per unit time per unit energy interval due to the Hawking radiation in the  phase-I is
            given by <cit.>^̣2 N_1jẸṭ   =   274π R_S^2 g_j2π^2E^2exp(E/)± 1 = 274g_j32 π^3(E/)^2exp(E/)± 1 
            where the Schwarzchild radius is given by 
            R_S=/(4π^2). 
            The sign ± is used for fermionic and bosonic particles, respectively.
            We can estimate the total energy emitted per unit time from a BH by integrating over Eq. (<ref>) as
            Ṇ_1jṭ =274ξ g_j  ζ(3)16π^3^2,   where  ξ=
                1   for bosons
            34   for fermions.
             
            In the  phase-II,
            the emission rate is reduced, affected by the quantum corrections :
            
            ^̣2  N_2jẸṭ = 1ł[S()]̊^k^̣2 N_1jẸṭ ,
             
            where the expression for ^̣2N_1j/(Ẹṭ) is given in Eq. (<ref>). After integrating over all the energy modes, the emission rate is given by 
            Ṇ_2jṭ =274ξ g_j  ζ(3)2^k16π^3^2+2k^1+2k.
             
            
            
            
            
            The total number of particles emitted from the evaporation of a BH can be obtained by integrating Eq. (<ref>) and Eq. (<ref>). 
            Depending on the mass of the emitted particle 
            relative to the PBH formation temperature ^ in, 
            one should distinguish two cases. 
            If m_j<^ in, the production of 
            particles happen throughout the lifetime of the PBH, i.e., from the initial time  to a final time . Indeed, the BH
            temperature  increasing with time, the relation 
            m_j< will always hold. On the other hand,
            if m_j>^ in, the evaporation will start from a time t_j where m_j=(t_j), the production being exponentially 
            suppressed before. This can happen either during  phase-I  (t_j<) or during  phase-II (t_j>). Combining Eq. (<ref>) and Eq. (<ref>) we 
            can calculate the value of t_j 
            t_j=
            ^0ł[1-^6^3 m_j^3]̊   for t_j<
               
            ^kł[1-ł(^2q m_j)̊^3+2k]̊   for t_j> ,
              
            where  is given by Eq. (<ref>) and we supposed t_j ≫.
            
            
            
            With the relations in hand let us proceed to calculate the number of particle emitted from the evaporation. First we shall consider the case when m_j<T__ BH^ in. To calculate the number in the first phase, we integrate Eq. (<ref>) within the limit [t_ in,t_ q] which leads to
             
            N_1j   =   15 ξ g_jζ(3)(1-q^2)g_∗()π^4^2^2 ,
             
            where we see the case of complete evaporation due to only Hawking radiation can be obtained by substituting q=0. During the second phase of evaporation, we shall integrate Eq. (<ref>) from  to a final time t, and the number of dark matter emitted is given by 
            
            
            
            
             
            N_2j   =   15 ξ g_jζ(3)g_∗()π^4q^2^2^2ł[1-ł(1-t-t_ ev)̊^23+2k]̊ .
             
            The total number of dark matter particles from the evaporation of a PBH can be obtained by adding N_1j and N_2j 
            from Eq. (<ref>) and Eq. (<ref>). For those BHs evaporated by today, t=≫ and we have the total numbers of DM particles emitted from a single BH
             
            N_j=N_1j+N_2j=15 ξ g_jζ(3)g_∗()π^4^2^2 ,
             
            which is the same as the total number of particles emitted from the Hawking evaporation. This is not surprising as the total number of emitted particles due to the evaporation of a PBH does not depend on the evaporation process.
            
            
            
            For the case m_j>^ in,
            we have two different scenarios: t_j<t_ q and t_j>. For t_j<, we get the number of particles emitted 
            during  phase-I by simply integrating Eq. (<ref>) in the interval [t_j,], no particles being produced before due to the Boltzmann suppression factor,
            
             
            N_1j= 15 ξ g_jζ(3)g_∗()π^4ł(^2m_j^2 -q^2^2^2)̊.
              
