# Gravitational Particle Production and the Hubble Tension
## 4 Gravitational Particle Production after Decoupling
### In and Out States  
One may take \(\omega_{k}\) approximately constant in a time interval \(\Delta\eta\) if  
\[\left|\frac{\Delta\omega_{k}}{\omega_{k}}\right|\,=\,\,\left|\frac{\Delta\eta \left(\frac{d\omega_{k}}{d\eta}\right)}{\omega_{k}}\right|\,\ll\,1. \tag{32}\]  
If Equation (32) is satisfied, then Equation (4) may be expressed in its Minkowski form in the interval \(\Delta\eta\), so the field may expanded as [24; 25]  
\[\tilde{\phi}_{(i)}(\vec{x},\eta)\,\simeq\,\int\,\frac{d^{3}\tilde{p}}{(2\pi) ^{\frac{3}{2}}\sqrt{2\omega_{p,(i)}}}\left[a_{p,(i)}^{-}\,e^{i\left(\tilde{p} \,\tilde{p}\,\tilde{\tau}-\omega_{p,(i)}(\eta-\eta_{i})\right)}\,+\,a_{p,(i)} ^{+}\,e^{i\left(-\tilde{p}\,\tilde{\tau}+\omega_{p,(i)}(\eta-\eta_{i})\right)}\right] \tag{33}\]  
where \({}_{(i)}\) refers to the \(i\)th time interval between the times \(\eta_{i}\) and \(\eta_{i+1}\) with \(\Delta\eta\,=\,\eta_{i+1}-\eta_{i}\). It has been shown in [25] that Equation (32) may be easily imposed for \(\Lambda\)CDM. Note that Equation (32) may be guaranteed by taking \(\Delta\eta\) sufficiently small once Equation (29) is imposed, i.e.,  
\[\frac{\omega_{k}^{\prime}}{\omega_{k}^{2}}\,=\,\delta\,=\,\frac{\Delta\eta\, \omega_{k}^{\prime}}{\Delta\eta\,\omega_{k}^{2}}\,=\,\left|\frac{\Delta\omega_{ k}}{\omega_{k}}\right|\frac{1}{\Delta\eta\,\omega_{k}}\,\ll\,1\quad\Rightarrow \quad\left|\frac{\Delta\omega_{k}}{\omega_{k}}\right|\,=\,\delta\,\Delta\eta \,\omega_{k}\,\ll\,1 \tag{34}\]  
provided that \(\Delta\eta\) is sufficiently small. (However, Equation (29) is not guaranteed by Equation (32)). In other words, Equation (32) is always satisfied provided that \(\Delta\eta\) is sufficiently small. For such a \(\Delta\eta\), the spacetime essentially may be considered as a Minkowski spacetime. However, \(\Delta\eta\) cannot be arbitrarily small. The de Broglie wavelength of the relevant modes must be significantly smaller than the size of \(c\,\Delta\eta\) so that the detectors at the \(in\) and \(out\) regions can detect them (as free modes in Minkowski spacetime) [25]. This puts a lower bound on the values of \(|\vec{k}|\) (for which this method is applicable) for a given \(c\,\Delta\eta\) (and vice versa) through  
\[|\vec{k}|\,>\,|\vec{k}|_{\Delta}\,=\,\frac{2\pi}{c\,\Delta\eta}. \tag{35}\]  
\(\Delta\eta\) s that satisfy Equation (32) can also be taken considerably wide in \(\Lambda\)CDM as shown below. Equation (32) implies that  
\[\Delta\eta\,\ll\,\frac{\omega_{k}}{\omega_{k}^{\prime}}\,=\,\frac{\hbar^{2} \vec{k}^{2}\,+\,m_{q}^{2}c^{2}a^{2}(\eta)}{a^{3}(\eta)\,m_{\phi}^{2}c^{2}H( \eta)}. \tag{36}\]  
The right-hand side of Equation (36) is minimum at \(|\vec{k}|\,=\,0\), and \(H\,\sim\,a^{-\frac{3}{2}}\,H_{0}\) at the time of decoupling \(a_{dec}\,\sim\,10^{-3}\). These imply that \(\Delta\eta\,\ll\,\frac{10^{2}}{H_{0}}\) after decoupling and \(\Delta\eta\,\ll\,\,\frac{1}{H_{0}}\) at present. This, in turn, implies that \(\Delta\eta\) values can be taken long enough to identify the \(in\) and \(out\) vacuum states of S-matrix formulation in the form of Equation (33) [25]. Note that \(\Delta\eta\,\sim\,\frac{10^{2}}{H_{0}}\) at the time of decoupling and \(\Delta\eta\,\sim\,\frac{1}{H_{0}}\) at present are upper bounds on \(\Delta\eta\) for which the spacetime can approximately be taken to be a Minkowski spacetime. As \(\Delta\eta\) becomes smaller, the approximation becomes a better one. However, the price to be paid for a smaller \(\Delta\eta\) is that more modes with shorter wavelengths (i.e., with higher \(|\vec{k}|\)) become excluded from the domain of the applicability of the approximation used in this study. This point will be discussed in the next subsection in more detail. A mode function corresponding to a ground state with minimum energy at time \(\eta_{0}\) has the form \(v_{k}(\eta_{0})\,=\,\frac{1}{\sqrt{\omega_{k}(\eta_{0})}}\,e^{i\eta_{k}(\eta_ {0})}\) where \(\sigma_{k}\) is an arbitrary function of \(|\vec{k}|\) and \(\eta_{0}\)[1].