The Quantum Primordial Black Holes, Dimensionless Small Parameter,Inflationary Cosmology and Non-Gaussianity

1. Introduction

The primordial black holes (pbhs) [1,2,3] in the Early Universe are due to gravitational collapse of the high-density matter [4]. In [5,6,7] a sufficiently accurate estimate of the mass pbhs $M\left(t_M\right)$ formed during the period of time t since the Big Bang has been obtained

$$M\left(t_M\right)\approx {\displaystyle \frac{c^3{t}_M}{G}}\approx {10}^{15}\left({\displaystyle \frac{t}{{10}^{-23}\mathrm{s}}}\right)g.$$

(1)

As seen, for small times close to the Planckian time $t_M=t_p\approx {10}^{-43}$s, the mass of pbhs is close to the Planck mass $M\left(t_M\right)\approx {10}^{-5}g$. The names of such black holes were varying with time: ”mini-black holes”,”micro-black holes”, and, e.g. in [8], they were referred to as ”ultralight primordial black holes”. The author of this paper uses for such pbhs the name quantum pbhs (or qpbhs) introduced in [9,10] and notes that quantum-gravitational effects for these objects could be significant. Of particular interest are pbhs arising in the preinflationary epoch. In [11] a semiclassical approximation was used to study the problem of scalar perturbations due to such pbhs. But, considering that all the processes in this case proceed at very high energies E close to the Planckian $E\simeq E_p$, the inclusion of quantum-gravity corrections qgcs for these black holes in this pattern is necessary if quantum gravity exists [12]. Despite the fact that presently there is no self-consistent theory of quantum gravity, a consensus is reached on correctness of some approaches to the theory, specifically, replacement of the Heisenberg Uncertainty Principle (HUP) by the Generalized Uncertainty Principle (GUP) on going to high (Planck’s) energies, used in this paper.

Within the scope of a natural assumption based on the results from [11], in the present work the author studies the problem, how the above-mentioned qgcs for quantum pbhs can shift the inflationary parameters and contribute to cosmological perturbations involved in the inflationary process. Moreover, it is shown that inclusion of qgcs:

a) increases the probability of the generation of pbhs;

b) leads to the enhancement of non-Gaussianity in cosmological perturbations.

In what follows the abbreviation qgcs is associated with the foregoing quantum-gravity corrections. Beginning a study of qgcs in [?], here the author, using the definitions from [?], presents much more general results referring to the emergence probability of quantum pbhs as well as of quantum fluctuations,perturbations and non-Gaussianity.

The paper is structured as follows.

Section 2 presents the instruments used to obtain the principal results. Section 3 shows how qgcs shift the inflationary parameters within a natural assumption from [11]. In Section 4 it is demonstrated that inclusion of qgcs increases the occurrence probability for such pbhs. In Section 5 the high-energy deformations of Friedmann Equations on the basis of qgcs are derived for different cases. Finally, Section 6 begins a study of the contributions made by qgcs into different cosmological perturbations under inflation and, due to the involvement of qgcs, demonstrates the enhancement of non-Gaussianity for different perturbations revealed as growing moduli of bispectrums.

In what follows the normalization $c=\hslash =1$ is used, for which we have $G=l_p^2$.

2. PBHs with the Schwarzschild-de Sitter Metric in the Early Universe

It should be noted that Schwarzschild black holes in real physics (cosmology, astrophysics) are idealized objects. As noted in (p.324,[13]): ”Spherically symmetric accretion onto a Schwarzschild black hole is probably only of academic interest as a testing for theoretical ideas. It is of little relevance for interpretations of the observations data. More realistic is the situation where a black hole moves with respect to the interstellar gas...”

Nevertheless, black holes just of this type may arise and may be realistic in the early Universe. In this case they are pbhs.

During studies of the early Universe for such pbhs the Schwarzschild metric [13,14]

$$d{s}^2=\left(1-{\displaystyle \frac{2MG}{r}}\right)d{t}^2-{\left(1-{\displaystyle \frac{2MG}{r}}\right)}^{-1}d{r}^2-r^{2}d{\Omega }^2,$$

(2)

for pbhs is replaced by the Schwarzschild-de Sitter (SdS) metric [11] that is associated with Schwarzschild black holes with small mass M in the early Universe, in particular in pre-inflation epoch

$$d{s}^2=-f\left(\tilde{r}\right)d{t}^2+{\displaystyle \frac{d{\tilde{r}}^2}{f\left(\tilde{r}\right)}}+{\tilde{r}}^{2}d{\Omega }^2$$

(3)

where $f\left(\tilde{r}\right)=1-2GM/\tilde{r}-\Lambda {\tilde{r}}^2/3=1-2GM/\tilde{r}-{\tilde{r}}^2/L^2,L=\sqrt{3/\Lambda }=H_0^{-1}$, M - black hole mass, $\Lambda$ – cosmological constant, and $L=H_0^{-1}$ is the Hubble radius.

In general, such a black hole may have two different horizons corresponding to two different zeros $f\left(\tilde{r}\right)$: event horizon of a black hole and cosmological horizon. This is just so in the case under study when a value of M is small [15,16]. In the general case of $L\gg GM$, for the event horizon radius of a black hole having the metric (3), $r_{\mathrm{H}}$ takes the following form (formula (9) in [17]):

$$r_{\mathrm{H}}\simeq 2GM\left[1+{\left({\displaystyle \frac{r_M}{L}}\right)}^2\right],where{r}_M=2MG.$$

(4)

Then, due to the assumption concerning the initial smallness of $\Lambda$, we have $L\gg r_M$. In this case, to a high accuracy, the condition $r_{\mathrm{H}}=r_M$ is fulfilled, i.e. for the considered (SdS) BH we can use the formulae for a Schwarzschild BH, to a great accuracy.

Remark 2.1.

Note that, because Λ is very small, the condition $L\gg GM$ and hence the formula of (4) are obviously valid not only for black hole with the mass $M\propto m_p$ but also for a much greater range of masses, i.e. for black holes with the mass $M\gg m_p$, taking into account the condition $L\gg GM$. In fact we obtain ordinary Schwarzschild black holes (2) with small radius.

Specifically, for the energies on the order of Plank energies (quantum gravity scales) $E\simeq E_p$, the Heisenberg Uncertainty Principle (HUP) [18]

$$\left(\delta X\right)\left(\delta P\right)\ge {\displaystyle \frac{\hslash }{2}},$$

(5)

may be replaced by the Generalized Uncertainty Principle (GUP) [19]

$$\left(\delta X\right)\left(\delta P\right)\ge {\displaystyle \frac{\hslash }{2}}〈exp\left({\displaystyle \frac{{\alpha }^2{l}_p^2}{{\hslash }^2}}{P}^2\right)〉.$$

(6)

Then there is a possibility for existence of Planck Schwarzschild black hole, and accordingly of a Schwarzschild sphere (further referred to as ”minimal”) with the minimal mass $M_0$ and the minimal radius $r_{min}$ (formula (20) in [19]) that is a theoretical minimal length $r_{min}$:

$$r_{min}=l_{min}={\left(\delta X\right)}_0=\sqrt{{\displaystyle \frac{e}{2}}}\alpha l_p,M_0={\displaystyle \frac{\alpha \sqrt{e}}{2\sqrt{2}}}{m}_p,$$

(7)

where $\alpha$ - model-dependent parameters on the order of 1, e - base of natural logarithms, and $r_{min}\propto l_p,M_0\propto m_p$.

In this case, due to GUP (6), the physics becomes nonlocal and the position of any point is determined accurate to $l_{min}$. It is impossible to ignore this nonlocality at the energies close to the Planck energy $E\approx E_p$, i.e. at the scales $l\propto l_p$ (equivalently we have $l\propto r_{min}=l_{min}$).

Actually, [19] presents calculated values of the mass M and the radius R for Schwarzschild BH with regard to the quantum-gravitational corrections within the scope of GUP (6).

