Cosmologists find themselves in this interesting situation where they have a set of hypotheses -- dark matter, dark energy, inflation -- that serve to make impressively precise predictions that have been tested against a wide variety of data, but presently lack a firm grounding in established physics. We don't know what exactly the dark matter is, what the dark energy is, or how inflation happened, if indeed it happened at all. So it behooves us to push at the boundaries a bit -- start with the simple models and tweak them in some way, and then check whether the new version still fits the data. How confident are we that the dark sector has the properties we think it does, or that inflation happened in a straightforward way? This was the philosophy that led Lotty Ackerman, Mark Wise and I to ask what the universe would look like if rotational invariance were violated during inflation -- if there were a preferred direction in space, which left some imprint on the cosmological perturbations that currently show up as large-scale structure and temperature fluctuations in the cosmic microwave background. I talked about how that paper came to be in a series of posts: one, two, three. And now there is even tantalizing evidence that our model fits the data! I don't get too excited about it, but it's something to keep an eye on as the data improve (e.g. when the Planck satellite gets results). Ever since then, Mark and I have toyed with the idea that once you've broken rotational invariance, your next step is obvious: violate translational invariance! Instead of imagining a preferred direction in space, imagine there were a preferred place in the universe. Not because you have some good reason to think there is, but because you want to quantify the level of confidence we have in the assumption that there is not. So we have now teamed up with Chien-Yao Tseng, another grad student here at Caltech, to do exactly that. The result is this paper:
Translational Invariance and the Anisotropy of the Cosmic Microwave Background Sean M. Carroll, Chien-Yao Tseng and Mark B. Wise Primordial quantum fluctuations produced by inflation are conventionally assumed to be statistically homogeneous, a consequence of translational invariance. In this paper we quantify the potentially observable effects of a small violation of translational invariance during inflation, as characterized by the presence of a preferred point, line, or plane. We explore the imprint such a violation would leave on the cosmic microwave background anisotropy, and provide explicit formulas for the expected amplitudes $langle a_{lm}a_{l'm'}^*rangle$ of the spherical-harmonic coefficients.
It took a while to put into equations what exactly was meant by "violating translational invariance" in an operational way. But once you figure it out, it's obvious, and there are three ways to do it: imagining that there is a preferred point, line, or plane in the universe. Then you hypothesize that the density fluctuations are very slightly modulated in a way that depends on your distance from that preferred place. Once you have that, it's just a matter of cranking out some monstrous equations. Thank goodness there are only three macroscopic dimensions of space, is all I can say.
So now we have some predictions to compare with data, so that we can understand exactly how well the cosmic microwave background really assures us that there is no special place in the universe. But aside from the general motivation of being careful to test all of our cherished assumptions, there is another reason for work like this: there are a handful of ways in which cosmological perturbations don't look completely the same in every direction. As we say in the paper:
There is another important motivation for studying deviations from pure statistical isotropy of cosmological perturbations: a number of analyses have found evidence that such deviations might exist in the real world. These include the ``axis of evil'' alignment of low multipoles, the existence of an anomalous cold spot in the CMB, an anomalous dipole power asymmetry, a claimed ``dark flow'' of galaxy clusters measured by the Sunyaev-Zeldovich effect, as well as a possible detection of a quadrupole power asymmetry of the type predicted by ACW in the WMAP five-year data. In none of these cases is it beyond a reasonable doubt that the effect is more than a statistical fluctuation, or an unknown systematic effect; nevertheless, the combination of all of them is suggestive. It is possible that statistical isotropy/homogeneity is violated at very high significance in some specific fashion that does not correspond precisely to any of the particular observational effects that have been searched for, but that would stand out dramatically in a better-targeted analysis.
In other words, we have a handful of anomalies, each of which might easily go away, but perhaps when they are taken together they imply that something is going on. Maybe there is some incredibly strong signal out there, and we just haven't been looking for it in the right way. We won't know until we understand better how such anomalies would show up in the observations -- and then go collect better data.
