Troubles understanding slow-roll conditions in Inflation

  • #1
I've been reading up on inflation, and have arrived at the so-called slow roll conditions $$\epsilon =-\frac{\dot{H}}{H^{2}}\ll 1\; ,\qquad\eta =-\frac{\ddot{\phi}}{H\dot{\phi}}\ll 1$$
I have to admit, I'm having trouble understanding a couple of points. First, how does ##\epsilon\ll 1## guarantee that ##H## is approximately constant (evolving slowly) during inflation? Doesn't this condition simply enforce ##\dot{H}\ll H^{2}## and not necessarily that ##\dot{H}\ll 1##? I can see in the limit that ##\epsilon\to 0## that this condition implies that ##\dot{H}=0##. Is the point that the relevant time-scale of the problem is the Hubble time ##t=H^{-1}##, and thus we want ##H## to not change appreciably over one Hubble time, such that the solution for ##H## will be well approximated by ##H\approx\text{constant}##?

As for the second condition, is this just to ensure that the equation of motion for the scalar field is dominated by the friction term ##3H\dot{\phi}## and the derivative of the potential ##V'(\phi)##? Again, how is it clear from this that ##\ddot{\phi}## is (in some sense) small during inflation.

Moving on to the "potential" slow-roll conditions, these follow from the above conditions: $$\epsilon_{V}=\frac{M_{\text{Pl}}^{2}}{2}\left(\frac{V'(\phi)}{V(\phi)}\right)^{2}\ll 1\;,\qquad\eta_{V}=M_{\text{Pl}}^{2}\frac{V''(\phi)}{V(\phi)}\ll 1$$ Analogously to the questions above, how do these condition guarantee that the potential is approximately constant during inflation? Don't they simply imply that the first and second derivatives of ##V## must be much less than ##V/M_{\text{Pl}}##? Is the point that one has ##\frac{1}{2}\dot{\phi}^{2}\ll V(\phi)##, which implies that ##p_{\phi}\approx -
\rho_{\phi}## and ##\rho_{\phi}\approx V(\phi)##, and so the continuity equation becomes ##\dot{\rho}_{\phi}\approx 0\Rightarrow V(\phi)\approx\text{constant}##?
I feel a bit dense asking this question, but I seem to be having a mental block on this.
 
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  • #2
Orodruin
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Neither ##\dot H## nor ##\ddot\phi## are dimensionless. Hence, an expression such as ##\dot H \ll 1## does not make sense. Instead, ##\dot H## needs to be compared to something that carries information about a typical scale for that quantity. Since H has dimension 1/T, ##\dot H## has dimension 1/T^2 and the typical time scale is set by 1/H so it needs to be compared to ##H^2##, not to 1. The key is in your parenthesis: ”in some sense”. You can only define the ”in some sense” relaive to other quantities with the same dimensions.
 
  • #3
Neither ##\dot H## nor ##\ddot\phi## are dimensionless. Hence, an expression such as ##\dot H \ll 1## does not make sense. Instead, ##\dot H## needs to be compared to something that carries information about a typical scale for that quantity. Since H has dimension 1/T, ##\dot H## has dimension 1/T^2 and the typical time scale is set by 1/H so it needs to be compared to ##H^2##, not to 1. The key is in your parenthesis: ”in some sense”. You can only define the ”in some sense” relaive to other quantities with the same dimensions.
Thanks for your response. I was editing my original post when your answer was posted. I'd appreciate it if you could have a look at the updated version and see if you agree with what I've written.

Why is the typical time scale set by ##1/H##? I've read that it's the time scale over which the scale factor changes appreciably. Is this why it is the relevant time scale, as it's the time-scale over which the universe changes noticeably? How can one see that this is the case? Is it simply that one can Taylor expand ##a(t)## about its value ##a(t_{1})## at some time ##t_{1}##, i.e., ##a(t_{1}+\Delta t)\approx a(t_{1})\left(1+\frac{\Delta t}{t_{H}} +\mathcal{O}\Big(\big(\frac{\Delta t}{t_{H}}\big
)^{2}\Big
)\right)##, where ##t_{H}## is the Hubble time at ##t=t_{1}##. Therefore ##a(t)## will only change appreciably from its value at ##t=t_{1}## over the time scale ##\Delta t\sim t_{H}##?
 