With the use of the normalization $G=l_p^2$ adopted in [19], temperature of a Schwarzschild black hole having the mass M (the radius R) [13] in a semi-classical approximation takes the form

$$T_H={\displaystyle \frac{1}{8\pi GM}}.$$

(8)

Within the scope of GUP (6),the temperature $T_H$ with regard to (qgc) is of the form ((23) and (25) in [19]))

$$\begin{array}{c}\hfill T_{H,q}={\displaystyle \frac{1}{8\pi MG}}exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)={\displaystyle \frac{1}{8\pi MG}}exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}\left({\displaystyle \frac{A_0}{A}}\right)\right)\right)=\\ \hfill ={\displaystyle \frac{1}{8\pi MG}}\left(1+{\displaystyle \frac{1}{2e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2+{\displaystyle \frac{5}{8e^2}}{\left({\displaystyle \frac{M_0}{M}}\right)}^4+{\displaystyle \frac{49}{48e^3}}{\left({\displaystyle \frac{M_0}{M}}\right)}^6+\dots \right),\end{array}$$

(9)

where A is the black hole horizon area of the given black hole with mass M and event horizon $r_M$ ,$A_0=4\pi {\left(\delta X\right)}_0^2$ is the black hole horizon area of a minimal quantum black hole from formula (7) and $W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)=W\left(-{\displaystyle \frac{1}{e}}\left({\displaystyle \frac{A_0}{A}}\right)\right)$ – value at the corresponding point of the Lambert W-function $W\left(u\right)$ satisfying the equation (formulae (1.5) in [20] and (9) in [19])

$$W\left(u\right)e^{W\left(u\right)}=u.$$

(10)

$W\left(u\right)$ is the multifunction for complex variable $u=x+yi$. However, for real $u=x,-1/e\le u<0$,$W\left(u\right)$ is the single-valued continuous function having two branches denoted by $W_0\left(u\right)$ and $W_{-1}\left(u\right)$ , and for real $u=x,u\ge 0$ there is only one branch $W_0\left(u\right)$ [20].

Obviously, the quantum-gravitational correction (qgc) (9) presents a deformation (or more exactly, the quantum deformation of a classical black-holes theory from the viewpoint of the paper [21] with the deformation parameter $A_0/A$):

$${\displaystyle \frac{A_0}{A}}={\displaystyle \frac{4\pi r_h^2}{4\pi r_M^2}}={\displaystyle \frac{l_{min}^2}{r_M^2}},$$

(11)

where $r_h=l_{min}$ is the horizon radius of minimal pbh from formula (7).

It should be noted that this deformation parameter

$$l_{min}^2/r_M^2\doteq {\alpha }_{r_M}$$

(12)

has been introduced by the author in his earlier works [22,23], where he studied deformation of quantum mechanics at Planck scales in terms of the deformed quantum mechanical density matrix. In the Schwarzschild black hole case ${\alpha }_{r_M}=l_{min}^2\mathcal{K}$ – Gaussian curvature $\mathcal{K}=1/{\alpha }_{r_M}$ of the black-hole event horizon surface [24].

It is clear that, for a great black hole having large mass M and great event horizon area A, the deformation parameter ${\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2$ is vanishingly small and close to zero. Then a value of $W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)$ Is also close to $W\left(0\right)$. As seen, $W\left(0\right)=0$ is an obvious solution for the equation (10). We have

$$exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)\approx 1.$$

(13)

So, a black hole with great mass $M\gg m_p$ necessitates no consideration of qgcs.

But in the case of small black holes we have

$$exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)>1.$$

(14)

In formulae above it is assumed that $M>M_0$, i.e. the black hole under study is not minimal (7).

We can rewrite the formula of (9) as follows:

$$\begin{array}{c}\hfill T_{H,q}={\displaystyle \frac{1}{8\pi M_{q}G}},M_q=Mexp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right);\\ \hfill R_q=2M_{q}G=Rexp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right),\end{array}$$

(15)

where $M_q$ and $R_q$ are respectively the initial black-hole mass and event horizon radius considering qgcs caused by GUP (6).

Note that in a similar way $M_q$ is involved in ([19] formula (26)) as a function of the black-hole temperature. But, instead of the small parameter $M_0/M$, the author uses the small parameter $T_H/T_H^{max}$, where $T_H^{max}$ is the black hole maximum temperature.

Remark 2.2

It is clear that the formula (15) with the substitution of $M\mapsto M_q$ is of the same form as formula (8), in fact representing (9),i.e. in the formula for temperature of a black hole the inclusion of qgcs may be realized in two ways with the same result: (a)the initial mass M remains unaltered and qgcs are involved only in the formula for temperature, in this case (9); (b) qgcs are involved in the mass-the above-mentioned substitution takes place $M\mapsto M_q$ (formula(15)). Such ”duality” is absolutely right in this case if a black hole is considered in the stationary state in the absence of accretion and radiation processes. Just this case is also studied in the paper.

A recent preprint [25] in the case (b) for the space-time dimension $D\ge 4$, using approaches to quantum gravity of the alternative GUP, gives a formula for the mass $M_q$ of a black hole with a due regard to qgc

$$M_q={\left[1-\eta exp\left(-{\displaystyle \frac{\pi r_0^{D-2}}{G_D}}\right)\right]}^{D-3}M.$$

(16)

Here in terms of [25] $r_0$ is the Schwarzschild radius of the primordial black hole with the mass M,$G_D$-gravitational constant in the dimension D, and $\eta =[0,1]$ is a parameter. In case under study this parameter, as distinct from cosmology, has no relation to conformal time. Obviously, for $\eta =0$ we have a semiclassical approximation and, as noted in [25], the case when $\eta =1$ corresponds to qgc as predicted by a string theory.

Remark 2.3

It should be noted that, with the expansion $exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)$ in terms of the small parameter ${\alpha }_{r_M}={(M_0/M)}^2$, in formula (9) one can easily obtain the small-parameter expansion of the inverse number $exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)$, and also of all the integer powers for this exponent, specifically for its square $exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)$.

3. Inflation Parameters Shifts Generated by QGCs

To this end in cosmology, in particular inflationary, the metric (3) is conveniently described in terms of the conformal time $\eta$ [11]:

$$d{s}^2=a^2\left(\eta \right)\left\{-d{\eta }^2+{\left(1+{\displaystyle \frac{{\mu }^3{\eta }^3}{r^3}}\right)}^{4/3}\left[{\left({\displaystyle \frac{1-{\mu }^3{\eta }^3/r^3}{1+{\mu }^3{\eta }^3/r^3}}\right)}^{2}d{r}^2+r^{2}d{\Omega }^2\right]\right\},$$

(17)

where $\mu ={(GM{H}_0/2)}^{1/3}$, $H_0$ – de Sitter-Hubble parameter and scale factor, a – conformal time function $\eta$:

$$a\left(\eta \right)=-1/\left(H_0\eta \right),\eta <0.$$

(18)

Here r satisfies the condition $r_0<r<\infty$ and a value of $r_0=-\mu \eta$ in the reference frame of (17) conforms to singularity of the back hole.

Due to (4), $\mu$ may be given as

$$\mu ={(r_M{H}_0/4)}^{1/3},$$

(19)

where $r_M$ is the radius of a black hole with the SdS Schwarzschild-de Sitter metric (3).

Remark 3.1.

As we consider quantum pbhs with the SdS-metric, in this case, as noted in (4), they to a high accuracy are coincident with the Schwarzschild black holes (2) and hence they have the identical formulae for qgcs: (15),(16)... Further we consider the contribution made by these qgcs into the quantities associated with inflation: inflationary parameters, cosmological perturbations etc. It should be made clear:

In [11] in general only the case $\mu =const$ is considered and, as noted in [11], for the case $\mu \ne const$ we can use only the pattern including the radiation processes of pbhs. However, the value of $\mu =const$ itself contains no information concerning the treatment of either the initial quantum pbh in a semiclassical approximation or the consideration with regard to qgcs. Obviously, in this case involvement of qgcs shifts all the parameters derived for a semiclassical (canonical) pattern.

Let us consider the following pattern related to that studied in [11]: it is supposed that, as the mass M of pbh may be changed due to the radiation process, the corresponding change takes place for $\mu$ – in the general case we have ($\mu \ne const$) in view of these processes. But further it is assumed that after termination of these processes $\mu$ is unaltered with regard to qgcs, i.e. in formula (19) we have $\mu ={(r_M{H}_0/4)}^{1/3}={(r_{M_q}{H}_{0,q}/4)}^{1/3}$, where $r_{M_q},H_{0,q}$ - values of $r_M,H_0$, respectively, with due regard for qgcs.

