Extremal noncommutative black holes as dark matter furnaces

TL;DR

This paper explores the potential role of extremal noncommutative black holes as dark matter relics that could act as cosmic furnaces, fueled by dark matter accretion, and examines their thermodynamic properties at the evaporation endpoint.

Contribution

It introduces the concept of extremal noncommutative black holes as stable dark matter relics and hypothesizes their thermal equilibrium state fueled by dark matter accretion.

Findings

Dark matter annihilation occurs near noncommutative black holes.

Hypothesized existence of a thermal equilibrium state of black hole relics.

Black hole relics could serve as dark matter furnaces.

Abstract

In this letter, we consider dark matter annihilation in the gravitational field of noncommutative black holes. At final stage of evaporation, we hypothesize the existence of a thermal equilibrium state composed of a burning black hole relics fueled by dark matter accretion.

Equations

$$[x^{\mu },x^{\nu }]=i\theta {ϵ}^{\mu \nu }.$$

$$\delta(\vec{r})\to\frac{1}{(4\pi\theta)^{3/2}}\exp\bigg{(}-\frac{r^{2}}{4\theta}\bigg{)}.$$

$$\displaystyle ds^{2}=\bigg{(}1-\frac{r_{g}(r)}{r}\bigg{)}c^{2}dt^{2}-\bigg{(}1-\frac{r_{g}(r)}{r}\bigg{)}^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$$

$$\displaystyle r_{g}(r)=\frac{4GM}{c^{2}\sqrt{\pi}}\gamma\bigg{(}\frac{3}{2},\frac{r^{2}}{4\theta}\bigg{)},$$

$$\gamma (s,x)\equiv {\int }_0^{x}dt{t}^{s-1}{e}^{-t}.$$

$$ct=\frac{E_0}{m{c}^2}\int \frac{dr}{(1-\frac{r_g}{r})\sqrt{(\frac{E_0}{m{c}^2}{)}^2-(1-\frac{r_g}{r})(1+\frac{L^2}{m^2{c}^2{r}^2})}},$$

$$\varphi =\int \frac{Ldr}{r^2\sqrt{\frac{E_0^2}{c^2}-(m^2{c}^2+\frac{L^2}{r^2})(1-\frac{r_g}{r})}},$$

$$E_{0}=\bigg{(}1-\frac{r_{g}}{r}\bigg{)}mc^{3}\dot{t},\qquad L=mr^{2}\dot{\phi}.$$

$$v_r=\sqrt{\frac{-g_{rr}}{g_{tt}}}\frac{dr}{dt}=\sqrt{\frac{r_g}{r}-\frac{{\alpha }^2}{r^2}(1-\frac{r_g}{r})},$$

$$v_t=\sqrt{\frac{-g_{\varphi \varphi }}{g_{tt}}}\frac{d\varphi }{dt}=\frac{\alpha }{r}\sqrt{1-\frac{r_g}{r}},$$

$$dF=4\pi r^2\mathcal{N}v\cos \theta d\tau d\Omega ,$$

$$\cos \theta d\Omega =\frac{\pi }{r_{g}r}(1-\frac{r_g}{r})d({\alpha }^2),v=\sqrt{v_r^2+v_t^2}=\sqrt{\frac{r_g}{r}},$$

$$dF=4{\pi }^2\mathcal{N}\frac{(r-r_g{)}^{3/2}}{\sqrt{r_g}r}d({\alpha }^2)dt$$

$$d{F}_{\infty }=\pi \frac{n_{\infty }}{v_{\infty }}d({\alpha }^2)dt,$$

$$\mathcal{N}=\frac{n_{\infty }}{4\pi v_{\infty }}\frac{\sqrt{r_g}r}{(r-r_g{)}^{3/2}},$$

$$J^k{}_{;k}=0⟶Nv\sqrt{-g}=c_1;$$

$$T_{i;k}^k=0⟶(P+\mathcal{E})g_{00}c\dot{t}v\sqrt{-g}=c_2.$$

$$\bigg{(}\frac{P+{\cal E}}{N}\bigg{)}\bigg{(}1-\frac{r_{g}}{r}+v^{2}\bigg{)}^{1/2}=\frac{c_{2}}{c_{1}}.$$

$$\frac{dv}{v}\big{[}V^{2}F(r,v)-v^{2}\big{]}-\frac{dr}{2r}\bigg{[}\frac{r_{g}(r)}{r}-r_{g}^{\prime}(r)-4V^{2}F(r,v)\bigg{]}=0,$$

