Gravitational radiation as the bremsstrahlung of superheavy particles in the early Universe

The number of superheavy particles with the mass of the Grand Unification scale with trans-Planckian energy created at the epoch of superheavy particle creation from vacuum by the gravitation of the expanding Universe is calculated. In later collisions of these particles gravitational radiation is radiated playing the role of bremsstrahlung for gravity. The effective background radiation of the Universe is evaluated.


Introduction
In paper published by one of the present authors (A.A.G.) together with S.G. Mamayev [1] finite results for particle creation in the early expanding Friedmann Universe were obtained. The important result was calculation of the finite density and finite total number of particles created in the Lagrange volume. Our results were obtained by using Fock quantization with vacuum defined as the ground state of the instantaneous Hamiltonian. These results have simple intuitive physical interpretation due to the work made on the virtual pair on the Compton length of the particle by the tidal forces due to gravity of the expanding Universe. It occurred that the number of particles depends on the mass of the particle and leads to observable numbers of visible and dark matter particles if this mass identified with the mass of the dark matter particles is equal to the number close to the Great Unification scale [2]. Possible explanation of the origin of ultra high energy cosmic rays due to the decay of such particles was made in [3]- [6].
So the main idea is that superheavy particles were created by gravitation, then some part of them decayed on visible particles at high energies but after the energy became smaller the decays were frozen and survived superheavy particles formed observed dark matter [3]. However mathematical calculation of particle creation makes possible calculation of the distribution function depending on the momenta i.e density not in coordinate but in momentum space. Surely finiteness of the particle density in coordinate space means that this function is going to zero at very high values of the momentum. This means that the larger the momentum the smaller will be the number of created particles.
In this paper we shall answer the question about the number of superheavy particles with Grand unification mass but with the energy close to the Planckian mass. How many trans-Planckian particles are created? Surely their number is much smaller than the general number of created particles but how much? For the case of inflationary models similar question was asked for same particles called wimpzillas [7]. Why this question and the answer on it are important? It is because in our recent paper [8] it was found that at trans-Planckean energies of colliding particles the gravitation wave radiation is produced. This radiation is much stronger than comparable electromagnetic radiation if it could exist. It plays the role of bremsstrahlung for superheavy particle and can lead to formation of structures for them as it is the case for electromagnetic radiation at small energies when Galaxies, stars etc. can be formed. In paper [9] it was also shown that collision of particles with trans-Planckean energies can lead to formation of mini black holes. The problem of the formation of primary galactic nuclei during phase transition in the early Universe was considered in [10].
In the book [11] it was mentioned that in Friedmann expanding Universe the number of created particles is proportional to the number of causally disconnected parts of the Universe at the Compton time of the existence of the Universe. However later these disconnected parts are united and collisions of particles and black holes occur. This makes possible the speculation of possible arising of the primordial black holes leading to active nuclei of Galaxies by this mechanism.