3.1. The Stationary Picture

From the start of creation, the primordial black hole, with the mass M and the event horizon area A, is considered in the absence of absorption and radiation processes. It may be assumed that such quantum pbh was generated immediately before the onset of inflation, when there were no absorption and radiation processes. On the other hand, 3.1 is completely consistent with the paradigm in [9,10] presuming the absence of Hawking radiation for quantum pbh, with the mass and the event horizon radius close to the Planckian values $M\approx m_p,r_M\approx l_p$ .

As $\mu =const$ and pbh we consider in the stationary state, then, due to Remark 2.2 with regard for qgcs, replacement $r_M\mapsto r_{M_q}$ in this formula leads to replacement of $H_0\to H_{0,q}$, due to Remark 3.1. meeting the condition

$$\mu ={(r_M{H}_0/4)}^{1/3}={(r_{M_q}{H}_{0,q}/4)}^{1/3}.$$

(20)

Here $r_{M_q}=R_q$ from the general formula (15).

Based on the last formula and formulae (9),(12),(15) it directly follows that

$$H_{0,q}=H_{0}exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)=H_{0}exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right).$$

(21)

Then qgc for the scale factor $a\left(\eta \right)$ (18)

$$\begin{array}{c}\hfill a\left(\eta \right)\to a{\left(\eta \right)}_q\doteq -1/\left(H_{0,q}\eta \right)=-1/(H_{0}exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)\eta )=\\ \hfill =-1/(H_{0}exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)\eta )=a\left(\eta \right)exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right),\eta <0,\end{array}$$

(22)

and for the Hubble parameter in general case $H\left(\eta \right)\doteq H$

$$\begin{array}{c}\hfill H=a^{\prime }\left(\eta \right)/a^2\left(\eta \right)\mapsto H_q\left(\eta \right)=a^{\prime }{\left(\eta \right)}_q/a^2{\left(\eta \right)}_q\end{array}$$

(23)

As directly follows from the last formula, in this pattern $H_0$ and H are identically transformed, i.e. we have

$$\begin{array}{c}\hfill H\mapsto H_q=Hexp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)\end{array}$$

(24)

Because the potential energy of inflation V is related to the Hubble parameter H by the Friedmann equation (formula (12.12) in [28]) $H^2=8\pi V/\left(3M_p^2\right)$,from (21) we can derive a ”shift” for V that is due to quantum-gravitational corrections for the primordial Schwarzschild black hole with the mass M as follows:

$$\begin{array}{c}\hfill [V=3M_p^2{H}^2/\left(8\pi \right)]\mapsto V_q=3M_p^2{H}_q^2/\left(8\pi \right)=\\ \hfill =3exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)M_p^2{H}^2/\left(8\pi \right)=exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)V,\end{array}$$

(25)

In a similar way, taking account of qgcs for quantum pbhs (formulae (9,(15), we can find these ”shifts” for all inflationary parameters, in particular

$$\begin{array}{c}\hfill (a\sim H^{-1})\mapsto a_q\sim H_q^{-1};\\ \hfill (V\sim H^2)\mapsto V_q\sim H_q^2;\\ \hfill (\dot{\varphi }=-{\displaystyle \frac{V^{\prime }\left(\varphi \right)}{3H}}\sim H)\mapsto {\dot{\varphi }}_q={(-{\displaystyle \frac{V^{\prime }\left(\varphi \right)}{3H}})}_q\sim H_q;\\ \hfill (\dot{H}={\displaystyle \frac{1}{2m_p}}{\left({\displaystyle \frac{8\pi }{3V}}\right)}^{1/2}{V}^{\prime }\left(\varphi \right)\dot{\varphi }\sim H^2)\mapsto {\dot{H}}_q={\displaystyle \frac{1}{2m_p}}{\left[{\left({\displaystyle \frac{8\pi }{3V}}\right)}^{1/2}{V}^{\prime }\left(\varphi \right)\dot{\varphi }\right]}_q\sim H_q^2;\\ \hfill (\ddot{\varphi }=-{\displaystyle \frac{m_p^2}{8\pi }}({\displaystyle \frac{V^{\prime \prime }}{V}}-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{V^{\prime }}{V}}\right)}^2)H\dot{\varphi }\sim H^2)\mapsto {\ddot{\varphi }}_q\sim H_q^2,...\end{array}$$

(26)

with retention of some part of them on the transformation $H\mapsto H_q$,for example it is easy to check retention of the deceleration parameters $ϵ,\tilde{\eta }$ [28]:

$$\begin{array}{c}\hfill ϵ=-{\displaystyle \frac{\dot{H}}{H^2}}=m_p{\displaystyle \frac{V^{\prime 2}}{V^2}}={ϵ}_q=-{\displaystyle \frac{\dot{H_q}}{H_q^2}}=m_p{\displaystyle \frac{V_q^{\prime 2}}{V_q^2}},\\ \hfill \tilde{\eta }={\displaystyle \frac{m_p^2}{8\pi }}{\displaystyle \frac{V^{\prime \prime }}{V}}={\tilde{\eta }}_q.\end{array}$$

(27)

Here in the last two formulae (26),(27) a point means differentiation with respect to t, but a prime means differentiation with respect to the field $\varphi$. Let us denote the second deceleration parameter $\tilde{\eta }$ instead of $\eta$ [28], to avoid confusion with the conformal time.

As we have $ϵ={ϵ}_q,\tilde{\eta }={\tilde{\eta }}_q$,the slow-roll conditions (formula (27) in [28]) in the inflationary scenario with regard to qgcs for qpbhs remains unaltered.

Due to formula (9), all the above-mentioned shifts of the inflationary parameters generated byqgcsfor quantumpbhsin the pre-inflationary period may be series expanded in terms of the small parameter ${(M_0/M)}^2$ (same ${\alpha }_{r_M}$).

3.2. Black Hole Evaporation and qgcs

(This case studied in [?] is given for completeness of the presentation.)

Also,black holes are associated with the process of Hawking radiation (evaporation). The primordial black holes are no exception. In the general case this process is considered only within the scope of a semiclassical approximation (without consideration of the quantum-gravitational effects). Because of this, it is assumed that a primordial black hole may be completely evaporated [13].

Still, in this pattern the situation is impossible due to the validity of GUP (6) and due to the formation of a minimal (nonvanishing) Planckian remnant as a result of evaporation (7) [19,29].

We can compare the mass loss for a black hole in this process when using a semiclassical approximation and with due regard for qgcs.

Let M be the mass of a primordial black hole. Then a loss of mass as a result of evaporation, according to the general formulae, takes the following form ([13],p.356):

$${\displaystyle \frac{dM}{dt}}\sim \sigma T_{\mathrm{H}}^4{A}_M,$$

(28)

where $T_{\mathrm{H}}$ - temperature of a black hole with the mass M,$A_M$ - surface area of the event horizon of this hole $A_M=4\pi r_M^2$, and $\sigma ={\pi }^2{k}^4/\left(60{\hslash }^3{c}^2\right)$ is the Stefan-Boltzmann constant.

Using this formula for the same black hole but with regard to qgcs, we can get the mass loss ${[dM/dt]}_q$ in this case

$${\left[{\displaystyle \frac{dM}{dt}}\right]}_q\sim \sigma T_{\mathrm{H},q}^4{A}_M,$$

(29)

where $T_{\mathrm{H},q}$ - temperature of a black hole with the same mass M, when taking into consideration qgcs (9).

For all the foregoing formulae associated with a random black hole having the mass M, the following estimate is correct ((10.1.19) in [13]):

$$-{\displaystyle \frac{dM}{dt}}\sim b{\left({\displaystyle \frac{M_p}{M}}\right)}^2{\left({\displaystyle \frac{M_p}{t_p}}\right)}^{2}N,$$

(30)

where $b\approx 2.59\times {10}^{-6}$, and N is the number of the states and species of particles that are radiated. The minus sign in the left part of the last formula denotes that the mass of a black hole diminishes as a result of evaporation, i.e. we have $dM/dt<0$.

Unfortunately, the last formula is hardly constructive as it is difficult to estimate the number N, especially at high energies $E\simeq E_p$.

