# Gravitational Particle Production and the Hubble Tension
## 4 Gravitational Particle Production after Decoupling
### Energy Density of Gravitationally Produced Particles
Increasing the mass of \(\phi\) results in an overall increase in the total energy density irrespective of the masses and the ratio of the particles in the background. Gravitational particle production is not specific to scalars. It is possible for all particles [33; 34; 31], but their contribution to total energy density is proportional to their masses in all cases. Hence, if the total mass of the scalar particles are taken to be very large, e.g., at order Planck mass while all other particle masses are taken to be much smaller, then the increase in total energy density will be determined by the total mass of the scalars. In such a case, the effective total energy density increases significantly, while the energy density of the background particles such as baryons virtually remain the same. This point will be important in the discussion in the paragraph after Equation (55) (i.e., in the argument that the number density of baryons essentially remains the same in such a case while the effective Hubble constant in direct measurements increases considerably). Moreover, the effective increase in the value of the Hubble parameter cannot be also attributed to a true physical production of \(\phi\) particlessince a true physical production of scalar particles would induce an energy density for a scalar field in Equation (48) that scales as that of a scalar field. (Note that, in principle, the energy density of a scalar field may mimic the energy density of any fluid, e.g., of \(\Lambda\)CDM while it cannot be exactly the same as that of that fluid for a finite time). In contrast, there is no energy density that scales as that of a scalar field in Equation (48) if \(\rho^{(ps)}\) is taken as the energy density of \(\Lambda\)CDM. This point, i.e., \(\rho^{(PP)}\) above should be identified as the effective energy density due to quasi-particles [31] rather than true particles, can also be seen in the following way. Identification of \(\rho^{(PP)}\) as (effective) energy density of true physical particles would lead to an inconsistency. If \(\rho^{(PP)}\) were a true energy density it would increase the total energy density, hence increase \(H\); this, in turn, would induce additional gravitational production of particles; this, in turn, would increase the total energy density further, and eventually the total energy density would be infinite. In other words such an argument would eventually result in \(\rho^{(PP)}\propto lim_{N\rightarrow\infty}\,(1-\gamma)^{-N}\rightarrow\infty\) where \(\gamma=1.8\,\times\,10^{-58}\times\,\left(\frac{m_{\phi}c^{2}}{eV}\right)^{2}\). In light of the above consideration, it is more conceivable and reliable to identify the effective increase in the Hubble parameter and the Hubble constant to be due to an effective increase in \(G\) as described in Equation (48). This effect may be significant, for example, for scalar particles that were present at extremely early times (e.g., at the time of inflation and then decayed wholly into other particles) with masses at the order of Planck masses. Another comment is that gravitational particle production does not modify the evolution of energy density, as is evident in Equation (48) (since the effect of gravitational particle production is to multiply the background energy density by an overall constant, as is evident in Equation (48). On the other hand, a true physical production of scalar particles would induce an energy density that scales as that of a scalar field. Therefore, as mentioned above, the effective energy density induced by gravitational particle production of \(\phi\) particles in the present context should be identified as the energy density of quasi-particles rather than that of physical particles. In fact, it is possible to consider the case where there is also a contribution to the energy density by physical \(\phi\) particles. In that case, there would also be a contribution to the Hubble parameter that scales as that of a scalar field.