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

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 sequence, and present our work step by step.

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

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

Fig. 8. This figure is to show the large-scale CMB structure difference between WMAP and us. *Left panel:* The CMB map of WMAP minus ours, smoothed to *Nside* = 8, so that it's dominated by large-scale differences. *Right panel:* The quadrupole component of the released WMAP CMB anisotropy map. They are amazingly similar, which casts doubt on the WMAP

Systematics in WMAP and Other CMB Missions 123

**CMB quadrupole** structure by WMAP (Fig. 8). This phenomena certainly worths further

The CMB power spectrum is calculated by spherical harmonic decomposition. The spherical harmonics *Ylm*(*θ*, *φ*) are known as a family of normalized, orthogonal and complete functions on the sphere, which is widely used to analyze problems on the sphere including the CMB

> *l* ∑ *m*=−*l*

Besides Equation 7, there are many further corrections in calculating the final CMB power spectrum; however, as illustrated in Section 2.2, we can confirm that we have appropriately applied all these corrections by the consistency between the crosses/squares in Fig. 6 (so we see how important it is to be consistent to WMAP first). Based on the confirmed consistency, we can conclude that the differences in both large and small scale power spectra (Fig. 9) are


*lm*(*θ*, *φ*) sin(*θ*)*dθdφ* , (6)

<sup>2</sup> (7)

*T*(*θ*, *φ*)*Y*∗

*Cl* <sup>=</sup> <sup>1</sup> 2*l* + 1

anisotropy. The spherical harmonic coefficients are calculated by:

*αlm* = 

and the CMB power spectrum is obtained by:

data.

consideration.

Fig. 6. We are now able to obtain fully consistent CMB result to the WMAP team using our own software but adopting their convention. This greatly simplify the search for the origin of the difference. *Square*: CMB power spectrum obtained from the WMAP official maps. *Cross*: CMB power spectrum obtained from our maps using their conventions.

Fig. 7. The Q1-band CMB temperature maps by the WMAP team (*left panel*) and by us (*right panel*). Both in the Galactic coordinate and the unit is mK. They look almost the same.

8 Will-be-set-by-IN-TECH

Fig. 6. We are now able to obtain fully consistent CMB result to the WMAP team using our own software but adopting their convention. This greatly simplify the search for the origin of the difference. *Square*: CMB power spectrum obtained from the WMAP official maps. *Cross*:

Fig. 7. The Q1-band CMB temperature maps by the WMAP team (*left panel*) and by us (*right panel*). Both in the Galactic coordinate and the unit is mK. They look almost the same.

CMB power spectrum obtained from our maps using their conventions.

Fig. 8. This figure is to show the large-scale CMB structure difference between WMAP and us. *Left panel:* The CMB map of WMAP minus ours, smoothed to *Nside* = 8, so that it's dominated by large-scale differences. *Right panel:* The quadrupole component of the released WMAP CMB anisotropy map. They are amazingly similar, which casts doubt on the WMAP data.

**CMB quadrupole** structure by WMAP (Fig. 8). This phenomena certainly worths further consideration.

The CMB power spectrum is calculated by spherical harmonic decomposition. The spherical harmonics *Ylm*(*θ*, *φ*) are known as a family of normalized, orthogonal and complete functions on the sphere, which is widely used to analyze problems on the sphere including the CMB anisotropy. The spherical harmonic coefficients are calculated by:

$$\mathfrak{a}\_{lm} = \int T(\theta, \phi) Y\_{lm}^\*(\theta, \phi) \sin(\theta) d\theta d\phi \,, \tag{6}$$

and the CMB power spectrum is obtained by:

$$C\_{l} = \frac{1}{2l+1} \sum\_{m=-l}^{l} |a\_{lm}|^2 \tag{7}$$

Besides Equation 7, there are many further corrections in calculating the final CMB power spectrum; however, as illustrated in Section 2.2, we can confirm that we have appropriately applied all these corrections by the consistency between the crosses/squares in Fig. 6 (so we see how important it is to be consistent to WMAP first). Based on the confirmed consistency, we can conclude that the differences in both large and small scale power spectra (Fig. 9) are

causes the calculated pointing vectors to be different for about half a pixel to them. This looks