            Where we used (t_j)=^2/m_j. 
            The number of particles emitted  N_2j in the  phase-II 
            would be the same as obtained in Eq. (<ref>). 
            On the other hand, if t_j>, corresponding to m_j>^2/q, 
            there is obviously no dark matter emission during  phase-I, i.e., N_1j=0.
            The number of particles emitted during the second phase is given by 
             
            N_2j=15ξ g_j ζ(3)g_∗()π^4q^2^2^2ł[ł(1-t_j-)̊^23+2k 
            -ł(1-t-)̊^23+2k]̊,
            
            where we have integrated Eq. (<ref>) in the time interval [t_j,t] with t_j given in Eq. (<ref>) for t_j>.
            In both cases after complete evaporation, we get the total number of DM particles emitted for
            m_j>^ in by setting t=> t_j≫, to be
             
            N_j=N_1j+N_2j=15ξ g_j ζ(3)g_∗()π^4^2m_j^2,
             
            which is again the number of particles 
            emitted by pure Hawking evaporation for m_j>^ in<cit.>.
            In the next 
            section, we shall discuss the case where the PBH evaporates completely,
            before studying the case where relic PBHs could contribute to the dark abundance, in addition to its decay products.
            
            
            
             §.§ Dark matter from the evaporation of PBHs before BBN 
            
            
              §.§.§ Computation of β_ c
            
            The memory burden effect increases the lifetime of the PBHs. Moreover, increasing 
            the value of k, such burden effect becomes stronger, and as a consequence, it limits the maximum value of 
            the PBH mass that evaporates before BBN, 
            which is also evident from Fig. <ref>. Note that in this figure, we assume 
            that the memory effect is effective after its half-life (q=1/2). 
            On the other hand, the minimum allowed mass
            for a PBH is set by the minimum
            horizon size at the end of inflation, and determined by the energy available of the inflaton. Assuming de-sitter-like inflation and taking the constraint on the 
            tensor-to-scale ratio r<0.036 from Planck combined with the latest BICEP/ Keck data <cit.>, the minimum allowed mass is roughly around ≳ 0.5 g. Interestingly, for this minimum PBH mass, one requires k≲3 to ensure it is evaporated completely before BBN.  
            
            
            
            
            In this section, we will consider two 
            possible scenarios for dark matter production, depending on the PBH parameters 
            (, β), where β = ρ_ BH^ in/ρ_R^ in is the ratio between the PBH energy density over radiation energy density at the point of formation. Initially, we will assume that the radiation energy density always dominates the background. Next, 
            we will suppose that for values of β above a critical value β_ c, 
            the PBH energy density dominates over the radiation energy 
            density at some time before the evaporation. 
            In both cases, we shall assume that the dark matter is produced solely from the evaporation of PBHs. Let's first determine 
            β_ c.
            
            
            The radiation energy density evolves as 
            ρ_R(a)    =   ρ_R^inł(a)̊^4,
             
            where ρ_R^in is the radiation energy density at the time of PBH formation, and  is the corresponding scale factor.
            We also know that PBHs behave like matter; therefore, the energy density of PBHs varies as ∼ a^-3. At any time, the energy density of PBH can be written in terms of the initial energy density as 
            ρ_ BH(a)= 
            ρ_ BH^ inł(a)̊^3    for  a< 
            
                q ρ__ BH^ inł(a)̊^3    for  a> ,
            
            where for a∼, we assume that the mass of the PBHs will be reduced to q instantly. There 
            always exists a β value, i.e., β_ c, above which it is possible 
            that after a time , the PBH 
            energy density dominates over the radiation energy density.
            Equating Eqs. (<ref>) and (<ref>), this happens for a scale factor /=1/q β .
            
            The domination of PBHs is possible if <,
            or
            β>/q= 1/q√(H_ ev/H_ in) = 1/q√(^k/2 H_ in) = β_ c ,
            
            or 
            β_ c=1/q^5/2+kł()̊^1+k√((3+2k)2^kϵ8 πγ).
            
            where we used Eqs. (<ref>) and (<ref>).
            If one assume q=1/2, we have
            β_ c≃ 5.5× 2^3k2 √((3+2k))(/)^1+k ,
            
            or
            β_ c^k=0≃ 4.0× 10^-5(1  g/) ,   β_ c^k=1≃ 6.5× 10^-10(1  g/)^2 ,   β_ c^k=2≃ 9.4× 10^-15(1  g/)^3 .
            
            As we can notice, the critical value of β, for a given ,
            decreases drastically with k. This is understandable because 
            larger k means a longer lifetime. In other words, smaller values 
            of β are necessary to ensure PBH domination.
            
