# Gravitational Particle Production and the Hubble Tension
## 2 Preliminaries  
Spacetime at cosmological scales may be described by the spatially flat Robertson-Walker metric  
\[ds^{2}\,=\,-dt^{2}\,+\,a^{2}(t)\Big{[}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d \varphi^{2})\Big{]}. \tag{1}\]  
We consider the following action for a scalar field \(\phi\) in this space  
\[S =\,\int\sqrt{-g}\ d^{4}x\,\frac{1}{2}\Big{[}-g^{\mu\
u}\partial_{ \mu}\phi\partial_{\
u}\phi\,-\,m_{\phi}^{2}\phi^{2}\Big{]}\,=\,\int\,d^{3}x\, d\eta\,\frac{1}{2}\Big{[}\tilde{\psi}^{\prime\,2}-(\vec{\
abla}\tilde{\phi})^{2}- \tilde{m}_{\phi}^{2}\tilde{\phi}^{2}\Big{]}\, \tag{2}\]  
where \(m_{\phi}\) is the mass of \(\phi\), prime denotes derivative with respect to conformal time \(\eta\)[1] (while an over-dot denotes the derivative with respect to \(t\)) and  
\[d\eta\,=\,\frac{dt}{a(t)}\,\ \tilde{\phi}\,=\,a(\eta)\ \phi\,\ \ \tilde{m}_{\phi}^{2}\,=\,m_{\phi}^{2}a^{2}-\frac{a^{\prime\prime}}{a}. \tag{3}\]  
The field \(\tilde{\phi}\) may be expressed as  
\[\tilde{\phi}(\vec{x},\eta)\,=\,\frac{1}{\sqrt{2}}\,\int\,\frac{d^{3}k}{(2\pi) ^{\frac{3}{2}}}\,\Big{[}e^{i\vec{k}\cdot\vec{x}}\,v_{k}^{*}(\eta)\,\hat{a}_{ \vec{k}}^{-}\,+\,e^{-i\vec{k}\cdot\vec{x}}\,v_{k}(\eta)\,\hat{a}_{\vec{k}}^{+} \Big{]} \tag{4}\]  
The mode function \(v_{k}(\eta)\) satisfies the equation of motion for \(\tilde{\phi}\)  
\[v_{k}^{\prime\prime}\,+\,\omega_{k}^{2}\,v_{k}=0 \tag{5}\]  
where  
\[\omega_{k}\,=\,\sqrt{\vec{k}^{2}\,+\,\tilde{m}_{\phi}^{2}} \tag{6}\]  
Vacuum state is the ground state with minimum energy. In curved space, in general, the ground state at an instant of time is not the ground state at another time. Hence, the annihilation operators corresponding to the corresponding vacuum state at a given time do not destroy the vacuum state at another time. Therefore, the annihilation and creation operators and the mode functions at different times are different in general in curved spaces [1]. The field \(\tilde{\phi}\) in another vacuum (other than the one specified in Equation (4)) with annihilation and creation operators \(\hat{b}_{\vec{k}}^{-}\) and \(\hat{b}_{\vec{k}}^{+}\) may be expanded as  
\[\tilde{\phi}(\vec{x},\eta)\,=\,\frac{1}{\sqrt{2}}\,\int\,\frac{d^{3}k}{(2\pi) ^{\frac{3}{2}}}\,\Big{[}e^{i\vec{k}\cdot\vec{x}}\,u_{k}^{*}(\eta)\,\hat{b}_{ \vec{k}}^{-}\,+\,e^{-i\vec{k}\cdot\vec{x}}\,u_{k}(\eta)\,\hat{b}_{\vec{k}}^{+} \Big{]} \tag{7}\]  
where \(u_{k}\) satisfies the same equation as Equation (5) and is related to \(v_{k}\) and \(v_{k}^{*}\) by  
\[u_{k}(\eta)\,=\,a_{k}\,v_{k}(\eta)\,+\,\beta_{k}\,v_{k}^{*}(\eta) \tag{8}\]where \(\alpha_{k}\), \(\beta_{k}\) are called Bogolyubov coefficients. In a similar fashion, \(\hat{b}_{\vec{k}}^{-}\) is related to \(\hat{a}_{\vec{k}}^{-}\) and \(\hat{a}_{\vec{k}}^{+}\) by  
\[\hat{b}_{\vec{k}}^{-}\,=\,\alpha_{k}\,\hat{a}_{\vec{k}}^{-}\,-\,\beta_{k}\,\hat{a }_{\vec{k}}^{+} \tag{9}\]  
Mode functions may be expressed in a WKB-approximation-like form  
\[v_{k}(\eta)\,=\,\frac{1}{\sqrt{W_{k}(\eta)}}\,\exp\bigg{[}i\int_{\eta_{0}}^{ \eta}\,W_{k}(\eta)\,d\eta\bigg{]} \tag{10}\]  
where Equation (5) implies that \(W_{k}(\eta)\) should satisfy  
\[S =\,\int\sqrt{-g}\ d^{4}x\,\frac{1}{2}\Big{[}-g^{\mu\
u}\partial_{ \mu}\phi\partial_{\
u}\phi\,-\,m_{\phi}^{2}\phi^{2}\Big{]}\,=\,\int\,d^{3}x\, d\eta\,\frac{1}{2}\Big{[}\tilde{\psi}^{\prime\,2}-(\vec{\
abla}\tilde{\phi})^{2}- \tilde{m}_{\phi}^{2}\tilde{\phi}^{2}\Big{]}\, \tag{2}\]0  
The following \(W_{k}\) that approximately satisfies Equation (11) may be adopted as an approximate solution  
\[S =\,\int\sqrt{-g}\ d^{4}x\,\frac{1}{2}\Big{[}-g^{\mu\
u}\partial_{ \mu}\phi\partial_{\
u}\phi\,-\,m_{\phi}^{2}\phi^{2}\Big{]}\,=\,\int\,d^{3}x\, d\eta\,\frac{1}{2}\Big{[}\tilde{\psi}^{\prime\,2}-(\vec{\
abla}\tilde{\phi})^{2}- \tilde{m}_{\phi}^{2}\tilde{\phi}^{2}\Big{]}\, \tag{2}\]1  
Equation (12) may be identified as adiabatic conditions [2; 22]. **3.