This is a post I started writing nearly a year ago and have only finally got around to publishing. Part of the reason for this are my doubts about whether I should simply hold back and write it up as a standard paper or not. Since I have not done that yet, I figure it is better to talk about it here and get the ball rolling.
One of the delightful things that often happens in physics is when two things that occupy disparate parts of your brain suddenly turn out to be intimately related (e.g. the Langlands program in mathematics). There are also many lovely examples in physics but one I worked on myself was the unexpected link between parametric resonance in preheating after inflation (see e.g. here or here*) and the band structure of metals. Turns out they are deeply connected because the time-dependent Klein-Gordon equation in Fourier space is mathematically equivalent to the time-independent 1-D Schrodinger equation with the interchange of time and space. The conduction bands of metals turn out to have eigenvalues that are dual to the wavelengths at which there is no resonant particle production during preheating. This means there is even a cosmological version of Anderson localisation and other exotica (see here and here) which I think is pretty cool.
Anyway, there is another example that I think is very interesting and which I have not seen discussed anywhere in the literature (though I may be wrong of course), other than a rather badly written conference proceedings that I put together during the first year of my PhD, available here (which may even be wrong in places, I confess I haven’t checked since reading ones early work is like going to the dentist). Since this still makes me smile I thought I would write a bit about it on the off chance that it may turn out to be a useful in some way.
To put the idea crudely: galaxy bias is intimately linked to isocurvature purturbations.
We usually define the bias between two fields (i,j) to be the (in general spacetime-dependent) factor , that relates their dimensionless density fluctuations: . Typically by the word “bias” we mean the bias factor between the baryon and dark matter fields, or perhaps between the number density of galaxies and the dark matter distribution.
So, to make the link more precise, note that a space and time-independent bias of unity, for two pressureless fluids, is precisely what relativistic cosmologists would normally call an adiabatic perturbation: the two fields go up and down in exactly the same way at each point in space and time.
If there is a relative isocurvature (also known as an entropy mode for added confusion) perturbation between two fields, then the following quantity does not vanish:
or its gauge-invariant generalisation (see e.g. the KS classic). Here as usual.
Hence we see that if we have a constant bias factor of . An immediate corollary of this is that if the bias is time or scale dependent, then it corresponds to a relative isocurvature perturbation for which a precise and very detailed theory in many different contexts exists (see e.g. here). I think it is pretty cool that these widely used concepts in two very different sub-communities are actually the same thing.
Why is this link not commonly discussed? I think that partly the reason is that galaxy bias is something observers tend to think about while isocurvature perturbations are primarily something theoretical cosmologists think about. There are deeper reasons too I expect. Galaxies are highly nonlinear objects which we then use as tracers of the underlying linear dark matter density field, so the link to standard linear cosmological perturbation theory is not clear, at least to me. Also, we are typically interested in the bias on intermediate and small scales, where linear perturbation theory has broken down.
Nevertheless, we can immediately see that when isocurvature modes decay (as happens in the standard LCDM model) we asymptotically find that the bias goes to any time the two fluids have the same equation of state (e.g. for baryons and CDM) and hence will become scale-independent too, a standard result. In the more general cases however, the bias between the two fluids will approach (1+w_i)/(1+w_j). However, note that on small scales the bias cannot be scale-independent even in linear theory since one cannot have exactly adiabatic fluctuations (since adiabatic modes source isocurvature modes on small scales).
Interestingly, in the case that the bias is scale-independent but is not unity, there is still an isocurvature perturbation in general, it is just highly correlated with the adiabatic mode since they are simply proportional to each other (). In the case where the bias is scale-dependent the correlation between the adiabatic and isocurvature modes degrades. It may be possible to use this in future, e.g. with the SKA and LSST to good effect by following the HI and dark matter separately.
What is interesting about this approach is the hope that one could use some of the machinery from multi-fluid perturbation theory at first and second order to make theoretical predictions for bias, especially in the case of new scenarios (e.g. dark energy – dark matter bias). This might be particularly useful for understanding the effects of bias on Baryon Acoustic Oscillation measurements which should be pretty linear on 100 Mpc scales.
Of course we mostly want to know about galaxy bias on nonlinear scales. Even here there is some hope. There is a very elegant formulation of “perturbation theory” in Relativity which is fully nonlinear (the covariant approach by Ellis and Bruni) and which has been worked out for multi-component fluids in this extensive review (section 2.4)**. This may allow for a fully nonlinear formulation of bias.
Is this link between bias and isocurvature modes useful otherwise? Well I had one insight that came from this way of thinking. I was considering the multi-tracer approach to reducing cosmic variance and trying to understand how it works. I realised that it works only when the different tracers are perfectly correlated with the dark matter, so that knowledge of the clustering of one gives knowledge of the clustering of the other. In this case we have three fields: the dark matter (D), and two tracers . Then the three isocurvature modes all satisfy .
I believe this method of removing the cosmic variance error will fail as soon as there is a general isocurvature mode between the two tracers or between the tracers and the dark matter, since then one cannot use knowledge of one to predict the other and hence cosmic variance could affect the tracers differently to each other and/or from the dark matter. Of course, how would we know if this true? How plausible such isocurvature modes are is difficult to access, but it would be interesting to find out. I would certainly want it checked out before I believed a claim of detection of primordial non-Gaussianity using this method. Perhaps this is a good project to raise at the upcoming SuperJEDI workshop…
* Not at all a representative list of references!
** UPDATED : 21/5/13 Thanks to Roy for this update which I wasn’t aware of.