1. Looking for non-Gaussianity Going beyond fNL
Robert Crittenden
Institute of Cosmology and Gravitation
University of Portsmouth
Work with D Pietrobon and D Wands

2. Characterizing maps of the sky
We usually focus on the power spectra, or two point moments, for describing maps of the CMB or galaxy distribution. 
As long as the maps are Gaussian, this is a great way of compressing the data, as the power spectrum contains all the information within the map. Most cosmological constraints are derived from the power spectrum alone.
However, you could easily be losing information if the maps are non-Gaussian in some way.

3. Beyond the power spectrum
While power spectra are good for learning about linear processes and seeing large signals, they can miss out on information if things become non-linear in some way.
We want to search for non-Gaussianity to discover more about possible non-linear physics, either in the early or late universe.
Unfortunately, there are innumerable ways that things can be non-Gaussian. Which do we look for?
What is the best way to look for non-Gaussianity?
Assuming we see signs of non-Gaussianity, how are we to interpret them, given that there are many possible sources of non-Gaussianity?
What other sources of non-Gaussianity might exist?

4. Apparent non-Gaussianity
Just looking at a given map, strange features might be seen, which can be might be evidence for non-Gaussianity.
Many anomalies in the WMAP data have been seen:
Asymmetry in power in North-South hemispheres
Axis of evil – alignments of low multi-poles
WMAP cold spots, hot spots?
Lack of correlations beyond 60 degrees?

5. WMAP anomalies North-South asymmetry – Civil war?
The power seems to be asymmetrically distributed between different hemispheres, for many different ranges of multipole.
The hemisphere choices which maximise this asymmetry are well aligned.
Hansen et al 2008.

6. WMAP anomalies ‘Axis of Evil’
With a particular choice of frame, most of the power can be focused within a single m value. (m=3 for these.)
What is more, the frames where this happens for the lowest multipoles (l=2-5) seems to be very highly aligned.
Land & Magueijo 2005

7. WMAP anomalies Anomalous cold and hot spots?
Wavelet analyses have noted regions which seem colder or hotter than typically expected from Gaussian realisations.
Cruz et al 2005, Pietrobon et al 2008

8. WMAP anomalies No large scale correlations?
There appears to be no large scale correlations beyond 60 degrees. This is especially true for the cut sky maps, where the foreground contamination is minimised.
While this is not necessarily a sign of non-Gaussianity, it seems to require particular cancelations of the various multipole contributions.
Spergel et al 2003, Copi et al 2008

9. Apparent non-Gaussianity
Just looking at a given map, strange features might be seen, which can be interpreted as evidence for non-Gaussianity.
Unfortunately, the significance of these is hard to interpret, as any given realization of a Gaussian map is bound to have some atypical features.
If we only look for things based on what we have seen, there is an inevitable a posteriori bias in the detection. We notice weird aspects of the maps, so it is not surprising that those things are unlikely in random maps.
There is also a reporting bias, because people may do many tests but not find non-detections interesting enough to report. But if you do enough tests, eventually you will find one which appears significant!
If you don’t know how many things were looked at, it is hard to judge the weirdness of those that stand out. How many ways are the maps actually ordinary?

10. But still useful: COBE example
Strange features may give indications that something is amiss with the data and it is worth looking for its origin.
NG was seen even in the COBE satellite data, in the bispectrum at low multipoles (Ferreira, Magueijo & Gorski, 1998).
This NG was never fully understood; however, Banday, Zaroubi & Gorski (1999) showed that when certain parts of the time stream data were removed, data originating when the satellite passed through the Earth’s shadow, the NG went away.
NG was thus useful for determining that some of the data was corrupt, so it was apparently a good indicator of systematic problems.

11. Targeted non-Gaussian searches
It is generally better to search for those types of non-Gaussianity which you have a reason to expect, either theoretically or through possible experimental systematics.
Many sources of non-Gaussianity are expected:
Inflation – local vs equilateral (DBI) types; defects? early magnetic fields?
Reheating or non-linear pre-heating
Recombination and reionisation
Non-linear structure growth
Gravitational lensing
Foregrounds (point sources, SZ, etc.)
Experimental effects
If we understand these, we can target them directly!

