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I Rescaling the equation of motion of inflation

  1. Jan 19, 2017 #1
    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)?
     
  2. jcsd
  3. Jan 20, 2017 #2
    [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?
     
    Last edited by a moderator: Jan 21, 2017
  4. Jan 21, 2017 #3
    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. [tex]w=p/\rho [/tex]

    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
     
    Last edited: Jan 21, 2017
  5. Jan 22, 2017 #4

    bapowell

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    The inflaton has dimension m in 3+1 dimensions.
     
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