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  • #4
kimbyd
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Why is the typical time scale set by ##1/H##? I've read that it's the time scale over which the scale factor changes appreciably. Is this why it is the relevant time scale, as it's the time-scale over which the universe changes noticeably?
Basically. It's a statement that the universe will have to expand quite a lot before there is a significant change in either ##H## or ##\dot{\phi}##. And if the universe has to expand a lot before there is a significant change in ##H##, then that is an indication that the rate of expansion is nearly exponential.
 
  • #5
Basically. It's a statement that the universe will have to expand quite a lot before there is a significant change in either ##H## or ##\dot{\phi}##. And if the universe has to expand a lot before there is a significant change in ##H##, then that is an indication that the rate of expansion is nearly exponential.
So the slow roll condition is that the change in ##H## is small per Hubble time then. Ss what I wrote in the couple of sentences at the end of my previous post correct at all (as to why the Hubble time is the appropriate time scale)?
 
  • #6
kimbyd
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So the slow roll condition is that the change in ##H## is small per Hubble time then. Ss what I wrote in the couple of sentences at the end of my previous post correct at all (as to why the Hubble time is the appropriate time scale)?
Not quite. ##a(t)## might change quite dramatically over that period. But it will behave close to ##a(t) = a_0 e^{H t}##. The expansion would be exactly exponential if ##H(t)## was exactly constant. The slow-roll conditions ensure that the behavior is close enough to this that it's useful to think of the expansion during inflation in terms of exponential expansion.

Mathematically the slow-roll conditions are chosen to make the equations of motion easier to solve. That requirement is ultimately why they take the precise form they do.
 
  • #7
Not quite.
But I thought the Hubble time set the time-scale over which the scale factor changes appreciably (at least that's what I've read in a set of notes)?!
 
  • #8
kimbyd
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But I thought the Hubble time set the time-scale over which the scale factor changes appreciably (at least that's what I've read in a set of notes)?!
After one Hubble time with exponential expansion, ##a## increases by one factor of ##e## (2.718), i.e. nearly triples. So yes, a Hubble time is related to the amount of time for there to be a significant amount of expansion. But that amount of expansion can be pretty large depending upon the contents of the universe.
 
  • #9
After one Hubble time with exponential expansion, ##a## increases by one factor of ##e## (2.718), i.e. nearly triples. So yes, a Hubble time is related to the amount of time for there to be a significant amount of expansion. But that amount of expansion can be pretty large depending upon the contents of the universe.
Okay, so can one say that, given the scale factor ##a(t)## at some time ##t##, then over a time interval ##\Delta t## of order the Hubble time (i.e. ##\Delta t\sim H^{-1}##) the universe will change significantly, however, for intervals much less than this (i.e. ##\Delta t\ll H^{-1}##), the scale factor at ##t+\Delta t##, ##a(t+\Delta t)## will not have changed significantly from its value at time ##t## (and thus the universe will not "look" much different than it did at time ##t##).
 
  • #10
kimbyd
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Okay, so can one say that, given the scale factor ##a(t)## at some time ##t##, then over a time interval ##\Delta t## of order the Hubble time (i.e. ##\Delta t\sim H^{-1}##) the universe will change significantly, however, for intervals much less than this (i.e. ##\Delta t\ll H^{-1}##), the scale factor at ##t+\Delta t##, ##a(t+\Delta t)## will not have changed significantly from its value at time ##t## (and thus the universe will not "look" much different than it did at time ##t##).
Correct, provided the ways in which the universe changes are dominated by expansion. One might imagine scenarios where other effects are dominant (e.g. gravitational collapse in a slowly-expanding universe). As inflation is inherently tied to the expansion, the time scale of expansion is the right time scale.
 

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