Hao Liu<sup>1</sup> and Ti-Pei Li2

<sup>1</sup>*Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences* <sup>2</sup>*Department of Physics and Center for Astrophysics, Tsinghua University, Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences China*

#### **1. Introduction**

#### **1.1 The success of WMAP**

When the WMAP team released their unprecedentedly precise CMB anisotropy maps in the year of 2003 (Fig. 1, Bennett et al. (2003)), everyone were celebrating the finally arrival of experimental foundation of precision cosmology. It did worth all praises, because the measured CMB power spectrum was so beautifully consistent with the theoretical expectation, and a clear scene of the birth and growth of a flat and ordered Universe seems to be right within the reach of our hands, as well as the first precision percentage estimation of its contents. Many people begin to believe that, we no longer need to be bored by one new cosmic model per day, and the Big Bang scene and ΛCDM model will be good enough to give us satisfactory explanation to anything we want to know.

It's not strange that there still remains some small disharmonious flaws, like an unexpectedly low quadrupole power (Bennett et al., 2003; Hinshaw et al., 2003b), alignment issues and NS asymmetry (Bielewicz et al., 2004; Copi et al., 2007; Eriksen et al., 2005; 2007; Hansen et al., 2004; 2006; Wiaux et al., 2006), non-Gaussianity including cold/hot spots (Copi et al., 2004; Cruz et al., 2005; 2007; Komatsu et al., 2003; Liu & Zhang, 2005; McEwen et al., 2006; Vielva et al., 2004; 2007), etc. Among all these flaws, the loss of the quadrupole power is the

Fig. 1. The CMB anisotropy map and power spectrum obtained by WMAP (Bennett et al., 2003).

Fig. 2. The 141◦ scan rings of 2000 hottest pixels on the CMB sky.

of a systematical error, not something cosmic or occasional.

**1.2.2 The** *T***-***N* **correlation issue**

in Liu & Li (2009a).

then we would be able to deduce which is true with more observations that involves more pixels; however, if something doesn't work well, then we may obtain a wrong guess for A is very cold, or maybe not that bad: A is just slightly colder than what is expected. If such an deviation does exist, then it can be detected by checking the average temperature of all pixels in a ring that is 141◦ away from the very hot pixel B. This is right what we have done

Systematics in WMAP and Other CMB Missions 117

In that work, we pick out 2000 hottest pixels from the CMB anisotropy map provided by the WMAP team, and select all pixels on the corresponding 141◦ rings which look like Fig. 2. The average temperature of these pixels are calculated for all WMAP bands Q, V and W, which are between −11 *μ*K and −13 *μ*K, and these values are 2.5 ∼ 2.7*σ* lower than expectation. This is not enough yet: Although the values are really colder than expected, it can still be something occasional. However, the problem will become more serious if such a phenomena appear to be stuck on 141◦, the man-made physical separation angle between the two antennas. It's possible to test this: Suppose that the final CMB map is perfect, then it must be completely "blind" to the physical separation angle, thus we can set a "guessed" value for the separation angle, then pick out each center pixel and the average temperature on the corresponding ring as a pair, and calculate the correlation coefficients between them to see if the 141◦ separation angle has an outstanding correlation strength. This is found to be true: By force the angular radius of the scan ring to change between 90◦ to 160◦, we discovered that the anti-correlation strength is really strongest around 141◦. Moreover, if the choice of center pixel is limited outside the foreground mask (so that the center pixel temperatures will not be very hot), then the correlation will be significantly weaker, indicating that these center pixels are less likely to arouse a cold ring effect (Fig. 3). With these self-consistent evidences, it's apparently more reasonable to deduce that the pixel-ring coupling is some kind

