Rescaling equations in Mathematica

[Whitehole]

Suppose I have a differential equation

$$\ddot \phi + 3H (1+Q) \dot \phi + V_{,\phi} = 0$$

where $\phi$ is the inflaton field. $H$ is the Hubble parameter, $Q$ is just a number, $V_{,\phi}$ is the derivative with respect to $\phi$, and initial conditions given by $\phi[0] = 2 M_p~,$ $\dot\phi[0] = 0.1 M_p$ .

Now, an example situation would be, $V = \frac{1}{2} m^2 \phi^2$ where $m$ is the inflaton mass. Parameters are given by: $~m = 10^{13} GeV~$, $M_p = 10^{18}$ (reduced Planck mass). Since these numbers are big, I want to rescale these equations using $M_p$

$d\tilde{t} = M_p dt \quad,$ $\tilde{H} = \frac{H}{M_p} \quad,$ $\tilde{\phi} = \frac{\phi}{M_p} \quad,$ $\tilde{m} = \frac{m}{M_p}$

So that,

$\dot\phi = M_p^2 \tilde{\phi'}~,$ $~\ddot\phi = M_p^3 \tilde{\phi''}~,$ $\phi[0] = 2~,$ $\dot\phi[0] = 0.1$

If I want to solve these, I would just input (let $\phi = y$)

NDSolve[ {$y''[t] + 3 H (1+Q) y'[t] + \tilde{m}^2 y[t] = 0, y[0] = 2, y'[0] = 0.1$}, {t,0,10^7}, PlotRange -> Full]

Is this correct? I think by entering the rescaled initial conditions for the variable y would rescale the equation in NDSOlve right? I think it's wrong to write for example,

NDSolve[ {$y''[t]/M_p^3 + 3 H/M_p (1+Q) y'[t]/M_p^2 + \tilde{m}^2 y[t]/M_p = 0, y[0] = 2 Mp, y'[0] = 0.1 Mp$}, {t,0,10^7}, PlotRange -> Full]

 

[Valkarie]

because this would mean that I am trying to solve the original equation (without rescaling). Am I correct?