Hot primordial regions with anomalous hydrogenless chemical composition

We study primordial nucleosynthesis in hypothetical hot regions that could be formed by the primordial density inhomogeneities. It is shown that the regions survived up to the present times acquire an abnormally high metallicity. This conclusion holds in wide range of initial parameters of such regions. We considered the thermonuclear reaction rates and estimated abundances of deuterium and helium-3 and -4 inside these areas. It has been established that all baryons tend to form helium-4, which is the thermonuclear link in the chain of formation of heavier elements.


Introduction
We suppose that stable hot regions can be formed in the early Universe. This hypothesis was put forward on the basis of the cosmic X-ray observations and IR background [1]. The cluster of primordial black holes (PBH) can be responsible for such regions. Formation of PBH clusters and their possible observational effects are now of special interest [2][3][4][5][6][7][8][9][10][11][12][13], but we do not constrain possibility of such regions appearance by PBH clusters only. PBH and their cluster formation can be the consequence of existence and breaking of some new symmetry in quantum field theory [2,14,15].
PBH cluster can be the seed of a quasar or galaxy formation [16][17][18]. Here we consider the matter trapped in this region, which can be protogalaxy or exist separately. So the prerequisites for the task in question are the regions decoupled from Hubble flow (and virialized) containing primordial plasma. Plasma must flow out to the surroundings by diffusion in the CMB field. If the region is big enough, it can survive to the present time, as it was obtained for the antimatter domain [19,20]. The region of the size ∼ 1 pc spreads over ambient matter after recombination (z ≈ 1000), when the structure forms already (see, e.g., Eq.(12) from [19]). By this time, heavy chemical elements, as we show, have time to be formed in a wide range of considered model parameters, therefore areas contaminated with heavy elements can be expected to exist even if the matter had spread outside primordial region. Moreover, cooling by conventional thermal (gammaray) radiation is ineffective for big regions. The escaping time of photons from the region interior (thermal time scale) at the taken parameters (given below) exceeds the modern age of the Universe. The matter inside the area can be additionally heated with respect to the surrounding one during its formation due to domain wall kinetic energy in the case of the respective mechanism of PBH cluster formation [2][3][4] including Higgs field [21,22].
We consider the chemical composition of such possible hot regions, whatever their origin is. The thermal evolution of such regions involves many factors. The matter inside areas can be heated or cooled by various processes acting at the same time. These processes include the neutrino cooling [23,24], inelastic reactions between elementary particles and nuclei, the radiation from star-forming hot plasma [1], the gravitational dynamics of the system, the shock waves and diffusion of matter during the region formation [25,26], energy transfer from collapsing walls mentioned above [14-16, 27, 28], accretion [1,26] and the Hawking evaporation [26,29,30]. We focus here on the pure effect of inelastic reactions between elementary particles and nuclei. They may play a dominant role within a wide range of the region parameters which are specified below. We have shown earlier [23,24], that neutrino emission can be decisive in the temperature evolution of such regions at the first stage. Here we extend consideration by involving reactions with the lightest element formation.
We use the results obtained in [16-18, 23, 31], where the mass of the detached region was supposed to range 10 4 − 10 8 M 1 . The following are the most important starting parameters: the area has a radius of R ∼ 1 pc, a mass of 10 4 M , and an initial temperature The goal of this work is to investigate certain reaction networks, which define the evolution of temperature and chemical composition of the regions in the early Universe. Light element abundance ratios (n d /n B , n3 He /n B and n4 He /n B ) are finally obtained, heavier element production is discussed.
Hypothesis on existence of the regions discussed can be supported by the evidences of cosmic infrared and X-ray background correlations [1], anomalous star existence [33,34], and can be probed in direct searches for large areas with abnormal chemical composition in future. Also, such sources of high temperature radiation at the pre-recombination stage can give specific observed patterns of CMB temperature variations (∆T/T) [35] because it is determined by the interaction of these fluctuations in matter density with the CMB during the Universe's expansion and cooling, which are not applicable to small scales. Section 2 is dedicated to the discussion on the main nuclear reactions. Subsection 2.1 contains the information about the region temperature, subsection 2.2 -about proton and neutron abundances, 2.3 -deuterium and helium-3, 2.4 -about abundances of helium-4 and heavier elements. A closing overview of the research is provided in section 3. We also include some useful information on reaction rates and cross-sections in appendix A.

