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 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 $G=l_p^2$.

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] 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 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]): 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.

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] may be replaced by the Generalized Uncertainty Principle (GUP) [19] 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 $r_{min}$: 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 Within the scope of GUP (6),the temperature $T_H$ with regard to (qgc) is of the form ((23) and (25) in [19])) 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]) $W\left(u\right)$ 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 $A_0/A$): where $r_h=l_{min}$ is the horizon radius of minimal pbh from formula (7).

It should be noted that this deformation parameter 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 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 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: 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.

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 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.

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

To this end in cosmology, in particular inflationary, the metric (3) is conveniently described in terms of the conformal time $\eta$ [11]: where $\mu ={(GM{H}_0/2)}^{1/3}$, $H_0$ – de Sitter-Hubble parameter and scale factor, a – conformal time function $\eta$: 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 where $r_M$ is the radius of a black hole with the SdS Schwarzschild-de Sitter metric (3).

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.

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: 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 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] 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: 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 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] which, according to qgc in the formula of (15), may be replaced by 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]) and with due regard for qgc 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 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.

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 we can obtain the Quantum Deformation (QD) [21] of the Friedmann Equation due to qgcs for pbh in the early Universe or 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]) within the foregoing (QD) are replaced by or It should be noted that the equation of the covariant energy conservation for the homogeneous background ((2.6) in [28]) 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 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 Or, equivalently, 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 or It is clear that, in this most general pattern, the covariant energy conservation for the homogeneous background ((2.6) in [28]) remains unaltered with replacement of $\rho \mapsto {\rho }_q,p\mapsto p_q$.