12. Local fNL non-Gaussianity
Non-linearities during inflation can produce a particular type of non-Gaussianity which is called a local-type, since it is local in real space.
Shape of NG:
In real space:
It is easy to see the this is NG, because it has a non-zero three point moment, arising from the four field expectation:

13. Measures of fNL
Much excitement has arisen from the recent measurements by Yadav and Wandelt, but these have been disputed.
Various groups have attempted to measure this with the WMAP CMB data:
27 < fNL < 147 (95% CL) Yadav & Wandelt WMAP3 data
-9 < fNL < 111 (95% CL) Komatsu et al. WMAP5 data
-178 < fNL < 64 (95% CL) Minkowski functionals
-4 < fNL < 80 (95% CL) Smith et al. Optimal
-50 < fNL < 110 (95% CL) Pietrobon et al. Needlet
-18 < fNL < 80 (95% CL) Curto et al. Wavelet
Large scale structure observations of the scale dependence of the quasar have recently given independent indications:
-29 < fNL < 70 (95% CL) Slosar et al. LSS data

14. Bispectrum of the field
The presence of the NG causes the three-point moment (called the bispectrum in Fourier space) to be non-zero.
First contribution is from four fields:
Since the power spectrum blows up at small k
Peaks for squeezed triangles in harmonic space k2 k1 k3

15. CMB bispectrum
We need to translate the predictions for the field into something we can actually measure, like the CMB maps or the density field.
For the CMB, we deal with the spherical harmonic amplitudes rather than the Fourier modes. Generically, assuming rotational invariance, the three point moments can be written as
Here, we have used the Wigner 3j symbols. The CMB angular bispectrum is related to the potential bispectrum by projecting and integrating over the CMB transfer functions arising from Boltzmann codes.
The bispectrum of the field for the local type NG given above can be translated into a CMB bispectrum, with various Bessel integrals and Tl (k) to account for the CMB physics.

16. Optimal estimators
Generically we want an estimator of the degree of non-Gaussianity which is unbiased and has the smallest error bar.
There are many techniques, usually based on three point moments (in real, Fourier or wavelet space) or on geometrical measurements (Mikowski functionals).
The most common are based on the bispectra. Any given bispectrum measurement can be used to estimate fNL, but as the individual measurements are very noisy, we usually add them all together.
Since we have an expectation for what the signal should look like, we can use a matched template technique:
Here, the indices span the possible data points (individual bispectra), t is the theoretical expectation or template (fNL =1,) d are the observed bispectra and C is the covariance between measurements.

17. Optimal estimators
In the case of data over the whole sky and isotropic noise properties, this is relatively straight forward.
Here the covariance between bispectrum measurements takes a simple form related to the CMB power spectra:
So the estimator takes a simple form,
Komatsu et al. further showed that this could be calculated quickly for the local NG type, by simply multiplying three maps together, which included a Wiener filtered estimate of the primordial potential field.

18. Nearly-optimal estimators
Unfortunately, in the real world the noise is not isotropic and large parts of the sky are either unobserved or dominated by foregrounds. In this case, calculating and inverting the covariance between bispectrum measurements is hard.
Cremenelli et al. found the variance of the estimator in this case could be improved by adding a linear correction to the cubic estimator.
Most initial measurements have been made using this corrected estimator, but using approximations to the true bispectra covariance matrix to speed up the calculations. This however results in expanded error bars.

19. Back to optimal
Recent work by Kendrick Smith et al. has attempted to do the full calculation, by finding fast methods to calculate the true covariance matrices.
This had led to the best constraints so far,
-4 < fNL < 80 (95% CL) Smith et al
While there is still plenty of room for non-Gaussianity, there is no strong evidence as yet.
Potential differences with Yadav & Wandelt 2008 could be due to additional data (3 year to 5 year), different mask choices, and foreground cleaning.
Smith et al. largely can reproduce their results, though differences at high multipoles.

