Rescaling the equation of motion of inflation

[shinobi20]

From the equation of motion of inflation, $$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + \frac{dV}{d\phi} = 0$$ Example: $V= \frac{1}{2}m^2\phi^2$
$$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + m^2\phi = 0$$
If I want to make the DE dimensionless then I let $~t = \frac{1}{H_o} \tilde t~$ and $~H = H_o \tilde H~$ then,
$$H_o^2 \frac{d^2\phi}{d\tilde t^2} + 3H_o^2\tilde H \frac{d\phi}{d\tilde t} + m^2\phi = 0$$
But the last term has $m^2$ in it, so how can I rescale this DE such that every term would be dimensionless? Also, what is the dimension of $~\phi~$(inflaton)?

 

[shinobi20]

[Moderator's note: moved from a separate thread to this one since the topic is the same. Also edited to delete duplicate content.]

Another question:

To solve this differential equations, we need two initial value conditions, $\phi(0) = ?\,$ and $\dot \phi(0) = ?\,$. But I don't know what they should be, I know that in the early stages of inflation, the potential $V$ should be dominant so I think $\dot \phi(0)$ should be small?

 

[Mordred]

Did you look at my reply in your other thread?

https://www.physicsforums.com/threads/different-forms-of-energy-density-in-inflation.900158/

you keep missing the detail that there is two potentials involved. One for kinetic the other for pressure. In order to get your dimension
less parameter which I assume is w you require both terms. $$w=p/\rho$$

Look at the equation's of state (Cosmology) see the section on scalar modelling.
https://en.m.wikipedia.org/wiki/Equation_of_state_(cosmology)

Here some additional examples see equations 1.36 to 1.39
http://www.google.ca/url?sa=t&source=web&cd=2&ved=0ahUKEwi67MqH0dPRAhVH4mMKHf9vBhgQFgggMAE&url=http://www3.imperial.ac.uk/pls/portallive/docs/1/56439.PDF&usg=AFQjCNFCbq4LLlR6366LhUvr8T_y6_f0eA&sig2=n5C7FRMAGfPcq5gYfq4hMw

The formulas showing action via those equation's are included

 

[bapowell]

The inflaton has dimension m in 3+1 dimensions.