}-D \delta^{ij} \partial_i u \partial_j u  -m^2_{\text{eff}}u^2 ]$$ where $$m^2_{\text{eff}} = - \frac{\sqrt{D}}{a} \left( \frac{a}{\sqrt{D}} \right)^{''} .$$ Till now the considerations of the gravitational waves has been purely classical. The next step is the quantisation of the classical gravitational waves what brings us the concept of gravitons. To quantise the field $u$ we need to firstly calculate conjugated momenta $$\pi(\tau,{\bf x})=\frac{\delta S_{\text{t}}}{\delta u'} = u '.$$ The procedure of quantisation is the simple change of fields $u$ and $\pi$ for the operators just adding hats and to introduce the relations of commutation. We decompose operators considered for the Fourier modes $$\begin{aligned}
\hat{u}(\tau,{\bf x} )   &=& \frac{1}{2(2\pi)^{3/2}} \int d^3{\bf k } \left[ \hat{u}_{{\bf k}}(\tau) e^{i{\bf k}\cdot {\bf x}} +  
\hat{u}_{{\bf k}}^{\dagger}(\tau) e^{-i{\bf k}\cdot {\bf x}}  \right],   \label{decomp1}       \\
\hat{\pi}(\tau,{\bf x} ) &=& \frac{1}{2(2\pi)^{3/2}} \int d^3{\bf k }
 \left[ \hat{\pi}_{{\bf k}}(\tau) e^{i{\bf k}\cdot {\bf x}} +  
\hat{\pi}_{{\bf k}}^{\dagger}(\tau) e^{-i{\bf k}\cdot {\bf x}}  \right],  \label{decomp2}        \end{aligned}$$ where the Fourier components fulfil the relations of commutation $$\begin{aligned}
\left[ \hat{u}_{{\bf k}}(\tau) ,\hat{\pi}_{{\bf p}}^{\dagger}(\tau) \right]  &=& i \delta^{(3)}({\bf k} - {\bf p}),\label{com1}\\
\left[ \hat{u}_{{\bf k}}(\tau)^{\dagger} ,\hat{\pi}_{{\bf p}}(\tau) \right]  &=& i \delta^{(3)}({\bf k} - {\bf p}),\label{com2} \\
\left[ \hat{u}_{{\bf k}}(\tau) ,\hat{\pi}_{{\bf p}}(\tau) \right]  &=& i \delta^{(3)}({\bf k} + {\bf p}), \label{com3}  \\
\left[ \hat{u}_{{\bf k}}(\tau)^{\dagger} ,\hat{\pi}_{{\bf p}}^{\dagger}(\tau) \right]  &=& i \delta^{(3)}({\bf k} + {\bf p}).   
\label{com4}\end{aligned}$$

To express the Fourier modes in terms of the annihilation and creation operators we need to solve the quantum Hamilton equations $$\begin{aligned}
\hat{u}^{'}   &=& i [ \hat{H}_{\text{t}}, \hat{u}   ],  \label{Ham1}  \\
\h