Musings

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

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

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,

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

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

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.)

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).