$$F(r,v)=1-\frac{r_g(r)}{r}+v^2,V^2=\frac{d\log (P+\mathcal{E})}{d\log N}-1.$$

$$v_{c}^{2}=\frac{1}{4}\bigg{(}\frac{r_{g}(r_{c})}{r_{c}}-r_{g}^{\prime}(r_{c})\bigg{)},\qquad V^{2}\big{|}_{r_{c}}=\frac{v_{c}^{2}}{1-3v_{c}^{2}+r_{g}^{\prime}(r_{c})}.$$

$$Q=\frac{mN}{⟨v^2{⟩}^{3/2}}$$

$$\mathcal{E}=mN{c}^2+\frac{P}{\gamma -1},P=\frac{1}{3}mN⟨v^2⟩c^2.$$

$$\frac{d(P+{\cal E})}{dN}=mc^{2}+\frac{5\gamma}{9(\gamma-1)}mc^{2}\bigg{(}\frac{N}{Q_{N}}\bigg{)}^{2/3}\,,$$

$$V^{2}\simeq\frac{2\gamma}{9(\gamma-1)}\bigg{(}\frac{N}{Q_{N}}\bigg{)}^{2/3}-\frac{2\gamma^{2}}{27(\gamma-1)^{2}}\bigg{(}\frac{N}{Q_{N}}\bigg{)}^{4/3}+\cdots\,.$$

$$\frac{c_{2}}{c_{1}}=mc^{2}+\frac{\gamma}{3(\gamma-1)}mc^{2}\bigg{(}\frac{N}{Q_{N}}\bigg{)}^{2/3}$$

$$\bigg{(}\frac{N_{c}}{N_{\infty}}\bigg{)}^{4/3}\simeq\frac{2(\gamma-1)}{\gamma}\langle v_{\infty}^{2}\rangle,\qquad\langle v_{c}^{2}\rangle\simeq\bigg{(}\frac{2(\gamma-1)}{\gamma}\langle v_{\infty}^{2}\rangle\bigg{)}^{\frac{1}{2}}\,.$$

$$r_c\simeq \frac{R_g}{4}\sqrt{\frac{\gamma }{2(\gamma -1)⟨v_{\infty }^2⟩}}.$$

$$\frac{d{M}_{bh}}{dt}=4\pi r_c^2\frac{T_0^1}{c}=\frac{\pi }{4}{r}_g(r{)}^{2}cQ,$$

$$\frac{d{M}_{bh}}{dt}=\frac{\pi }{4}{R}_g^{2}cQ-c^{-2}(4\pi r_H^2)ϵ\sigma T^4,$$

$$T_{rem}=\frac{1}{2}\bigg{(}\frac{c^{3}Q}{\sigma}\bigg{)}^{\frac{1}{4}}\,.$$

$$Q\approx \frac{3.51\times 10^{-9}}{M_{11}^{1.54}}{M}_{\odot }p{c}^{-3}k{m}^{-3}{s}^3,$$

Full text

I Introduction

The Dark Matter is estimated to contribute about $26.8\%$ content of our universe and abundant in galaxies as halos. The Dark Matter Particles are believed to accrete and affect various annihilation channels significantly on the supermassive black hole at the center of each galaxyBaushev:2008yz . There are estimated certain amount of primordial black holes (PBHs) remained since their creation in the very early universe. Those tiny relics may possess mass range from $10^{14}$ to $10^{23}$kg and their possible role as a Dark Matter candidate has been widely discussed111Readers are directed to Green:2014faa ; Gelmini:2015zpa for a review.. However, regardless any chance to be dark matter candidate, the fate of those tiny black holes remains unclear due to the lack of full knowledge of Planck scale physics in curved spacetime. The conventional thermal description of black holes seems doomed to failure due to the unphysical infinite Hawking temperature for infinitesimal mass. Several proposals to modify the fundamental properties of spacetime, such as General Uncertainty Principle (GUP) and noncommutative geometry (NG), is to introduce a new scale when the black hole could stop evaporating about Planck size. In particular, a NG or GUP inspired Schwarzschild black hole would only reach up to sub-Planck temperature before it cools down to its extremal state of Planck massNicolini:2005vd ; Isi:2013cxa . It was proposed this extremal state as a candidate of dark matterKovacik:2015yqa . In this letter, we consider a different scenario that dark matter annihilation in the gravitational field of noncommutative black holes. At final stage of evaporation, we hypothesize the existence of a thermal equilibrium state composed of a burning black hole relics fueled by dark matter accretion. Note that it has been discussed that the accretion of primordial black holes may be significant in the radiation-dominated era for some types of modified theories of gravity and the extension of the lifetime is estimated PBH_Accretion . In contrast, we consider much late time equilibrium states of PBHs in which the effect of noncommutativity might be significant.