            As we increase k, the β_ c shifted towards smaller values, such as for k=0 and =1 g, β_ c∼ 4× 10^-5,whereas for k=1, β_ c∼ 6× 10^-10.
            We show in Fig. <ref> the evolution of β_ c
            as function of  for different choices of k. Note that even for very light PBHs ∼ 10 g, a value of β as low as 10^-17 is sufficient for the black holes to dominate the evolution of the Universe before BBN for k=2.
            
            
            
            
            
            To find the relic abundance of dark matter of the species j
            today we use <cit.>, 
            Ω_jh^2= 1.6× 10^8 g_0/g_ ev n_ j()/^3 m_j/GeV ,
            
            where n_j() is the number density of DM at the end of evaporation, whereas g_ ev∼106.75 and g_0=3.91 are the effective numbers of light species for entropy at the end of evaporation and present-day, respectively. 
            
            Once the PBH evaporated, we can write n_j()=N_j (), where N_j be the total number of particles emitted from the evaporation of a single PBH and () is the PBH energy density at the point of evaporation. Let us first discuss the case  β>β_ c.
            
            
              §.§.§ β>β_ c
            
            For β>β_ c, there will be a PBH domination  before PBH decays, which means that the reheating
            would then be produced entirely through 
            the evaporation. In this case, if one supposes an instant thermalization, the reheating temperature would be the temperature of the decay products given by
                  
            H^2()=ρ_R()/3M_P^2
            =α^4/3 ^2
            =4  (^k)^2/9 ,
            
                  ⇒=ł(43α)̊^1/4ł[(3+2k)2^kϵł(q )̊^3+2k]̊^1/2 .
            
            Since PBH decay happens in a matter-dominated universe,
            H()=2/(3 ). 
            
            From Eq. (<ref>), using
            ρ_ BH()=ρ_R()≃()× q, we have the PBH number density at the evaporation point
             ()=43^3 (3+2k)^22^2kϵ^2ł(q )̊^7+4k .
            
            Combining Eq. (<ref>) with Eqs. (<ref>) and (<ref>), writing n_j=× N_j,
            we obtain for m_j<^ inΩ_jh^20.12= 2.85×10^6 ξ g_j/q^2 ł(2^k(3+2k))̊^1/2ł(q )̊^2k+1/2 m_j/GeV .
            
            Doing the same exercise for m_j>^ in, Eq. (<ref>) gives
            Ω_jh^20.12= 1.64×10^43 ξ g_j ł(2^k(3+2k))̊^1/2ł(q )̊^2k+5/2 GeV/m_j .
            
            
            
            We show in Fig. <ref> the exclusion parameter 
            due to an overabundance of dark matter in the shaded regions, 
            for different values of k and q=1/2 (on the left)
            or k=1 and different values of q (on the right). 
            We easily recognize the two 
            allowed regions, for light masses, corresponding to m_j≲^ in, Eq. (<ref>), and for large mass, when the time of allowed decay (when  reaches m_j)
            is sufficiently short to avoid the overproduction of dark matter, 
            Eq. (<ref>). The allowed region widens for larger values of k. This is easily understandable by the fact that 
            the memory burden effect extends the lifetime of PBH, diluting them further and then decaying. More diluted black holes imply a more diluted dark matter production, which requires a larger mass m_j
            to fit with the relic abundance constraints. The effect is the opposite if one decreases q because, in this case, the memory burden effect is  delayed, and a larger population of PBH decay
            follows the semiclassical approximation as we can see in Fig. <ref> (right). In the limit q→ 0, we recover exactly the semiclassical limit. Moreover,
            Note that once β>β_ c, the relic abundance no longer depends on β.
            
            
            
              §.§.§ β<β_ c
            
            
            For β<β_ c, PBHs evaporate during radiation domination, so there is no PBH domination. 
            The Hubble parameter at the point of evaporation is then given by
            
            H^2()=ρ_R()/3^2
            =g_ *π^2 ^4/90 ^2
            = (^k)^2/4 ,
            
            which gives the evaporation temperature 
            = ł(34α)̊^1/4ł[(3+2k)2^kϵł(q )̊^3+2k]̊^1/2.
             
