# Gravitational Particle Production and the Hubble Tension
## 5 Impact of Gravitational Particle Production on the Hubble Tension
\(\tilde{H_{0}}\) is obtained from the number density of baryons \(n_{b}\) that is unaffected by gravitational production (as seen in Equation (54)), while \(H_{0}\) is obtained from Equation (48) which includes the effect of GPP. No new model is introduced in this paper. The model employed here is just the standard \(\Lambda\)CDM model (where the effect of GPP is included). The effect of the GPP, as is evident from Equation (48), is to multiply the Hubble parameter of the background by an overall constant. Therefore, no new data analysis (in addition to that of \(\Lambda\)CDM) is needed for CMB and BAO datasets (unlike the extensions of the \(\Lambda\)CDM model [39]). The values obtained from these datasets (with \(\Lambda\)CDM adopted) remain applicable here. The point here is that the values of the Hubble constant obtained by the use of the CMB and BAO anisotropy data versus the corresponding formula Equations (50) and (51) are employed for the best fit value of \(z_{*}\) or \(z_{d}\) which in turn are determined by \(n_{b}\), and so by \(\tilde{H_{0}}\). Hence, \(\tilde{H_{0}}\) corresponds to the values of the Hubble constant obtained in CMB and BAO observations. In the second paragraph after Equation (48), the effective increase in the Hubble constant is identified as an effective increase in [PERSON]'s gravitational constant \(G\), rather than an effective increase in the total energy density. It should be remarked that the approach to Hubble tension in the present study is quite different from the models with a jump in the value of \(G\) at very small redshifts [11; 13]. Those types of models need a rigorous theoretical motivation and do not solve the Hubble tension wholly (while they ameliorate it) [12], and data seem not to support the prediction of those models that \(H_{0}\) should vary when obtained in different redshift bins [40]. The gravitational constant \(G\) in those studies varies with redshift, while the gravitational constant in the present study does not vary with redshift. Moreover, the model we employ is the standard model of cosmology \(\Lambda\)CDM and no new physics is used. Only the effect of gravitational particle production (that is an element of the standard established physics which is overlooked in the previous studies) is taken into account. The inclusion of this effect explains why the values of the Hubble constant in the direct measurements and in the CMB and BAO calculations are different. No additional numerical simulations are needed. What is done is just the usual \(\Lambda\)CDM data analysis that was carried out by CMB and BAO collaborations. In other words, what is done in this paper is to give an explanation for having two different values of the Hubble constant obtained from direct measurements and CMB and BAO collaborations rather than proposing a new model. The model employed in this paper is just \(\Lambda\)CDM (both at the background and at the level of perturbations) since it just amounts to multiplying the Newton constant by an overall constant, as is evident from Equation (48). This is also different from the case in some models (such as dark energy dark matter coupling models) where the evolution of the background is the same [41] or almost the same as the one in \(\Lambda\)CDM [14], while their predictions differ at the level of the evolution of cosmological perturbations [15]. Instead, the evolution of the Hubble parameter before and after inclusion of the effect of gravitational particle production is the same as that of \(\Lambda\)CDM.