Black hole quantum spectrum

Abstract

Introducing a black hole (BH) effective temperature, which takes into account both the non-strictly thermal character of Hawking radiation and the countable behavior of emissions of subsequent Hawking quanta, we recently re-analysed BH quasi-normal modes (QNMs) and interpreted them naturally in terms of quantum levels. In this work we improve such an analysis removing some approximations that have been implicitly used in our previous works and obtaining the corrected expressions for the formulas of the horizon’s area quantization and the number of quanta of area and hence also for Bekenstein–Hawking entropy, its subleading corrections and the number of micro-states, i.e. quantities which are fundamental to realize the underlying quantum gravity theory, like functions of the QNMs quantum “overtone” number n and, in turn, of the BH quantum excited level. An approximation concerning the maximum value of n is also corrected. On the other hand, our previous results were strictly corrected only for scalar and gravitational perturbations. Here we show that the discussion holds also for vector perturbations.

The analysis is totally consistent with the general conviction that BHs result in highly excited states representing both the “hydrogen atom” and the “quasi-thermal emission” in quantum gravity. Our BH model is somewhat similar to the semi-classical Bohr’s model of the structure of a hydrogen atom.

The thermal approximation of previous results in the literature is consistent with the results in this paper. In principle, such results could also have important implications for the BH information paradox.

Appendix. Derivation of the quasi-normal modes equation in non-strictly thermal approximation

Being frequencies of the radial spin-j=0,1,2 perturbations ϕ of the Schwarzschild space-time, QNMs are governed by the Schrodinger-like equation [4, 5, 34]

$$ \biggl(-\frac{\partial^{2}}{\partial x^{2}}+V(x)-\omega^{2} \biggr)\phi . $$

(A.1)

In strictly thermal approximation one introduces the Regge–Wheeler potential

$$ V(x)=V \bigl[x(r) \bigr]= \biggl(1-\frac{2M}{r} \biggr) \biggl( \frac{l(l+1)}{r^{2}}+2\frac{(1-j^{2})M}{r^{3}} \biggr). $$

(A.2)

We recall that j=0,1,2 for scalar, vector and gravitational perturbation, respectively.

The relation between the Regge–Wheeler “tortoise” coordinate x and the radial coordinate r is [4, 5, 34]

$$ \begin{aligned} &x=r+2M\ln\biggl(\frac{r}{2M}-1 \biggr) \\ & \frac{\partial}{\partial x}= \biggl(1-\frac{2M}{r} \biggr)\frac{\partial}{\partial r}. \end{aligned} $$

(A.3)

These states are analogous to quasi-stationary states in quantum mechanics [34]. Thus, their frequency is allowed to be complex [34]. They must have purely outgoing boundary conditions both at the horizon (r=2M) and in the asymptotic region (r→∞) [34]

$$ \phi(x)\sim c_{\pm}\exp(\mp i\omega x )\quad \mbox{for}\ x=\pm \infty. $$

(A.4)

Considering the non-strictly thermal behavior of BHs, one substitutes the original BHs M in Eqs. (A.1) and (A.2) with the effective mass of the contracting BH defined in Eq. (6) [4, 5]. Hence, Eqs. (A.2) and (A.3) are replaced by the effective equations [4, 5]

$$ V(x)=V \bigl[x(r) \bigr]= \biggl(1-\frac{2M_{E}}{r} \biggr) \biggl( \frac{l(l+1)}{r^{2}}+2\frac{(1-j^{2})M}{r^{3}} \biggr) $$

(A.5)

and

$$ \begin{aligned} & x=r+2M_{E}\ln\biggl(\frac{r}{2M_{E}}-1 \biggr) \\ & \frac{\partial}{\partial x}= \biggl(1-\frac{2M_{E}}{r} \biggr)\frac{\partial }{\partial r}. \end{aligned} $$

(A.6)

In order to streamline the formulas, here we also set

$$ 2M_{E}=r_{E}\equiv1\quad \mbox{and}\quad m\equiv n+1. $$

(A.7)

As the Planck mass m _p is equal to 1 in Planck units one rewrites (8) as

$$ \frac{\omega_{m}}{m_{p}^{2}}=\frac{\ln3}{4\pi}+\frac{i}{2}\biggl(m- \frac{1}{2}\biggr)+\mathcal{O}\bigl(m^{-\frac{1}{2}}\bigr),\quad \mbox{for}\ m \gg1, $$

