[Cosmo-torun] 16.00 Fri 20 March - Boud Roukema - the missing fluctuations problem + hot pixels

Boud Roukema boud w astro.uni.torun.pl
Sob, 21 Mar 2009, 01:43:53 CET

hi cosmo-torun

   Next talk: Agnieszka will talk about something (maybe a journal
   article?)  next week 16.00 Fri 27.03.2009.

In this message, i'll try to clarify what got confused about Table 2
of Copi et al. 0808.3767. Partly this was my fault for not
understanding what they did carefully enough. As it turns out, what
they did is simpler than what i thought, but the results are
consistent with what i summarised.

In short, the statistical isotropy in independent spherical
harmonics assumption is at least violated for 2 \le l \le 5
if we take the "best estimates" based on those assumptions;
and if we use the best-fit infinite flat model instead of
the "best estimates" directly, then we require l=2,3 to violate
the same assumptions and extremely low C_2, C_3 values.

Either from the observations "directly" or from the best-fit
model, the statistical isotropy in independent spherical
harmonics assumption is invalidated for either 4 or 2 l values

Details of how Copi et al. get to these conclusions:

On Thu, 19 Mar 2009, Boud Roukema wrote:

> Witam cosmo-torun
> On Wed, 18 Mar 2009, Boud Roukema wrote:
>> hi cosmo-torun,
>> It's my turn to give a talk this Friday @16.00.
> i'll present and stimulate a discussion about:
> (1) the missing fluctuations problem - Copi et al.
> http://arXiv.org/abs/0808.3767  - new paper!

http://arXiv.org/abs/0808.3767v1  - let's look at v1, in case later
there is v2 and someone reads this email on the mailman archive.

Table 2  page 8.

The infinite, flat, gaussian-fluctuations-in-spherical-harmonics model 
says that the different l's are statistically independent. People often put 
error bars on the different C_l's and show a plot of the angular
power spectrum with infinite-flat model vs observations + error bars, and
then it looks like only the quadrupole is significantly in disagreement
with the model.

A key point of Copi et al. is that this interpretation is correct
*only if* the statistical assumptions are correct. If the statistical
assumptions are wrong, then the error bars cannot be interpreted as
being independent from one another. Since the most common
presentations of this data assume statistical independence and
gaussianity in the spherical harmonics to extrapolate from high
galactic latitudes to the "galactic plane" (e.g. KQ75 mask), these
presentations are based on wrong assumptions.  So what can be done to
get a valid statistical statement?

One answer is in Table 2 (bits in Table 1) and the accompanying
discussion. Here's a simplified explanation.

Let us *assume* that statistical isotropy and independence of C_l's 
are correct assumptions.  (*)

(1) Consider the line labelled "WMAP". Let us suppose that the "best"
estimates of the C_l's are correct. Using assumptions (*), we have 
Eq.(3) and Eq.(6) of Copi et al., i.e. we can analytically calculate
S using these C_l's.  For example, this gives  S = 8833 muK^4  in the
2nd last line of Table 1.  Comparing this to 1152 muK^4 says that 
these C_l's have a lot of "power" in the galactic plane, which is 
the most suspicious part of the data set.   But maybe just a few C_l's
are "biased" by the galactic plane, and could be "removed" ?

(2) Now, since we consider the galactic plane area to most likely to
have systematic error, we decide to violate (*) for the lowest l's,
and we find out how many C_l's can be chosen in a correlated ("tuned",
"dependent") way in order for the full set  of C_l's (tuned + untuned)
to give an S that is consistent with the S for the high galactic
latitudes (1152 muK^4).

Naively, it might seem that we can set C_2 = 0, C_3 = 0, ..., C_i = 0
as "tuned" values, and then calculate S using the C_{i+1}, ..., C_8
untuned (same as the same "best" estimates), and we should get a lower
S. However, Copi et al decide to allow even more "tuning", since it
might happen that negative or positive "tuned" low C_l's are better
than zero values for "cancelling" the high C_l's in order to get low
S. [The P_l(cos theta)'s include negative values.] So they allow arbitrary
violation of statistical isotropy/gaussianity in just a few C_l's,
in order to try to "save" the statistical isotropy model.  Then it is
possible to argue that just a few C_l's are extreme events, and the other
C_l's are OK.

(3) Let's start with the first column, still in the row "WMAP" of Table 2.
So the first number is 8290 muK^4.  This means that allowing arbitrary
values for C_2, either zero or negative or positive, it is *not* possible
to get S anywhere near  1152 muK^4.

(4) Next column, row "WMAP", we have 2530 muK^4. This means that for
some "tuned" values (NOT given in the table) of C_2 and C_3, we can
get S calculated from C_2, ..., C_8, ... C_infinity (value of
"infinity" is probably not stated in the paper?) to be 2530 muK^4, but
not any lower.  This is still more than twice the high galactic
latitude value of 1152.