Nucleosynthesis
Consider the reaction between two nuclei 1 and 2. The reaction rate is proportional to the mean lifetime τ of the nuclear species in the stellar plasma. The number density change rate of nucleus 1 caused by reactions with nucleus 2 can be expressed as [36,37] Here r 12 is the rate of interaction, δ 12 is the Kronecker symbol equals one if 1 = 2 and zero if 1 = 2, n 1 and n 2 are the number densities of nuclei of type 1 and type 2 (having the atomic numbers Z 1 and Z 2 , as well as the mass numbers A 1 and A 2 ), and σv 12 represents the product of the reaction cross section and the interacting nuclei's relative velocity v. The case of identical initial nuclei is taken into account by the presence of the Kronecker symbol.
We will look at how the neutrons, protons, 2 H, 3 He and 4 He abundances change over time due to the reactions of mostly proton-proton chain. The n + p and p + p reaction produces 2 H, which is then destroyed by the d + p and d + γ reactions, whereas the d + p reaction produces 3 He, which is then destroyed by the 3 He + 3 He reaction producing 4 He. We consider neutrinos to be able to leave the region freely and therefore cool it down. The essential reactions of light elements and neutrinos produced are the following: We neglected energy releases of less than 1 MeV. The initial number densities are approximately described as Here η = n B /n γ ≈ 0.6 × 10 −9 is the baryon to photon relation in the modern universe, g B ∼ 1 is the correction factors of that relation due to entropy re-distribution, n γ (T ) = 2ζ(3) π 2 T 3 and n eq e (T ) = 3ζ(3) 2π 2 T 3 are the equilibrium photon and electron number densities respectively, ∆m = m n −m p = 1.2 MeV. The forms of Equations (11) and (12) for number densities are chosen to fit their asymptotics in the case of thermodynamic equilibrium.
We consider all densities to be independent on space coordinates within the region. The equations (12) are also used to calculate electron and positron current number densities with T instead of T 0 and total electric charge instead of n p inside of ∆n e .
The rates per unit volume, γ i ≡ Γ i /V , for reactions listed above are respectively γ ep = n e − n p σv ep , γ en = n e + n n σv en , γ ee = n e − n e + σv ee , γ n = n n τ n , γ np = n n n p σv np , γ dp = n d n p σv dp , γ3 Here n i is the concentration of the respective species, σv ij is the reaction rate of interacting particles i and j, v is their relative velocity, for reactions (2) -(4) v 1 and τ n ≈ 1000 s is the neutron lifetime. The electron-electron, electron-proton and electronneutron cross section are given by Eqs. (33) and (34) of Appendix.
The temperature balance is defined by the first law of thermodynamics where ∆Q and δU are the heat and inner energy gains (in fact, a decrease) of the matter inside the heated area, respectively. Expanding all the values one obtains where Q i is energy release of the respective reaction, E ν ∼ T is the energy of outgoing neutrino, b = π 2 /15 is the radiation constant. Using Eq. (1) and (20) and reactions (4) -(10), we can compose the following system of differential equations.
d(n n ) dt = n e − n p σv e − p + n γ n d σv γd − n n τ n − n n n p σv np − n e + n n σv e + n (21) d(n p ) dt = n e + n n σv e + n + n n τ n + n γ n d σv γd + (n3 He ) 2 σv 3 He 3 He − n e − n p σv e − p − n 2 p σv pp − n d n p σv dp (22) d(n d ) dt = n 2 p 2 σv pp + n n n p σv np − n d n p σv dp − n γ n d σv γd (23) d(n3 He ) dt = n d n p σv dp − (n3 He ) 2 σv 3 He 3 He (24) The initial number densities of deuterium and helium are considered to be zero inside the region.
As can be seen from the equations, we do not consider any reactions of heavy elements production for the sake of simplicity. Evidently, some parts of 4 He will be transformed into heavier elements subsequently, so our estimations of its number density effectively show the number density of 4 He together with all heavier elements.

Temperature evolution
The temperature evolution (Eq.(26)) follows from the equation system above. It is dominated by the cooling due to the reaction (4). Figure 1 shows the time dependence of Figure 1: The time behaviour of the temperature inside the heated area.
the temperature for different initial temperatures T 0 .

Abundances of free protons and neutrons
We can estimate the abundance of (free) neutrons and protons numerically using Eqs. (21) and (22). Figure 2 shows the evolution of the number densities, while Figure  3 shows the fraction of protons (left) or neutrons (right) from the initial baryon number density.  One can explain qualitatively these figures. There are five processes considered to affect the neutron number density. However, while the production (Eq. (8)) and destruction (Eq. (7)) of the deuterium are generally the two most active of them, their reaction rates have almost negligible difference in the most cases. Therefore, the neutron abundance is defined by neutron decays (Eq. (5)) with the combined effect of electron-proton and positron-neutron reactions (Eqs. (2) and (3)). At the higher initial temperatures the latter starts as dominant, slowly decreasing its effect with the fall of the temperature, until it reaches the level of the neutron decays somewhere below 1 MeV. After that, the combination of all three of these reactions causes the slow and gradual fall of neutron abundance. At low initial temperature neutron decays start as dominant process, causing the exponential drop at around 10 3 seconds, until the decay rate matches the one of the e-p and e-n combination. After that, the neutron abundance remains stable for a long time until the temperature starts having noticeable changes, affecting the reaction rates and causing the neutron number density to have a slow and gradual fall, as in the case of high initial temperatures.
The rise in proton number density is caused by neutron decays (slowed down due to the processes described above). This effect is more visible at the high initial temperatures, as the neutrons constitute a higher part of the baryons there. The consequent fall in the proton number density is caused by the irreversible transition of the baryons to the 3 He and 4 He. The abundance of helium-3 is growing most of the time, as the rate of its production is greater than the rate of its destruction into 4 He. For the high initial temperatures, at late time, this situation reverses due to the decrease in temperature and number densities of protons and, subsequently, deuterium, and 3 He starts falling.

