Seeing neutrinos in the CMB

One of the really cool things about cosmology is the fact that you can look at the sky and use the cosmic microwave background to determine that there must be at least three generations of neutrinos, without having directly observed them in any ground-based experiments.

Well, actually we already knew that there were at least three neutrino generations from ground-based experiments before we had anywhere near the technological capability to determine this from the CMB, but it's arguably still pretty cool. What would inarguably be pretty amazing is if we could use CMB observations to prove that there were further, yet-undiscovered, generations of neutrinos that we could then go and look for elsewhere. Or, for a particular type of person, it might be quite satisfying to use the CMB to prove that there definitely weren't any more than the three generations we already know about, thus killing a lot of other people's pet theories stone dead.

All this is (somewhat) topical because of the release of new data from three experiments measuring the CMB: the Wilkinson Microwave Anisotropy Probe (WMAP) satellite, the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT), that could in principle do something like this. Given my various recent blogging delays, however, I haven't been particularly quick about writing about this, and there are other good discussions elsewhere on the web, especially at Résonaances, that many readers will already have seen. I'm also afraid I don't have time for a layperson-level introduction to the topic right now. But there's a bit of a jumble of information from the different collaborations, some of it mildly contradictory, and in this post I'd like to – partly simply as a note for myself – summarise it all and sort it into a logical order.

In case you'd like to just see the executive summary, it is that – in my opinion – the new data doesn't tell us anything very interesting. For a fuller justification of this statement, keep reading.

Firstly, there's a bit of a mess of CMB data available, which needs a bit of sorting out. The WMAP collaboration released the result of 7 years of observations and discussed the cosmological implications in a paper by Komatsu et al. back in 2010; call this data WMAP7. Their latest data release in December last year, with 9 years of data, is discussed in Hinshaw et al.; call this data WMAP9. ACT released some data in 2011 – call this ACT11 – and then updated this last month to the latest release – ACT13 – using which they obtained cosmological constraints in Sievers et al.  Similarly, SPT had a previous data release in late 2011 (SPT11, Reichardt et al.), updated just after Christmas to SPT12 (Hou et al.).

The cosmic microwave background temperature anisotropy power spectrum as measured by WMAP, ACT and SPT. WMAP9 data are in black, ACT11 data in orange, and SPT11 data in blue. The grey line is the theoretical prediction from the standard $\Lambda$CDM model with parameters fit to WMAP9 alone. Figure from arXiv:1212.5226 About 7 acoustic peaks can be seen, which is pretty awesome in itself; with the newer SPT data it is possible to do even better!

A few complications in the analysis of this data arise because of overlaps in the timings of the various releases. So when Hinshaw et al. use CMB data in their analysis, they mean WMAP9+ACT11+SPT11. On the other hand, Sievers et al. use WMAP7+ACT13, and Hou et al. use WMAP7+SPT12. Of course to be really up-to-date we'd like to see fits to WMAP9+ACT13+SPT12, but none of the three collaborations did this (I'll come back to this later).

Now, in addition to this data from the CMB, in order to be able to place useful constraints on the number of neutrinos, one also needs to input some data from other sources. The most important is a measurement of the Hubble constant $H_0$: here all three collaborations use the measured value of $H_0=73.8\pm2.4\;{\rm km s}^{-1}\,{\rm Mpc}^{-1}$ from Reiss et al., though there is also another more recent measurement with a higher value and smaller error bars (I'm not sure I completely trust the error bars on these measurements, as I might explain some other time).

Then there are also baryon acoustic oscillations (BAO) from large surveys of hundreds of thousands of galaxies: this requires a very careful measurement of a small peak in the correlation function determined from the clustering and at the end of the day all this information is compressed into a single number ($D_V(z)/r_s$; the ratio of an effective distance measure at redshift $z$ to the sound horizon at recombination) characterizing the survey or sub-sample from the survey. This number is what is compared to theoretical predictions.

There are several different measurements of this number at different redshifts, from different surveys: one from luminous red galaxies in the Sloan Digital Sky Survey (SDSS), one from the more up-to-date Baryon Oscillation Spectroscopic Survey (BOSS) and three from different sub-samples of the WiggleZ survey. There's also another low-redshift measurement from the 6dFGRS survey, which Hinshaw et al. and Sievers et al. use, but for some reason the SPT analysis doesn't.

(As an aside: if you think about it, it seems kind of strange to reduce a measured set of coordinates for tens or hundreds of thousands of galaxies to a single data point. The equivalent with the CMB data would be to throw away all the information about the different $C_l$ values and simply keep the location of the first peak, and then use this as one additional data point while fitting the galaxy power spectrum. The reason we don't do it this way is because the CMB is so much better than the messy galaxy survey data.)

Putting all of this information together (in various different permutations) allows us to check the effect on the fit of the model to the data of varying $N_{\rm eff}$, the number of extra relativistic species present at the time of recombination. For 3 neutrino generations, $N_{\rm eff}$ should be $3.04$ (I won't go into the reason for the extra $0.4$ here). Since it is primarily the damped tail of the CMB power spectrum that is sensitive to $N_{\rm eff}$, the SPT and ACT data is what allows us to place tighter constraints on this number – though SPT and ACT are not much use by themselves without the anchoring provided by WMAP at low-$l$.