Systematics in WMAP and Other CMB Missions 125

In most cases, when you find the origin of the problem, the case will soon be totally solved. However, for us, discovering the antenna pointing vector issue is just the beginning of the

When we study the key of the difference: the interpolation parameter change carefully, we feel that WMAP is probably correct, because they have used the center of each observation interval as the effective antenna pointing vector for that observation, but we have used the start of it. Since WMAP is continuously receiving the microwave signal, the center of the observation is apparently a better choice. Thus the prospect of our finding seem to be dim at

However, an unconvinced fact still remain in our mind: When adopting our interpolating settings, nearly 90% of the CMB quadrupole (the component with *l* = 2) will disappear. If this doesn't look suspicious enough, then from another point of view we will see what it actually

The antenna pointing vector difference affects the CMB quadrupole via the Doppler signal subtraction. In the WMAP observation, the strongest contamination to the CMB signal is the Doppler signal caused by the motion of the spacecraft towards the CMB test frame. This

where *T*<sup>0</sup> is the 2.73 K CMB monopole, *c* is the speed of light in vacuum, **V** is the velocity of the spacecraft relative to the CMB rest frame, and **nA**, **nB** are the antenna pointing vectors. We can see clearly that, if the antenna pointing vectors are slightly different, either on **nA** or **nB** or

It's interesting that, from Equation 9 we can see that we don't have to know any CMB information in calculating Δ*d*. Therefore, the deviation upon the CMB quadrupole caused by possible antenna pointing vector error (no matter what reason) can be calculated independently of the CMB maps or CMB detection. This has been done by us (Liu, Xiong & Li 2010), and the result is again almost same to the released WMAP CMB quadrupole, as shown in Fig. 10. Now we can see the real puzzle: The "CMB" quadrupole has been reproduced without any observation, which is absurd. Theoretically speaking, this is not impossible, but a much more reasonable explanation is that the claimed "CMB" quadrupole is actually

**V** · (**nA** − **nB**), (8)

**V** · **Δn**, **Δn** = **ΔnA** − **ΔnB**. (9)

*<sup>d</sup>* <sup>=</sup> *<sup>T</sup>*<sup>0</sup> *c*

both, then the calculated Doppler dipole signal *d* will be consequently different:

<sup>Δ</sup>*<sup>d</sup>* <sup>=</sup> *<sup>T</sup>*<sup>0</sup> *c*

really a trivial thing, but it's right the key of all differences shown in Fig. 8 and 9.

**4. Zigzag**

first.

means.

zigzag pursuit.

**4.1 WMAP seem to be right?**

**4.2 A doubtful point: evidence against WMAP**

particular signal can be calculated by:

a systematical error.

Fig. 9. *Left Panel:*The CMB power spectra derived with our software from our new map (*dash line*), with our software but from the WMAP official maps (*dotted line*), and directly released by the WMAP team (*solid line*). By the difference between the solid line and dash line, we can see that the small scale CMB power spectra derived by us and WMAP are apparently different. *Right Panel:* The large scale CMB power spectra by us (*dot*) and by WMAP (*asterisk*). We can see that only the quadrupole is significantly different.

not due to the power spectra estimation, but due to a possible issue in the map-making from the raw data.

#### **3.2 The origin of the difference**

We took a long time to search for the origin of the differences. Finally we found that it's due to an antenna pointing vector issue. This had been noticed by us before, but was ignored at first because it's really too small.

The WMAP spacecraft doesn't record the antenna pointing vector for each observation: The pointing vectors are much less frequently recorded than the science data in order to reduce the data file size. Therefore in raw data processing, we need to interpolate the available antenna pointing vector to calculate the unrecorded pointing vectors for each observation. In our data processing program, one of the interpolation parameters is slightly different to WMAP, which

causes the calculated pointing vectors to be different for about half a pixel to them. This looks really a trivial thing, but it's right the key of all differences shown in Fig. 8 and 9.

