Does dark energy or cosmic inflation explain flatness?

George Jones
Staff Emeritus
Gold Member
Yes, good point. And a density only 30% of critical (much less only 5% of critical) would not be anywhere close enough to critical for this.
But this is where things stood in 80s, before dark energy in the late 90s. From page 18
https://arxiv.org/abs/2201.04741

"By 1983, inflation had become the driving force in cosmology because of the power of its three big predictions: (i) flat Universe (##\Omega_0=1##); (ii) almost scale-invariant spectrum of nearly-Gaussian density (curvature) perturbations;21 and (iii) almost scale-invariant spectrum of gravitational waves (134).

The first prediction made inflation both bold and falsifiable: the observational evidence at the time was ##\Omega_0 \sim 0.1##, and so there must be something in the Universe in addition to baryons"

The article at the link above is a detailed history of modern cosmology by one of giants, Mike Turner, and is technical but largely non-mathematical. I "discovered" it a couple of hours ago.

RogerWaters and PeroK
PeterDonis
Mentor
the observational evidence at the time was ##\Omega_0 \sim 0.1##, and so there must be something in the Universe in addition to baryons"
Yes, but even without inflation, we already knew that the total density had to be more than that. If that really were all that was in the universe, then, as @kimbyd pointed out, no structure would have formed. So while inflation certainly did predict that "there must be something in the Universe in addition to baryons", I don't think inflation was necessary to obtain that particular prediction--just the observation that there is structure in the universe was sufficient for that.

Inflation doesn't "create" the stress-energy density; the stress-energy density is already there during inflation, stored in the inflaton field. The process of inflation, exponential expansion at a huge rate, makes the stress-energy density stored in the inflaton field equal to the critical density--although it does that by changing the critical density, not the inflaton field density (since the latter is constant during the inflation epoch). So at the end of inflation, the critical density is equal to the stress-energy density in the inflaton field, and then that stress-energy density gets transferred from the inflaton field to the Standard Model fields (quarks, leptons, gauge bosons) in a process called "reheating" (which is a misnomer since the Standard Model fields had never been "heated" before that). Since the density was equal to critical before reheating, and reheating happens basically instantly, the density is equal to critical right after reheating.
I've been reading this over a few times. How did inflation change the critical density, such that it become equal to the mass-density of the universe (which was then the stress-energy density of the inflaton field)?

As a side note, you say (and I have commonly read) that the stress-energy stored in the inflaton field got transferred into matter at the end of inflation. Given matter (including dark) represents around 30 percent of the mass-energy of the universe, does this mean only 30 percent of the inflaton stress-energy turned into matter? And similarly did 70 percent of the inflaton stress-energy come to be dark energy (I understand that as space expands, dark energy density remains constant and so I assume its contribution to the mass-density of the universe was there from the beginning)?

Last edited:
PeterDonis
Mentor
How did inflation change the critical density, such that it become equal to the mass-density of the universe (which was then the stress-energy density of the inflaton field)?
Because that's what that kind of stress-energy (i.e., any stress-energy that has the same equation of state as a cosmological constant, with ##p = - \rho##) does. In fact, in de Sitter spacetime, which is a spacetime with a positive cosmological constant and no other stress-energy, the actual density (i.e., the density associated with the cosmological constant) and the critical density are always exactly equal. Any stress-energy with the same equation of state as a cosmological constant rapidly pushes the critical density towards the same equality with the actual density that exists all the time in de Sitter spacetime.

Because that's what that kind of stress-energy (i.e., any stress-energy that has the same equation of state as a cosmological constant, with ##p = - \rho##) does. In fact, in de Sitter spacetime, which is a spacetime with a positive cosmological constant and no other stress-energy, the actual density (i.e., the density associated with the cosmological constant) and the critical density are always exactly equal. Any stress-energy with the same equation of state as a cosmological constant rapidly pushes the critical density towards the same equality with the actual density that exists all the time in de Sitter spacetime.
I'm sure it just comes out of the math that way, but are there any analogies or thought experiments that might help one understand why inflation does this? I am supposing the critical density for flatness to be a mass-energy (or stress-energy) density of the universe (or inflaton field) which balances the rate of expansion such that curvature does not occur (but this might be conflating the critical density for expansion vs collapse with the critical density for curvature). It boggles me that the nature of that stress-energy could change the amount of mass/energy needed for flatness, as opposed to being an input into whether the universe is, in fact, flat or not given a certain expansion rate (if that makes sense) .