             During radiation dominated era, taking H=1/(2 t) and a∝ t^1/2, using
            ()=()(/)^3=βρ_R^ in/(/)^3
            ,   /=√(H()/)=√(^k/2) ,
             
             and  given by Eq. (<ref>),
             we get the number density of the PBH to be
             ()= 3 β ^3 
            ł(π γ2)̊^1/2ł((3+2k)2^k ϵq^3+2k)̊^3/2ł()̊^6+3k.
              
             
            Finally Combining Eq. (<ref>) together with Eqs. (<ref>) and (<ref>), we have DM abundance today for m_j<^ in
                Ω_jh^20.12=2.54×10^6 β ξ g_j ł()̊^1/2 m_j/GeV≃ξ g_j(β/10^-20)(/1 g)^1/2(m_j/8.2× 10^10 GeV)
                 .
            
            On the other hand, by doing the same analysis, one can arrive at the following expression for m_j>T_ BH^ in
                Ω_jh^20.12= 1.47×10^43 β ξ g_jł()̊^3/2 GeV/m_j≃ξ g_j(β/10^-20)(1 g/)^3/2(1.3× 10^15 GeV/m_j)
                 .
            
            From the above expressions for two different cases Eqs. (<ref>) and (<ref>), it is interesting to note that the DM relic abundance is independent of the memory burden parameter k. This is easily understandable as, in this case, the total number of produced dark matter particles is the same irrespective of the physics behind the PBH decay, and further, the dilution effect is exclusively due to the radiation background.
            
            
            
            
            
            
            
            We show in Fig. <ref> the relic abundance constraint for q=1/2 and two different values of β=10^-19
            and 10^-21, well below β_ c as one can see from 
            Fig. <ref>. We note that for the case β < β_ c the parameter space follows a triangular exclusion zone 
            in the center limited by the cases m_j < ^ in and m_j > ^ in. However, we indeed see that such shape of the constraints region does not 
            depend on the burden parameter k anymore except for setting the maximum value  represented by red dashed lines. As stated earlier, this is due to the fact that the total number of dark matter particles produced is the same for both semiclassical and quantum-corrected PBH evaporation processes in conjunction with the radiation background. 
            On the other hand, as expected, there exists a strong dependence
            of the relic density on β, for this parameter determines the relative amount of PBHs present in the thermal plasma, which will eventually decay 
            (proportionally to β) into dark matter.
            
            
            
            
             §.§ Dark matter from the stable PBHs with Hawking evaporation ( phase-I) before BBN 
            
            
            
            Let us now look into the detail of the situation where standard Hawking evaporation takes place  before BBN 
            while the memory burden effect delays the complete evaporation, making the PBHs decay  after 
            the present time, rendering them stable on the cosmological scale. 
            In this case, the total dark matter relic (Ω_ DM) has two contributions: the dark matter as stable fundamental particles (Ω_j) produced from PBH evaporation computed in the previous sections 
             plus the stable PBHs Ω_ PBH that contribute
            also to the dark component of the Universe :
            Ω_ DM=Ω_j+ Ω_ PBH.
            
            This is attributed to the allowed region
            in white tagged as “Stable PBHs" in Fig. <ref>.
            
            
            
            To compute the dark matter abundance
            generated by the evaporation during  phase-I, Ω_j, we can use the expression for present-day DM relic due to evaporation, Eq. (<ref>),
            taken at , the time duration of  phase-I :
            
            Ω_jh^2= 1.6× 10^8 g_0/g_ q n_ j()/T^3() m_j/GeV ,
            
            where g_ q is the effective number of degrees of freedom for the entropy at the end of 
             phase-I, which we assume has the same value[Generalization for any g_ q is straightforward.] as g_ ev. 
            The ratio n_j()/T^3() is given by[We supposed here β < β_c as when stable PBHs contribute to the entire dark matter, PBH domination starts roughly at the standard radiation-matter equality.]n_j()T^3()=
            N_j ()^3=
            N_jβα / ,
             
            where  is the radiation temperature at the point of formation, and we supposed the Universe 
            is radiation-dominated during the semiclassical 
            phase (β<β_ c). Following Eq. (<ref>) we have
            
            =(48π^2γ^2/α)^1/4 √(/)  ,
            
            Now, upon substitution of Eqs. (<ref>) and (<ref>) into (<ref>), the final expression for DM relic from evaporation 
            Ω_j h^20.12≃1.5× 10^9N_j βł()̊^3/2m_j GeV ,
             
            Where the total number of particles emitted per PBHs from  phase-I, are given by Eqs. (<ref>)
            and (<ref>) 
            N_j= 15 ξ  g_j ζ(3)g_∗() π^4
                (1-q^2)^2^2,     for m_j<^ in
            ^2m_j^2-q^2 ^2^2,     for ^ inq > m_j>^ in.
             