In fact, the effect of noncommutativity in the early universe has been considered in various contexts; for example, via the density fluctuation and the CMB power spectrumNC_CMB . This study, on the other hand, aims to shed light on a different aspects of the effect of NG to cosmology; namely the late time effects through stabilized primordial black holes. Since it is urged to find any kind of imprinting of Planck scale physics in cosmology, it is worth considering another scenario here.

This paper is organized as follows: in the section II, we review the fate of noncommutative geometry inspired Schwarzschild black hole (NCGS). In the section III, we calculate the dark matter distribution in the vicinity of an extremal NCGS black hole. In the section IV, we discuss the accretion model for a polytropic type of dark matter. In the section V, we discuss the stable configuration of near-extremal NCGS black hole. At last, we have comments and discussion in the section VI.

II The fate of Schwarzschild black holes in noncommutative space

For a noncommutative space, one expects its coordinates do not commute and satisfy following relation:

[TABLE]

The formulation of coherent state in a complexified plane suggests a position measurement gives a smearing Gaussian distribution instead of the delta functionSmailagic:2003rp , that is

[TABLE]

Incorporate this smearing effect into mass distribution in the usual General Relativity, a noncommutative geometry inspired Schwarzschild (NCGS) black hole was constructedNicolini:2005vd and has the metric

[TABLE]

where the lower incomplete Gamma function is defined as

[TABLE]

It is straightforward to show that the Schwarzschild radius $r_{g}\to\frac{2GM}{c^{2}}\equiv R_{g}$ is recovered as commutative limit, $\theta\to 0$, is taken. That the noncommutative black hole has two horizons while its mass is larger than a critical value $M_{c}=1.904\sqrt{\theta}c^{2}/G$ shows an interesting similarity to the Reisner-Norström black holeKim:2008vi . The thermal behavior of a NCGS black hole starts to gradually deviate from that of a Schwarzschild black hole when the horizon $r_{H}\lesssim 6\sqrt{\theta}$. The final stage of a Schwarzschild black hole could have been violent and unpredictable according to Hawking’s relation $T_{H}=c^{3}/8\pi GM$. It is more likely that the Einstein’s theory of general relativity would be replaced by a UV finite theory of quantum gravity toward the Planck scale. Though a fully comprehensive theory of quantum gravity is still unavailable, its effect on spacetime might be captured by some effective theories. With the naive help of GUP, the evaporation could be stopped at finite but still high temperatureAdler:2001vs ; Scardigli:2010gm . In contrast, the NCGS black holes cool down and reach its extremal state at $M=M_{c}$. These cold relics may not stay completely quiet since they could trap the surrounding abundant dark matter particles (DMP) and were ignited by the accretion.

III Dark Matter Particle Phase-Space Distribution

In this section, we would like to derive the dark matter phase-space distribution in the vicinity of a extremal NCGS black hole. The distribution around the Schwarzschild black hole has been discussed in Baushev:2008yz and here we generalized it to the case of extremal NCGS black hole. We will assume those are non-interacting and non-relativistic DMPs and their speed are estimated a few hundred kilometers per second in the Galaxies. The trajectory of a DMP of mass $m$ in the gravitational field of a NCGS black hole can be summarized in the following equations:

[TABLE]

where the conserved total energy $E_{0}\simeq mc^{2}$ and angular momentum $L$ are define as

[TABLE]

The dot derivative is with respect to the proper time. The radial speed component $v_{r}$ and tangent one $v_{t}$ can be calculated from (III):

[TABLE]

where we have adopted the natural unit $c=1$ and defined $\alpha\equiv\frac{L}{mc}$ for convenience, and DMP is also assumed to be non-relativistic. Now we define the flux as number of particles crossing a sphere of fixed radius $r$ in unit proper time and unit solid angle, that is

[TABLE]

where ${\cal N}$ is the particle density. After substituting

[TABLE]

One obtains

[TABLE]

Assume our DMP detector locates at far distance $r_{\infty}$ where the space is asymptotic flat, this flux becomes

[TABLE]

for the number density per radius $n_{\infty}=\int{d\Omega}{\cal N}$ and DMP speed $v_{\infty}$ found at $r_{\infty}$. To proceed, we will assume the flux per angular momentum and per time $\frac{dF}{d(\alpha^{2})dt}$ remains constant at arbitrary distance. This determines the DMP density distribution per solid angle

[TABLE]

for each given $n_{\infty}$ and $v_{\infty}$ found at our detector. We remark that from (III) only those DMPs with angular momentum $\alpha<2.408$ can reach the horizon and be caught by the extremal NCGS black hole.