Abundances of deuterium and helium-3
The Figure 4 shows that deuterium and helium-3 have very low abundances, making them very likely undetectable. Nonetheless, they play a significant role in the synthesis of heavier elements due to their high reaction rates.

Abundance of helium-4 with heavier elements
We can estimate the helium abundance (n4 He /n B ) numerically from Eq. (25). Figure  5 shows the obtained results. As already stated above, 4 He abundance here effectively stands for not only helium-4 itself, but also for heavier elements. While our assumptions do not allow us to estimate the metallicity of such a region, we can still make an interesting conclusion that for most of the considered initial temperatures, the dominant part of baryons will be transformed into helium-4 and subsequent elements, leaving the region with almost no hydrogen.

Conclusion
We considered the possible existence of stable hot areas formed in the early Universe. Their origin could be related to the formation of PBH clusters. There are many factors that affect the evolution of such regions, we focus here on the nuclear reactions inside them. The neutrinos produced in these processes carry away energy, what is found to play a decisive role in temperature change under our approximation (considering nuclear reactions only with the given density and reaction rate dependencies). The considered nuclear reactions tend to form heavy elements, depleting the hydrogen content. The absence of hydrogen in such areas may be a distinguishing feature for their possible search. It will be possible to relate the observed chemical composition to its initial temperature and can account for the existence of anomalous stars.

A Reaction rates and cross-sections
Here we calculate the thermonuclear reaction rates. Maxwell-Boltzmann distributions are assumed for interacting nuclei at thermodynamic equilibrium, therefore it follows that the relative velocities between the two species of nuclei will also be Maxwellian in nature [37]. We may write for the Maxwell-Boltzmann distribution where m 12 = m 1 m 2 /(m 1 + m 2 ) is the reduced mass (Boltzmann constant is assumed to be 1). With E = m 12 v 2 /2 and dE/dv = m 12 v, we may write the velocity distribution as an energy distribution, For the reaction rate we obtain [38] σv 12 = ∞ 0 v σ(v) P (v)dv = 8 πm 12 The rate of the reaction is significantly dependent on the cross section σ, which varies for each nuclear reaction.
The reaction rates can be calculated using either numerical integration or analytical formulas used in this section. At this stage, we define the astrophysical S-factor [39], S(E), as Remember that the Gamow factor e −2πη is just a rough approximation for the s-wave transmission probability for energies considerably below the Coulomb barrier height. We write for the reaction rate using the S-factor definition. n 1 n 2 σv 12 = 8 πm 12 where Z i is the charges of target and projectile. The energy dependency of the integrand is remarkable. The term e −E/T , derived from the Maxwell Boltzmann distribution, approaches zero for high energy, whereas the term e −1/ √ E , derived from the Gamow factor, approaches zero for low energies. The most significant contribution to the integral will come from energies where the product of both terms is near its maximum.
Correction is required here since the S-factor for many reactions is not constant but changes with energy. In most situations, just expanding the experimental or theoretical S-factor into a Taylor series around E = 0 is acceptable [40,41].
where the primes are derivatives with regard to E. Substituting this expansion into Eq.(31) results in a sum of integrals, each of which may be extended into powers of 1/τ (τ ≡ E 0 /(T )). Table 1 shows the values of the astrophysical S-factor, S(E), for three reactions in proton-proton chains (ppI chain), that will be investigated in the next sections of this paper. For reaction with participation of e ± the following approximate formulas are used σ en = σ ee = σ w , σ ep = σ w exp − Q T , σ w ∼ G 2 F T 2 , Q = m n − (m e + m p ) = 0.77 MeV.
They effectively take into account the threshold effect in respective reaction, G F = 1.166 × 10 −5 GeV −2 is the Fermi constant. Such an estimation of cross section has accuracy factor 3 of in the range where it is relevant. The reaction rates of the light elements are calculated by Eq.(31). Figure 6 shows reaction rates of p(n,γ)d, d(p,γ) 3 He and p(p,e + ν)d with temperature, which calculated by Eq.(31).