(A.8)

where now m _p ≠1. Putting

$$ \tilde{\omega}_{m}\equiv\frac{\omega_{m}}{m_{p}^{2}}, $$

(A.9)

Eqs. (8), (A.1), (A.5), and (A.6) become

$$ \tilde{\omega}_{m}=\frac{\ln3}{4\pi}+\frac{i}{2}\biggl(m- \frac{1}{2}\biggr)+\mathcal{O}\bigl(m^{-\frac{1}{2}}\bigr),\quad \mbox{for}\ m \gg1, $$

(A.10)

$$ \biggl(-\frac{\partial^{2}}{\partial x^{2}}+V(x)-\tilde{\omega}^{2} \biggr) \phi, $$

(A.11)

$$ V(x)=V \bigl[x(r) \bigr]= \biggl(1-\frac{1}{r} \biggr) \biggl( \frac{l(l+1)}{r^{2}}-\frac{3(1-j^{2})}{r^{3}} \biggr) $$

(A.12)

and

$$ \begin{aligned} &x=r+\ln(r-1 ) \\ & \frac{\partial}{\partial x}= \biggl(1-\frac{1}{r} \biggr)\frac{\partial}{\partial r} \end{aligned} $$

(A.13)

respectively. Although M _E and r _E (and consequently the tortoise coordinate and the Regge–Wheeler potential) are frequency dependent, Eq. (A.7) translates such a frequency dependence into a continually rescaled mass unit in the following discussion. We will show at the end of this appendix that such a rescaling is extremely slow and always included within a factor 2. Thus, it does not influence the following analysis.

We emphasize that hereafter we closely follow ref. [34]. The solution of Eq. (A.11) can be expanded as [34]

$$ \phi(r)= \biggl(\frac{r-1}{r^{2}} \biggr)^{i\tilde{\omega}}\exp\bigl[-i {{\tilde{\omega}}}(r -1) \bigr]\sum _{m}a_{m} \biggl(\frac{r-1}{r} \biggr)^{m}. $$

(A.14)

The pre-factor has to satisfy the boundary conditions (A.4) both at the effective horizon (r=1) and in the asymptotic region (r=∞) [34]:

• 1. One needs $\exp[-i{{\tilde{\omega}}}(r-1) ]$ for the correct leading evolution at r→∞

• 2. $(r-1 )^{i\tilde{\omega}}$ fixes the evolution at r→1⁺

• 3. $(\frac{1}{r^{2}} )^{i\tilde{\omega}}$ arranges the subleading evolution at r→∞ which arises from the logarithmic term in Eq. (A.13).

The power series (A.14) converges for $\frac{1}{2}<r\leq \infty$ assuming that the boundary conditions at r=∞ are preserved [34]. On the other hand, Eq. (A.1) is equivalent to the recursion relation [34]

$$ c_{0}(m,\tilde{\omega})a_{m}+c_{1}(m,{ {\tilde{\omega}}})a_{m}-1+c_{2}(m,{ {\tilde{\omega}}})a_{m}-2=0. $$

(A.15)

One can extract the coefficients $c_{k}(m,\tilde{\omega})$ from Eq. (A.15) and rewrite them in a more convenient way [34]:

$$ c_{0}(m,{{\tilde{\omega}}})=m(m+2i{{ \tilde{\omega}}}) $$

(A.16)

$$ c_{1}(m,\tilde{\omega})=-2\biggl(m+2i{{\tilde{ \omega}}}-\frac{1}{2}\biggr)^{2}+j^{2}- \frac{1}{2} $$

(A.17)

$$ c_{2}(m,\tilde{\omega})=(m+2i{{\tilde{ \omega}}}-1)^{2}-j^{2}. $$

(A.18)

We see that, except for c ₀, the coefficients c _k depend on $m,\tilde{\omega}$ only through their combination $m+2i {{\tilde{\omega}}}$. The preservation of the boundary conditions at r=1 is guaranteed by the initial conditions for the recursion relation [34]. Such conditions are a ₀=1 (in general any non-zero constant that we set equal to the unity for the sake of simplicity), and a ₋₁=0 (that, together with Eq. (A.15), implies a _−m =0 for all positive m). One also defines [34]

$$ R_{m}=-\frac{a_{m}}{a_{m-1}}, $$

(A.19)

where we choose the minus sign in agreement with [34]. By using Eq. (A.15), one gets [34]

$$ c_{1}(m,\tilde{\omega})-c_{0}(m,{{\tilde{ \omega}}})R_{m}=\frac{c_{2}(m,\tilde{\omega})}{R_{m-1}} $$