IV Accretion versus Radiation

Now one considers the model of dark matter accreted to the NCGS black hole. The accretion model was derived in Michel:1972 and applied to many situations including the supermassive black holesChakrabarti:1996cc ; Peirani:2008bu . We assume the DMP behaves like an ideal fluid, i.e., $T_{\mu}^{\nu}=(P+{\cal E})u^{\nu}u_{\mu}-P\delta_{\mu}^{\nu}$. The conservation of mass flux and energy-momentum flux give

[TABLE]

Together we obtain

[TABLE]

The differential relation between $v$ and $r$ is captured by the (solar) wind equation:

[TABLE]

where

[TABLE]

The critical point flow ($v$ monotonically increases or decreases along the trajectory) occurs where both bracketed factors vanish simultaneously, namely

[TABLE]

It is convenient to use the indicator of phase space density $Q$ for a self-similar radial infalling DMPHenriksen:2006gi , where $Q\propto r^{-\beta}$. A typical $\beta\simeq 1.87$ was found by Taylor:2001bq for cluster-size halo. The indicator is conserved during expansion of the universe and defined as

[TABLE]

for velocity dispersion $\langle v^{2}\rangle$. To be general, let us consider a polytropic gas of index $1/(\gamma-1)$ with total energy density and pressure:

[TABLE]

One can further obtain the differential:

[TABLE]

for $Q_{N}\equiv Q/m$. After Taylor expansion at large $Q_{N}$, equation (17) becomes

[TABLE]

At far away from the black hole where $v_{\infty}\to 0$ for $r_{\infty}\gg r_{g}$, one obtains

[TABLE]

and222We notice a mistake in Peirani:2008bu for the power in the ratio $N_{c}/N_{\infty}$ due to picking up a subleading term in the Taylor’s expansion.

[TABLE]

where we have assumed that $r_{g}^{\prime}(r_{c})$ is negligible. Since $r_{g}^{\prime}(r)$ becomes exponentially small for $r>\sqrt{\theta}$, this is justified if $r_{c}$ is somehow large compared the location of degenerate horizon, $r\simeq 3.02\sqrt{\theta}$. At last, $r_{c}$ is solved from the first relation of(18). Again, we assume that $r_{g}^{\prime}(r_{c})$ is negligible and $r_{g}(r_{c})$ is taken as its asymptotic value $r_{g}(\infty)=\frac{2GM}{c^{2}}\equiv R_{g}$, and the solution is found to be

[TABLE]

The accretion rate becomes

[TABLE]

where $\langle v_{\infty}^{2}\rangle\ll 1$ and $r_{g}^{\prime}(r_{c})\ll 1$ are assumed. Taking account of Hawking radiation

[TABLE]

where the Stefan-Boltzmann constant $\sigma=\frac{\pi^{2}k_{B}^{4}}{60\hbar^{3}c^{2}}$ and assuming $\epsilon=1$ for perfect blackbody. $r_{H}$ is the location of the outer horizon, a solution of $r_{g}(r_{H})=r_{H}$. To the leading order calculation, we can approximate $r_{H}\simeq R_{g}$. A careful treatment with subleading correction is given in the next section. This implies a unique temperature in the thermal equilibrium

[TABLE]

We have following remarks: at first, the fact that equilibrium temperature (28) is independent of $\theta$ implies that we simply obtained the solution of Schwarzschild counterpart. Namely the difference between a noncommutative black hole and an ordinary black hole is hardly observed at this equilibrium point. Secondly, this NCGS black hole immersed in the dark matter halo with typical phase-space density $Q\approx 3.51\times 10^{-9}M_{\odot}pc^{-3}km^{-3}s^{3}$ would burn at a temperature333The phase-space density of halo is well fitted by the relationPeirani:2007cj

