Flatness Problem: Unstable Ω-1 Parameter in Dark Energy Dominated Universe?

[michael879]

Ok, so I was trying to explain the flatness problem and I noticed a flaw in the logic I can't explain.. The wikipedia page (http://en.wikipedia.org/wiki/Flatness_problem) covers the basic problem pretty well, with the added fact that
$\rho \approx a^{-3(1+w)}$
where w=0 for matter, 1/3 for radiation, and -1 for dark energy, and
$a_m(t) \approx \dfrac{2}{3}t$
$a_r(t) \approx \dfrac{1}{2}t$
$a_Λ(t) \approx e^{\sqrt{1/3Λt}}$
for matter, radiation, or dark energy dominated universes. Now for a matter or radiation dominated universe, it is very clear why the (Ω-1) parameter is unstable at 0, since it grows as a power of time. However, if you look at the case of a dark energy dominated universe you find that (Ω-1) goes to 0 exponentially! Since our universe now is 70% dark energy, how can anyone claim its obvious that (Ω-1)=0 is unstable?? Clearly I've used the usual over-approximations so its certainly possible that if you make various assumptions about the history of the universe you can get an answer. However, I can't find a single source that mentions this point, and all they say is that inflation is the only way to fix it! Naively it seems possible that (Ω-1) grew until some point where it went back down to the 0 we observe today right?

 

[PeterDonis]

if you look at the case of a dark energy dominated universe you find that (Ω-1) goes to 0 exponentially!

Yes, but with what time constant? With the dark energy density we have now, it would take much, much longer than the current age of the universe. With the energy density in the inflaton field during inflation, it would happen in a few times $10^{-35}$ seconds (or something of that general order of magnitude).

all they say is that inflation is the only way to fix it!

Because in this respect, inflation works just like dark energy, but with a much, much larger energy density and hence a much, much shorter time constant.

 

[michael879]

Yes, but with what time constant? With the dark energy density we have now, it would take much, much longer than the current age of the universe. With the energy density in the inflaton field during inflation, it would happen in a few times $10^{-35}$ seconds (or something of that general order of magnitude).

Hey Peter, thanks again for the quick response! I get that inflation would do the job quicker, but how do we know its NEVER been higher than 0? From the energy density evolution of the universe (ignoring inflation) I would expect it to go up a little slower than linear, until the dark energy content of the universe kicked in. At this point it would drop exponentially back to 0.

*edit* Just to clarify, I'm not arguing against inflation I know it solves a lot of separate problems. I'm just writing my dissertation right now and I'm trying to justify inflation. And this argument doesn't seem very strong to me right now

 

[PeterDonis]

how do we know its NEVER been higher than 0?

Why do you think we know that? Inflation does not claim it was always zero; it only claims that it was zero at the end of inflation (because exponential expansion had driven it there with a very short time constant).

I would expect it to go up a little slower than linear, until the dark energy content of the universe kicked in. At this point it would drop exponentially back to 0.

But with a time constant much, much longer than the age of the universe. So this model does not predict that we should see $\Omega - 1 \approx 0$ now; it only predicts that we would see that some time in the very, very far future.

 

[michael879]

Hey Peter, thanks again for the quick response! I get that inflation would do the job quicker, but how do we know its NEVER been higher than 0? From the energy density evolution of the universe (ignoring inflation) I would expect it to go up a little slower than linear, until the dark energy content of the universe kicked in. At this point it would drop exponentially back to 0.

Sorry I think you misunderstood me. When I said never I was referring to the post-inflation era. I'm trying to justify inflation so even mentioning it makes the argument circular! There are two scenarios we're talking about:
1) Inflation occurred very early on and drove it down so close to 0 that its still almost 0 now
2) It gradually increased until dark energy became dominant at which point it was driven back down to 0

The second scenario would involve a fairly long period where the density deviates from the critical density, but it would still (potentially) result in it being 0 today

 

[PeterDonis]

When I said never I was referring to the post-inflation era.

I understand that. You're not understanding my response. See below.