(A.20)

which can be rewritten as

$$ R_{m-1}=\frac{c_{2}(m,\tilde{\omega})}{c_{1}(m,\tilde{\omega })-c_{0}(m,{{\tilde{\omega}}})R_{m}}. $$

(A.21)

Therefore, one can write R _m in terms of a continued fraction. The condition a ₋₁=0 becomes [34]

$$ R_{0}=\infty \rightarrow c_{1}(1,\tilde{ \omega})-c_{0}(1,{{\tilde{\omega}}})R_{1}=0. $$

(A.22)

As Eq. (A.14) converges at r=∞, there is a particular asymptotic form of R _m for large m (with |R _m |<1 for very large m) [34]. Thus, the boundary conditions require Eq. (A.22). One can write down Eq. (A.22) in terms of continued fractions [34]

$$ 0=c_{1}(1,\tilde{\omega})-c_{0}(1,{{ \tilde{\omega}}})\frac{c_{2}(2,\tilde{\omega})}{c_{1}(2,\tilde{\omega })-c_{0}(2,{{\tilde{\omega}}})\frac {....}{....}}\cdots , $$

(A.23)

which is a condition for the existence of the quasi-normal modes.

As R _m →−1 for large m and the changes of R _m slow down for large $|\tilde{\omega}|$, assuming that R _m changes adiabatically is an excellent approximation and one gets [34]

$$ \frac{R_{m}}{R_{m-1}}\simeq1+\mathcal{O}\bigl(\tilde{\omega}^{-\frac{1}{2}} \bigr). $$

(A.24)

This approximation works for both $\operatorname{Re} (m+2i{{\tilde {\omega}}} )>0$, when one computes R _m recursively from R _∞=−1, and for $\operatorname{Re} (m+2i{{\tilde{\omega}}} )<0$, when one starts to compute from R ₀=∞ [34]. If one inserts R _m−1=R _m in Eq. (A.20), one gets a quadratic equation, having the solutions (at the leading terms for large m) [34]

$$ R_{m}^{\pm}=\frac{-(m+2i{{\tilde{\omega}}})\pm\sqrt {2i{{\tilde{\omega}}}(m+2i{{\tilde {\omega}}})}}{m}+\mathcal{O} \bigl(m^{-\frac{1}{2}}\bigr). $$

(A.25)

The approximated solution (A.25) can be carefully checked by using Mathematica [34]. The issue that R _m must satisfy Eq. (A.25) for one of the signs is a necessary condition [34]. One needs a more deep discussion in order to see if that condition is also sufficient [34]. When $\operatorname{Re} (m+2i{{\tilde{\omega}}} )<0$, the sign arises from the condition R ₁ is small. Two terms in Eq. (A.25) are approximately deleted. The sign for $\operatorname{Re} (m+2i{{\tilde{\omega}}} )>0$ arises from the condition |R _m |<1 for very large m [34].

When $|\tilde{\omega}|$ is very large (but minor than the total mass of the black hole), one chooses an integer N such that [34]

$$ 1\ll N\ll|\tilde{\omega}|. $$

(A.26)

For the values of N in Eq. (A.26), Eq. (A.25) can be used to determine $R_{ [-2i\tilde{\omega} ]\pm N}$ [34] and only the second term in the RHS of Eq. (A.25) results to be relevant (the symbol for the integer part $[-2i\tilde {\omega} ]$ represents an arbitrary integer differing from $-2i\tilde{\omega}$ by a number much smaller than N which is assumed to be even) [34]. Such a relevant term in Eq. (A.25) implies [34]

$$ R_{ [-2i\tilde{\omega} ]+x}\propto\pm\frac{i\sqrt{x}}{\sqrt{-2i\tilde {\omega}}}\quad \mbox{for}\ 1\ll x\ll|\tilde{ \omega}|, $$