            
            We then need to consider the contribution from stable PBHs. Their relic abundance can be written as
            
            
            Ω_ PBH h^2=1.6× 10^8 g_0/g_ qρ_ BH()/T^3() 1/GeV .
            
            Connecting the end of  phase-I with the formation point, one can find the ratio
            ρ_ BH()/T^3()= q ρ_ BH()/T^3()=q β α  ,
            
            where we assumed that the relativistic degrees of freedom associated with radiation at the point of formation and at the end of the semiclassical phase are the same. Finally, combining Eq. (<ref>)
            with Eq. (<ref>), and substituting into Eq. (<ref>), we obtain the density of stable PBHs to be 
            Ω_ PBH h^20.12= 3.5 × 10^27 q βł()̊^1/2≃ q (β/1.4× 10^-25)
            (1  g/)^1/2 .
             
            Note that the quantity on the left side is also described as =Ω_ PBH h^2/0.12, which is the fraction of total dark matter that comes from stable PBHs today. 
            Finally, the total dark matter relic would be the sum of the contribution from evaporation in  phase-I, Eq. (<ref>) and from the sable PBHs which acts as dark matter, (<ref>), which gives
            
            Ω_ DM h^20.12= 3.5 × 10^27 βł()̊^1/2ł[q+N_jm_j]̊ ,
             
            which is the key equation of our paper. It summarizes the amount of dark components due to stable PBH  and its dark products. We expect domination of the dark products for small , whereas PBH relics 
            dominates for larger .
            
            
            
            
            
            
            Indeed, to summarize, we show for comparison 
            in Fig. <ref>
            the parameter space allowed by the relic abundance constraint in the plane (, β), 
            in three cases : 
            without taking into account the burden effect (k=0), and with burden effect for k=1,2 and 3 for q=1/2
            and m_j=10^6, 10^9 and 10^12 GeV.
            We first distinguish the two white regions, for low  and larger , corresponding to the “Evaporate before BBN" and “Stable PBHs" of Fig. <ref>. For k=0, in the mass range
            1  g≲≲ 10^9  g
            the PBHs decay completely before BBN, and the correct relic abundance is accomplished, depicted by the brown lines
            by their decay product to a fundamental dark component in the first panel of Fig. <ref>.
            They follow the behavior β∝^-1/2, see Eq. (<ref>) when 
            m_j < ^ in (< 10^13 GeV/m_j g), and β∝^3/2, see Eq. (<ref>)
            when m_j >^ in (> 10^13 GeV/m_j g).
            
            We recognize the same region for k=1 but with a restricted mass range, and the restriction is due to longer PBHs lifetime attributed to the memory burden effect, rendering  PBHs of mass 10^4   <  < 10^7 to decay after BBN and before the present time. In addition to that, the mass range from 10^7   <  < 10^9 are stable PBHs with the first phase before BBN but still restricted by the constraints from the extragalactic γ rays and CMB anisotropy shown in the orange and green shaded regions of Fig. <ref>. On the other hand, for k=2, a new unconstrained PBH mass window opens for large mass within 10^5  <  < 10^9, which survives at present due to the quantum effect and populates as total DM component,
            while in the mass range 1  <  < 20, still keeping the possibility of the DM to be populated by the dark matter product from PBH decay.
            In the case of stable PBH, the relic abundance is given by β∝^1/2 as one can see from Eq. (<ref>). Finally, only PBH as a dark matter is allowed for k≥ 3, with PBH relic densities not depending on k. This set of 4 figures nicely summarizes our work. Note that we set the q value at 1/2 to do this analysis.
            