Also, the back-story is Komatsu et al. found that WMAP7+BAO+H$_0$ (the BAO and $H_0$ that was available in 2010, not the new values) gave $N_{\rm eff}=4.34^{+0.86}_{-0.88}$, which was high enough to get particle physicists all a-flutter, though with rather large error bars – understandably so, since there was no high-$l$ SPT or ACT data.

If you are still with me after this barrage of information, good. Now we come to the more interesting stuff.

The first apparently interesting thing was that Hinshaw et al. initially claimed that the combination WMAP9+ACT11+SPT11+BAO favoured a much lower value, i.e., $N_{\rm eff}=2.96\pm0.36$, which brought the particle physicists right back down to earth. The number looked suspicious though, because simply removing the BAO data jumped it back up to $3.89\pm0.67$, and adding the $H_0$ prior also took it upwards, to $3.2\pm0.35$. And indeed it turned out that the numbers in the first draft of the paper were wrong due to some bug in the code; the correct value for WMAP9+ACT11+SPT11+BAO+H$_0$ is actually $N_{\rm eff}=3.84\pm0.40$ – a bit on the high side, but still perfectly consistent with the "vanilla" expectation. So that wasn't a very interesting thing after all.

In any case, WMAP9 is all very well, but since the real constraining power comes from SPT and ACT, and the significantly improved versions of these data were not included by Hinshaw et al., the two later analyses were probably always going to be more important.  But then we discover that SPT12 and ACT13 appear to be mildly inconsistent with each other: WMAP7+ACT13+BAO+H$_0$ pulls $N_{\rm eff}$ downwards a bit ($2.79\pm0.56$), whereas WMAP7+SPT12+BAO+H$_0$ leaves it pretty much where it was ($3.71\pm0.35$).

The difference between the results from these two experiments is not something to get too worried about just yet, as it isn't very dramatic. A more recent paper on the arXiv shows that replacing WMAP7 with WMAP9 makes the numbers fit together a bit better, but there is still a small discrepancy. The problem could be due to the fact that the two telescopes look at two different (and relatively small) patches of the sky, or that one or the other has some undetected source of systematic error. If it is the latter, my gut instinct – based on not much – is to trust SPT rather than ACT. Either way, the release of the all-sky Planck data next month ought to clear this up.

Another point that the SPT analysis brought up is that combining the BAO and $H_0$ data with the CMB doesn't work very well. In particular, adding the BOSS data point to WMAP7+SPT12+H$_0$ worsens the model fit rather more than it should do (an increase in $\chi2$ of $5.1$ for one additional degree of freedom), so that the $2\sigma$ likelihood contours no longer overlap.

The discrepancy between constraints from SPT12+WMAP7 CMB measurements and from the BOSS BAO measurement and the $H_0$ prior, if $N_{\rm eff}=3.04. The right-hand panel is the most relevant. Figure taken from arXiv:1212.6267.

To reconcile the CMB measurements with the two priors (i.e., to get the blue-green contours and the grey contours to overlap), we would either need a larger sound horizon value, a lower $H_0$ value, or some theoretical modification, such as an $N_{\rm eff}$ larger than $3.04$. Essentially the main reason that the SPT analysis favours a larger value of $N_{\rm eff}$ even to the degree it does is just this: it allows us to deform the blue-green contours sufficiently to get them to overlap with the grey contours and improve the overall fit to the data. That's it. The BOSS data shown in the figure is the most problematic; the other BAO measurements would already involve some overlap of the contours without any further adjustments.

For the benefit of any particle theorists who might be reading this, let me now state my strong Bayesian prior: I think it far more likely that the the error bars have been understated on either the BAO sound horizon measurements or the $H_0$ value than that we have more than three generations of neutrinos! It's actually really nice to see the reason for the higher $N_{\rm eff}$ values in the fits nailed down like this, and one of the several pleasing aspects of Hou et al.

There are similar arguments to be made for the evidence, or lack thereof, for non-zero neutrino masses, but this post is already long enough so I think I might skip them for now.

I should now come back to the original statement I made about the new data not being very interesting. Actually, that was deliberately a little over the top, because I do find the hints of tension between ACT and SPT, and between the BAO and $H_0$ values very interesting, if still inconclusive. But these are probably observational issues rather than theoretical ones. Specifically on the question of whether there are more than three generations of neutrinos, I don't think we have learned anything. The $N_{\rm eff}$ value is still a little bit high, though not more than $2\sigma$ away from the Standard Model expectation. So from one perspective there is no need to add any extra relativistic species or other exotic ingredients to your model; the data does not require it. On the other hand, if you are a particle theorist who particularly likes models with extra neutrino generations and suchlike, the data can't rule your model out either.

So sadly, in this instance it is a case of carry on as you were for now. Neither the eternal fantasist nor the killjoy theory-crusher can take much solace from this. At least until the Planck data is released soon ...