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 [1004.3995] Induced CMB quadrupole from pointing offsets
 Authors: Adam Moss, Douglas Scott, Kris Sigurdson Abstract: Recent claims in the literature have suggested that the {\it WMAP} quadrupole is not primordial in origin, and arises from an aliasing of the much larger dipole field because of incorrect satellite pointing. We attempt to reproduce this result and delineate the key physics leading to the effect. We find that, even if real, the induced quadrupole would be smaller than claimed. We discuss reasons why the {\it WMAP} data are unlikely to suffer from this particular systematic effect, including the implications for observations of point sources. Given this evidence against the reality of the effect, the similarity between the pointing-offset-induced signal and the actual quadrupole then appears to be quite puzzling. However, we find that the effect arises from a convolution between the gradient of the dipole field and anisotropic coverage of the scan direction at each pixel. There is something of a directional conspiracy here -- the dipole signal lies close to the Ecliptic Plane, and its direction, together with the {\it WMAP} scan strategy, results in a strong coupling to the $Y_{2,\,-1}$ component in Ecliptic co-ordinates. The dominant strength of this component in the measured quadrupole suggests that one should exercise increased caution in interpreting its estimated amplitude. The {\it Planck} satellite has a different scan strategy which does not so directly couple the dipole and quadrupole in this way and will soon provide an independent measurement. [PDF] [PS] [BibTex] [Bookmark]

Author Message
Molly Swanson

Joined: 22 Jun 2009
Posts: 9
Affiliation: University College London

 Posted: April 26 2010 This is a new contribution to the discussion of the WMAP quadrupole. 1003.1073 makes the claim that the quadrupole is induced by a timing offset in the recording of the time-ordered data, which causes the dipole to be subtracted incorrectly. Here, Moss, Scott, and Sigurdson build a toy model of this effect, and find that the induced quadrupole signal looks similar the observed signal, but is a factor of ~three smaller in amplitude, so the signal wouldn't be large enough to explain the claimed effect. They also discuss the reason why a timing offset induces a quadrupole: basically the WMAP scanning strategy is such that at a particular angle, it always scans across the sky in the same direction relative to the dipole direction. They argue that this could generically induce a quadrupole not just for the effect in question, but also for a whole host of other possible systematic errors: "Although the pointing offset effect appears not to be real, one can think of other systematics which could potentially couple to the large dipole field in a similar way. These include an asymmetric beam or a time constant in the instrument response." The main punchline is "Wait for Planck", as it has a different scanning strategy and shouldn't be subject to the same systematics.
Boud Roukema

Joined: 24 Feb 2005
Posts: 82
Affiliation: Torun Centre for Astronomy, University of Nicolaus Copernicus

Posted: April 29 2010

 Molly Swanson wrote: Here, Moss, Scott, and Sigurdson build a toy model of this effect, and find that the induced quadrupole signal looks similar the observed signal, but is a factor of ~three smaller in amplitude, so the signal wouldn't be large enough to explain the claimed effect.

I am a little confused regarding the quantitative estimates of
amplitudes of the dipole difference effect. Here I refer to offsets
relative to that chosen by the WMAP team, i.e. relative to what
the WMAP team believe are the midpoints of observing intervals.

(1) Quantitative amplitudes at −25.6 or about −32 ms

The second last paragraph of Section II of Moss et al. states an
amplitude of 6.5 μ K, presumably meaning $\sqrt{(3/\pi)C_2} = 6.5 \mu$K.

Section 5.3 of Liu & Li 0907.2731, for the V+W bands, i.e. on
average about −32 ms [half-interval offsets in each case: (25.6+38.4)/2 = 32 ],
gives $\sqrt{(3/\pi)C_2} = \sqrt{37.3} = 6.1$ μK vs $\sqrt{108.7} = 10.4$ μK from the official WMAP5 map.
WMAP7 Jarosik et al. (Section 4.1.1, 1001.4744) ILC/KQ85
estimate $\sqrt{(3/\pi) C_2} \sim 14 \mu$K.

It seems to me that 6.5 μK for −25.6 ms is consistent with 6.1
μK for approx −32 ms, so Liu et al and Moss et al seem to estimate
roughly about the same amplitude. In fact, if C2 is linear with the offset, then
Moss et al.'s estimate would give 6.5(32 / 25.6) = 8.125μK at −32 ms.
So Moss et al. estimate an amplitude about 33% higher than that of
Liu & Li 0907.2731.

(2) Estimates for −51.2 ms (Q band half-observation-interval)

Liu et al. 2010 Table 1 estimate 7.23 μ K for the r.m.s. of the
difference between the two maps in Q, i.e. for −51.2 ms. They do not
state $\sqrt{(3/\pi)C_2}$ for either map directly. If it's linear with
the offset, as the r.m.s. seems to be, then having about $\sqrt{(3/\pi)C_2}=6.1 \mu$K at −32 ms (from Liu & Li 0907.2731)
would imply having about 9.7 μK at −51.2 ms, i.e. about 69%
of the WMAP7/ILC/KQ85 amplitude.

If we take Moss et al's estimate instead, we get 6.5 * (51.2 / 25.6) = 13μK.

(3) Word/colour image description in Moss et al.