Another anomaly is much easier to understand: In any physical experiment, the most often adopted way to increase the accuracy and to suppress the noise is to apply more observations. With increasing number of observation, the result should be closer and closer to the true value, and converge at a accuracy level of 1/√*Nobs*, but it's never expected that the result should subsequently increase or decrease with *Nobs*. In other words, there should be no correlation between the number of observation and the derived values. If this is seen with

sharpest one: We can see clearly from the right panel of Fig. 1 that the first black dot starting from the left (which represents the measured CMB quadrupole power) is much lower than theoretical expectation (the red line). However, in several years, these flaws or doubts don't really shake the success of WMAP, partly because they have a general difficulty: it's hard to distinguish the exact cause of these phenomenons, whether it's just something occasional, or due to a cosmic issue, or due to a measurement error. Given different answer to the cause, the corresponding treatment would be significantly different: For something occasional, nothing need to be done, and a cosmic issue need only to be solved in the frame of cosmic theories, but for a measurement error, it must be corrected by improving the measurement or data processing before considering any further theoretical efforts. Although, in several years, there are no strong evidences for us to find out the true cause, the cosmic issue is apparently preferred, because it's really difficult for a researcher outside the WMAP team to deal with the mission details. Especially, almost nobody really believes that the WMAP team, as such a large, experienced and professional group, could make a significant mistake in such an important experiment. Therefore, unless there appears something more evidently suggesting a potential measurement error, the effort of rechecking the WMAP detecting and data processing system will probably never be seriously considered, and the WMAP cosmology will never be questioned from the technical side.

#### **1.2 Two early discovered anomalies**

We had found such evidences between 2008 and 2009 (Li et al., 2009; Liu & Li, 2009a), and these findings had directly driven us to explore the WMAP raw data to find the reason for them. It's interesting that, till now we haven't found explanations that are satisfactory enough for the two early discovered anomalies, but instead, we have found anomalies that seem to be much more important than them.

#### **1.2.1 The pixel-ring coupling issue**

The first anomaly is related to a pixel-ring coupling issue on the CMB maps. To understand this, we need to know a few things about the way that the WMAP spacecraft measures the anisotropy of the CMB. The CMB anisotropy is extremely weak: about 10−<sup>5</sup> of the well known nearly symmetric 2.73 K blackbody CMB emission. To detect such a weak signal, they have found a nice way to enhance the anisotropy relatively: The spacecraft receives CMB signals coming from two different directions by two highly symmetric antennas, and records only the difference between them to cancel the uniform 2.73 K blackbody CMB emission. This also help to counteract some of the systematical effects that affects the two antennas in the same way. In such an observational design, the separation angle between the two antennas is an important factor, which is 141◦ in the WMAP mission. It's apparently important to ensure that, when the measured differential signal is transformed into full-sky anisotropy maps by a sophisticated map-making process, the 141◦ separation angle, as an artificial thing, leaves completely no trace on the final map. In other words, there should be no clue for us to "guess" the man-made 141◦ separation angle from the final natural CMB map.

Intuitively, we would feel that, when the two antennas point at pixel A with a small deviation to 2.73 K (either higher or lower) and pixel B with very high temperature above 2.73 K respectively, the recorded differential data will have a large value; however, since we actually don't know either A or B temperature a priori, we will have at least two equally reasonable guesses: A is very cold or B is very hot. If the map-making from differential data is reliable, 2 Will-be-set-by-IN-TECH

sharpest one: We can see clearly from the right panel of Fig. 1 that the first black dot starting from the left (which represents the measured CMB quadrupole power) is much lower than theoretical expectation (the red line). However, in several years, these flaws or doubts don't really shake the success of WMAP, partly because they have a general difficulty: it's hard to distinguish the exact cause of these phenomenons, whether it's just something occasional, or due to a cosmic issue, or due to a measurement error. Given different answer to the cause, the corresponding treatment would be significantly different: For something occasional, nothing need to be done, and a cosmic issue need only to be solved in the frame of cosmic theories, but for a measurement error, it must be corrected by improving the measurement or data processing before considering any further theoretical efforts. Although, in several years, there are no strong evidences for us to find out the true cause, the cosmic issue is apparently preferred, because it's really difficult for a researcher outside the WMAP team to deal with the mission details. Especially, almost nobody really believes that the WMAP team, as such a large, experienced and professional group, could make a significant mistake in such an important experiment. Therefore, unless there appears something more evidently suggesting a potential measurement error, the effort of rechecking the WMAP detecting and data processing system will probably never be seriously considered, and the WMAP

We had found such evidences between 2008 and 2009 (Li et al., 2009; Liu & Li, 2009a), and these findings had directly driven us to explore the WMAP raw data to find the reason for them. It's interesting that, till now we haven't found explanations that are satisfactory enough for the two early discovered anomalies, but instead, we have found anomalies that seem to be