Also I edited by last post to include this further question, but probably after you saw it:

As a side note, you say (and I have commonly read) that the stress-energy stored in the inflaton field got transferred into matter at the end of inflation. Given matter (including dark) represents around 30 percent of the mass-energy of the universe, does this mean only 30 percent of the inflaton stress-energy turned into matter? And similarly did 70 percent of the inflaton stress-energy come to be dark energy (I understand that as space expands, dark energy density remains constant and so I assume its contribution to the mass-density of the universe was there from the beginning)?

Last edited:
PeterDonis
Mentor
are there any analogies or thought experiments that might help one understand why inflation does this?
Do you understand why de Sitter spacetime (i.e., positive cosmological constant and no other stress-energy) has the actual density equal to the critical density always? I would suggest thinking about that first. Once you understand that, the idea that that de Sitter state is a "fixed point" towards which any inflation model will drive the universe should be pretty easy to grasp.

Do you understand why de Sitter spacetime (i.e., positive cosmological constant and no other stress-energy) has the actual density equal to the critical density always? I would suggest thinking about that first. Once you understand that, the idea that that de Sitter state is a "fixed point" towards which any inflation model will drive the universe should be pretty easy to grasp.
Nope not at all.

PeterDonis
Mentor
I am supposing the critical density for flatness to be a mass-energy (or stress-energy) density of the universe (or inflaton field) which balances the rate of expansion such that curvature does not occur (but this might be conflating the critical density for expansion vs collapse with the critical density for curvature).
If the only stress-energy in the universe is matter (##p = 0##) or radiation (##p = \rho / 3##), then the two concepts of "critical density" (expansion vs. collapse and zero spatial curvature) coincide. But in the presence of stress-energy with the equation of state of a cosmological constant (##p = - \rho##), they don't. What cosmologists call the "critical density" for our best current model of the universe, which includes a positive cosmological constant, is the "flatness" one, although many cosmologists are not clear about that and will refer to the "expansion vs. collapse" definition without clarifying that that definition doesn't really apply to our actual best current model of the universe. (Believe it or not, Wikipedia actually gets this right in its "critical density" article.)

RogerWaters
PeterDonis
Mentor
Nope not at all.
Mathematically it's easily seen. The critical density is ##\rho_c = 3 H^2 / 8 \pi##, and the first Friedmann equation for the de Sitter case says ##H^2 = \Lambda / 3##. To convert ##\Lambda## to its equivalent density we have ##\rho_\Lambda = \Lambda / 8 \pi##, so we have ##H^2 = 8 \pi \rho_\Lambda / 3##. Plugging that into the ##\rho_c## equation gives ##\rho_c = \rho_\Lambda##.

Physically, the best intuition I know of is the one usually given for inflation models, that exponential expansion dilutes spatial curvature, and the greater the exponential factor the greater the dilution. Since typical inflation models have some 60 or so e-foldings, i.e., an exponential factor of ##e^{60}##, which is huge, we would expect them to hugely dilute any pre-existing spatial curvature. So the fixed point of exponential expansion would be expected to be zero spatial curvature, i.e., actual density equal to critical density. And that is what we see in de Sitter spacetime, which is the fixed point of exponential expansion.

PeroK
Physically, the best intuition I know of is the one usually given for inflation models, that exponential expansion dilutes spatial curvature, and the greater the exponential factor the greater the dilution. Since typical inflation models have some 60 or so e-foldings, i.e., an exponential factor of ##e^{60}##, which is huge, we would expect them to hugely dilute any pre-existing spatial curvature. So the fixed point of exponential expansion would be expected to be zero spatial curvature, i.e., actual density equal to critical density. And that is what we see in de Sitter spacetime, which is the fixed point of exponential expansion.
Right, I read this frequently and understand the intuition as far as resulting in flatness but not mass-energy density being at the critical value. I appreciate now that these are one and the same thing, as opposed to critical mass-energy density being a side effect. However, exponential expansion of, say, a massive balloon may all but flatten local sections of it (to an observer living on the surface) but it won’t change the critical density needed for flatness (I don’t think?)- I guess this is where analogy breaks down and you need to do the physics.