#### **4. Zigzag**

10 Will-be-set-by-IN-TECH

2000

 *l(l+1)Cl /2*π *(*

μ

K2)

 4000

 6000

50 100 200 500

Multiple moment *l*

Fig. 9. *Left Panel:*The CMB power spectra derived with our software from our new map (*dash line*), with our software but from the WMAP official maps (*dotted line*), and directly released by the WMAP team (*solid line*). By the difference between the solid line and dash line, we can

different. *Right Panel:* The large scale CMB power spectra by us (*dot*) and by WMAP (*asterisk*).

not due to the power spectra estimation, but due to a possible issue in the map-making from

We took a long time to search for the origin of the differences. Finally we found that it's due to an antenna pointing vector issue. This had been noticed by us before, but was ignored at

The WMAP spacecraft doesn't record the antenna pointing vector for each observation: The pointing vectors are much less frequently recorded than the science data in order to reduce the data file size. Therefore in raw data processing, we need to interpolate the available antenna pointing vector to calculate the unrecorded pointing vectors for each observation. In our data processing program, one of the interpolation parameters is slightly different to WMAP, which

see that the small scale CMB power spectra derived by us and WMAP are apparently

We can see that only the quadrupole is significantly different.

the raw data.

**3.2 The origin of the difference**

first because it's really too small.

In most cases, when you find the origin of the problem, the case will soon be totally solved. However, for us, discovering the antenna pointing vector issue is just the beginning of the zigzag pursuit.

#### **4.1 WMAP seem to be right?**

When we study the key of the difference: the interpolation parameter change carefully, we feel that WMAP is probably correct, because they have used the center of each observation interval as the effective antenna pointing vector for that observation, but we have used the start of it. Since WMAP is continuously receiving the microwave signal, the center of the observation is apparently a better choice. Thus the prospect of our finding seem to be dim at first.

#### **4.2 A doubtful point: evidence against WMAP**

However, an unconvinced fact still remain in our mind: When adopting our interpolating settings, nearly 90% of the CMB quadrupole (the component with *l* = 2) will disappear. If this doesn't look suspicious enough, then from another point of view we will see what it actually means.

The antenna pointing vector difference affects the CMB quadrupole via the Doppler signal subtraction. In the WMAP observation, the strongest contamination to the CMB signal is the Doppler signal caused by the motion of the spacecraft towards the CMB test frame. This particular signal can be calculated by:

$$d = \frac{T\_0}{c} \mathbf{V} \cdot (\mathbf{n\_A} - \mathbf{n\_B}) \, \tag{8}$$

where *T*<sup>0</sup> is the 2.73 K CMB monopole, *c* is the speed of light in vacuum, **V** is the velocity of the spacecraft relative to the CMB rest frame, and **nA**, **nB** are the antenna pointing vectors. We can see clearly that, if the antenna pointing vectors are slightly different, either on **nA** or **nB** or both, then the calculated Doppler dipole signal *d* will be consequently different:

$$
\Delta d = \frac{T\_0}{c} \mathbf{V} \cdot \Delta \mathbf{n}, \quad \Delta \mathbf{n} = \Delta \mathbf{n}\_\mathbf{A} - \Delta \mathbf{n}\_\mathbf{B}. \tag{9}
$$

It's interesting that, from Equation 9 we can see that we don't have to know any CMB information in calculating Δ*d*. Therefore, the deviation upon the CMB quadrupole caused by possible antenna pointing vector error (no matter what reason) can be calculated independently of the CMB maps or CMB detection. This has been done by us (Liu, Xiong & Li 2010), and the result is again almost same to the released WMAP CMB quadrupole, as shown in Fig. 10. Now we can see the real puzzle: The "CMB" quadrupole has been reproduced without any observation, which is absurd. Theoretically speaking, this is not impossible, but a much more reasonable explanation is that the claimed "CMB" quadrupole is actually a systematical error.