(A.27)

while the first term in Eq. (A.25) is $\propto\frac {x}{\tilde{\omega}}$ which is subleading [34]. Neglecting such a first term is equivalent to neglecting $c_{1}(m,\tilde{\omega})$ in the original equation (A.20). In fact, this term is irrelevant for all the m for large $|\tilde{\omega}|$, with the possible exclusion of some purely imaginary frequencies where c ₀(m) or c ₂(m) vanish [34]. The ratio $R_{ [-2i\tilde{\omega} ]\pm N}$ is computed from Eq. (A.25) like the ratio of $\sqrt{x}$ for x=±N [34]

$$ \frac{R_{ [-2i\tilde{\omega} ]+N}}{R_{ [-2i\tilde{\omega} ]-N}}=\pm i+\mathcal{O}\bigl(\tilde{\omega}^{-\frac{1}{2}} \bigr). $$

(A.28)

The assumptions that permitted to obtain Eq. (A.25) break down when $|m+2i{{\tilde{\omega}}}|\sim1$ [34]. In that case, the coefficients c ₀(m),c ₁(m),c ₂(m) contain terms of order 1 that cannot be neglected [34]. They also strongly depend on m. The adiabatic approximation breaks down in this region and the quantities $R_{ [-2i\tilde{\omega} ]\pm N}$ have to be related through the original continued fraction. Below, we will calculate the continued fraction exactly in the limit of very large $|\tilde{\omega}|$. The continued fraction will give the same result of Eq. (A.28) like the adiabatic argument. In fact, the two solutions will eventually “connect” [34]. Such a connection will release a non-trivial constraint on $\tilde{\omega}$.

As $R_{ [-2i\tilde{\omega} ]+x}\propto\frac{1}{\sqrt{\tilde{\omega}}}$, c ₁(m) in the denominator of Eq. (A.21) is negligible when compared to the other term (which results $\propto\sqrt{\tilde {\omega}}$ ) [34]. By fixing N, one understands that the effect of c ₁(m) in Eq. (A.21) vanishes for large $|\tilde {\omega}|$[34]. An exception should appear when c ₀(m) and/or c ₂(m) →0 for some m, but we will show that this exception cannot occur when $\operatorname{Re}(\tilde{\omega})\neq0$.

We note that the orbital angular momentum l is irrelevant because c ₁(m) does not affect the asymptotic frequencies [34]. This is not surprising as it is in agreement with Bohr correspondence principle [21–23, 25, 26]. One can replace the factor m in Eq. (A.16) with $[-2i\tilde{\omega} ]$. In fact, $|\tilde{\omega}|$ becomes extremely large when one studies only a relatively small neighborhood of $m\sim[-2i\tilde{\omega} ]$ [34]. We also see that the continued fraction is simplified into an ordinary fraction. If one inserts Eq. (A.21) recursively into itself 2N times, one gets [34]

$$\begin{aligned} &R_{ [-2i\tilde{\omega} ]-N} \\ &\quad {} =\prod_{k=1}^{N} \biggl( \frac{c_{2}( [-2i\tilde{\omega} ]-N+2k-1)c_{0}( [-2i\tilde{\omega} ]-N+2k)}{c_{0}( [-2i\tilde{\omega} ]-N+2k-1)c_{2}( [-2i\tilde{\omega} ]-N+2k)} \biggr) \\ &\qquad {}\times R_{ [-2i\tilde{\omega} ]+N}. \end{aligned}$$

(A.29)

The dependence of the c _k (m) on the frequency is suppressed. Thus, one can combine Eqs. (A.28) and (A.29) to eliminate the other R _m . We require that the generic solution (A.25), which holds almost everywhere, is “patched” with the solution (A.29), which holds for $m\sim [-2i\tilde{\omega} ]$ and for $|\tilde{\omega}|$ extremely large [34].

One can express the products of c ₀(m) and c ₂(m) in terms of the gamma function Γ, which is an extension of the factorial function to real and complex numbers [34]. As c ₂(m) is bilinear, the four factors of Eq. (A.29) lead to a product of 6 factors, i.e. (2+1+1+2). Each of those equals a ratio of two Γ functions. In other words, one gets a ratio of 12 Γ functions [34].

