This blog post was written by Andrew Pontzen and originally published by astrobites. Hiranya Peiris will be writing a response exploring why she believes that both rapid and slow expansion are confusing, and giving her take on explaining the inflationary picture to non-specialists – watch this space!

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Cosmic inflation is a hypothetical period in the very early universe designed to solve some weaknesses in the big bang theory. But what actually happens during inflation? According to wikipedia and other respectable sources, the main effect is an ‘extremely rapid’ expansion. That stock description is a bit puzzling; in fact, the more I’ve tried to understand it, the more it seems like inflation is secretly all about slow expansion, not rapid expansion. The secret’s not well-kept: once you know where to look, you can find a note by John Peacock that supports the slow-expansion view, for example. But with the rapid-expansion picture so widely accepted and repeated, it’s fun to explore why slow-expansion seems a better description. Before the end of this post, I’ll try to recruit you to the cause by means of some crafty interactive javascript plots.

A tale of two universes

There are many measurements which constrain the history of the universe. If, for example, we combine information about how fast the universe is expanding today (from supernovae, for example) with the known density of radiation and matter (largely from the cosmic microwave background), we pretty much pin down how the universe has expanded. An excellent cross-check comes from the abundance of light elements, which were manufactured in the first few minutes of the universe’s existence. All-in-all, it’s safe to say that we know how fast the universe was expanding all the way back to when it was a few seconds old. What happened before that?

Assuming that the early universe contained particles moving near the speed of light (because it was so hot), we can extrapolate backwards. As we go back further in time, the extrapolation must eventually break down when energies reach the Planck scale. But there’s a huge gap between the highest energies at which physics has been tested in the laboratory and the Planck energy (a factor of a million billion or so higher). Something interesting could happen in between. Inflation is the idea that, because of that gap, there may have been a period during which the universe didn’t contain particles. Energy would instead be stored in a scalar field (a similar idea to an electric or magnetic field, only without a sense of direction). The Universe scales exponentially with time during such a phase; the expansion rate accelerates. (Resist any temptation to equate ‘exponential’ or ‘accelerating’ with ‘fast’ until you’ve seen the graphs.) Ultimately the inflationary field decays back to particles and the classical picture resumes. By definition, all is back to normal long before the universe gets around to mundane things like manufacturing elements. For our current purposes, it’s not important to see whether inflation is a healthy thing for a young universe to do (wikipedia lists some reasons if you’re interested). We just want to compare two hypothetical universes, both as simple as possible:

  1. a universe containing fast-moving particles (like our own once did);
  2. as (1), but including a period of inflation.

Comparisons are odorous

There are a number of variables that might enter the comparison:

  • a: the scalefactor, i.e. the relative size of a given patch of the universe at some specified moment;
  • t: the time;
  • da/dt: the rate at which the scalefactor changes with time;
  • or if you prefer, H: the Hubble rate of expansion, which is defined as d ln a / dt.

We’ll take a=1 and t=0 as end conditions for the calculation. There’s no need to specify units since we’re only interested in comparative trends, not particular values. There are two minor complications. First, what do we mean by ‘including inflation’ in universe (2)? To keep things simple it’ll be fine just to assume that the pressure in the universe instantaneously changes. (Click for a slightly more specific description.) The change will kick in between two specified values of a — that is, over some range of ‘sizes’ of the universe. In particular, taking the equation of state of the universe to be pressure = w × density × c2, we will assume w=1/3 except during inflation, when w= –1. The value of will switch instantaneously at a=a0, and switch back at a=a1. (Click for details of the transition.)The density just carries over from the radiation to the inflationary field and back again (as it must, because of energy-momentum conservation). In reality, these transitions are messy (reheating at the end of inflation is an entire topic in itself) – but that doesn’t change the overall picture.