still take effect in the data calibration. Although such an possibility seems to be minor, it was

Systematics in WMAP and Other CMB Missions 127

Three months later after Roukema's previous work, by checking the median per map of the temperature fluctuation variance per pixel, he discovered that, in a very high significance level, there does seem to be an antenna pointing vector error in the raw data, possibly due to the data calibration (Roukema, 2010b). We have confirmed this finding with different method (Liu, Xiong & Li, 2011), and the obtained results are very well consistent to Roukema (2010b). This seems to be puzzling: How can the same problem, the antenna pointing issue,

To answer this question, we need to have a look at the antenna pointing issues again. The

All these three effects can cause antenna pointing vector errors; however, we have confirmed that the 1st and 2nd effects can further cause both the blurring effect and power spectrum deviation, and the 3nd causes only the power spectrum deviation. Thus the evidences found in Roukema (2010a) doesn't in principle conflict with Roukema (2010b) and Liu, Xiong & Li (2011). Moreover, all these three articles have also pointed out that the antenna pointing error can possibly take effect in the calibration stage without blurring the final image, which lead

Now we have a clear logical sequence: The three reasons listed above (possibly more) cause antenna pointing vector errors (real or equivalent), and the antenna pointing vector errors further cause all the CMB deviations discussed in this article. The antenna pointing vector

It's easy to understand the first factor, thus we introduce no more discussion for it. The second factor is due to the rotation of the spacecraft: In order to scan the sky, the spacecraft must rotate continuously, thus we must know the angular velocity and the local time to calculate the antenna pointing vectors. If there is a timing error, then the derived antenna pointing vectors will certainly be mistaken. In Liu, Xiong & Li (2010), we have discovered that the WMAP spacecraft attitude data are asynchronous to the CMB differential data, thus existence of the timing error is apparently possible. This is further confirmed in Liu, Xiong & Li (2011)

Both the first and the second factors are called real pointing vector errors, because the antenna pointing vectors will be substantially in error due to them. However, besides the real pointing vector error, there is also equivalent pointing vector error caused by the third factor. In this case, the antenna pointing vectors might be accurate, but the recorded data are twisted, as if

soon confirmed by following works by him and us.

be confirmed and rejected at the same time?

*• Timing error*

*• Sidelobe uncertainty*

to the same conclusion.

and Roukema (2010b).

there were a pointing vector error.

**4.5.1 Real pointing vector error**

**4.5 Real and equivalent pointing vector errors**

error is hence the node of the entire problem.

*• Direct pointing error (e.g., the antennas are misplaced).*

antenna directions are affected by at least three independent factors:

Fig. 10. *Left Panel:* The expected deviation on final CMB temperature map caused by Δ*d* (see Equation 9). Note that there is no inclusion of any CMB signal in obtaining this figure. *Right Panel:* The claimed CMB quadrupole by WMAP.

The results in Fig. 10 has been independently reproduced by Moss et al. (2010) and Roukema (2010a), and has been regarded as an important evidence in questioning the WMAP cosmology (Sawangwit & Shanks, 2010). The software used to obtain this result is the same software written by us for re-processing the WMAP TOD, which is publicly available on the websites of the cosmocoffee forum <sup>3</sup> or the Tsinghua Center for Astrophysics4.

#### **4.3 WMAP is supported again?**

Since the focus of the problems seem to be upon the antenna pointing vectors, a third party test on this is soon applied by Roukema (2010a), in which he believe that, if there is an antenna pointing vector error, then the generated CMB maps will be blurred, thus by checking the image sharpness of the CMB maps, we can decide whether there really exists such an error. His conclusion is that, the WMAP official maps are apparently less affected by the blurring effect compared to ours. This is a strong supporting evidence for WMAP being correct, and the prospect of our finding seems to be dim again.