Nevertheless, using the terminology and symbols of this paper, and also the results from [19], the formula (30) for the mass loss by a black hole with regard to qgcs may be written in a more precise and constructive form. Really, according to formula (45) in [19], within the scope of GUP (6) we will have

$$\begin{array}{c}\hfill {\displaystyle \frac{dM}{dt}}=-{\displaystyle \frac{{\gamma }_1}{M^2{l}_p^4}}exp\left(-2W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)\times \\ \hfill \times \left(1-{\displaystyle \frac{8{\gamma }_2}{e{\gamma }_1}}{\left({\displaystyle \frac{M_0}{M}}\right)}^{2}exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)\right),\end{array}$$

(31)

where ${\gamma }_1={\displaystyle \frac{{\pi }^2}{480}},$${\gamma }_2={\displaystyle \frac{{\pi }^2}{16128}}$.

The minus sign in the right side of the last formula means the same as the minus sign in the left side of formula (30).

Due to (12), formula (31) is of the following form:

$$\begin{array}{c}\hfill {\displaystyle \frac{dM}{dt}}=-{\displaystyle \frac{{\gamma }_1}{M^2{l}_p^4}}exp(-2W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right))\times \\ \hfill \times \left(1-{\displaystyle \frac{8{\gamma }_2}{e{\gamma }_1}}{\alpha }_{r_M}exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)\right).\end{array}$$

(32)

We can expand the right sides of formulae (31) and (32) into a series in terms of the small parameter $e^{-1}{(M_0/M)}^2=e^{-1}{\alpha }_{r_M}$ (formula (46) in [19]) that, proceeding from the deformation parameter ${\alpha }_{r\left(M\right)}$, takes the form

$$\begin{array}{c}\hfill {\displaystyle \frac{dM}{dt}}=-{\displaystyle \frac{{\gamma }_1}{M^2{l}_p^4}}\left(1+{\displaystyle \frac{2}{e}}{\alpha }_{r_M}+{\displaystyle \frac{4}{e^2}}\left(1-{\displaystyle \frac{2{\gamma }_2}{e{\gamma }_1}}\right){\alpha }_{r_M}^2+{\displaystyle \frac{25}{3e^3}}\left(1-{\displaystyle \frac{72{\gamma }_2}{25e{\gamma }_1}}\right){\alpha }_{r_M}^3+\dots \right).\end{array}$$

(33)

Neglecting the last equation due to the time interval chosen, e.g., due to $▵t=t_{infl}-t_M$,where $t_{infl}$–time of the inflation onset and $t_M$– time during which the black hole under study has been formed, formula (1), the mass loss for a black hole with regard to qgcs by the inflation onset time may be given as

$$\begin{array}{c}\hfill {\Delta }_{Evap,q}M(t_M,t_{infl})\doteq \underset{t_M}{\overset{t_{infl}}{\int }}{\displaystyle \frac{dM}{dt}}=\\ \hfill =-\underset{t_M}{\overset{t_{infl}}{\int }}{\displaystyle \frac{{\gamma }_1}{M^2{l}_p^4}}\left(1+{\displaystyle \frac{2}{e}}{\alpha }_{r_M}+{\displaystyle \frac{4}{e^2}}\left(1-{\displaystyle \frac{2{\gamma }_2}{e{\gamma }_1}}\right){\alpha }_{r_M}^2+{\displaystyle \frac{25}{3e^3}}\left(1-{\displaystyle \frac{72{\gamma }_2}{25e{\gamma }_1}}\right){\alpha }_{r_M}^3+\dots \right).\end{array}$$

(34)

Next, we can determine the mass of a black hole after its evaporation until the inflation onset with regard to qgcs

$$\begin{array}{c}\hfill M_{Evap,q}(t_{M_q},t_{infl})\doteq M+{\Delta }_{Evap,q}{M}_q(t_{M_q},t_{infl}).\end{array}$$

(35)

In the pattern of a semiclassical approximation the above-mentioned formulae are greatly simplified because in this case ${\alpha }_{r_M}=0$ due to the absence of a minimal black hole.

Then in a semiclassical pattern formula (35), with the use of the suggested formalism, takes the following form:

$$\begin{array}{c}\hfill M_{Evap}(t_M,t_{infl})\doteq M+{\Delta }_{Evap}M(t_M,t_{infl}),\end{array}$$

(36)

where

$$\begin{array}{c}\hfill {\Delta }_{Evap}M(t_M,t_{infl})=\underset{t_M}{\overset{t_{infl}}{\int }}{\displaystyle \frac{dM}{dt}}=-\underset{t_M}{\overset{t_{infl}}{\int }}{\displaystyle \frac{{\gamma }_1}{M^2{l}_p^4}}.\end{array}$$

(37)

Accordingly, for the radii $M_{Evap}(t_M,t_{infl}),M_{Evap,q}$ we can get

$$\begin{array}{c}\hfill r\left(M_{Evap}\right)=2G{M}_{Evap}(t_M,t_{infl}),\\ \hfill r\left(M_{Evap,q}\right)=2G{M}_{Evap,q}(t_M,t_{infl}).\end{array}$$

(38)

In accordance with Remark 3.1, we have

$$\begin{array}{c}\hfill {\mu }_{Evap}\doteq {(r_{M_{Evap}}{H}_{0,Evap}/4)}^{1/3}={(r_{M_{Evap},q}{H}_{0,Evap,q}/4)}^{1/3};\\ \hfill H_{0,Evap,q}={\displaystyle \frac{r_{M_{Evap}}}{r_{M_{Evap},q}}}{H}_{0,Evap}.\end{array}$$

(39)

The right side of the last line in formula (39) gives the ”quantum-gravitational shifts” (abbreviated as qgs) of the de Sitter Hubble parameter $H_0$ for black holes evaporation process.

Substituting $H_{0,Evap,q}$ from (39) into formulae (25)–(27) and so on, we can obtain qgsc for all cosmological parameters in the inflationary scenario when a primordial black hole evaporates before the inflation onset.

Remark 3.2.

By the present approach we can consider the case of the particle absorption by a pbh. Let the Schwarzschild-de Sitter pbh of the mass M has the event horizon area A.In [26,27] ”a minimal increment” of the event horizon area for the black hole absorbing a particle with the energy E and with the size R:${\left(\Delta A\right)}_0\simeq 4l_p^2\left(ln2\right)ER$ has been estimated within the scope of the Heisenberg Uncertainty Principle HUP. In quantum consideration we have $R\sim 2\delta X$ and $E\sim \delta P$. Within the scope of GUP this ” minimal increment” is replaced by ${\left(\Delta A\right)}_{0,q}$ as follows (formula (27) in [19]):

$$\begin{array}{c}\hfill {\left(\Delta A\right)}_{0,q}\approx 4l_p^{2}ln2exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\displaystyle \frac{A_0}{A}}\right)\right)=\\ \hfill 4l_p^{2}ln2exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)=4l_p^{2}ln2exp\left(-{\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)\end{array}$$

(40)

Assuming that an arbitrary increment of the event horizon area A (same with the mass M) may be represented as a chain of ”minimal increments” (for quantum pbhs with the mass close to that of the Planck’s such an assumption is fairly justified) for $\mu =const$ and any absorption we can compare in the given approach the values of all cosmological parameters in the semiclassical approximation and their ”shifts” generated by qgcs.

4. Quantum-Gravity Corrections for Appearance Probabilities PBHs in the Pre-Inflationary Era

There is the problem of estimating the probability of occurrence for pbh with Schwarzschild-de Sitter SdS metric (3) in the pre-inflation epoch.

This problem has been studied in [11] without due regard for qgcs. Let us demonstrate that consideration of qgcs in this case makes the probability of arising such pbhs higher.

Similar to [11], it is assumed that in pre-inflation period non-relativistic particles with the mass $m<M_p$ are dominant (Section 3 in [11]). For convenience, let us denote the Schwarzschild radius $r_M$ by $R_S$.

When denoting, in analogy with [11], by $N(R,t)$ the number of particles in a comoving ball with the physical radius $R=R\left(t\right)$ and the volume $V_R$ at time t, in the case under study this number (formula (3.9) in [11]) will have by qgc:

$$\begin{array}{c}\hfill N(R,t)\mapsto N{(R,t)}_q;\\ \hfill (\langle N(R,t)\rangle ={\displaystyle \frac{m_p^2{H}^2{R}^3}{2m}})\mapsto (\langle N{(R,t)}_q\rangle ={\displaystyle \frac{m_p^2{H}_q^2{R}^3}{2m}}).\end{array}$$

(41)

Here the first part of the last formula agrees with formula (3.9) in [11], whereas $H,H_q$ in this case are in agreement with formulae (23),(24). And from (24) it follows that

$$\langle N{(R,t)}_q\rangle =\langle N(R,t)\rangle exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right).$$

(42)

According to (15), it is necessary to replace the Schwarzschild radius $R_S$ by

$R_{S,q}=R_{S}exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)$.