20. Could these estimators be improved?
These are the optimal three-point statistics, but could things improve if we look beyond, to four point statistics or higher?
While it appears that there may be more information in the higher order moments, or perhaps in other tests like the topological Minkowski functionals, the three point estimators already have the minimum variance possible.
Cramer-Rao bound
The full likelihood could be calculated in principle by inverting the maps to solve for the underlying Gaussian field. This would provide a likelihood function for fNL and through its curvature, we can discover how well any estimator might do, known as the Cramer-Rao bound.
While the full likelihood calculation is difficult, its expectation has been calculated and the three point estimator and other techniques saturate this bound. (Babich 2005;Creminelli et al. 2007)

21. Interpreting the results
How can we be sure when we measure something, that it is really the kind of non-Gaussianity we were looking for?
One problem is that the optimal approaches only provide one number, and simply looking at this one has no confirmation that the NG you have assumed is actually the shape of NG on the sky.
It could have a much different shape, which just happens to respond to the estimator you tried. Potentially if you tried the correct shape, you could get a much more significant detection.
One really needs more estimators:
For a range of different kinds of non-Gaussianity.
Multiple (sub-optimal) estimators of the same type of NG, using different data, to provide cross checks and confirmation of a detection.

22. Overlapping non-Gaussianity
Even if its tuned to a particular type, any estimator may respond to a variety of sources.
In the isotropic case, the expectation of the estimator if there is a different source for the non-Gaussianity is
If you know how orthogonal various sources are, this estimator can be used to estimate different kinds of NG, but it could be far from optimal.
The overlap amplitudes have recently been calculated in a nice paper by Ferguson & Shellard (archiv: ) for a large number of types of NG, including local, equilateral and warm inflation models.

23. Overlapping non-Gaussianity
Ferguson and Shellard have recently calculated the overlap for a few families of intrinsic non-Gaussianity and began to develop a basis for NG types.
Some kinds of NG are highly correlated with each other, so similar estimators can be used with nearly optimal results.
Also edge dominated models.
DBI
Local

24. A new flavour of non-Gaussianity
Its worth searching for alternative sources of non-Gaussianity, from early and late universe, or systematic sources.
One type, motivated by multi-field inflationary models, has a separate field adding quadratically to the potential (e.g. Alabidi & Lyth, 06):
This new field is assumed to be Gaussian and uncorrelated with the other field for simplicity. It may have a significantly different spectral index from the main field, and deviate significantly from scale invariant.
In addition, these perturbations could be either adiabatic or isocurvature, so the transfer function could be significantly different.
Because they are not correlated with the primary field, the limits are expected to be much weaker than in the usual local case.

25. Curvaton example
Such non-Gaussianities can arise in models with multiple fields, such as the curvaton model, where there is a curvaton as well as the usual inflaton.
In the δN formalism, where N is the number of e-folds, one can expand the curvature perturbation as a Taylor series in the inflaton and curvaton perturbations,
Some of the terms involving the inflaton could be suppressed by slow roll conditions; if the inflaton dominates the linear contribution, we get this type of non-Gaussianity. Otherwise, it leads to the local type.
The curvaton can produce a combination of adiabatic and isocurvature perturbations.

26. Normalisation of second field
Expressed like this, FNL and the amplitude of the χ field are degenerate. In order to make contact with the usual case, we assume that ϕ and χ have similar power spectrum amplitudes:
These may have different spectral indices, so we normalise on a particular scale. We use a scale comparable to the one on which they are observed today.
In this way, comparable FNL and fNL imply comparable amplitudes of non-Gaussianity (though not necessarily comparable bispectra.)
Again, the second field could be adiabatic or isocurvature, or some combination of both.

27. Three point moment
It is easy to calculate the three point moment in this case, which is dominated by a six field term rather than a four field term:
Thus there are three power spectra appearing rather than two, and a convolution over momentum which complicates things.
Surprisingly, perhaps, if one assumes a power law form for the power spectrum which is sufficiently red, one can do the convolution producing an expression very similar to the local NG case.
In this limit, the usual estimator (with more freedom allowed for spectral index and transfer functions) might be good enough.