One can write down the resulting condition in terms of a shifted frequency defined by $-2if\equiv-2i\tilde{\omega}- [-2i\tilde{\omega} ]$ [34]:

$$ \pm i= \begin{array} {c} \frac{\varGamma(\frac{N+2if+j}{2} )\varGamma(\frac{N+2if-j}{2} )\varGamma(\frac{-N+2if+1}{2} )}{\varGamma(\frac{-N+2if+j}{2} )\varGamma(\frac{-N+2if-j}{2} )\varGamma(\frac{+N+2if+1}{2} )} \\ \\ \frac{\varGamma(\frac{-N+2if+j+1}{2} )\varGamma(\frac{-N+2if-j+1}{2} )\varGamma(\frac{+N+2if+2}{2} )}{\varGamma(\frac{N+2if+j+1}{2} )\varGamma(\frac{N+2if-j+1}{2} )\varGamma(\frac{-N+2if+2}{2} )} \end{array} $$

(A.30)

Considering Eq. (A.18), the factors m result cancelled. Six of the Γ functions in Eq. (A.30) show an argument with a huge negative real part. Hence, they can be converted into Γ of positive numbers by using the formula [34]

$$ \varGamma(x)=\frac{\pi}{\sin(\pi x )\varGamma(1-x)}. $$

(A.31)

Thus, the π factors cancel like the Stirling approximations for the Γ functions which have a huge positive argument [34]

$$ \varGamma(m+1)\approx\sqrt{2{{\pi}}n} \biggl(\frac {m}{e} \biggr)^{m}, $$

(A.32)

while the factors with sin(x) survive. The necessary condition for regular frequencies for large m (the frequencies for which the analysis is valid. We will return on this point later) reads [34]

$$ \pm i=\frac{\sin[\pi(if+1) ]\sin[\pi(if+\frac{j}{2}) ]\sin[\pi (if-\frac{j}{2}) ]}{\sin[\pi(if+\frac{1}{2}) ]\sin[\pi(if+\frac {1+j}{2}) ]\sin[\pi(if+\frac{1-j}{2}) ]}. $$

(A.33)

Choosing $N\in4\mathbb{Z}$, one erases N from the arguments of the trigonometric functions. One also replaces f by $\tilde {\omega}$ again as the functions in Eq. (A.33) are periodic with the right periodicity and the number $[-2i\tilde{\omega} ]$ can be chosen even. We can use the terms $\frac{\pi}{2}$ in the denominator in order to convert the sin functions into cos. Multiplying Eq. (A.33) by the denominator of the RHS and expanding the sin functions in terms of the exponentials (one has to be careful about the signs) the result is (ϵ(y) is the sign function) [34]

$$ \exp\bigl[\epsilon \operatorname{Re} (\omega)\cdot4\pi\tilde{\omega} \bigr]=-1-2\cos (\pi j). $$

(A.34)

Hence, for scalar or gravitational perturbations the allowed frequencies are [34]

$$ \tilde{\omega}_{m}=\frac{i}{2}\biggl(m-\frac{1}{2} \biggr)\pm\frac{\ln3}{4\pi}+\mathcal{O}\bigl(m^{-\frac{1}{2}}\bigr), $$

(A.35)

while for vector perturbations one gets

$$ \tilde{\omega}_{m}=\frac{im}{2}+\mathcal{O}\bigl(m^{-\frac{1}{2}} \bigr). $$

(A.36)

We note that, as we are in the large m regime, one gets $\tilde{\omega }_{m}\simeq\frac{im}{2}$ independent of j. This implies the correctness of Eq. (9) and the important issue that the quantum of area (37) is an intrinsic property of Schwarzschild BHs. Again, this is in full agreement with Bohr correspondence principle [21–23, 25, 26].

In order to finalize the analysis one has to resolve the question marks concerning the special frequencies where the used approximation, which neglects c ₁(m), breaks down [34].

The first step is to argue that the “regular” solutions must exist [34]. In fact, if one can relate the remainders $R_{ [-2i\tilde{\omega} ]\pm N}$ by using the continued fraction, where c ₁(m) can be neglected, one can also extrapolate them to Eq. (A.25) [34].

From the boundary conditions R ₀=∞, R _∞=−1⁺, one sees that a specific sign of the square root in Eq. (A.25) has to be separately chosen for $m< [-2i\tilde{\omega} ]$ and for $m> [-2i\tilde{\omega} ]$ [34]. But one finds that the signs agree with the signs of ±i that automatically lead to the solutions [34].

The condition for $\tilde{\omega}$ is both necessary and sufficient [34]. These kinds of solutions are only the ln(3) solutions from Eq. (A.10). The existence of those solutions is guaranteed [34].

As one needs to find all irregular solutions, let us recall two useful points [34].

