# Gravitational Particle Production and the Hubble Tension
## 4 Gravitational Particle Production after Decoupling
### Energy Density of Gravitationally Produced Particles  
The energy density corresponding to Equation (43) is  
\[\rho^{(PP)}\,=\,\frac{1}{(2\pi)^{3}\,a^{3}}\,\int\,d^{3}k\,E_{k}\,|\beta_{k}|^{ 2}\,=\,\frac{\hbar}{96\,\pi\,c}\,\Big{(}\frac{m_{\phi}\,c}{\hbar}\Big{)}^{2}H^ {2} \tag{46}\]  
where \(E_{k}\,=\,\sqrt{\hbar^{2}\Big{(}\frac{\bar{k}}{\bar{a}}\Big{)}^{2}c^{2}+m_{ \phi}^{2}c^{4}}\) is the physical energy of a \(\phi\) particle with physical momentum \(\frac{1}{\bar{a}}\vec{k}\) (while \(\vec{k}\) is the comoving coordinate momentum of the particle). The total effective energy density \(\rho^{eff}\) (that is the sum of the energy density of background \(\rho^{(bg)}\) and the energy density of gravitationally produced particles \(\rho^{(PP)}\)) relates to Hubble parameter \(H\) as  
\[\frac{3\,c^{2}}{8\pi\,G}\,H^{2} =\,\rho^{eff}\,=\,\rho^{(PP)}\,+\,\rho^{(bg)}\,=\,\frac{\hbar}{96 \,\pi\,c}\,\sum_{i}\left(\frac{m_{i}\,c}{\hbar}\right)^{2}H^{2}\,+\,\rho^{(bg)} \tag{47}\] \[\simeq \rho^{(bg)}\,+\,1.8\,\times\,10^{-58}\times\,\sum_{i}\left(\frac{ m_{i}\,c^{2}}{eV}\right)^{2}\frac{3\,c^{2}}{8\pi\,G}\,H^{2}\] \[\text{i.e.,}\;\;3\,H^{2}\,=\,\frac{8\pi}{c^{2}}\left[\frac{G}{1\, -\,1.8\,\times\,10^{-58}\times\,\sum_{i}\left(\frac{m_{i}\,c^{2}}{eV}\right) ^{2}}\right]\rho^{(bg)} \tag{48}\]  
where \(\rho^{(bg)}\) is identified as the total energy density, and \(m_{i}\) denotes the mass of the \(ith\) scalar particle (that contributes to \(\rho^{(PP)}\)). Equation (48) implies that gravitational particle production has a significant contribution to the effective Hubble parameter if \(\sum_{i}\left(\frac{m_{i}\,c^{2}}{eV}\right)^{2}\) is not extremely smaller than \(10^{58}\). For example, if there are ten (scalar) particles with masses of the order of the the Planck mass \(M_{Planck}\,\simeq\,1.22\,\times\,10^{28}\,\text{eV}/c^{2}\) (that were, for example, present at the beginning of the universe and later may had decayed wholly into standard model particles), then \(G\) would be multiplied by an overall factor \(\sim\)1.37 in Equation (48). Note that ultra-heavy particles are extensively studied in references [27; 28; 29]. A comment is in order here. The present study is applicable for the times after the time of decoupling till the present. We have checked the applicability of Equation (29) in this interval (which is in the order of \(\frac{1}{H_{0}}\)). In a similar way, we found in in Section 4.2 that identification of the \(in\) state in an unambiguous way requires \(\Delta\eta\) to be smaller than \(\frac{\sqrt{10}}{H_{0}}\), and identification of the \(out\) state in an unambiguous way requires \(\Delta\eta\) to be smaller than \(\frac{1}{H_{0}}\). Therefore, the present analysis is valid for wavelengths smaller than \(\frac{1}{H_{0}}\). This corresponds to a lower bound on the relevant modes, namely, \(\hbar\,|\vec{k}|\,c\,\sim\,10^{-33}\,\text{eV}\). For modes with lower \(|\vec{k}|\), the validity of the approximation cannot be guaranteed. However, the contribution of such low momentum modes in Equation (46) is small (unless there is a drastic change in the form of \(\left|\beta_{k}\right|^{2}\) in Equation (43) at such low values of \(|\vec{k}|\)). Therefore, Equations (45) and (48) may be taken as decent approximate expressions. In fact, the same formulas Equations (45) and (46) are obtained in references [30; 31; 32]. It is evident from Equation (48) that gravitational production of \(\phi\) particles results in an effective overall increase in the value of the Hubble parameter, hence in the value of the Hubble constant. This increase, at first sight, may be attributed either to an effective increase in [PERSON]'s gravitational constant \(G\) or to an effective increase in the total energy density. However, such an effective increase in the total energy density cannot be considered to be due to a physical increase in the energy density of background particles (e.g., baryons).