Moss et al. in their Fig 1 caption state that a −25.6 ms offset gives
larger than the WMAP7/ILC/KQ85 estimate of Jarosik et al. (above),
which could be explained by not masking, for example.

The last paragraph of Section II states that Moss et al. find 1/3 the
amplitude of "that claimed in [Liu et al. 2010]". This means that Moss
et al. interpret something in Liu et al. 2010 to mean that Liu et
al. estimate 19.5μK for −25.6 ms. I do not see where Liu et
al. state that they estimate 19.5μK for −25.6 ms.

In the same paragraph, Moss et al. infer that "if the claimed effect
was real it could not have a large enough amplitude to explain the
WMAP result". i'm not really sure what "explain" means here. It seems
to be a response to Liu et al's rather strong words, e.g. in the Liu
et al. 2010 abstract (v2), saying that the official quadrupole can be
"exactly generated" from the dipole difference effect, and that the
CMB quadrupole "disappears" and is "almost completely artificial and
the real quadrupole of the CMB anisotropy should be near zero". I
agree that Liu et al's words here are too strong compared to what they
have stated quantitatively.

So let's focus on the numbers. Unless I have made some errors above
(which is possible :), it seems to me that Moss et al. get a slightly
higher amplitude than Liu & Li 0907.2731 and get 90% of the WMAP7
amplitude for a −52.1 ms offset. So Moss et al's toy model gives a
(numerically) much stronger result than the authors realised.

Also, they don't seem to refer at all to the fact that the offset is
written in the TOD files - as Liu et al. 2010 say in their footnote
and on cosmocoffee. The adjective "claimed" is incorrect for the
existence of an offset between the Meta and Science Data Table times:
there is definitely a 25.6 ms offset internally in each file that I
have checked, and this offset is not documented in
the WMAP Explanatory Supplement as of 21 April 2010. What is speculative (or claimed) is
whether or not this had any effect at any of the processing steps by
the WMAP team, and whether or not the true error
was just the offset written in the data files or a multiple of it.
Since the offset is half of a W observing interval,
a true error of a full W observing interval sounds like
a natural error to make. Moss et al. seem to show with
their toy model that the full W interval, −52.1 ms,
would give $\sqrt{(3/\pi)C_2}=13 \mu$K.

Joined: 30 Sep 2005
Posts: 3
Affiliation: University of British Columbia

 Posted: April 30 2010 Hi Boud, sorry for the confusion regarding our estimates. In the paper I referred to the peak amplitude, but really should have given the RMS. We obtain [(3/π) C2]1/2 = 4.5 μ K for a 25.6 ms offset, i.e about 1/3 of the WMAP value. Adam
Boud Roukema

Joined: 24 Feb 2005
Posts: 82
Affiliation: Torun Centre for Astronomy, University of Nicolaus Copernicus

Posted: April 30 2010

you can check that i've got things clear now:

 Code: offset/ms  source   \sqrt{3/pi C_2} /\mu K   details 25.6       MSS       4.5                    stated (cosmocoffee) MSS 1004.3995 25.6       LL0907    4.9                    inferred linearly from (32, 6.1) 32         LL0907    6.1                    stated Section 5.3 LL 0907.2731 51.2       MSS       9.0                    inferred linearly from (25.6, 4.5) 51.2       LL0907    9.8                    inferred linearly from (32, 6.1)            WMAP5     10.4                   stated Section 5.3 LL0907.2731            WMAP7     14                     Jarosik et al. (Section 4.1.1, 1001.4744) ILC/KQ85            WMAP7     $\sim$ 13.5            inferred from MSS et al. Fig 1 caption, WMAP/ILC 76.8       MSS       13.5                   inferred linearly from (25.6, 4.5) 76.8       LL0907    14.7                   inferred linearly from (32, 6.1)

I would say that the MSS and LL0907 estimates for the effect look
compatible - a 10% difference given the different methods is small.
The −76.8 ms offset would be for one Q observing interval
minus half a W observing interval. For someone wishing to check
these further, remember that LL0907 is for WMAP5, not WMAP7.
The complication in analysing WMAP5 or WMAP7 calibrated TOD is that
they are given as calibrated TOD, but not as calibrated, filtered TOD,
while WMAP3 is available as calibrated, filtered TOD.

So my word summary would be: "LL0907 and MSS10 estimate about the same
amplitude of the effect, which is about 1/3 to 1/2 of the WMAP(5 or 7)
estimate for −25.6 ms, and within ± 10% or so of equality with WMAP(5 or 7)
for −51.2 ms or 76.8 ms, depending on which of various combinations of the
last 7 lines in the above table are chosen." Is this a fair summary?

For the benefit of people who haven't scribbled a diagram on paper to
show the relations between major frames, frames, and observations in
the TOD, here's a time diagram, using metatime and scitime
from the Meta and Science tables,
respectively. The diagram assumes
that the times written in the two Data Tables should be interpreted
literally. This literal interpretation would imply (LL0907, MSS10) that about
1/3 to 1/2 of the WMAP(5 or 7) CMB quadrupole is an artefact. The Explanatory Supplement (Sections 3.1, 3.2)
does not seem to refer to the 25.6 ms offset.

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