Suppose I have a differential equation(adsbygoogle = window.adsbygoogle || []).push({});

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

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]

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# Mathematica Rescaling equations in Mathematica

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