# Gravitational Particle Production and the Hubble Tension
## 5 Impact of Gravitational Particle Production on the Hubble Tension
The general aspects of recombination may be studied with the Saha equation [35]  
\[X\,(1\,+\,S\,X)\,=\,1 \tag{52}\]  
where \(X\,=\,\frac{n_{p}}{n_{p}+n_{1s}}\,=\,\frac{n_{e}}{n_{p}+n_{1s}}\) is the fraction of protons or electrons to the total number of baryons (i.e., protons plus neutral hydrogen atoms), and  
\[S\,=\,0.76\,n_{b}\left(\frac{m_{e}k_{B}\,T}{2\pi\hbar^{2}}\right)^{-\frac{3}{ 2}}\,\exp\frac{B_{1}}{k_{B}T}. \tag{53}\]  
Here \(n_{b}\), \(m_{e}\), \(k_{B}\) and \(B_{1}\) are the number density of baryons (at temperature T), electron mass, Boltzmann constant and the binding energy of hydrogen atom in its ground state, respectively. The decoupling of photons from baryons took place at a sufficiently small value of \(X\), say at \(X_{*}\,\ll\,1\). It is evident from Equation (52) that the value of \(X\) is determined by the value of \(S\) which is related to \(n_{b}\) by Equation (53). \(n_{b}\) is related to the number density at the present time \(n_{b0}\) by \(n_{b}\,=\,n_{b0}\left(\frac{T}{T_{b0}}\right)^{3}\) where \(T_{\gamma 0}\simeq 2.73\) K is the present day temperature of CMB. \(n_{b0}\) is calculated by using  
\[n_{b0}\,=\,\frac{3\Omega_{b}\,H_{0}^{2}}{8\pi\,G^{(effective)}\,m_{N}}\,=\,\frac {3\Omega_{b}\,H_{0}^{2}}{8\pi\,G\,m_{N}}\,=\,1.121\,\times\,10^{-5}\,\Omega_{b }\,\hbar^{2}\,\,\mathrm{nucleons/cm^{3}} \tag{54}\]where \(\Omega_{b}\) is the density parameter for baryons. Here, essentially Equation (48) is used where \(G\) in [35] is replaced by its effective value \(G^{(effective)}\), and \(H_{0}\) is identified as the effective value of the Hubble constant in the Friedmann equation (i.e., in Equation (48)) that includes the contribution due to gravitational particle production. \(\bar{H_{0}}\) is the value of the Hubble constant before inclusion of the effect of gravitational particle production, and \(\bar{h}\,=\,\frac{\bar{H_{0}}}{100\,\,\mathrm{km\,s^{-1}\,Mpc^{-1}}}\). It is evident from Equation (54) that the parameter that determines the evolution of the photon-baryon plasma before decoupling is \(\bar{h}\) rather than \(h\). Although the Saha equation is enough to give the basic elements of the evolution the photon-baryon plasma, it has some important shortcomings. The first shortcoming is that it does not specify the exact value of \(z_{*}\). The second is that the Saha equation is derived by assuming the chemical equilibrium in the scattering \(e^{-}+p\leftrightarrow H+\gamma\) (where H denotes hydrogen atom), while the chemical equilibrium is not applicable at the time of decoupling. Finally, the Saha equation describes the evolution of the background, while CMB anisotropies and BAO calculations are at the level of cosmological perturbations. These shortcomings may be removed by using the Boltzmann equation corresponding to this case. The photon-baryon system at the time of recombination has kinetic equilibrium (while not necessarily chemical equilibrium) and the electrons are non-relativistic. The corresponding Boltzmann equation is [36]  
\[\frac{dX}{dt}\,=\,\left[<\sigma\,v>\,\left(\frac{m_{e}k_{B}\,T}{2\pi\hbar^{2}} \right)^{\frac{3}{2}}(1\,-\,X)\exp\left\{-\left(m_{e}+m_{p}-m_{H}\right)c^{2} /(k_{B}T)\right\}-\,<\sigma\,v>\,n_{b}\,X^{2}\right] \tag{55}\]  
where \(n_{e}<\sigma\,v>\) is thermally averaged rate for the decrease of electrons in \(e^{-}+p\leftrightarrow H+\gamma\). Note that \(n_{b}\) in Equation (55) is related to \(n_{b0}\) in Equation (54) (that depends on \(\bar{h}\) rather than \(h\)). Equation (55) may be integrated numerically to have a detailed evolution of \(X\), and \(z_{*}\) (for given values of \(\bar{h}\) and the density parameters). \(z_{*}\) may be determined by finding the value of \(z\) where there is a sharp decrease in \(X\), i.e., by finding \(X_{*}\,\ll\,1\) where \(X\) drops sharply. Hence, the best fit values of \(\bar{h}\) and the density parameters may be determined by using Boltzmann codes such as CAMB [4]. In fact, this is how the Hubble constant is determined in CMB and BAO calculations.