Computing Our Universe 1999


Can Bias Mask Primordial Non-Gaussianity?
Part III: Statistics and Conclusions

Author: Bryan Gmyrek
Overview:  First, it is important to look at the particle plots at a redshift of 3,
or about 3/4 of the age of the universe ago,
which is the earliest time that we analyzed this data visually.

Compare this to the particle distribution at z=0 (today):
Things to notice:
  1. Compare all of the other (S+,S-,B) models to the Gaussian.
  2. Note the lack of voids in the Skew+ model, this should be expected.
  3. Note the voids in the Skew- model, but note there are not many dense regions.
  4. Notice that the Broad model has characteristics of both S+ and S-.

Unfortunately, these are not the best quality due to the compression
that I used in order to make the files a reasonable size.  But you get the
idea, over the billions of years from z=3 to z=0, gravity acts on the
particles in this simulation and causes them to become more clumped.

As you can see, none of the other models really succeds in mimicking
the Gaussian plot, which is what we want.  Mabye the Skew+ looks
relatively similar to the Gaussian (in the z=0 plot), but the lack of underdense
regions in the S+ plot is an obvious flaw.

Here are three more particle plots that show the biased, sampled version of
the model in the upper left, the sampled in the upper right, the Gaussian
sampled in the lower left, and the full version of the model in the lower
right.

Please compare by eye the upper left (biased S+, S-, or Broad) to the lower
left (the sampled Gaussian).  If our biasing sheme were really masking
primordial non-Gaussianity, then the two plots on the left would be
identical.  However, as you can check for yourself, they are different.
This is our first piece of evidence that biasing is NOT masking the primordial
non-Gaussianity.

SKEW POSITIVE:

SKEW NEGATIVE:

BROAD:

Statistics:

Now on to some statistics to illustrate the effects of biasing graphically.
 

First, we look at the correlation function.  The correlation function is
defind in the Weinberg and Cole paper as:
 

"If you start from a randomly chosen galaxy, this (the correlation function)
gives the excess probability of finding a second galaxy at a distance r."
 

In simpler terms, it is a measure of the density as a function of r.
Here is the correlation function graph for the three models, with
the lower right graph containg all three plus the gaussian plotted
together.

log Xi(r) vs log r

Notice that biasing does not help the models mimic the Gaussian.
Also notice how poorly the Skew Negative model, biased or unbiased,
matches the Gaussian.  One might expect this because the Skew Negative
model does not tend to make things clump, but rather just pushes particles
out of voids, thus there is not much correlation between neighboring particles.
 

Next, we examine the group multiplicity function. This statistic is simpler
than it sounds.  Basically, we assign galaxies to groups using a friends
of friends algroithm (which uses a linking length of .2 times the mean
interparticle separation.)  This basically makes clusters of galaxies.  Then
we plot the # of groups vs. the # of members in the group.  In this way
we can see how many galaxy clusters there are as a function of the number
of members in the cluster.

log Ngroups vs log Nmembers

As you can see, again, the S+ and Broad don't quite do the job, but they
are certainly closer to the Gaussian than the S- case.  It is interesting that
bias actually helps out the S- case a lot, but still not enough to really mask
the primordial non-Gaussianity.
 

Another important statistic is the cumulative "temperature funciton"
This is important because it can be used to extimate the potential depth, or
"virial temperature" and is measured observationally.  We plot here:
log Ngroups (>Sigmav) vs Sigmav .  This is the number density of groups
with 1D velocity dispersion greater than Sigmav .

log Ngroups (>Sigmav) vs Sigmav .

Notice that in this case, biasing maybe helps a little bit, especially in the
S- case, but not very much.  This statistic shows an undesirable aspect
of the S+ model which, at least in the author's opinion, has been the
closest to mimicking Gaussian out of all three distorted models.
 

The last statistic that we will examine is the underdense probability funciton.
This one is a little confusing at first, but it is just basically a measure
of the underdensity as a function of radius.  the strict definition is that the UPF
is the "Probability that the average density in a randomly placed sphere
of radius r is more than 80% below the mean density."  Just remember that
it is basically a measure of the underdensity on different scales.

log P80 vs Radius

Notice here, that biasing helps a lot in the S+ case, but not at all in the others.
However, we have already seen undesirable aspects of the S+ model so, again,
we are led to conclude that bias was not able to mask the primordial non-
Gaussianity.

CONCLUSION

Our analysis has shown that using a monotonic, local biasing
scheme on Skew Positive, Skew Negative, and Broad models
produces results that are distinguishable from the Gaussian model.
 
 

Since Gaussian initial fluctiuations fit the observed data far better
than any of the other models (even when they are biased) there
is no reason to believe that the S+, S-, or Broad models could
have produced the large scale structure that we see today.
 


In terms of how this result relates to the rest of cosmology, this means that, based on our research, there is no compelling reason to believe that theories predicting non-Gaussian initial density fluctuations are correct.
 

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