The first anomaly is related to a pixel-ring coupling issue on the CMB maps. To understand this, we need to know a few things about the way that the WMAP spacecraft measures the anisotropy of the CMB. The CMB anisotropy is extremely weak: about 10−<sup>5</sup> of the well known nearly symmetric 2.73 K blackbody CMB emission. To detect such a weak signal, they have found a nice way to enhance the anisotropy relatively: The spacecraft receives CMB signals coming from two different directions by two highly symmetric antennas, and records only the difference between them to cancel the uniform 2.73 K blackbody CMB emission. This also help to counteract some of the systematical effects that affects the two antennas in the same way. In such an observational design, the separation angle between the two antennas is an important factor, which is 141◦ in the WMAP mission. It's apparently important to ensure that, when the measured differential signal is transformed into full-sky anisotropy maps by a sophisticated map-making process, the 141◦ separation angle, as an artificial thing, leaves completely no trace on the final map. In other words, there should be no clue for us to "guess"

Intuitively, we would feel that, when the two antennas point at pixel A with a small deviation to 2.73 K (either higher or lower) and pixel B with very high temperature above 2.73 K respectively, the recorded differential data will have a large value; however, since we actually don't know either A or B temperature a priori, we will have at least two equally reasonable guesses: A is very cold or B is very hot. If the map-making from differential data is reliable,

cosmology will never be questioned from the technical side.

the man-made 141◦ separation angle from the final natural CMB map.

**1.2 Two early discovered anomalies**

much more important than them.

**1.2.1 The pixel-ring coupling issue**

Fig. 2. The 141◦ scan rings of 2000 hottest pixels on the CMB sky.

then we would be able to deduce which is true with more observations that involves more pixels; however, if something doesn't work well, then we may obtain a wrong guess for A is very cold, or maybe not that bad: A is just slightly colder than what is expected. If such an deviation does exist, then it can be detected by checking the average temperature of all pixels in a ring that is 141◦ away from the very hot pixel B. This is right what we have done in Liu & Li (2009a).

In that work, we pick out 2000 hottest pixels from the CMB anisotropy map provided by the WMAP team, and select all pixels on the corresponding 141◦ rings which look like Fig. 2. The average temperature of these pixels are calculated for all WMAP bands Q, V and W, which are between −11 *μ*K and −13 *μ*K, and these values are 2.5 ∼ 2.7*σ* lower than expectation.

This is not enough yet: Although the values are really colder than expected, it can still be something occasional. However, the problem will become more serious if such a phenomena appear to be stuck on 141◦, the man-made physical separation angle between the two antennas. It's possible to test this: Suppose that the final CMB map is perfect, then it must be completely "blind" to the physical separation angle, thus we can set a "guessed" value for the separation angle, then pick out each center pixel and the average temperature on the corresponding ring as a pair, and calculate the correlation coefficients between them to see if the 141◦ separation angle has an outstanding correlation strength. This is found to be true: By force the angular radius of the scan ring to change between 90◦ to 160◦, we discovered that the anti-correlation strength is really strongest around 141◦. Moreover, if the choice of center pixel is limited outside the foreground mask (so that the center pixel temperatures will not be very hot), then the correlation will be significantly weaker, indicating that these center pixels are less likely to arouse a cold ring effect (Fig. 3). With these self-consistent evidences, it's apparently more reasonable to deduce that the pixel-ring coupling is some kind of a systematical error, not something cosmic or occasional.

#### **1.2.2 The** *T***-***N* **correlation issue**

Another anomaly is much easier to understand: In any physical experiment, the most often adopted way to increase the accuracy and to suppress the noise is to apply more observations. With increasing number of observation, the result should be closer and closer to the true value, and converge at a accuracy level of 1/√*Nobs*, but it's never expected that the result should subsequently increase or decrease with *Nobs*. In other words, there should be no correlation between the number of observation and the derived values. If this is seen with

Fig. 4. The distribution of the T-N correlation coefficients (*solid*) compared with expectation and corresponding error bar given by 50,000 simulation (*dash*). It's very easy to see the