Then from the general formula $N(R_S,t)=\langle N(R_S,t)\rangle +\delta N(R_S,t)$, used because of the replacement of $R_S\mapsto R_{S,q}$, we obtain an analog of (3.12) from [11]

$$\begin{array}{c}\hfill \delta N>\delta N_{\mathrm{cr},\mathrm{q}}\doteq {\displaystyle \frac{m_p^2{R}_{S,q}}{2m}}-\langle N{(R_S,t)}_q\rangle ={\displaystyle \frac{m_p^2{R}_{S,q}}{2m}}[1-{\left(H{R}_S\right)}^2]=\\ \hfill ={\displaystyle \frac{m_p^2{R}_S}{2m}}[1-{\left(H{R}_S\right)}^2]exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)=\delta N_{\mathrm{cr}}exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right).\end{array}$$

(43)

In the last formula in square brackets we should have ${\left(H_q{R}_{S,q}\right)}^2$ instead of ${\left(H{R}_S\right)}^2$ but, as we consider the case $\mu =const$, these quantities are coincident.

It should be noted that here the following condition is used:

$$H{R}_S<1,$$

(44)

i.e. Schwarzschild radius $R_S$ less than Hubble radius, $R_S<R_H=1/H$.

As we have $exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)<1$, then

$$\delta N_{\mathrm{cr},\mathrm{q}}<\delta N_{\mathrm{cr}}.$$

(45)

Considering that for the formation of a Schwarzschild black hole with the radius $R_S$ it is required that, due to statistical fluctuations, the number of particles $N(R_S,t)$ with the mass m within the black hole volume $V_{R_S}=4/3\pi R_S^3$ be in agreement with the condition [11]

$$N(R_S,t)>R_S{M}_p^2/\left(2m\right),$$

(46)

which, according to qgc in the formula of (15), may be replaced by

$$N(R_{S,q},t)>R_{S,q}{M}_p^2/\left(2m\right)=exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)R_S{M}_p^2/\left(2m\right).$$

(47)

As follows from these expressions, with regard to qgc for the formation of pbh in the pre-inflation period, the number of the corresponding particles may be lower than for a black hole without such regard, leading to a higher probability of the formation.

Such a conclusion may be made by comparison of this probability in a semi-classical consideration (formula (3.13) in [11])

$$P\left(\delta N(R_S,t)>\delta N_{\mathrm{cr}}(R_S,t)\right)={\int }_{\delta N_{\mathrm{cr}}}^{\infty }d\left(\delta N\right)P\left(\delta N\right)$$

(48)

and with due regard for qgc

$$P\left(\delta N(R_{S,q},t)>\delta N_{\mathrm{cr}}(R_{S,q},t)\right)={\int }_{\delta N_{\mathrm{cr},\mathrm{q}}}^{\infty }d\left(\delta N\right)P\left(\delta N\right).$$

(49)

Considering that in the last two integrals the integrands take positive values and are the same, whereas the integration domain in the second integral is wider due to (45), we have

$$\begin{array}{c}\hfill {\int }_{\delta N_{\mathrm{cr},\mathrm{q}}}^{\infty }d\left(\delta N\right)P\left(\delta N\right)=\\ \hfill ={\int }_{\delta N_{\mathrm{cr},\mathrm{q}}}^{\delta N_{\mathrm{cr}}}d\left(\delta N\right)P\left(\delta N\right)+{\int }_{\delta N_{\mathrm{cr}}}^{\infty }d\left(\delta N\right)P\left(\delta N\right)>{\int }_{\delta N_{\mathrm{cr}}}^{\infty }d\left(\delta N\right)P\left(\delta N\right).\end{array}$$

(50)

As follows from the last three formulae, in the case under study the probability that the above-mentioned pbh will be formed is higher with due regard for qgc.

5. High Energy Deformations of Friedmann Equations

Based on the obtained results, it is inferred that there is the deformation (having a quantum-gravitational character) of the Schwarzschild-de Sitter metric and Friedmann Equations due to these qgsc. Indeed,substituting the expression $a{\left(\eta \right)}_q$ instead of a into the Friedmann Equation ((2.4) in [28]) without term with curvature

$${\displaystyle \frac{a^{\prime 2}}{a^4}}={\displaystyle \frac{8\pi }{3}}G\rho ,$$

(51)

we can obtain the Quantum Deformation (QD) [21] of the Friedmann Equation due to qgcs for pbh in the early Universe

$${\displaystyle \frac{a_q^{\prime 2}}{a_q^4}}={\displaystyle \frac{a^{\prime 2}}{exp\left(W\left(-\frac{1}{e}{\alpha }_{r_M}\right)\right)a^4}}={\displaystyle \frac{8\pi }{3}}G\rho$$

(52)

or

$$\begin{array}{c}\hfill {\displaystyle \frac{a^{\prime 2}}{a^4}}={\displaystyle \frac{8\pi }{3}}G\rho exp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)\doteq {\displaystyle \frac{8\pi }{3}}G{\rho }_q,\\ \hfill {\rho }_q\doteq \rho exp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)<\rho .\end{array}$$

(53)

The last line in (53) is associated with the fact that the Lambert W-function $W\left(u\right)$ is negative for $u<0$.

Similarly, $\left(ij\right)$-components of the Einstein equations ((2.5) in [28])

$$2{\displaystyle \frac{a^{\prime \prime }}{a^3}}-{\displaystyle \frac{a^{\prime 2}}{a^4}}=-{\displaystyle \frac{8\pi }{3}}Gp$$

(54)

within the foregoing (QD) are replaced by

$$\begin{array}{c}\hfill 2{\displaystyle \frac{a_q^{\prime \prime }}{a_q^3}}-{\displaystyle \frac{a_q^{\prime 2}}{a_q^4}}=-{\displaystyle \frac{8\pi }{3}}Gp\end{array}$$

(55)

or

$$\begin{array}{c}\hfill 2{\displaystyle \frac{a^{\prime \prime }}{a^3}}-{\displaystyle \frac{a^{\prime 2}}{a^4}}=-{\displaystyle \frac{8\pi }{3}}Gpexp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)=-{\displaystyle \frac{8\pi }{3}}G{p}_q,\\ \hfill p_q\doteq pexp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)<p.\end{array}$$

(56)

It should be noted that the equation of the covariant energy conservation for the homogeneous background ((2.6) in [28])

$$\begin{array}{c}\hfill {\rho }^{\prime }=-3{\displaystyle \frac{a^{\prime }}{a}}(\rho +p)\end{array}$$

(57)

remains unaltered with replacement of $\rho \mapsto {\rho }_q,p\mapsto p_q$.

So, in the pattern of 3.1 (the stationary pattern), taking into consideration of qgcs for pbhs in the pre-inflationary era increases the initial values of the density $\rho$ and of the pressure p in Friedmann equations.

