# Mathematica Rescaling equations in Mathematica

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1. May 5, 2017

### 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]

2. May 10, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.