Systematics in WMAP and Other CMB Missions 119

In producing output full sky CMB temperature maps from the raw time-order differential data (TOD), the WMAP team have provided a beautiful formula (Hinshaw et al., 2003a): Let the CMB temperature anisotropy be a single-column matrix **T** with *Npix* rows (for *Npix* pixels on the sky), and the observation course be represented by a matrix **A** with *N* rows, *Npix* columns (*N* is the total number of observations, which is much higher than *Npix*) that is mostly zero, except for one 1 and one -1 in each row, then the TOD **D** can be described by a matrix

where *A* and *B* stand for the two antennas, and *j* = 1, ··· , *N*. The difficulty in solving for the CMB temperatures **T** is that **A** is not a square matrix. In linear algebra, this means either Equation 1 has no solution, or it contains many redundant rows, because *N* is much higher than *Npix*. However, we can multiply the transpose of **A** to both side and obtain a square

Equation 4 seems to be simple and beautiful, but it's incorrect in linear algebra, because **ATA** is a singular matrix and (**ATA**)−**<sup>1</sup>** doesn't exist. Actually, even if (**ATA**)−**<sup>1</sup>** exists, there is no way to exactly solve for such a huge matrix with millions of rows and columns. An approximation

**AT** = **D** (1)

*<sup>j</sup>* = *Dj* , (2)

(**ATA**)**T** = **ATD** (3)

**T** = (**ATA**)−**1ATD** (4)

deviation of real data from expectation.

multiplication:

matrix **ATA**:

**2.1 Some basic thing about the map-making**

The *j*th line of Equation 1 is actually a simple sub-equation:

Then the solution can be formally obtained by:

*T<sup>A</sup> <sup>j</sup>* <sup>−</sup> *<sup>T</sup><sup>B</sup>*

Fig. 3. The dependence of the correlation coefficients upon the separation angle. *Solid square*: The center pixels are limited within the hottest pixels on the CMB sky. *Empty square*: The center pixels being all pixel on the CMB sky. *Triangle*: The center pixels are limited outside the foreground mask.

enough significance, then it's almost certainly that a problem will exist in the measurement system, because the number of observation has nothing to do with the theoretical issues.

We did see this in the WMAP data (Li et al., 2009): the average absolute magnitude of the correlation coefficients between the WMAP CMB anisotropy temperatures and corresponding *Nobs* are 4.2*σ* ∼ 4.8*σ* higher than expectation, and their distribution also significantly deviates from expected Gaussian distribution (Fig. 4), indicating that there is very likely a systematical problem in the WMAP mission, which has remarkably contaminated the final CMB anisotropy result.

When we see the river is muddy and believe it should not be like this, we will certainly go upstream to see where comes the mud. These two anomalies give rise to a doubt on the WMAP data, and will certainly drive us upstream for the origin, although the way is really cliffy.

#### **2. Reprocessing the raw data**

It's a tedious story how we explored the raw data and saw many false "differences" to the WMAP team. In brief, for many times when we worked on the WMAP raw data, we had seen this or that kind of "little surprises" smashed quickly by following tests, that we nearly despaired of finding anything worth noticing. This illustrated from another side how excellently had the WMAP team done. But at last we saw something that was not negligible on the problem of the CMB quadrupole. It's funny that it also started with a mistake: we actually got a much higher quadrupole value than WMAP at first, which was certainly welcome by theorist, but like usual, in a few days we found this is just another mistake. However, the turning point came silently before we realized it: When we corrected the mistake, the CMB quadrupole didn't come back and disappeared almost completely. We thought this was just one more fragile "little surprise", but we were wrong again.

4 Will-be-set-by-IN-TECH

Fig. 3. The dependence of the correlation coefficients upon the separation angle. *Solid square*: The center pixels are limited within the hottest pixels on the CMB sky. *Empty square*: The center pixels being all pixel on the CMB sky. *Triangle*: The center pixels are limited outside

enough significance, then it's almost certainly that a problem will exist in the measurement system, because the number of observation has nothing to do with the theoretical issues.