• 1. The continued fractions of Eq. (A.23) depend on the coefficients c ₀(m) and c ₂(m+1) only through their product c ₀(m)c ₂(m+1) [34].

• 2. If one finds zeroes in Eq. (A.29) exclusively in the numerator or in the denominator, the ratio $\frac{R_{ [-2i\tilde {\omega} ]-N}}{R_{ [-2i\tilde{\omega} ]+N}}$ can be only either zero, or infinite. When one takes into account c ₁(m), “zero” or “infinity” results to be replaced by a negative or positive power of $|\tilde {\omega}|$, respectively [34].

Point 2 means that one can obtain irregular solutions only by finding ω such that Eq. (A.29) becomes an indeterminate form $\frac{0}{0}$ [34].

As c ₀(m) can be null at most for one value of m, one finds that there is at least one value of m where c ₂(m) vanishes [34]. Thus, Eq. (A.18) implies that one between $2 (i\tilde{\omega}+1 )$ and $2 (i\tilde{\omega}-1 )$ (maybe both) must be integer [34]. As the two conditions are equivalent, both numbers $2 (i\tilde{\omega}\pm1 )$ must be integers to give a chance to exist to the quasi-normal frequency [34]. Thus, the two numbers differ by an even number. Then, both the vanishing factors of c ₂(m) must appear in the numerator of Eq. (A.29), or, alternatively, both must appear in the denominator of such an equation [34]. One can assume, for example, that they appear in the denominator without loss of generality [34]. Thus, one finds the indeterminate form only if the vanishing c ₀(m) appears in the numerator [34]. Clearly, $2 (i\tilde{\omega}\pm1 )$ and $2i\tilde{\omega}$ are different modulo two. Thus, the effect of c ₁(m) gives the desired result, confirming that the regular states of Eq. (A.35) are the only solutions [34]. By using Eqs. (A.9) and (A.7) one easily returns to Planck units and obtains Eq. (8).

Now, we show that the continually rescaled mass unit in the above discussion, which is due to the frequency dependence of M _E and r _E , did not influence the analysis. We note that, although $\tilde {\omega}$ in the analysis can be very large because of definition (A.9), ω must instead be always minor than the BH initial mass as BHs cannot emit more energy than their total mass. Inserting this constrain in Eq. (6) we obtain the range of permitted values of M _E (|ω _n |) as

$$ \frac{M}{2}\leq M_{E}\bigl(|\omega_{n}|\bigr)\leq M. $$

(A.37)

Thus, setting 2M _E (|ω _n |)=r _E (|ω _n |)≡1(|ω _n |) one sees that the range of permitted values of the continually rescaled mass unit is always included within a factor 2. On the other hand, we recall that the countable sequence of QNMs is very large, see Sect. 3 and [4]. Thus, the mass unit’s rescaling is extremely slow. Hence, one can easily check, by reviewing the above discussion step by step, that the continually rescaled mass unit did not influence the analysis.

Another argument which remarks the correctness of the analysis in this appendix is the following. One can choose to consider M _E as being constant within the range (A.37). In that case, it is easy to show that such an approximation is indeed very good. In fact, Eq. (A.37) implies that the range of permitted values of T _E (|ω _n |) is

$$ T_{H}=T_{E}(0)\leq T_{E} \bigl(|\omega_{n}|\bigr) \leq2T_{H}=T_{E} \bigl(|\omega_{n_{\mathrm{max}}}|\bigr), $$

(A.38)

where T _H is the initial BH Hawking temperature. Therefore, if one fixes $M_{E}=\frac{M}{2}$ in the analysis, the approximate result is

$$ \omega_{n}\simeq2\pi in\times2T_{H}. $$

(A.39)

On the other hand, if one fixes M _E =M (thermal approximation), the approximate result is

$$ \omega_{n}\simeq2\pi in\times T_{H}. $$

(A.40)

As both the approximate results in correspondence of the extreme values in the range (A.37) have the same order of magnitude, fixing 2M _E =r _E ≡1 does not change the order of magnitude of the final (approximated) result with respect to the exact result. In particular, if we set $T_{E}=\frac{3}{2}T_{H}$ the uncertainty in the final result is 0.33, while in the result of the thermal approximation (A.40) the uncertainty is 2. Thus, even considering M _E as constant, our result is more precise than the thermal approximation of previous literature and the order of magnitude of the total emitted energies (10) is correct.