The above calculations are correct if, from the start, we assume that a black hole (i.e., its event-horizon radius) is invariable until the onset of inflation. But such a situation is idealized because this period is usually associated with the radiation and absorption processes

Then again for $\mu =const$ from formulae (19),(18) we have

$$\begin{array}{c}\hfill H_{0,q}=H_0{\displaystyle \frac{r_{M_{orig}}}{r_{M_{orig,q}}}},\\ \hfill a{\left(\eta \right)}_q=a\left(\eta \right){\displaystyle \frac{r_{M_{orig,q}}}{r_{M_{orig}}}}.\end{array}$$

(58)

Substituting the expression $a{\left(\eta \right)}_q$ from formula (58) in all formulae (52)–(57) we obtain analogues of these formulae in the general case. In particular, for formula (52) we have

$${\displaystyle \frac{a_q^{\prime 2}}{a_q^4}}={\displaystyle \frac{r_{M_{orig}}^2}{r_{M_{orig,q}}^2}}{\displaystyle \frac{a^{\prime 2}}{a^4}}={\displaystyle \frac{8\pi }{3}}G\rho$$

(59)

Or, equivalently,

$$\begin{array}{c}\hfill {\displaystyle \frac{a^{\prime 2}}{a^4}}={\displaystyle \frac{8\pi }{3}}G\rho {\displaystyle \frac{r_{M_{orig,q}}^2}{r_{M_{orig}}^2}}={\displaystyle \frac{8\pi }{3}}G{\rho }_q\\ \hfill {\rho }_q\doteq {\displaystyle \frac{r_{M_{orig,q}}^2}{r_{M_{orig}}^2}}\rho .\end{array}$$

(60)

In the same way as for formula (55), in this pattern for the general quantum deformation $\left(ij\right)$-components of Einstein equations by substitution of the value for $a{\left(\eta \right)}_q$ from the formula (58) we obtain

$$\begin{array}{c}\hfill 2{\displaystyle \frac{a_q^{\prime \prime }}{a_q^3}}-{\displaystyle \frac{a_q^{\prime 2}}{a_q^4}}=-{\displaystyle \frac{8\pi }{3}}Gp\end{array}$$

(61)

or

$$\begin{array}{c}\hfill 2{\displaystyle \frac{a^{\prime \prime }}{a^3}}-{\displaystyle \frac{a^{\prime 2}}{a^4}}=-{\displaystyle \frac{8\pi }{3}}Gp{\displaystyle \frac{r_{M_{orig,q}}^2}{r_{M_{orig}}^2}}=-{\displaystyle \frac{8\pi }{3}}G{p}_q,\\ \hfill p_q\doteq {\displaystyle \frac{r_{M_{orig,q}}^2}{r_{M_{orig}}^2}}p.\end{array}$$

(62)

It is clear that, in this most general pattern, the covariant energy conservation for the homogeneous background ((2.6) in [28])

$$\begin{array}{c}\hfill {\rho }^{\prime }=-3{\displaystyle \frac{a^{\prime }}{a}}(\rho +p)\end{array}$$

(63)

remains unaltered with replacement of $\rho \mapsto {\rho }_q,p\mapsto p_q$.

6. The Quantum Fluctuations and Cosmological Perturbations Corrections Generated by qgcs for qpbhs. Non-Gaussianity Enhancement. The Onset

6.1. General Remarks

It is known that inflationary cosmology is characterized by cosmological perturbations of different nature (scalar, vector, tensor) [28,30,31], though vector perturbations are usually ignored as they die out fast.

It is clear that, as qgcs for pbhs in the early Universe cause shifts of the inflationary parameters, they inevitably lead to corrections of the cosmological perturbations on inflation. (14) The following Remark, with the inferences used in the previous section, holds true:

Remark 6.1

From the above formulae it immediately follows that these shifts are arising only in the monomials, where the total power of the scale factor a (or Hubble parameter H) and of any derivatives with respect to η is not equal to 0. In this case the corresponding shifts are calculated using formulae (21)–(26) and their analogues.

6.2. Commentary on Corrections for Quantum Fluctuations and Cosmological Perturbations

Specifically, in the case of scalar cosmological perturbations consideration of the indicated qgcs for the rest of the Einstein equations (formulae (2.74)–(2.76) in [28]) in case 3.1 (the stationary picture) in virtue of Remark 6.1 gives

$$\begin{array}{c}\hfill \Delta \mathsf{\Phi }-3{\displaystyle \frac{a^{\prime }}{a}}{\mathsf{\Phi }}^{\prime }-3{\displaystyle \frac{a^{\prime 2}}{a^2}}\mathsf{\Phi }=4\pi G{a}^{2}exp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)·\delta {\rho }_{tot};\\ \hfill {\mathsf{\Phi }}^{\prime }+{\displaystyle \frac{a^{\prime }}{a}}\mathsf{\Phi }=-4\pi G{a}^{2}exp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)·{\left[(\rho +p)\upsilon \right]}_{tot};\\ \hfill {\mathsf{\Phi }}^{\prime \prime }+3{\displaystyle \frac{a^{\prime }}{a}}{\mathsf{\Phi }}^{\prime }+(2{\displaystyle \frac{a^{\prime \prime }}{a}}-{\displaystyle \frac{a^{\prime 2}}{a^2}})\mathsf{\Phi }=4\pi G{a}^{2}exp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)·\delta p_{tot}.\end{array}$$

(64)

Here in the right sides of all lines in the last formula the scale factor a is taken with regard to qgcs from formula (22), i.e., $a=a{\left(\eta \right)}_q$. In the left sides of these lines additional factors of the type

$exp\left({\displaystyle \frac{1}{2}}W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right),exp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right),...$ are cancelled out because they are independent of $\eta$. This is so in the general case when taking in consideration qgcs for the pbhs formed in the pre-inflationary era (for all types of the cosmological perturbations, not only for those of the scalar type).

According to this remark, under the linearized form of the gauge transformations (formulae (2.31) in [28]), spatial components of the metric perturbation transform are retained due to inclusion of qgcs ([28],p.30):

$$\begin{array}{c}\hfill {\tilde{h}}_{ij}=h_{ij}-2{\partial }_i{\partial }_j\sigma -{\displaystyle \frac{a^{\prime }}{a}}{\delta }_{ij}{\sigma }^{\prime }.\end{array}$$

(65)

And qgcs deform correspondingly the metric with scalar perturbations in the conformal Newtonian gauge (formulae (2.69) in [28]):

$$\begin{array}{c}\hfill \{d{s}^2=a^2\left(\eta \right)[(1+2\mathsf{\Phi })d{\eta }^2-(1+2\Psi )d{\mathbf{x}}^2]\}\mapsto a^2{\left(\eta \right)}_q[(1+2\mathsf{\Phi })d{\eta }^2-(1+2\Psi )d{\mathbf{x}}^2]=\\ \hfill =a^2\left(\eta \right)exp\left(W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right[(1+2\mathsf{\Phi })d{\eta }^2-(1+2\Psi )d{\mathbf{x}}^2].\end{array}$$

(66)

At the same time, the well-known Mukhanov–Sasaki equation [32,33,34]

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{z}}{\displaystyle \frac{d^{2}z}{d{\eta }^2}}=2a^2{H}^2(1+ϵ-{\displaystyle \frac{3}{2}}\tilde{\eta }+{\displaystyle \frac{1}{2}}{\tilde{\eta }}^2-{\displaystyle \frac{1}{2}}ϵ\tilde{\eta }+{\displaystyle \frac{1}{2H}}{\displaystyle \frac{dϵ}{dt}}-{\displaystyle \frac{1}{2H}}{\displaystyle \frac{d\tilde{\eta }}{dt}}),\end{array}$$

(67)

holds for these qgcs.

This is inferred from the fact that formula (26) may be completed with two lines

$$\begin{array}{c}\hfill \dot{ϵ}\sim H;\\ \hfill \dot{\tilde{\eta }}\sim H.\end{array}$$

(68)

As usual, in (67) $z\equiv a\dot{\varphi }/H$ [34].

Then, for convenience to emphasize the use of quantum pbh with the mass M from the start, we introduce the following designations:

$$\begin{array}{c}\hfill a{\left(\eta \right)}_M\doteq a{\left(\eta \right)}_q,H_M\doteq H_q;V_M\doteq V_q,...\end{array}$$

(69)

Let us consider, as in [35], quantum fluctuations of a generic scalar field during the de Sitter stage (p.121 in [35]). Similar to section 2.3 of [35], we represent the scalar field ${\varphi }_0$ in the form $\chi (\tau ,\mathbf{x})$ as follows:

$$\begin{array}{c}\hfill \chi (\tau ,\mathbf{x})=\chi \left(\tau \right)+\delta \chi (\tau ,\mathbf{x}),\end{array}$$

(70)

where $\chi \left(\tau \right)$ is the homogeneous classical value of the scalar field, $\delta \chi$ are its fluctuations and, in line with [35], $\tau$ is the the conformal time instead of $\eta$.