(5) Continue to columns C_4, C_5. It is only after we have "tuned" all
of C_2, C_3, C_4, and C_5 *arbitrarily* in order to minimise S, that
we can get the integral for C_2, ... C_infinity (which assumes (*)
statistical isotropy) to give something as low as 1152 muK^4.  In
other words, we need a very special relationship between C_2, C_3, C_4
and C_5 in order for these plus the higher l observational C_l
estimates to give a value of S \le 1152 \muK^4.

Informally, we could say that this shows an "amplitude alignment"
between all these four multipoles and the galactic plane, not just the
quadrupole and the octopole with each other. This is not an alignment
in orientation, it's a tuning of the amplitudes C_l, so "amplitude
alignment" is probably a bad way to say this, but let's use this just
for this discussion, where "amplitude alignment" is temporarily
defined to mean the procedure done to get Table 2.

Is it cosmologically reasonable to claim that the different C_l's are
statistically independent from one another but that C_2, C_3, C_4, and
C_5 are all "amplitude aligned" with each other and the Galactic

(6) Now let's go to the line "Theory". This says that if we use the
"best-fit" infinite flat model and then forget the fact that the
actual measurements should be closer to the truth than the model,
we do not have to do as much "tuning". It is enough to tune just
C_2 and C_3, i.e. we get 922 muK^4 if we can choose some arbitrary
values of C_2 and C_3 and then use C_4, ... C_infinity from the
best-fit model (forgetting the measured values).

So, we could say that we only need to "amplitude align" the quadrupole
and octopole in order to get S \le 1152 muK^4, if we accept the
best-fit infinite flat model and then forget about the measured
values. However, i think what is meant in paragraph 3, page 7,
2nd column, is that the two C_l's which give this minimisation of S are 
6 C_2/2\pi = 149 muK^2 and 12 C_3/2\pi = 473 muK^2.  (i don't
see any other reasonable interpretation of the sentence - i think it's just a
typo between "Table 2" and "Table 1".)

So, in order to "amplitude align" the quadrupole and octopole in order
to get S \le 1152 muK^4 using the best-fit infinite flat model, the
l(l+1)C_l/(2\pi) 's we get are 149 and 473 muK^2, as the true values.
For l \ge 4 we have independence of C_l's and gaussian distributions,
but for l=2,3, we have a special selection with these two values.
These are very low values, see e.g. the last line of Table 1 - they
should be about 1207 and 1114  muK^2 according to the infinite flat
model with independent gaussian distributions in the spherical 
harmonics, not 149 and 473 muK^2.  (Look at these on a typical C_l
plot, e.g. Figs 3,4 Caillerie et al. arXiv:astro-ph/0705.0217.)

So now let's get back to the question of statistical isotropy and
statistically independent l's. Either for the observational estimates
or the best-fit infinite flat model "estimates", we need to choose
special values of the C_l's in order to get  S  \le 1152 muK^4.

This means that representing the WMAP sky map in terms of statistically
independent C_l's gives a representation of the data inconsistent with
statistical isotropy - i.e. the galactic plane should not be special.

One key point in Copi et al's conclusion is that they recommend that
any attempted physical or statistical error or other explanation of
the "low l" problems in the WMAP data should better concentrate on
explaining the nearly zero two-point correlation function for theta >
60 degrees, rather than trying to explain the large length
scale/angular scale problems in terms of low l spherical harmonics,
since use of the low l spherical harmonics requires assumptions which
are inconsistent with the properties of the observational data.

A few things which may help if you are going to re-read this and
the relevant bits of the paper in order to get things clear in
your mind:

* Table 2 does *not* show the tuned (correlated) C_l's which give
the minimal S values. The table *only* shows the S values. One comment
in the text (see above) gives what seem to be two C_l values in one
case. Probably the table would have been clearer if C_l values had
been listed too. In principle, it should not be difficult to calculate

* The sum in Eq.(3) and the integral in Eq.(6) are taken over theta,
*not* over 4 pi steradians. So it is invalid to say that the S values
for the full sky and the cut sky calculated using Eqs (3) and (6) 
from the full sky and cut sky C_l values should be different unless
statistical isotropy is violated.

* If we consider the galactic plane region to have a serious
systematic error (for the obvious reason), then we do not have a
violation of statistical isotropy, and we don't have a
quadrupole-octopole-ecliptic-plane alignment problem ("not readily
testable"), but we do have C_theta which is unusually close to zero
given the infinite flat statistically isotropic gaussian,
k^1 spatial power spectrum model as a whole.  In other words,
rather than a violation of statistical isotropy, we have 
"The New Isotropy Problem" - the COBE and WMAP skies are *too*
isotropic on scales greater than one present-day matter-horizon
radius, i.e. 60 degrees in angular scale.


> +
> (2) the hot pixel correction for WMAP data - Aurich et al.
> http://arXiv.org/abs/0903.3133 - thanks to Bartek for
> pointing me to this :)
> pozdr
> boud
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