We did see this in the WMAP data (Li et al., 2009): the average absolute magnitude of the correlation coefficients between the WMAP CMB anisotropy temperatures and corresponding *Nobs* are 4.2*σ* ∼ 4.8*σ* higher than expectation, and their distribution also significantly deviates from expected Gaussian distribution (Fig. 4), indicating that there is very likely a systematical problem in the WMAP mission, which has remarkably contaminated the final CMB anisotropy

When we see the river is muddy and believe it should not be like this, we will certainly go upstream to see where comes the mud. These two anomalies give rise to a doubt on the WMAP data, and will certainly drive us upstream for the origin, although the way is really

It's a tedious story how we explored the raw data and saw many false "differences" to the WMAP team. In brief, for many times when we worked on the WMAP raw data, we had seen this or that kind of "little surprises" smashed quickly by following tests, that we nearly despaired of finding anything worth noticing. This illustrated from another side how excellently had the WMAP team done. But at last we saw something that was not negligible on the problem of the CMB quadrupole. It's funny that it also started with a mistake: we actually got a much higher quadrupole value than WMAP at first, which was certainly welcome by theorist, but like usual, in a few days we found this is just another mistake. However, the turning point came silently before we realized it: When we corrected the mistake, the CMB quadrupole didn't come back and disappeared almost completely. We thought this was just

the foreground mask.

**2. Reprocessing the raw data**

one more fragile "little surprise", but we were wrong again.

result.

cliffy.

Fig. 4. The distribution of the T-N correlation coefficients (*solid*) compared with expectation and corresponding error bar given by 50,000 simulation (*dash*). It's very easy to see the deviation of real data from expectation.

#### **2.1 Some basic thing about the map-making**

In producing output full sky CMB temperature maps from the raw time-order differential data (TOD), the WMAP team have provided a beautiful formula (Hinshaw et al., 2003a): Let the CMB temperature anisotropy be a single-column matrix **T** with *Npix* rows (for *Npix* pixels on the sky), and the observation course be represented by a matrix **A** with *N* rows, *Npix* columns (*N* is the total number of observations, which is much higher than *Npix*) that is mostly zero, except for one 1 and one -1 in each row, then the TOD **D** can be described by a matrix multiplication:

$$\mathbf{AT} = \mathbf{D} \tag{1}$$

The *j*th line of Equation 1 is actually a simple sub-equation:

$$T\_j^A - T\_j^B = D\_{\dot{j}} \, \tag{2}$$

where *A* and *B* stand for the two antennas, and *j* = 1, ··· , *N*. The difficulty in solving for the CMB temperatures **T** is that **A** is not a square matrix. In linear algebra, this means either Equation 1 has no solution, or it contains many redundant rows, because *N* is much higher than *Npix*. However, we can multiply the transpose of **A** to both side and obtain a square matrix **ATA**:

$$(\mathbf{A}^\mathsf{T}\mathbf{A})\mathbf{T} = \mathbf{A}^\mathsf{T}\mathbf{D} \tag{3}$$

Then the solution can be formally obtained by:

$$\mathbf{T} = (\mathbf{A}^T \mathbf{A})^{-1} \mathbf{A}^T \mathbf{D} \tag{4}$$

Equation 4 seems to be simple and beautiful, but it's incorrect in linear algebra, because **ATA** is a singular matrix and (**ATA**)−**<sup>1</sup>** doesn't exist. Actually, even if (**ATA**)−**<sup>1</sup>** exists, there is no way to exactly solve for such a huge matrix with millions of rows and columns. An approximation

**2.2 Be consistent to WMAP first**

*• f* −1 *noise (an equipment feature)*

*• The Sun velocity uncertainty (likewise)*

*• Window function correction (likewise)*

sequence, and present our work step by step.

**3.1 The difference in the CMB result using the same raw data**

*• TOD flag issue (likewise)*

*•* ···

spectrum.

Although our emphasis is the difference to the WMAP team, we would like to illustrate at the very beginning that we are now able to obtain fully consistent results to the WMAP team using our own map-making software, as shown in Fig 6. The reason is: Unless we are able to do so, the crucial reason of the **difference** between our results and WMAP will never be determined, because we will be lost in countless technical details, each seems to be able to cause some specious "differences" in this or that way, as briefly but incompletely listed below:

Systematics in WMAP and Other CMB Missions 121

*• Usage of the processing mask (determines how many raw data are unused)*

*• Problem in resolving the spacecraft velocity (affects the Doppler signal and calibration)*

*• Beam function correction (which can greatly suppress the small scale anisotropy)*