28. Three point moment
Three point is very similar for red spectra (n < 1), but changes can be significant for bluer spectra.
Local
Ratio
Additional field

29. Four point moment
Similarly, the connected four-point moment is dominated by an eight field term rather than a six field term.
Uncorrelated case:
Local case (Byrnes, Sasaki & Wands):

30. Order of magnitude arguments
Power spectrum:
Bispectrum upper bound:
These both imply:
Trispectrum upper bound:
For the same bispectrum, there could be a larger trispectrum!

31. Do these dominate?
If there is some subdominant linear contribution of the second field, it is correlated and may dominate the non-Gaussianity, looking like a local non-Gaussianity.
How big could the linear contribution be without it dominating?
Assume:
Three point:
Four point:
Same condition as for the three point!

32. Optimal estimator
Easy to right down, but hard to calculate in practice!
As we know the exact three point moments we expect, it is straight forward to project this and wrap on the CMB transfer functions to find the CMB bispectrum prediction.
Thus, we can use the same matched filter technique as before to calculate the optimal estimator. We can even use the same trick of multiplying maps in real space!
However, the extra convolution still remains, which seems to make the calculation of the optimal estimator very expensive to calculate, even assuming a simple covariance matrix.

33. Looking for shortcuts
While the ideal calculation is difficult, there may be a nearly optimal solution which is tractable.
For example, in the equilateral NG, seeded by DBI models, the expected NG does not factorise, so the Komatsu-Spergel trick of writing the estimator as a product of three maps doesn’t work.
In order to use this fast method, Creminelli et al. found a form which factorised which nearly matched the theoretical expectation, and based their estimator on that instead.
We are looking at a number of approaches:
Understand what present estimators imply for our NG
Simplify the form using approximate shapes, like the local type.
Focus on those geometries where the effect is biggest.
Other groups have been trying to approach this using Minkowski functionals.

34. Overlap with fNL estimator
First approach is to calculate the overlap:
In principle this is straight forward to calculate analytically, since we have ‘simple’ expressions for both bispectra, and this is what we are now calculating.
The other approach is to make simulated maps of our non-Gaussianity, and apply the usual estimator to them. This allows us to try to take into account more of the systematic issues, like non-isotropic noise and point source contamination.

35. Early magnetic fields
Other sources of non-Gaussianity include magnetic fields from the early universe, which might be created by inflation or cosmological defects.
Even Gaussian magnetic fields have very non-Gaussian densities, and so even if their effects are sub-dominant in the power spectrum, they could dominate the non-Gaussian signal.
We are presently working to propagate these NG’s into the CMB, and find tests which are tuned to it.
Also contain interesting tensor and vector modes.
(Brown & Crittenden, in preparation.)

36. The way forward?
Power spectra are passé. Non-Gaussian features have significant power to probe the most interesting non-linear physics and should be exploited.
Its better to try to search for things which you might expect than to blindly search for strange features, as these signals can be difficult to interpret. However, the latter can be useful to discover the unexpected!
We need to develop a range of tests for different types of non-Gaussianity, so we can tell which explains the observations the best.
Starting to make progress on two fronts:
Additional scalar fields (curvature or isocurvature).
Early universe magnetic fields.
Non-Gaussian tests are not orthogonal, so we need to understand the degree of overlap of different estimators.
We need a way of confirming what these tests are telling us, so we should develop multiple independent tests for a given type of non-Gaussianity.

37. Local fNL non-Gaussianity
Non-linearities during inflation can produce a particular type of non-Gaussianity which is called a local-type, since it is local in real space.
Heuristic local NG explanation:
In inflation, the amplitude of the quantum fluctuations depends on the Hubble constant.
As a mode leaves, it produces small fluctuations in the local Hubble constant, which in turn determine the amplitude of the later fluctuation spectrum.
Thus the power on smaller scales depends on local large scale variations of the density.
This only creates small NG (of order 1); other mechanisms are required to get large enough NG we could detect today. (E.g. curvaton, new ekpyrotic scenarios.)

38. Local fNL non-Gaussianity
Large modes affect the local Hubble expansion. Since the quantum perturbations depend on the Hubble expansion as the modes leave the horizon, this couples small perturbations with large ones.
Small perturbations in the background density
k1-1 separation
H-1 patches
k2-1
k3-1
This perturbed background helps determine the power generated later, on smaller scales.