Then, specifically for quantum fluctuations of a generic scalar field during the de Sitter stage (section 2.3 in [35]), the Klein–Gordon equation, which gives in an unperturbed FRW Universe (formula (40) in [35])

$${\chi }^{\prime \prime }+2\mathcal{H}{\chi }^{\prime }=-a^2{\displaystyle \frac{\partial V}{\partial \chi }},$$

(71)

due to Remark 6.1, remains unaltered with deformation (21)–(26),(69)

$$\{{\chi }^{\prime \prime }+2{\mathcal{H}}_M{\chi }^{\prime }=-a_M^2{\displaystyle \frac{\partial V_M}{\partial \chi }}\}\equiv \{{\chi }^{\prime \prime }+2\mathcal{H}{\chi }^{\prime }=-a^2{\displaystyle \frac{\partial V}{\partial \chi }}\}.$$

(72)

Here in the usual way we have $\mathcal{H}=a^{\prime }/a=a_M^{\prime }/a_M={\mathcal{H}}_M$.

But the above-mentioned deformation generated by qgcs for quantum pbhs not always retains the physical quantities arising in this consideration.

Because in this case such an important quantity as the power-spectrum acquires multiplicative increments. In particular, in the case of the de Sitter phase and of a very light scalar field $\chi$, with $m_{\chi }\ll 3/2H$ (formula (64) in [35]), the power-spectrum on superhorizon scales in this pattern is changed

$$\begin{array}{c}\hfill \{{\mathcal{P}}_{\delta \chi }\left(k\right)={\left({\displaystyle \frac{H}{2\pi }}\right)}^2{\left({\displaystyle \frac{k}{aH}}\right)}^{3-2{\nu }_{\chi }}\}\Rightarrow \{{\mathcal{P}}_{\delta \chi ,M}\left(k\right)={\left({\displaystyle \frac{H_M}{2\pi }}\right)}^2{\left({\displaystyle \frac{k}{a_M{H}_M}}\right)}^{3-2{\nu }_{\chi }}\}=\\ \hfill =exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right){\mathcal{P}}_{\delta \chi }\left(k\right).\end{array}$$

(73)

In the last formula we use the obvious equality to the condition $aH=a_M{H}_M$ and the fact that the quantity ${\nu }_{\chi }$ satisfies the condition (formula (57) in [35])

$$\begin{array}{c}\hfill {\displaystyle \frac{3}{2}}-{\nu }_{\chi }\simeq {\eta }_{\chi }=(m_{\chi }^2/3H^2).\end{array}$$

(74)

As for the effective mass $m_{\chi }$ of the scalar field $\chi$ we have the equality $m_{\chi }^2={\partial }^{2}V/{\chi }^2$, from formula (25) it follows that the exponent $3-2{\nu }_{\chi }$ in formula (73) is also retained with this deformation. Since the quantity $exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right)>1$, with due regard for qgcs in the de Sitter stage, magnitude of the power-spectrum is growing.

Similar results are true also for the case of quantum fluctuations of a generic scalar field in the quasi–de Sitter stage (Section 2.4 in [35]). We obtain an analog of the shift in formula (73)

$$\begin{array}{c}\hfill \{{\mathcal{P}}_{\delta \chi }\left(k\right)\simeq {\left({\displaystyle \frac{H}{2\pi }}\right)}^2{\left({\displaystyle \frac{k}{aH}}\right)}^{3-2{\nu }_{\chi }}\}\Rightarrow \{{\mathcal{P}}_{\delta \chi ,M}\left(k\right)\simeq {\left({\displaystyle \frac{H_M}{2\pi }}\right)}^2{\left({\displaystyle \frac{k}{a_M{H}_M}}\right)}^{3-2{\nu }_{\chi }}\simeq \\ \hfill \simeq exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right){\mathcal{P}}_{\delta \chi }\left(k\right)\}.\end{array}$$

(75)

Though, as distinct from 74), formula (72) in [35] is true

$$\begin{array}{c}\hfill {\displaystyle \frac{3}{2}}-{\nu }_{\chi }\simeq {\eta }_{\chi }-ϵ,\end{array}$$

(76)

where $ϵ$ is the deceleration parameter from formula (27) characterized by $ϵ={ϵ}_M\doteq {ϵ}_q$.From this it follows that formula (75) is valid, whereas the power-spectrum for quantum fluctuations of a generic scalar field in the quasi–de Sitter stage is growing similar to the de Sitter stage.

Remark 6.2

From Remark 2.3 it directly follows that right sides of formulae (73),(75)are series expanded in terms of the small parameter ${\alpha }_{r_M}$:

$$\begin{array}{c}\hfill {\mathcal{P}}_{\delta \chi ,M}\left(k\right)\simeq exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right){\mathcal{P}}_{\delta \chi }\left(k\right)=\\ \hfill =(1+{\omega }_{\mathbf{1}}{\alpha }_{r_M}+{\omega }_{\mathbf{2}}{\alpha }_{r_M}^2+{\omega }_{\mathbf{3}}{\alpha }_{r_M}^3+...)P_{\delta \chi }\left(k\right),\end{array}$$

(77)

where the first (leading) term in this series corresponds to the initial value of this power-spectrum (without the indicated qgcs), whereas the other terms – to the corrections of different orders for these qgcs, which are the greater the lower the size of the initial pbh, i.e. the lower the mass M (or same the radius $r_M$).

From formula (9) we can easily derive explicit values of the coefficients ${\omega }_i,i\ge 1$. Indeed, because

$$\begin{array}{c}\hfill (1+{\omega }_{\mathbf{1}}{\alpha }_{r_M}+{\omega }_{\mathbf{2}}{\alpha }_{r_M}^2+{\omega }_{\mathbf{3}}{\alpha }_{r_M}^3+...)={\left(1+{\displaystyle \frac{1}{2e}}{\alpha }_{r_M}+{\displaystyle \frac{5}{8e^2}}{\alpha }_{r_M}^2+{\displaystyle \frac{49}{48e^3}}{\alpha }_{r_M}^3+\dots \right)}^2,\end{array}$$

(78)

we have ${\omega }_{\mathbf{1}}=1/e,{\omega }_{\mathbf{2}}=3/2e^2,...$.

In the right side of (78) we have the squared series from the right side of formula (9) in terms of the small parameter ${\alpha }_{r_M}$.

It is clear that we can also express as a series in terms of a dimensionless small parameter ${\alpha }_{r_M}$ the right sides in other formulae too, specifically in (52)–(56),(64),(66), etc. In every case, a part of this series going after the leading term detects the deviation from the semi-classical approximation mode in the approach to quantum gravity within the scope of GUP (6).

6.3. Enhancement of Non-Gaussianity

It is understandable, when non-Gaussianities are involved in the process of inflation, the above-mentioned qgcs for quantum pbhs should change the parameters of these non-Gaussianities. It is required to know what are the changes. Deviation from the Gaussian distribution for the random field $g\left(\mathbf{x}\right)$ is a nonzero value of the three-point correlator ((6.30) in [34])

$$\begin{array}{c}\hfill \langle g_{{\mathbf{k}}_{\mathbf{1}}}{g}_{{\mathbf{k}}_{\mathbf{2}}}{g}_{{\mathbf{k}}_{\mathbf{3}}}\rangle ={\left(2\pi \right)}^3{\delta }_{{\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}+{\mathbf{k}}_{\mathbf{3}}}^3{\mathit{B}}_g(k_1,k_2,k_3),\end{array}$$

(79)

where $g_{{\mathbf{k}}_{\mathbf{i}}},i=1,2,3$ represent the Fourier transform $g\left(\mathbf{x}\right)$ in the momentum $k_i$ and the quantity ${\mathit{B}}_g$, named the bispectrum, is a measure of non-Gaussinity for the random field $g\left(\mathbf{x}\right)$.

We can show that, with due regard for the foregoing qgcs, the absolute value of the bispectrum ${\mathit{B}}_g$ in the inflation pattern for different random fields is growing.

Let us consider several examples:

6.3.1. Non-Gaussianity qgcs-Correction of Field Perturbations in Different Patterns

Non-Gaussianity correction from the self-interaction of the field

In this case the non-Gaussianity arises from the self-interaction of the field ([36] and 24.4.2 in [34]). Neglecting the metric perturbation at the initial stage and considering the effect of keeping the quadratic term in the perturbed field equation, we can obtain (formula (24.39) in [34]) the following:

$$\begin{array}{c}\hfill \ddot{\delta }\varphi +3H_{*}\dot{\delta }\varphi -a^{-2}{\nabla }^2\delta \varphi +{\displaystyle \frac{1}{2}}{V}_{*}^{\prime \prime \prime }{\left(\delta \varphi \right)}^2=0.\end{array}$$

(80)

Here the asterisk denotes a value of the corresponding quantity during inflation which is taken to be constant, and the dot, as usual, denotes $\partial /\partial t$.