**3. The origin of the difference between our results and WMAP**

*• Incomplete sky coverage (some sky regions are unused in calculating the CMB power spectrum)*

Needless to say, nothing valuable can be obtained before we climb out of such a bottomless list. However, once we get Fig. 6, we will then be able to clearly identify which item in this list makes the major contribution to the difference between WMAP and us. This can be done by changing one item one time, and see how it takes effect on the final CMB maps and power

In time sequence, the difference in the CMB anisotropy maps and power spectrum is discovered first, and then the origin of the difference is found and confirmed as illustrated in Section 2.2. This is not the end of the problem, because we need to check whether WMAP or us is more likely to be correct. Soon after that, several evidences supporting our results are obtained, and we also found reasonable explanations to why WMAP could be wrong. Such explanations is not only valuable in improving the WMAP result, but also valuable in preventing future CMB detecting mission from making the same mistake. We will follow this

After tottering through the raw data processing, we finally get a tentative CMB result in the end of 2008 (Liu & Li, 2009b), which looks very similar to the WMAP official release, as shown in Fig. 7 However, if we subtract our map from WMAP and smooth the result, the difference can be clearly seen, which is a four-spot structure, two hot and two cold. This is a typical quadrupole structure. More interestingly, such a difference is almost the same to the claimed

*• Antenna imbalance (the antennas are not absolutely symmetric)*

*• Dipole signal subtraction (remove the unwanted Doppler signal) • Foreground subtraction (remove the unwanted foreground emission)*

*• Map making convergence (does the iteration converges well?) • The antenna pointing vectors (affected by various factors)*

Fig. 5. The expected residual map-making error from Fig. 2 of Hinshaw et al. (2003a), which is negligible, and can be even reduced by more iterations. However, the observation noise is not taken into account here.

for (**ATA**)−**<sup>1</sup>** is a *Npix* <sup>×</sup> *Npix* diagonal matrix1:

$$\mathbf{N}^{-1} = \begin{pmatrix} 1/N\_1 & 0 & \cdots \\ 0 & 1/N\_2 & \cdots \\ \cdots & \cdots & \cdots \end{pmatrix} \tag{5}$$

where *Ni* is the total number of observation for the *i*th map pixel, *i* = 1, ··· , *Npix* and ∑*<sup>i</sup> Ni* = *N*. Based on this approximation, an iterative solution can be preformed to estimate the final CMB temperatures **T**.

The iterative solution is simple: we give an initial all-zero guess **T<sup>0</sup>** to the CMB anisotropy map **T**, and, based on that, use the TOD **D** to improve the guess by *T*1,*iA <sup>j</sup>* <sup>=</sup> *<sup>T</sup>*0,*iB <sup>j</sup>* <sup>+</sup> *Dj* or *<sup>T</sup>*1,*iB <sup>j</sup>* = *T*0,*iA <sup>j</sup>* − *Dj* (see also Equation 2), where *Tj* means the *j*th temperature estimation corresponding to the *j*th observation, and *iA* stands for the A-side pixel in this observation, so do *iB*. In this way, each *Dj* gives two improved estimations *<sup>T</sup>*1,*iA <sup>j</sup>* and *<sup>T</sup>*1,*iB <sup>j</sup>* for the positive/negative sides one the map respectively. The improved **T<sup>1</sup>** is the assembly of all *Tj*, averaged at each sky pixel *i* respectively: *T*<sup>1</sup> *<sup>i</sup>* = (∑*<sup>j</sup> <sup>T</sup>*1,*iA*=*<sup>i</sup> <sup>j</sup>* <sup>+</sup> <sup>∑</sup>*<sup>j</sup> <sup>T</sup>*1,*iB*=*<sup>i</sup> <sup>j</sup>* )/*Ni* (*Ti* mean the averaged temperature of the *i*th pixel on the sky), and the final CMB anisotropy map is produced by 50 to 80 iterations like this2.

Although a strict solution to **T** is impossible, the iterative solution works quite well, at least according to simulation: As presented by the WMAP team (Fig. 5), discarding the observation noise and the uncertain monopole, the residual full sky map-making error is negligible. This has been confirmed by us, but we have found that for such an excellent convergence in Fig. 5, the average of odd and even iteration rounds should be used, which was not mentioned in the WMAP documents.