Finding the plot

The Friedmann equations (or equivalently, the Einstein equations) take our history of the contents of the universe and turn it into a history of the expansion (including the exponential behaviour during inflation). But now the second complication arises: such equations can only tell you how the Hubble expansion rate H (or, equivalently, da/dt) changes over time, not what its actual value is. So to compare universes (1) and (2), we need one more piece of information – the expansion rate at some particular moment. Since we never specified any units, we might as well take H=1 in universe (1) at t=0 (the end of our calculation). Any other choice is only different by a constant scaling. What about universe (2)? As discussed above, the universe ends up expanding at a known rate, so really universe (2) had better end up expanding at the same rate as universe (1). But, for completeness, you’ll be able to modify that choice below and have universe (1) and (2) match their expansion  rate at any time. All that’s left is to choose the variables to plot. I’ve provided a few options in the applet below. It seems they all lead to the conclusion that inflation isn’t ultra-rapid expansion; it’s ultra-slow expansion. By the way, if you’re convinced by the plots, you might wonder why anyone ever thought to call inflation rapid. One possible answer is that the expansion back then was faster than at any subsequent time. But the comparison shows that this is a feature of the early universe, not a defining characteristic of inflation. Have a play with the plots and sliders below and let me know if there’s a better way to look at it.

This plot shows the Hubble expansion rate as a function of the size of the universe. A universe with inflation (solid line) has a Hubble expansion rate that is slower than a universe without (dashed line). Inflation is a period of slow expansion! In the current plot, sometimes the inflationary universe (solid line) is expanding slower, and sometimes faster than the universe without inflation (dashed line). But then, you’ve chosen to make the Hubble rate exactly match at an arbitrary point during inflation, so that’s not so surprising. Currently it looks like the inflationary universe (solid line) is always expanding faster than the non-inflationary universe (dashed line). But the inflationary universe ends up (at a=1) expanding much faster than H=1, which was our target based on what we know about the universe today. So there must be something wrong with this comparison.


This plot shows the size of the universe as a function of time. Inflationary universes (solid line) hit a=0 at earlier times. In other words, a universe with inflation (solid line) is always older than one without (dashed line) and has therefore expanded slower on average. Inflation is a period of slow expansion! With the current setup you’re not matching the late-time expansion history in the inflationary universe against the known one from our universe; to make a meaningful comparison, the dotted and solid lines must match at late times (t=0). So the plot can’t be used to assess the speed of expansion during inflation.


This plot shows the Hubble expansion rate of the universe. The universe with an accelerating period (solid line) is always expanding at the same rate or slower than the one without (dashed line). Inflation is a period of slow expansion! Currently it looks like the inflationary universe (solid line) may expand faster than the non-inflationary universe (dashed line). But the inflationary universe ends up (at t=0) expanding much faster than H=1, which was our target based on what we know about the universe today. So there must be something wrong with this comparison.


This plot shows the rate of change of scalefactor (da/dt) as a function of time before the present day. The universe with an accelerating period (solid line) is always expanding at the same rate or slower than the one without (dashed line). Inflation is a period of slow expansion! With the current setup you’re not matching the late-time expansion history in the inflationary universe (solid line) against the known one from our universe (da/dt does not match at t=0, for instance). So the plot can’t be used to assess the speed of expansion during inflation.


First select the range of scalefactors over which inflation occurs by dragging the two ends of the grey bar. Currently, a0=X and a1=X

In realistic models of inflation, this range would extend over many orders of magnitude in scale, making the effects bigger than the graphs suggest.

Note: you’ve set inflation to start at a=0. That corresponds to eternal inflation in which the big bang is actually pushed back to t=–∞. That’s fine but the plots will be truncated at a finite negative t. (Fancy one extra thought?)In the light of this, you might also wonder what it means when people say that inflation happened a tiny fraction of a second after the big bang. Inflation itself changes the timeline — it could have happened any length of time after the big bang. The normal quoted time is an unphysical lower limit.
Note: you’ve set inflation to end at a=1. That corresponds to a period of exponential expansion at recent times, so it’s more like playing with dark energy than with inflation.
Note: you’ve set inflation to occur over all a. This corresponds to a de Sitter universe. It makes a little hard to make the connection with our own universe clearly.

Now select the scalefactor at which the expansion rate is matched between universe (1) and (2).

At the moment amatch=X: you’re matching after inflation is complete. That makes sense because various observations fix the expansion rate at this time.

At the moment amatch=X: you’re matching before or during inflation. Look at the Hubble rate at the end of inflation and you’ll find it disagrees between the two universes. That means they can’t both match what we know about the universe at late times, so the comparison isn’t really going to be fair.


Pedro Ferreira, Tom Whyntie, Nina Roth, Daniela Saadeh, Steven Gratton, Nathan Sanders, Rob Simpson, and Jo Dunkley made helpful comments on an early version. Rob Simpson and Brooke Simmons pointed me to the javascript libraries d3, rickshaw and numeric.


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