# Gravitational Particle Production and the Hubble Tension
## 5 Impact of Gravitational Particle Production on the Hubble Tension  
The effective increase of \(G\) in Equation (48) causes an increase in the overall value of the Hubble parameter, and thus an increase in the Hubble constant, namely,  
\[H_{0}^{2}\,=\,\left[\frac{1}{1\,-\,1.8\,\times\,10^{-58}\,\times\,\sum_{i} \left(\frac{m_{i}c^{2}}{eV}\right)^{2}}\right]\,\bar{H}_{0}^{2}\;. \tag{49}\]  
where the subscript \(0\) stands for the present time, and \(\bar{H}_{0}=\sqrt{\frac{8\pi G}{3}\,\rho_{0}}\) is the value of the Hubble constant without the effect of gravitational particle production included, while \(H_{0}\) is the value of the Hubble constant after inclusion of the effect of gravitational particle production. Note that, by Equation (48), \(H_{0}\) is the value of the Hubble constant determined in direct measurements. The Hubble constant may also be determined from the imprints of baryon acoustic oscillations on CMB or large-scale structure anisotropies by measuring the angle \(\theta\) subtended by sound horizon  
\[\theta\,=\,\frac{r_{s}}{D_{A}} \tag{50}\]  
where \(r_{s}\) is the comoving size of the sound horizon, \(D_{A}\) is the comoving angular diameter distance to the observed position. Here [6,8]  
\[r_{s}\,=\,\int_{z_{a}}^{\infty}\,\frac{v_{s}(z)\,dz}{H_{0}\,E(z)}\,\quad D_{A}\,=\,c \int_{0}^{z_{b}}\,\frac{dz}{H_{0}\,E(z)} \tag{51}\]  
where \(z\) denotes redshift; \(c\) is the speed of light; \(v_{s}(z)\) is the speed of the sound waves in baryon-photon fluid; \(a\,=\,^{*}\) or \(d\) stand for recombination or drag epoch (for the imprint of the acoustic oscillations on CMB radiation or on galaxy autocorrelation function, repectively); \(b\,=\,^{*}\) or \(\,obs\) denote the redshifts of recombination or of the observed galaxies; \(E(z)\,=\,\sqrt{\Omega_{\Lambda}\,+\,\Omega_{M}\,(1+z)^{3}\,+\,\Omega_{R}\,(1+ z)^{4}}\) in \(\Lambda\)CDM with \(\Omega_{\Lambda}\), \(\Omega_{M}\) and \(\Omega_{R}\) being the density parameters for cosmological constant, dust and radiation, respectively. Let us assume (unlike the early or late time solutions of the Hubble tension) that the evolution of the universe before and after the recombination are described by the (unmodified) standard model (i.e., \(\Lambda\)CDM). (In fact, we have expressed Equation (51) in a form that is more suitable for this case.) One observes that \(\theta\) in Equation (50) is unaffected by the values of \(H_{0}\) in the arguments of \(r_{s}\) and \(D_{A}\). However, the value of the Hubble constant affects \(r_{s}\) and \(D_{A}\) by its effect on \(z_{a}\) by affecting recombination, as we will see below. The effects of a change in \(z_{a}\) on \(r_{s}\) and \(D_{A}\) are not the same since the value of \(r_{s}\) is dominated by the value of \(E(z)\) at values of \(z\) close to \(z_{a}\), while the value of \(D_{A}\) is dominated by the value of \(E(z)\) at values of \(z\) close to \(z=0\). Hence, a variation in the Hubble constant varies \(\theta\) by its effect on \(z_{a}\). Thus, the observational value of the Hubble constant may be determined after finding the best fit values for the Hubble constant and the density parameters corresponding to the observed \(\theta\). Below, we will see that the Hubble constant determined in this way is its value without the contribution of gravitational particle production, i.e., \(\bar{H_{0}}\) (while the value of the Hubble constant that is determined in direct measurements is \(H_{0}\)). First, we will present the argument in the context of the Saha equation to see the situation in an easier way at a conceptual level. Then, we will reconsider the situation at the level of the corresponding Boltzmann equation to obtain essentially the same result in more concrete terms in a more rigorous way.