$Q\approx\frac{3.51\times 10^{-9}}{M_{11}^{1.54}}M_{\odot}pc^{-3}km^{-3}s^{3},$

(29)

for halo mass $M_{11}$ in the units of $10^{11}M_{\odot}$. It should also be noted that since we have used a dimensionless $\langle v^{2}\rangle$ to derive (28), we need to put an extra $c^{3}$ in the parenthesis of the formula. $T_{rem}\approx 1.17\times 10^{5}\,\mathrm{K}$, which could look like a hot spot in the present cosmic microwave background but difficult to be identified due to its small size, about $10^{-9}$m.444Now we can use the formula for conventional black holes. The equilibrium temperature is

$\displaystyle T_{H}=\frac{T_{P}}{4\pi}\frac{\ell_{P}}{R_{g}}=1.17\times 10^{5}K\,.$

(30)

By use of $T_{P}=1.42\times 10^{32}$K and $\ell_{P}=1.62\times 10^{-35}$m, the size of the black hole is $R_{g}=9.64\times 10^{25}\ell_{P}\simeq 1.56\times 10^{-9}$m. This is huge compared to Planck size, but microscopic in the context of cosmology. The total mass of the black hole is $M=4.82\times 10^{25}m_{P}=1.05\times 10^{18}$kg, or equivalent $10^{-12}M_{\odot}$. This black hole has longer life time than the age of universe, and in the range of proposed primordial black hole as dark matter candidate. At last, a conventional Schwarzschild may also reach a thermal equilibrium with surrounding dark matter at the temperature predicted in (28), when the radiation and accretion rate are the same. However, this equilibrium is unstable for the following reason: while the mass of a black hole decreases, the accretion rate also decreases. As a result the Hawking temperature increases for its negative specific heat. In contrast, the temperature of NCGS black hole may behave like a conventional black hole at large mass, but it starts to decrease after reaching the maximum temperature $T_{\text{max}}\simeq 0.015/\sqrt{\theta}$ and then cool down as a remnant. We then expect to have an additional stable equilibrium point for NCGS black hole at a much smaller mass. The detailed analysis will be carried out in the section V, and there it will be found that the other equilibrium takes place at $T\simeq 10^{7}$K, very close to its extremal state. Therefore at this equilibrium point, the Planck-size NCGS black hole has a much cooler temperature than its Schwarzschild companion. The total flux is also much smaller compared the previous equilibrium case. From the discussion above, this equilibrium is expected to be stable against evaporation.

V Another equilibrium point near the extremal one

In the previous section, we have considered the equilibrium configuration of PBH and dark matter halo through Hawking radiation and accretion. In this section, we look at an equilibrium point which is very close to the extremal point of NCBH. We will keep $r^{\prime}(r_{c})$ and also the difference between $r_{H}$ and $R_{g}$ (the gravitational radius for the total mass), which are neglected in the previous analysis.

With keeping $r^{\prime}(r_{c})$ in deriving (24), we find a correction,

[TABLE]

$r_{c}$ is defined as a solution of the following relation,

[TABLE]

where we have taken the first order in $\theta$. When $r_{c}$ is sufficiently large compared to $2\sqrt{\theta}$, we may take $r_{g}^{\prime}(r_{c})\leq{\cal O}(\theta)$, and

[TABLE]

where $r_{c}^{(0)}=R_{g}/4v_{c}^{2}$ is $\theta$ independent leading order solution. Thus, with keeping the correction terms, the accretion rate is given by

[TABLE]

Now we move on to the correction to Hawking radiation. The location of the horizon, a solution of $r_{g}(r_{H})=r_{H}$, is approximately

[TABLE]

Hawking temperature $T_{H}$ is given by Nicolini:2005vd

[TABLE]

Then the condition for equilibrium is

[TABLE]

and the solution is

[TABLE]

The extremal NCBH is realized if the mass coincides with the extremal mass, $M_{\text{ext}}=1.904m_{P}\sqrt{\theta\ell_{P}^{-2}}$ where $m_{P}$ and $\ell_{P}$ are Planck mass and length respectively, and the location of the degenerate horizon is $r_{0}=3.02\sqrt{\theta}$. We now examine the case in which the total mass of the NCBH is very close to $M_{\text{ext}}$. By setting $r_{H}=r_{0}+\epsilon$, the Hawking temperature is T_{H}=T_{P}\tilde{\epsilon}\tilde{\theta}^{-1/2}\big{(}0.0224-0.00744\tilde{\epsilon}+{\cal O}(\tilde{\epsilon}^{2})\big{)} where we have introduced dimensionless $\tilde{\epsilon}$ and $\tilde{\theta}$ as $\tilde{\epsilon}=\epsilon/\sqrt{\theta}$ and $\tilde{\theta}=\theta\ell_{P}^{-2}$, and $T_{P}$ is Planck temperature. As seen from this expression, the extremal limit $r_{H}\rightarrow r_{0}$ ($\epsilon\rightarrow 0$) and the commutative limit $\theta\rightarrow 0$ are not commuting. By using the leading order part of $T_{H}$ and also dropping the correction term for the accretion rate, the equilibrium condition is

