## March 28, 2005

### Superhorizon Fluctuations and Dark Energy?

There’s been a lot of buzz about Kolb et al’s suggestion that superhorizon fluctuations can mock-up the effect of a cosmological constant (current observations suggest $\Omega_\Lambda=0.7$). I haven’t commented, because the calculations are a bit beyond me. They involve intricacies of second-order perturbation theory about FRW, and an infrared divergence which implies that — even though the amplitude of fluctuations at any individual wavelength is small, $\epsilon=\delta\rho/\rho\sim 10^{-4}$ — if there have been enough e-foldings of inflation, the contributions from all superhorizon modes may be large enough to actually dominate the energy density today.

Éanna Flanagan has a very interesting critique, which is simple enough that even I have a chance of understanding it.

Consider a gedanken-universe in which the initial spectrum of perturbations was such that there are no sub-horizon perturbations today. An observer in such a universe can measure the redshift, $z$ and luminosity distance, $\mathcal{L}$ of nearby events. In a conventional FRW universe, these are related by

(1)$\mathcal{L}(z) = H_0^{-1} z + H_0^{-1} (1- q_0) z^2/2 + \dots$

But, since we won’t assume local isotropy, we have some more general angle-dependent relation,

(2)$\mathcal{L}(z,\theta,\phi) = A(\theta,\phi) z + B(\theta,\phi) z^2 + \dots$

and one reconstructs $H_0$ and $q_0$ as some angular averages of $A$ and $B$. The cosmological fluid has a stress tensor, $T_{\alpha\beta} = (p+\rho)u_\alpha u_\beta + p g_{\alpha\beta}$. One can expand the four-velocity in the usual way,

(3)$\nabla_\alpha u_\beta = \frac{1}{3} \theta (g_{\alpha\beta} + u_\alpha u_\beta) +\sigma_{\alpha\beta} +\omega_{\alpha\beta} - u_\alpha a_\beta$

where $\theta$, $\sigma_{(\alpha\beta)}$, $\omega_{[\alpha\beta]}$ and $a_\alpha$ are the expansion, shear, vorticity and four-acceleration. Assuming matter domination and no dark energy, $p\sim 0$ and hence $\nabla_\alpha T^{\alpha\beta}=0$ implies $a_\alpha=0$.

At this point, Flanagan uses a local Taylor series expansion to compute $H_0$ and $q_0$ in terms of the density and the four-velocity and its gradients. The result is that the Hubble constant

(4)$H_0 = \frac{1}{3}\theta$

measures the local expansion of the fluid and the deceleration parameter,

(5)$q_0 = \frac{4\pi}{3H_0^2}\rho +\frac{1}{3H_0^2} \left[\frac{7}{5} \sigma_{\alpha\beta}\sigma^{\alpha\beta}-\omega_{\alpha\beta}\omega^{\alpha\beta}\right]$

The first term is positive. In a spatially-flat, matter-dominated FRW universe, we would have $q_0=1/2$. Here, our ansatz allows for local spatial curvature, so $q_0\neq1/2$, but, in a spatially-curved, matter-dominated FRW universe, $q_0$ is nonetheless positive. The second and third terms involve the shear and vorticity of the cosmic fluid. Sure enough, we could get $q_0\lt 0$, provided the vorticity is large enough.

But, and this is Flanagan’s key observation, these are local observables of the cosmic fluid. We can estimate them, just knowing typical magnitude of peculiar velocities, and the fact that, in the absense of sub-horizon fluctuations, the scale over which the velocity varies, $l\gtrsim H_0$. The upshot is that these “second-order” contributions to the deceleration parameter, $\delta q_0\sim (\delta v)^2\sim\epsilon\sim 10^{-4}$.

That is, they’re tiny compared to the zeroth-order contribution, and can’t possibly give $q_0\sim -0.5$, to account for the observed cosmic acceleration.

Turning this around, Flanagan observes,

Thus, while an order-unity renormalization of $q_0$ from second order effects is possible in principle, our analysis implies that such a renormalization would also require second order contributions to the fluid velocity that violate observational bounds. (This also implies that the results of [Kolb et al] should yield an upper limit on the number of e-folds of inflation.)

#### Update:

Geshnizjani, Chung & Afshordi have an even shorter paper out today, in which they argue that the entire effect of Kolb et al can be seen to be a renormalization of the local spatial curvature (i.e. that $H_0^2\neq 8\pi\rho/3$). I’m a little confused, as their answer (equation (15) of their paper) is missing a term relative to the corresponding expression in equation (36) of Barausse et al. Might it correspond to the shear and vorticity effects considered by Flanagan? Whether it does, or not, is little relevant to Kolb et al. Their infrared-divergent term, $\varphi \nabla^2\varphi$, is, apparently, part of the contribution to the spatial curvature. So, even if they are right that the infrared divergence enhances its effect beyond the naïve expectaction for second-order perturbation theory ($\sim\epsilon^2\sim 10^{-8}$), it cannot push the deceleration parameter negative. Indeed, since there are pretty good observational bounds on the spatial curvature, this is another way of saying that Kolb et al’s results put an upper bound on the number of number of e-foldings of inflation.

Luboš has some more comments, but he ends somewhat glibly:

At any rate, Éanna assumes locality, and with this assumption, it seems clear that the paper of Kolb et al. cannot be correct without the need for complicated calculations such as those of Éanna.