The second scenario would involve a fairly long period where the density deviates from the critical density, but it would still (potentially) result in it being 0 today

No, it wouldn't, because, once more, the time constant for dark energy to drive $\Omega - 1$ to zero, given its density in the post-inflation era, is much, much longer than the current age of the universe. So this model, once more, does not predict that we would observe $\Omega - 1 \approx 0$ today. It only predicts that we would observe that in the far, far future.

 

[michael879]

I understand that. You're not understanding my response. See below.
No, it wouldn't, because, once more, the time constant for dark energy to drive $\Omega - 1$ to zero, given its density in the post-inflation era, is much, much longer than the current age of the universe. So this model, once more, does not predict that we would observe $\Omega - 1 \approx 0$ today. It only predicts that we would observe that in the far, far future.

Gotcha! Sorry for the confusion, I've been staring at this stuff for too long... Thanks again for the help :) I guess I should've worked out the time it would take dark energy to drive it to 0

*on a side note, what do you mean by "time constant"? I think that was causing a lot of my confusion. If you just remove the word constant I read it as I think you meant it

 

[PeterDonis]

what do you mean by "time constant"?

The time constant $\tau$ in the exponential $e^{-t / \tau}$, i.e., the time it takes for whatever is exponentially getting smaller to decrease in magnitude by a factor of $1 / e$.

 

[michael879]

Yea sorry, after getting some sleep I realized what you meant :)

 

[michael879]

Hey Peter, one more question. I did the math and for a universe dominated by the cosmological constant,
$\Omega^{-1}-1 \sim e^{-2t\sqrt{\Lambda/3}}$
Which given the measured value of $\Lambda\sim10^{-122}$ in Planck units, produces a time scale of τ~15 billion years which is shockingly close to the age of the universe. I get that this only applies to a universe that's been dominated by dark matter for that long (which we don't think is the case here), and would probably need a few multiples of τ to drive Ω to 1 depending on its initial value. However, I got the impression from your replies that the time scale for dark energy to drive the universe for its critical density was orders of magnitude away from the age of the universe.

Did I screw something up here? Because it seems plausible that some modification of the ΛCDM model (especially if you remove inflation, which is so critical to its success) could extend the dark energy dominated period and the age of the universe by a factor of about 1-10. So I'm finding it hard to make a solid argument that the flatness problem is actually a catastrophic problem for a universe without inflation...

*edit* actually, I just realized it would take about 4τ to get the number to 1% of its initial value. Since we know it is 1.00 to within around 1% that does make it a lot more difficult to explain a dark energy cause (it would have had to been very close to 1 at the time dark energy took over). It still bugs me that it's the same order of magnitude though...

 

[PeterDonis]

I got the impression from your replies that the time scale for dark energy to drive the universe for its critical density was orders of magnitude away from the age of the universe.

$\tau$ based on the current dark energy density should be around 17 billion years (marcus has a number of threads going in Cosmology discussing this value and what it means), but remember that that is just the time for the quantity to decrease in value by a factor of $1 / e$. How many factors of $1 / e$ would it take to reduce the value to something undetectably different from zero? Obviously that depends on where the value started, i.e,. how different from the critical density the actual density could get during radiation-dominated or matter-dominated phases that were not preceded by inflation. We don't really have any good way of estimating that that I'm aware of, since it should depend fairly sensitively on our assumption about initial conditions.

However, you can approach this question from the opposite direction: given the known time constant (about 17 billion years) and the known length of time during which the universe has been dark energy dominated (only a few billion years), how close to critical would the density have had to be at the start of the dark energy dominated era in order for $\Omega - 1$ to be undetectably different from zero now. The answer is, basically, still undetectably different from zero (since you are multiplying an undetectably different from zero value now by a factor of less than $e$, i.e,. a factor of order unity). And since we know that $\Omega - 1$ should increase, not decrease, during the radiation and matter dominated eras, that tells us that $\Omega - 1$ at the beginning of those eras must also have been undetectably different from zero. Which is, of course, just the flatness problem: how did it get that way?

I just realized it would take about 4τ to get the number to 1% of its initial value.

Yes, which means the dark energy dominated era would have had to last at least 17x4 = 68 billion years, whereas it has actually lasted less than one-tenth of that. So my "many orders of magnitude" was an overstatement, yes; but it's still a significant difference.