Creation of Superheavy Particles with Planckian Energy in the Early Universe
Consider creation of scalar and spinor particles in the early homogeneous isotropic Friedmann Universe with metric where dl 2 is the metric of an 3-dimensional space of constant curvature K = 0, ±1.
In theory of quantum effects in expanding curved space-time one usually takes the following equation (in the system of units in which Planck constant and light velocity are equal to one: = c = 1) for scalar field of mass m corresponding to the Lagrangian where g = det(g ik ) and R is the curvature scalar [12]. For ξ = ξ c ≡ 1/6 the scalar field is called conformal coupled. Then the equation (2) is conformally invariant in massless case [2]. However the nonconformal case is also important because 1) "gravitons" [13], 2) longitudinal components of vector bosons [2] are nonconformal. Minimal coupling ξ = 0 is popular in inflation theories [14].
We can find a complete set of solution for the equation (2) in following form where the prime denotes a derivative with respect to conformal time η, and indices J is numbering the eigenfunctions of Laplace-Beltrami operator ∆ in the space x with the metric dl 2 .
According to the Hamiltonian diagonalization method [2] the functions g λ (η) satisfy initial conditions: The number of pairs of scalar particles created up to the moment t in Lagrangian volume a 3 (t) of homogeneous isotropic Universe with flat space sections where Function s λ (η) defines the distribution in dimensionless momentum λ particles created up to the moment η. The "physical" momentum is λ/a. For ultrarelativistic particles λ ≫ ma.
To find the number of created ultrarelativistic particles one must find the asymptotic of the function s λ (η) if λ → ∞.
Using (5), (8) one can see that function s λ (t) satisfies integral Volterra equation where To find the asymptotic s λ (η) if λ → ∞ one can confine oneself to the first iteration of the integral equation (11) and to take into account that Θ(η 2 , η 1 ) → λ(η 1 − η 2 ) if λ → ∞. Then (11) has the form Integrating by parts the integral (13) So For λ → ∞ one has that w ∼ λ −2 , so s λ ∼ λ −6 and the integral in (9) is convergent. So in the method of Hamiltonian diagonalization the number of created by gravitation scalar particles is finite as for conformal as for nonconformal coupling with curvature. Evaluate the number of created particles with the energy E ≥ E b , where E b ≫ mc 2 . The cyclic frequency Ω for the scalar field with conformal coupling with curvature is In this case Let us consider the situation when a ′ a is increasing in time and a ′ (η 0 )a(η 0 ) ≪ a ′ (η)a(η). Then Find what limitations on background matter of the Universe arise due to this condition. Einstein's equations for homogeneous isotropic space in metric (1) are where ε and p are the energy density and pressure for background matter. From (21), (22) one has For realistic models of the Universe in the epoch when the effects of particle creation are important (the Compton time of the particle [2]) one can neglect the space curvature and consider |a ′ /a| ≫ 1. So and a ′ a is increasing if p < ε. Important case of dust p = 0, a = a 1 η = a 0 t 2/3 and radiation dominated Universes p = ε/3, a = a 1 η = a 0 √ t are in this family. Here t is the coordinate time dt = adη. The limiting case with scale factor a = a 1 √ η = a 0 t 1/3 corresponds to the most rigid state equation p = ε. Using (19) one obtains that the number of particle pairs with momenta larger then some λ b , created in volume a 3 (t) of homogeneous isotropic Universe is equal to Going to usual unit system is made by where the dot above symbols is the derivative with respect to time t. For ultrarelativistic particles with E b = λ b c/a and so (25) in usual units is For the scale factor a ∼ t α one obtains For Planckian energy E b = E P l ≡ c 5 /G, where G is the gravitational constant, and the Compton time t = t C ≡ /(mc 2 ) one obtains where l C = /(mc) is the Compton length of the particle, l P l = G/c 3 is the Planckian length. For radiation dominated case (α = 1/2) one obtains Note that this result is valid not only for conformal but also for nonconformal scalar particles because for the radiation dominated case R = 0 and formula (16) for the frequency (6) is correct for any value ξ of the parameter of connection of the scalar field with curvature.
For observable Friedmann radiation dominated Universe one obtains a(t P l ) ≈ 10 −5 m so that for the scale of Grand Unification m = 10 15 GeV one has 10 67 particle with the energy of the Planckian order. The general number of all superheavy particles created in the early Universe is close to the Eddigton number 10 80 [2]. So the number of trans-Planckian scalar particles is relatively small. Now consider the case of creation fermion particles. In this case the number of created pairs is [2] but the expression for s (1/2) λ does not coincide with the formula (10). However the first iteration in integral equation for function s λ (η) coincides with formular (13) if w for the spinor field is For large values λ The number of pairs of spinor particles with momenta larger then some λ b , created in volume a 3 (t) of homogeneous isotropic universe is equal to in usual units For the scale factor a ∼ t α one obtains that the number of the fermion created pairs is For Planckian energy E b = E P l and the Compton time one obtains in particular, for radiation dominated case (α = 1/2) one obtains For observable Friedmann radiation dominated Universe one obtains a(t P l ) ≈ 10 −5 m so that for the scale of Grand Unification m = 10 15 GeV one has 10 75 fermion particle with the energy of the Planckian order.

Gravitational Radiation in Collisions of Trans-Planckian Superheavy Particles
In our paper [8] there was obtained the following result for the gravitational radiation energy in two-particle collision with the energy E c.m. < E P l in the center of mass system where M P l = c/G = 2.18·10 −8 kg is Planck mass, m 1 and m 2 are the masses of colliding particles. Note that even for such light particles as electrons one has ln(M P l /m) < 52. So one can see from (39) that if E c.m. ≪ E P l , then gravitational radiation is small However for E c.m. ≥ E P l the result is different. One has This means that the role of gravitational radiation becomes large and it can play the role of bremsstrahlung in electromagnetic radiation. In [8] it was shown that if colliding particles have nonzero electric charges then bremsstrahlung due to electromagnetism is much smaller then the gravitational one. Let us evaluate the order of the energy of this gravitational radiation Evaluate the gravitational background radiation of these particles supposing that the energy of gravitation is radiated in collisions at Compton epoch t ≈ /mc 2 of the expanding Universe. Taking into account the decrease of the energy of gravitational waves due to the expansion of the Universe one obtains in the modern epoch t mod ≈ 10 18 s the value of the energy as 10 67 E P l t C /t mod ≈ 10 48 J for scalar particles and ≈ 10 56 J for fermion particles. So the density of this energy of radiation ≈ 10 −30 J/m 3 for case of scalar particles or ≈ 10 −22 J/m 3 for case of fermion particles is much less then the energy of electromagnetic background radiation equal to ≈ 4 · 10 −14 J/m 3 .