# Gravitational Particle Production and the Hubble Tension
## 5 Impact of Gravitational Particle Production on the Hubble Tension
One may obtain further insight into the problem by analytic formulas that express \(z_{*}\) and \(z_{d}\) in terms of \(\bar{h}^{2}\Omega_{M}\) and \(\bar{h}^{2}\Omega_{b}\)[37]  
\[z_{*}\,=\,1048\left[1+0.00124\left(\Omega_{b}\bar{h}^{2}\right)^{-0.738}\right] \left[1+g_{1}\Big{(}\Omega_{M}\bar{h}^{2}\Big{)}^{S2}\right] \tag{56}\]  
\[z_{d}\,=\,1315\,\frac{\left(\Omega_{M}\bar{h}^{2}\right)^{0.251}}{1\,+\,0.659 \left(\Omega_{M}\bar{h}^{2}\right)^{0.828}}\left[1+b_{1}\Big{(}\Omega_{b}\bar{ h}^{2}\Big{)}^{b_{2}}\right] \tag{57}\]  
where \(h\) in [37] is replaced by \(\bar{h}\) (since the dependence of Equation (55) on the Hubble constant is through \(n_{b}\) which is unaffected by gravitational production of \(\phi\) s). Here, \(g_{1}\), \(g_{2}\) are some functions of \(\Omega_{b}\bar{h}^{2}\) and \(g_{1}\), \(g_{2}\) are some functions of \(\Omega_{M}\bar{h}^{2}\) whose explicit forms may be found in [37]. The effect of \(n_{b}\) on \(z_{*}\) and \(z_{d}\) (through its dependence on \(\Omega_{b}\)) is evident in Equations (56) and (57). Note that Equations (56) and (57) are functions of \(\Omega_{M}\bar{h}^{2}\) and \(\Omega_{b}\bar{h}^{2}\) rather than being functions of \(\Omega_{M}\), \(\Omega_{b}\), \(\bar{h}\). \(D_{A}\) in Equation (51) may be expressed in terms of \(\Omega_{M}\bar{h}^{2}\) and \(\Omega_{A}\bar{h}^{2}\) (where the contribution of radiation may be neglected since the value of \(D_{A}\) is dominated by low redshift contributions) and \(r_{s}\) in Equation (51) may be expressed in terms of \(\Omega_{M}\bar{h}^{2}\) and \(\Omega_{r}\bar{h}^{2}\) (where the contribution of the cosmological constant may be neglected since the value of \(r_{s}\) is dominated by the redshifts close to \(z_{*}\)). As we remarked in the discussion after Equation (51), although the Hubble constants in \(H(z)\) of \(D_{A}\) and \(r_{s}\) cancel in Equation (50), \(z_{*}\) remains dependent on \(\Omega_{M}\bar{h}^{2}\). Therefore, we may express \(D_{A}\) and \(r_{s}\) in terms of the density parameters times \(\bar{h}^{2}\). This implies that what we obtain through data fit for \(\theta\) are \(\Omega_{M}\bar{h}^{2}\), \(\Omega_{b}\bar{h}^{2}\). Therefore, by observing \(\theta\) one cannot obtain the value of \(\bar{h}\) separately. However, one may use a phenomenological rule observed by [38], namely, in a spatially flat universe \(\Omega_{M}\bar{h}^{p}\) (where \(p=3.4\) in the original paper, while \(p\) is found to be 3 by [PERSON]) may be determined from the positions of the acoustic peaks (while \(\Omega_{M}\tilde{h}^{2}\) may be directly determined from data analysis for best fits). This information may be used to determine \(\Omega_{M}\), \(\tilde{h}\) (and \(\Omega_{b}\)) separately [4; 38]. To summarize, \(\tilde{H_{0}}\) is the value obtained by [PERSON] [4] (for the [PERSON] dataset) and does not contain a contribution from gravitational particle production (GPP), while \(H_{0}\) is the directly measured value of the Hubble constant that has contributions from GPP. The difference between \(H_{0}\) and \(H_{0}\) may be wholly attributed to GPP if the value of \(\sum_{i}\left(\frac{m_{i}z^{2}}{e\sqrt{e}}\right)\) is taken accordingly. In any case, GPP ameliorates the Hubble tension. It should be remarked that no new physics is employed in the present study. The standard \(\Lambda\)CDM model (without any extension) is employed here. The only difference between this study and the other studies in the past that employed the \(\Lambda\)CDM model is the inclusion of GPP that is neglected in the other studies. What has been done here is not the of introduction of a new model. What has been done here is to give an explanation for observing two different values of the Hubble constant in direct and indirect measurements. It has been shown here that GPP modifies the directly measured value of the Hubble constant \(H_{0}\), while it leaves the value of the Hubble constant in CMB measurements \(\tilde{H_{0}}\) intact.