            However, to complete our analysis, we also show the dependence on
            q in Fig. <ref>, for β=10^-19 (left) and 
            β=10^-21 (right), corresponding to the values used in Fig. <ref> for comparison. The shadowed region is excluded because it corresponds to parameter space, where there is an overproduction of dark matter. The limit q=0 on the left region corresponds to the semiclassical approximation, i.e., the memory burden effect never occurs. This limit corresponds to 
            the situation where ∼ 100% of the dark matter is composed of the decay products of PBH because there are no stable PBHs in this mass range.
            We can recognize the peak of the triangle shape of Fig. <ref> 
            appearing at ≃ 10^6 g 
            (≃ 100 g) for β=10^-19 (β=10^-21).
            For larger value of q, we know from Fig. <ref>
            that for β=10^-19, PBH cannot be a dark matter candidate, as they would overpopulate the Universe. That explains why all the regions are shadowed in Fig. <ref> left. However, for β=10^-21, we can 
            evaluate form Fig. <ref> bottom–left that  
            should be larger than ∼ 10^7 g for q=1/2. But we see from Fig. <ref> right that a larger region is allowed, up to the range 10  g≲≲ 10^9  g
            for q=5 × 10^-4. Indeed, for a fixed β value, smaller values of q induce a latter burden effect, and a smaller stable PBH mass is required to satisfy the total DM relic. In summary, for stable PBHs with q values q≤ 10^-2, there is a possibility that DM from PBH decay also contributes to the total DM relic, depending on the mass of the DM. However, for higher q values q≥ 10^-2, the dark matter from stable PBHs always dominates over the decay products.
            
            
            
            
            
            
            
            § CONCLUSIONS
            
            
            We computed the relic abundance in the presence of PBH beyond the semiclassical approximation. Indeed, when the mass of a BH reaches a certain fraction q of its initial value, the backreaction can not 
            be ignored, and it can potentially reduce the evaporation rate by the inverse power law of its entropy S^-k.
            Taking into account this effect due to the memory burden effect, 
            we added the dark component produced by the decay of PBHs during the early stage to the contribution of stable PBHs whose lifetime 
            is extended due to the quantum corrections. Since the lifetime of the BH is extended depending on the memory burden parameters k 
            and q, for the scenario of PBH domination, the duration of the PBH domination is also extended. As a result, the allowed region in m_j versus  parameter space is extended as we increase k for a fixed q or increase  q for a fixed k; that is also reflected in Fig. <ref>. Whereas, 
            in the case of PBH decay in a background dominated by radiation, the memory burden parameters limit the highest permitted BH mass 
            that completes the decay process before BBN and has no effect on the dark matter mass range, which we can see in Fig. <ref>. Depending on the strength of the burden effect, we show 
            that PBH decay products can satisfy the relic density constraints for PBH masses below ≲ (6× 10^3, 20, 1) g for k=(1, 2, 3), respectively. Moreover, with an increase in k > 3, there is no DM parameter space from PBH decay. Note that all of these mentioned values are quoted for considering phase-I 
            being complete after half-life, q=1/2. On the other hand, PBH mass in the range 10^3-10^9 g can be sufficiently stable and 
            numerous to fulfill the relic abundance of DM for k=3. The range of allowed masses even extends to 1  g-10^9  g for k≥5, see Fig. <ref>. 
            We also show that decreasing the value of q down to ∼10^-2
            reopens the dark matter parameter space from PBH decay, then we need to consider the contribution from both PBH decay and stable PBHs. 
            Our results are thus nicely summarized in Figs. <ref>, <ref> and <ref>, with Eq. (<ref>).
            
            
            
            
            § ACKNOWLEDGMENTS
            
            The authors want to warmly thank Donald Kpatcha, Simon Clery, Mathieu Gross, and Lucien Heurtier for extremely useful discussions during the Astro@Paris-Saclay symposium 2023.
            This project has received support from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 860881-HIDDeN and the CNRS-IRP project UCMN. 
            MRH wishes to acknowledge support from the Science and Engineering Research Board (SERB), Government of India (GoI), for the SERB National Post-Doctoral fellowship, File Number: PDF/2022/002988. 
            SM wishes to thank the Indian Institute of Technology (IIT) Madras,
            Chennai, India, for support through the Exploratory Research Project RF22230527PHRFER008479. SM also wishes to acknowledge the support from project SB22231259PHETWO008479 for the travel grand for Astro@Paris-Saclay symposium 2023.
            DM wishes to acknowledge support from the Science and Engineering Research Board
            (SERB), Department of Science and Technology (DST), Government of India (GoI), through the Core Research Grant CRG/2020/003664. 
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            JHEP