Then the quantity ${\mathit{B}}_g$ in this case (formula (24.43) in [34]) takes the form

$$\begin{array}{c}\hfill {\mathit{B}}_{\delta \varphi }^{self}\sim H_{*}^2{V}_{*}^{\prime \prime \prime }.\end{array}$$

(81)

However, considering the qgcs under study, due to formulae (24),(26) and (11),(12), the quantities in the right side of the last formula are transformed as follows:

$$\begin{array}{c}\hfill H_{*}^2\to H_{*,M}^2=exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)H_{*}^2;\\ \hfill V_{*}^{\prime \prime \prime }\to V_{*,M}^{\prime \prime \prime }=exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right))V_{*}^{\prime \prime \prime }.\end{array}$$

(82)

So, considering these qgcs, we find that the bispectrum ${\mathit{B}}_{\delta \varphi ,M}^{self}$ in this case is

$$\begin{array}{c}\hfill {\mathit{B}}_{\delta \varphi ,M}^{self}=exp\left(-2W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right){\mathit{B}}_{\delta \varphi }^{self}.\end{array}$$

(83)

As $exp\left(-W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)>1$,from the last relation it follows that

$$\begin{array}{c}\hfill |{\mathit{B}}_{\delta \varphi ,M}^{self}|>|{\mathit{B}}_{\delta \varphi }^{self}|.\end{array}$$

(84)

This means that with regard to qgcs for quantum pbhs non-Gaussianity is enhanced.

According to Remark 6.2, $exp\left(-2W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right)$ can also be series expanded in terms of the dimensionless small parameter ${\alpha }_{r_M}$ by squaring of the left side in the formula (78) and by calculating the coefficients for the corresponding powers ${\alpha }_{r_M}$. As noted in Remark 6.2, nontrivial terms of this series generate qgcs to the semiclassical bispectrum ${\mathit{B}}_{\delta \varphi }^{self}$.

The Non-Gaussianity Correction of Field Perturbations from Gravitational interactions

In the above analysis of non-Gaussianity there was no metric perturbation because the gravitational interaction was”excluded”. When it is taken into consideration, the non-Gaussianity in this case arises from the metric perturbation and the bispectrum takes the following form ([37] and formula (24.46) in [34]):

$$\begin{array}{c}\hfill {\mathit{B}}_{\delta \varphi }^{grav}\sim -{\displaystyle \frac{1}{8}}{H}_{*}^4\left({\displaystyle \frac{V^{\prime }}{V}}\right).\end{array}$$

(85)

With regard to qgcs, from formula (24) we can derive

$$\begin{array}{c}\hfill {\mathit{B}}_{\delta \varphi }^{grav}\to {\mathit{B}}_{\delta \varphi ,M}^{grav}=exp\left(-2W\left(-{\displaystyle \frac{1}{e}}{\left({\displaystyle \frac{M_0}{M}}\right)}^2\right)\right){\mathit{B}}_{\delta \varphi }^{grav}.\end{array}$$

(86)

Note that (86) is valid only in the case when qgcs-correction of the metric is not taken into consideration or such correction is considered to be vanishingly small. In this case from formulae (83),(86) we can obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{{\mathit{B}}_{\delta \varphi ,M}^{self}}{{\mathit{B}}_{\delta \varphi }^{self}}}={\displaystyle \frac{{\mathit{B}}_{\delta \varphi ,M}^{grav}}{{\mathit{B}}_{\delta \varphi }^{grav}}}=exp\left(-2W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right).\end{array}$$

(87)

Based on the last formula, it can be concluded that

growth of non-Gaussianity of the field perturbations in inflation generated byqgcsfor a primordial black hole in the pre-inflation era is independent of the pattern (with or without regard for the metric perturbation) considered ifqgcs-correction of the metric may be neglected.

6.3.2. Non-Gaussianity qgcs-Correction for the Tensor Primordial Perturbations

As seen, the right side of (87) arises with regard to the qgcs-correction as a factor of the enhanced non-Gaussianity and the tensor primordial perturbations.

Actually, denoting in this case bispectrum as ${\mathit{B}}_{h_{ij}^{TT}}$, where by formula (5.43) in [28] we have

$$\begin{array}{c}\hfill h_{ij}^{TT}(\eta ,\mathbf{k})=\sum _{\begin{array}{c}A=+,\times \end{array}}{e}_{ij}^{\left(A\right)}{h}^{\left(A\right)}(\eta ,\mathbf{k}),\end{array}$$

(88)

in the left side the symmetric transverse traceless tensor $h_{ij}^{TT}$ ((2.42) in [28]) has helicity 2, whereas the right side represents expansion of this tensor by the sum over polarization.

According to formulae (1.1) in [38] and (24.63) in [34], we have

$$\begin{array}{c}\hfill {\mathit{B}}_{h_{ij}^{TT}}\sim {\displaystyle \frac{H_{*}^4}{m_p^4}}.\end{array}$$

(89)

Whence, in analogy with formula (87), it directly follows that

$$\begin{array}{c}\hfill {\displaystyle \frac{{\mathit{B}}_{h_{ij}^{TT},M}}{{\mathit{B}}_{h_{ij}^{TT}}}}=exp\left(-2W\left(-{\displaystyle \frac{1}{e}}{\alpha }_{r_M}\right)\right).\end{array}$$

(90)

From the start In this section it has been assumed that this quantum pbh in the pre-inflationary era is considered in the state 3.1. the stationary pattern. Still, we can obtain similar results for other processes proceeding before the onset of inflation, specifically for 3.3 black hole evaporation.

7. Conclusion and Further Steps

In this way it has been demonstrated that, within the scope of natural assumptions, the qgcs calculated for pbhs arising in the pre-inflationary epoch contribute significantly to the inflation parameters, enhancing non-Gaussianity in the case of cosmological perturbations. Besides, withy due regard for these qgcs, the probability of arising pbhs is higher.

Based on the results of this paper, the following steps may be planned to study the corrections of cosmological parameters and cosmological perturbations due to qgcs for pbhs in the pre-inflationary era:

7.1 Comparison of the results obtained in Section 6 with the experimental data accumulated by space observatories: (Planck Collaboration), (WMAP Collaboration) [39,40,41].

7.2 Elucidation of the fact, how closely the author’s results are related to general approaches to inclusion of the quantum-gravitational effects in studies of inflationary perturbations (for example, [42,43,44,45]);

7.3 Elucidation of the possibility to extend the obtained results to other types of quantum pbhs, in particular to pbh with the mass M, with the electric charge Q but without rotation in the Reissner-Nordström (RN) Metric (for the normalization $c=\hslash =G=1$) [8,13]

$$d{s}^2=\left(1+{\displaystyle \frac{2M}{r}}+{\displaystyle \frac{Q^2}{r^2}}\right)d{t}^2-{\left(1-{\displaystyle \frac{2M}{r}}+{\displaystyle \frac{Q^2}{r^2}}\right)}^{-1}d{r}^2-r^2{d}^2\Omega .$$

(91)

IN this case the event horizon radius $r_{\pm ,RN}$ and the temperature $T_{RN}$ of such a hole respectively take the following forms: [8,13]

$$r_{\pm ,RN}=M\pm \sqrt{M^2-Q^2}.$$

(92)

and

$$T_{RN}={\displaystyle \frac{{\left(M^2-Q^2\right)}^{1/2}}{2\pi {\left(M+{\left(M^2-Q^2\right)}^{1/2}\right)}^2}}.$$

(93)

Selecting in formula (92) for the radius $r_{RN}$ of such pbh the value $r_{RN}=r_{+}$ and assuming, similar to [46], that $Q/M\ll 1$, we can obtain that the (RN) metric (91) for small M represents, to a high accuracy, the Schwarzschild metric (2) and the Schwarzschild-de Sitter metric (3) with the corresponding formulae for the event horizon radius $r_{RN}$ and for the temperature $T_{RN}$, which are close to the corresponding Schwarzschid’s from Section 2. In this way for the quantum PBH with (RN) metric derivation of the results similar to those given in this work (inclusion of qgcs) is relevant, at least for the case $Q/M\ll 1$.