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. WisePrimordial 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.
…And where are the puppies?
If the data indicate a special place, I presume there would be puppies there.
Sometimes you have to read between the lines.
How did you find time to work on this during the election? 🙂
> Once you have that, it’s just a matter of cranking out some monstrous equations.
After looking at the paper I think that monstrous is a monstrous understatement. Are there actually people out there who are going to “read” that?
Can the puppy calculations be simplified by using the dog show groupings instead of the individual breeds? (i.e. the hound group, the working group) Doesn’t this make things easier for first order effects?
e.
Brian: no. But there are people out there who are going to type our equations into a program that compares them against CMB data.
Interesting post, Sean…I will have to take a look at the paper.
But I am wondering the following. Translational invariance says that the laws of physics are the same anywhere in space. But we also invoke Lorentz invariance: the laws of physics are the same in any inertial reference frame. However, the presence f the CMBR seems to imply that there is a preferred reference frame in the universe (the frame in which there is no dipole component to the CMBR).
So how about it – does the universe as a whole have a preferred rest frame? (Not that the laws of physics there need be any different…the CMBR looks a certain way is all.)
I am not so sure about this, but then again who knows.
L. C.
Just curious: If you break a global symmetry you get Goldstone modes. In particular: if you break translational invariance, as happens for instance in a crystal, you get density fluctuations (phonons in the crystal). Do you imagine something similar could happen on cosmological scales by breaking these symmetries? If so, do you have an idea what the Goldstone modes might be ? Somehow the analog seems to be a gravitational wave if you break translational invariance….How about rotational invariance?
John and Erik– There are different levels at which you can break a spacetime symmetry. What we usually think of as “spontaneous symmetry breaking” really refers to symmetry breaking in the vacuum, and that’s when you get Goldstone modes. In cosmology, matter and radiation certainly do pick out a preferred rest frame (or more rigorously, a set of spacelike slices), but the universe is not in the vacuum, so we don’t really think of it as spontaneous symmetry breaking. (But it is still a very good question — what picked out that rest frame over all others?) Likewise, in this paper we are imagining a violation of translational invariance during inflation, which leaves an imprint on the CMB, but the laws of physics (and the vacuum state) are still translationally invariant. Imagine instead a “speck of dust” that was in our Hubble patch during inflation, and left its mark on the perturbations.
Hi Sean,
Have you taken a look at the non-gaussianities for such a model? I.e. the leading non-Gaussian term would be
~ (x-z)/d^2 P_k^2
if we assume delta_1 is some form of expansion in delta, i.e. delta_1 ~ delta_0^2.
This looks suspiciously like the f_NL bound that you might just grab off from WMAP5 results (roughly).
errr, the first equation sohuld be
< delta_0 delta_0 delta_1 > ~ (x-z)^2/d^2 P_k^2
(P_k is the power spectrum)
Don’t we already have tight constraints on these symmetries from tests of the conservation laws they induce via Noether’s theorem? What am I missing?
It is neat in a special way to read, even if I don’t understand the essence of what’s going on, science being done in this special part of the blogosphere.
Sean, thank you for your willingness to post this kind of stuff! I like your political ideas too!
You can keep the puppies. They leave too much biological dirt!
Eugene, you could be right — I wasn’t even thinking about non-gaussianities. I’ll have a look.
Interesting.
But I’m really not too impressed that one can make “impressively precise predictions” with the current hypothesis. The current set of hypothesis gives one so many degrees of freedom that practically anything can be “predicted”. E.g., you want to explain the distribution and velocities of matter in a galaxy. How do you do that in the current regime? Well, you design a distribution of “dark matter” specifically to explain the observations, overlay it on the galaxy, and — surprise! — it predicts your observations with impressive precision. Doesn’t seem very satisfying to me.
The eminent R. Munroe weighs in with a theory of his own:
http://xkcd.com/502/
Two remarks. First on the salesmanship – you can list all the ‘anomalies’ you like, but no-one has found any way of putting them together coherently. Of course it would be great if one found that (say) the ‘dark flow’ was aligned with the ‘axis of evil’ and the possible quadrupole power asymmetry, but this doesn’t seem to be the case (eg ‘dark flow’ is apparently more or less along the galactic plane as we see it…)
Or perhaps you are waiting until the next paper to show that the results of the monstrous equations do exactly that.
Anyway there is one CMB anomaly you don’t mention that seems to _disfavour_ extra sources of non-isotropy – namely the vanishing of the correlation at large angular scales. If some weird symmetry violation were going on at large scales you could naively expect it to boost the correlation not flatten it.
Second, you seem to be implying (eq.2) that the translation non-invariance is characterised by a fixed _comoving_ length scale. So the zone of influence of the ‘speck of dust’ (or piece of string, etc) that sources the violation effectively grows along with the scale factor. While this may be convenient for considering things occurring at one given era, it seems physically counterintuitive. If the new physics that produces translation non-invariance has a certain length scale, which generally implies an energy scale, I would not expect this scale to change hugely with time. This is more or less what you are arguing anyway in comparing the inflationary energy density to the Planck energy: you are thinking that new physics kicks in with a certain fixed physical (not comoving) energy and length scale.
Of course the imprints which it leaves at any given time will grow with the Universe’s expansion, so the actual effect on the calculations might not be very large.
This could have some bearing on the question of whether you need to assume that ‘the violation vanishes after the inflationary era’. If the _physical_ length scale of the violation is fixed, say somewhere round the GUT scale, then it doesn’t have to be turned off by hand – it just gets diluted to insignificance once the Hubble length gets much bigger.