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 respectively.
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 Plane?
(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 them.
* 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.
pozdr boud
(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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