That’s far from clear. Both papers today find that superhorizon fluctuations alter the expansion rate and deceleration parameter. The question is whether they alter the deceleration parameter enough to push it negative. This clearly can’t happen with spatial curvature alone (Geshnizjani et al). It is, however, technically possible, though it would require unphysically-large values of the vorticity (Flanagan).

#### Update (3/30/2005):

As Aaron points out, Hirata and Seljak do an even more thorough debunking job in a longer (and hence more readable) paper today.
Posted by distler at March 28, 2005 1:42 AM

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### Re: Superhorizon Fluctuations and Dark Energy?

According to Geshnizjani’s paper nonperturbative effects or some pathology in the perturbative series could also in principle push it negative. Thats a little bit troubling.

While im no expert, it seems to me there is a lot of gauge ambiguties at work here

Posted by: Haelfix on March 29, 2005 4:36 PM | Permalink | Reply to this

### Re: Superhorizon Fluctuations and Dark Energy?

Sean expresses the opinion that superhorizon fluctuations should be able to change our local observed quantities, but those should still be related to each other by our local Einstein equations, so in that sense we already have all the information in our local patch. That may be the more precise idea to which Lubos refers as “locality”.

Flannagan paper seems to me, if I understand correctly, to agree with that intuition- after finding the
effect of the superhorizon flcutuations on the decceleration, he found they are also accompanied by other effects (vorticity), which can be then considered to be, in the local description, the “cause” of the acceleration. Of course such a cause is then heavily constrained by observations.

The idea that unobservable physics can influence us is disturbing, one at least hopes it is wrong, for reasons that go beyond a particular calculational scheme.
I am not sure though that such reasoning exists…

Posted by: Moshe Rozali on March 29, 2005 6:50 PM | Permalink | Reply to this

### Re: Superhorizon Fluctuations and Dark Energy?

I don’t think anyone’s arguing that there doesn’t exist a local description. The whole point of the horizon is that things outside of it don’t affect us, after all.

Now I will follow the great tradition of blogging – and many hallway conversations – by commenting on Flannagan’s paper without having read it. My problem is that I don’t understand how we get away with just ditching the pressure term. Isn’t that begging the question?

Obviously (I hope) the matter in the universe cannot cause and is not causing the acceleration. So, if we assume matter domination, we’ve assumed what we wanted to prove, right?

Posted by: Aaron Bergman on March 29, 2005 9:40 PM | Permalink | Reply to this

### Re: Superhorizon Fluctuations and Dark Energy?

So, Hirata and Seljak seem to agree with what I mentioned at lunch – that the Raychaudhuri equations says that, you can’t get acceleration without SEC violation or vorticity.

Now, again speaking from a total lack of knowledge about the paper, I thought that Kolb et al had to have a SEC violating field floating around somewhere. If he doesn’t, then I withdraw all my objections to the debunking.

Hirata and Seljak also claim to have identified the error in the calculation of Kolb et al.

Posted by: Aaron Bergman on March 30, 2005 12:57 AM | Permalink | Reply to this

### Begging the question

My problem is that I don’t understand how we get away with just ditching the pressure term.

I should have written down Flanagan’s expression for the deceleration parameter before setting $p$ to 0.

(1)$q_0 = \frac{4\pi}{3H_0^2}(\rho+3p) +\frac{1}{3H_0^2}\left(a_\alpha a^\alpha +\frac{7}{5} \sigma_{\alpha\beta}\sigma^{\alpha\beta} - \omega_{\alpha\beta}\omega^{\alpha\beta} -2\nabla_\alpha a^\alpha\right)$

If the Strong Energy Condition holds, the 1st term is positive, even if the pressure is nonzero.

So, Hirata and Seljak seem to agree with what I mentioned at lunch – that the Raychaudhuri equations says that, you can’t get acceleration without SEC violation or vorticity.

That much, I think, we agreed upon at lunch, as is pretty plain from the above formula (which, I think, corresponds to Hirata and Seljak’s “$q_3$” definition of the deceleration parameter). They consider several definitions, and ultimately prove some theorems about “$q_4$”.

I really like their paper. Much more readable than the others (perhaps because it’s 4 times as long).

Posted by: Jacques Distler on March 30, 2005 1:37 AM | Permalink | PGP Sig | Reply to this

### Re: Superhorizon Fluctuations and Dark Energy?

Aaron,

I thought the point was exactly that superhorizon fluctuations can influence our measurments, and in particular can induce acceleration (or at least change in q), in the absence of any local source for such effect.

I find this idea troubling, but I don’t think one can throw it away on general principles. On a case by case basis, once well-measured quantities are related through local equations, the game is effectively over, and all nonlocal physics can be safely ignored. Of course one can ask what local quantities can give rise to acceleration (vorticity,violation of SEC etc.) but that is a different question.

Posted by: Moshe Rozali on March 30, 2005 7:42 AM | Permalink | Reply to this

### Re: Superhorizon Fluctuations and Dark Energy?

Superhorizon fluctuations can still act as local sources. Just because the wavelength is longer than the horizon doesn’t mean there’s no stress-energy inside the horizon.

Posted by: Aaron Bergman on March 30, 2005 10:24 AM | Permalink | Reply to this
Read the post Friedmann fights back
Weblog: Preposterous Universe
Excerpt: For those of you interested in the attempt by Kolb, Matarrese, Notari, and Riotto to do away with dark energy, some enterprising young cosmologists have cranked through the equations and come out defending the conventional wisdom.
Tracked: March 30, 2005 10:31 AM

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