[TABLE]

If we adopt the same typical $Q$ value in the main part, we find

[TABLE]

This value has to be very close to the extremal mass $M_{\text{ext}}$ under the assumption we take, which implies $\tilde{\epsilon}\simeq 4.14\times 10^{-25}\sqrt{\tilde{\theta}}$. If we choose $\tilde{\epsilon}$ to this value, the Hawking temperature is $T_{H}\simeq 0.927\times 10^{-25}T_{P}=1.31\times 10^{7}$K and the total mass of NCBH is $M\simeq M_{\text{ext}}$.

We remark that a thermally stable black hole relics cannot exist without the noncommutativity. Otherwise, accretion would be impossible due to the furious circumference heated up by a usual Schwarzschild black hole of Planck size.

VI Discussion

So far, our simple model only discussed the equilibrium between accretion of dark matter halo and the Hawking radiation. For the stable equilibrium, the mass of black hole is almost Planck mass $10^{-8}$kg and its size is of almost Planck size, while for the unstable equilibrium (namely for the conventional Schwarzschild black hole), the mass is about $10^{18}$kg ($10^{-6}$ of Earth mass or $10^{-4}$ of the mass of the Moon) and the size is of order $10^{-9}$m. Though the gravitational field near the horizon is not so small (about $10^{13}$ stronger compared to typical neutron stars for the unstable case, $10^{40}$ for the stable case), their total gravitational field and the flux are quite small. For the stable configuration, it seems to not form a dark matter halo around them and the equilibrium may be realized when they happen to locate in a region of high energy density. As for the unstable case, the situation is subtle and it would be necessary to reexamine the applicability of the same formula of accretion rate.

A lower bound of $\theta$ could be estimated by noticing that the highest temperature which can be reached by a noncommutative black hole is given by $T_{max}=\frac{0.015}{\sqrt{\theta}}$. Further asking the wavelength of Hawking radiation should not be shorter than the Planck length, one could estimate the lower bound $\theta\geq 10\l_{p}$Nicolini:2005vd .

The production rate for primordial black holes are considered in Carr:2005bd , but they mainly focused on the ones heavier than $10^{15}$g, as light ones have already evaporated in the current universe. Since the noncommutativity makes black holes stable, and then if the production rate for light primordial black holes are sufficiently high, we would have a upper limit for possible $\theta$. The mass of the extremal black hole is given by $1.9\sqrt{\theta}/G$ which is of order Planck mass ($10^{-8}$g).

Recent study has suggested possible connection between the Cosmic Infrared Background (CIB) and primordial black hole Kashlinsky:2016sdv . Those near-IR fluctuation of wavelength $2-5\mu m$ could have been majorly contributed from objects with temperature $580-1450$K according to Wien’s displacement law, which could be ordinary primordial black holes with mass among $4-11\times 10^{-11}M_{\odot}$. If some of those unidentified sources were from extremal black hole remnants, they should have larger number density but much lighter individual mass. For instance, the radiation flux of an ordinary primordial black hole of mass $M_{h}$ $\propto(2GM_{h})^{2}(8\pi GM_{h})^{-4}$; but if same amount mass were made of extremal NCGS black hole at same temperature, the flux $\propto(M_{h}/M_{ext})(2GM_{ext})^{2}(8\pi GM_{h})^{-4}$. The latter is much weaker than the former by a factor $M_{ext}/M_{h}\sim 10^{-26}$. The abundance of these extremal remnant might modify our resolutions toward some puzzles in the modern cosmology. Radiation come from those warm remnants may accelerate local reionization of hydrogens after the Big Bang to clear the foggy universe. Those remnants may also help early galaxies formation by serving as seeds. Inspection of the abovementioned phenomenological application is under progress and will be reported in a separated paper.

Acknowledgements.

We are grateful to Pei-Ming Ho and Je-An Gu for his helpful discussion. This work is supported in parts by the Chung Yuan Christian University, the Taiwan’s Ministry of Science and Technology (grant No. 102-2112-M-033-003-MY4) and the National Center for Theoretical Science.