<sup>1</sup> Since (**ATA**)−**<sup>1</sup>** doesn't exist, "approximation for (**ATA**)−**1**" means this matrix multiplies **ATA** gives an output matrix that is almost unitary.

<sup>2</sup> The map-making processes described here are simplified for ideally symmetric antennas with perfectly stable responses. For equations including necessary fine corrections, please refer to Hinshaw et al. (2003a)

## **2.2 Be consistent to WMAP first**

6 Will-be-set-by-IN-TECH


> 1/*N*<sup>1</sup> 0 ··· 0 1/*N*<sup>2</sup> ··· ··· ··· ···

where *Ni* is the total number of observation for the *i*th map pixel, *i* = 1, ··· , *Npix* and ∑*<sup>i</sup> Ni* = *N*. Based on this approximation, an iterative solution can be preformed to estimate the final

The iterative solution is simple: we give an initial all-zero guess **T<sup>0</sup>** to the CMB anisotropy map

*<sup>j</sup>* − *Dj* (see also Equation 2), where *Tj* means the *j*th temperature estimation corresponding to the *j*th observation, and *iA* stands for the A-side pixel in this observation, so do *iB*. In this

one the map respectively. The improved **T<sup>1</sup>** is the assembly of all *Tj*, averaged at each sky

the *i*th pixel on the sky), and the final CMB anisotropy map is produced by 50 to 80 iterations

Although a strict solution to **T** is impossible, the iterative solution works quite well, at least according to simulation: As presented by the WMAP team (Fig. 5), discarding the observation noise and the uncertain monopole, the residual full sky map-making error is negligible. This has been confirmed by us, but we have found that for such an excellent convergence in Fig. 5, the average of odd and even iteration rounds should be used, which was not mentioned in

<sup>1</sup> Since (**ATA**)−**<sup>1</sup>** doesn't exist, "approximation for (**ATA**)−**1**" means this matrix multiplies **ATA** gives an

<sup>2</sup> The map-making processes described here are simplified for ideally symmetric antennas with perfectly stable responses. For equations including necessary fine corrections, please refer to Hinshaw et al.

*<sup>j</sup>* <sup>+</sup> <sup>∑</sup>*<sup>j</sup> <sup>T</sup>*1,*iB*=*<sup>i</sup>*

*<sup>j</sup>* and *<sup>T</sup>*1,*iB*

⎞

⎠ , (5)

*<sup>j</sup>* for the positive/negative sides

*<sup>j</sup>* <sup>+</sup> *Dj* or *<sup>T</sup>*1,*iB*

*<sup>j</sup>* =

*<sup>j</sup>* <sup>=</sup> *<sup>T</sup>*0,*iB*

*<sup>j</sup>* )/*Ni* (*Ti* mean the averaged temperature of

not taken into account here.

CMB temperatures **T**.

pixel *i* respectively: *T*<sup>1</sup>

the WMAP documents.

output matrix that is almost unitary.

*T*0,*iA*

like this2.

(2003a)

for (**ATA**)−**<sup>1</sup>** is a *Npix* <sup>×</sup> *Npix* diagonal matrix1:

way, each *Dj* gives two improved estimations *<sup>T</sup>*1,*iA*

*<sup>i</sup>* = (∑*<sup>j</sup> <sup>T</sup>*1,*iA*=*<sup>i</sup>*

**N**−**<sup>1</sup>** =

**T**, and, based on that, use the TOD **D** to improve the guess by *T*1,*iA*

⎛ ⎝ Although our emphasis is the difference to the WMAP team, we would like to illustrate at the very beginning that we are now able to obtain fully consistent results to the WMAP team using our own map-making software, as shown in Fig 6. The reason is: Unless we are able to do so, the crucial reason of the **difference** between our results and WMAP will never be determined, because we will be lost in countless technical details, each seems to be able to cause some specious "differences" in this or that way, as briefly but incompletely listed below:


Needless to say, nothing valuable can be obtained before we climb out of such a bottomless list. However, once we get Fig. 6, we will then be able to clearly identify which item in this list makes the major contribution to the difference between WMAP and us. This can be done by changing one item one time, and see how it takes effect on the final CMB maps and power spectrum.