If the only stress-energy in the universe is matter (##p = 0##) or radiation (##p = \rho / 3##), then the two concepts of "critical density" (expansion vs. collapse and zero spatial curvature) coincide. But in the presence of stress-energy with the equation of state of a cosmological constant (##p = - \rho##), they don't. What cosmologists call the "critical density" for our best current model of the universe, which includes a positive cosmological constant, is the "flatness" one, although many cosmologists are not clear about that and will refer to the "expansion vs. collapse" definition without clarifying that that definition doesn't really apply to our actual best current model of the universe. (Believe it or not, Wikipedia actually gets this right in its "critical density" article.)
Thanks for clearing this up. I’m reading ‘heart of darkness’ by Ostriker and Mitton which is in between popular science and actual physics, but closer to popular science - it is a fantastic historical account of the development of LCDM model of cosmology, but the chapter on dark energy is a bit of a dog’s breakfast and suddenly looses readability. Part if this is a sudden shift from focusing on expansion vs contraction critical density to the conditions needed for flatness.

Last edited:
PeterDonis
Mentor
exponential expansion of, say, a massive balloon may all but flatten local sections of it (to an observer living on the surface) but it won’t change the critical density needed for flatness
If we're talking about exact exponential expansion, i.e., a cosmological constant with nothing else present, the critical density and the actual density are already the same. Just look at the math I gave earlier. Note carefully that for exact exponential expansion, the Hubble constant ##H## is literally constant--it has the same value everywhere in the spacetime.

If we're talking about inflation, asuming the scalar inflaton field is not exactly constant with time (or our current dark energy-dominated universe, which is also driving itself towards exact flatness, though much more slowly than inflation did), then the Hubble "constant" ##H## is not literally constant; it does change with time. And that is equivalent to the critical density changing with time. Again, just look at the math: the critical density is the Hubble constant ##H##, with some multipliers that are just fixed numbers.

Perhaps it might be helpful to consider several different cases and look at how the Hubble value ##H## changes with time in those cases:

Closed universe, zero cosmological constant: ##H## starts out positive, decreases to zero, and goes negative (because the universe recollapses).

Open universe, zero cosmological constant: ##H## starts out positive and decreases forever, asymptotically approaching a finite positive value which is determined by the curvature term in the Friedmann equation.

Flat universe, zero cosmological constant: ##H## starts out positive and decreases forever, asymptotically approaching zero.

Flat universe, positive cosmological constant, nothing else present (de Sitter): ##H## is always constant, at the value determined by the cosmological constant.

Universe being driven towards flat; either inflaton scalar field, or positive cosmological constant (dark energy) with other matter/radiation also present: ##H## starts out positive and decreases forever, asymptotically approaching a finite positive value which is determined by the cosmological constant (or the inflaton scalar field in the inflation case).

Now let's rewrite the above in terms of the actual density vs. the critical density:

Closed universe, zero cosmological constant: actual density is always greater than critical density; actual density starts out positive, decreases until maximum expansion, then increases again.

Open universe, zero cosmological constant: actual density is always less than critical density; actual density starts out positive and decreases forever, asymptotically approaching zero.

Flat universe, zero cosmological constant: actual density is always exactly equal to critical density.

Flat universe, positive cosmological constant, nothing else present (de Sitter): actual density is constant, at the value determined by the cosmological constant. This is also the critical density, so actual density is always exactly equal to the critical density.

Universe being driven towards flat; either inflaton scalar field, or positive cosmological constant (dark energy) with other matter/radiation also present: actual density starts out positive and decreases forever, asymptotically approaching the density determined by the cosmological constant (i.e., everything that isn't cosmological constant (or inflaton scalar field in the inflation case) gradually dilutes to zero). Since that density is also the critical density, the actual density and the critical density are driven towards the same value.

Note that, while the flat universe with zero cosmological constant looks different in terms of ##H## than the de Sitter case, in terms of actual density vs. critical density they are the same.

Klystron
Gold Member
Here is Krauss on page 86 of A Universe from Nothing: "[70 percent] is, remarkably, what is needed in order to make a flat universe consistent with the fact that only 30 percent of the required mass exists in and around galaxies and clusters". Here is Brian Greene in The Fabric of the Cosmos: "Since the early days of general relativity, physicists have realized that the total matter and energy in each volume of space - the matter/energy density - determine the curvature of space.... for a very special amount of matter/energy density - the critical density - space will... be perfectly flat: that is, there will be no curvature". Here is Ostriker and Mitton in Heart of Darkness: Unravelling the Mysteries of the Invisible Universe:
Consider also reading Alan Guth's popular science publications:
Amazon carries the former listing in hardcover and paperback. As a professor Guth is quite readable.

RogerWaters