- #1
shinobi20
- 277
- 20
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)?
$$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + m^2\phi = 0$$
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)?