#### **4.4 Not the end: more details discovered**

In fact, even in Roukema (2010a), the author didn't conclude that there will surely be no problem in the pointing vectors. He pointed out that the antenna pointing vector issue might

<sup>3</sup> http://cosmocoffee.info/viewtopic.php?t=1541

<sup>4</sup> http://dpc.aire.org.cn/data/wmap/09072731/release\_v1/source\_code/v1/

still take effect in the data calibration. Although such an possibility seems to be minor, it was soon confirmed by following works by him and us.

Three months later after Roukema's previous work, by checking the median per map of the temperature fluctuation variance per pixel, he discovered that, in a very high significance level, there does seem to be an antenna pointing vector error in the raw data, possibly due to the data calibration (Roukema, 2010b). We have confirmed this finding with different method (Liu, Xiong & Li, 2011), and the obtained results are very well consistent to Roukema (2010b). This seems to be puzzling: How can the same problem, the antenna pointing issue, be confirmed and rejected at the same time?

#### **4.5 Real and equivalent pointing vector errors**

To answer this question, we need to have a look at the antenna pointing issues again. The antenna directions are affected by at least three independent factors:


12 Will-be-set-by-IN-TECH

Fig. 10. *Left Panel:* The expected deviation on final CMB temperature map caused by Δ*d* (see Equation 9). Note that there is no inclusion of any CMB signal in obtaining this figure. *Right*

The results in Fig. 10 has been independently reproduced by Moss et al. (2010) and Roukema (2010a), and has been regarded as an important evidence in questioning the WMAP cosmology (Sawangwit & Shanks, 2010). The software used to obtain this result is the same software written by us for re-processing the WMAP TOD, which is publicly available on the

Since the focus of the problems seem to be upon the antenna pointing vectors, a third party test on this is soon applied by Roukema (2010a), in which he believe that, if there is an antenna pointing vector error, then the generated CMB maps will be blurred, thus by checking the image sharpness of the CMB maps, we can decide whether there really exists such an error. His conclusion is that, the WMAP official maps are apparently less affected by the blurring effect compared to ours. This is a strong supporting evidence for WMAP being correct, and

In fact, even in Roukema (2010a), the author didn't conclude that there will surely be no problem in the pointing vectors. He pointed out that the antenna pointing vector issue might

websites of the cosmocoffee forum <sup>3</sup> or the Tsinghua Center for Astrophysics4.

*Panel:* The claimed CMB quadrupole by WMAP.

the prospect of our finding seems to be dim again.

**4.4 Not the end: more details discovered**

<sup>3</sup> http://cosmocoffee.info/viewtopic.php?t=1541

<sup>4</sup> http://dpc.aire.org.cn/data/wmap/09072731/release\_v1/source\_code/v1/

**4.3 WMAP is supported again?**

*• Sidelobe uncertainty*

All these three effects can cause antenna pointing vector errors; however, we have confirmed that the 1st and 2nd effects can further cause both the blurring effect and power spectrum deviation, and the 3nd causes only the power spectrum deviation. Thus the evidences found in Roukema (2010a) doesn't in principle conflict with Roukema (2010b) and Liu, Xiong & Li (2011). Moreover, all these three articles have also pointed out that the antenna pointing error can possibly take effect in the calibration stage without blurring the final image, which lead to the same conclusion.

Now we have a clear logical sequence: The three reasons listed above (possibly more) cause antenna pointing vector errors (real or equivalent), and the antenna pointing vector errors further cause all the CMB deviations discussed in this article. The antenna pointing vector error is hence the node of the entire problem.

#### **4.5.1 Real pointing vector error**

It's easy to understand the first factor, thus we introduce no more discussion for it. The second factor is due to the rotation of the spacecraft: In order to scan the sky, the spacecraft must rotate continuously, thus we must know the angular velocity and the local time to calculate the antenna pointing vectors. If there is a timing error, then the derived antenna pointing vectors will certainly be mistaken. In Liu, Xiong & Li (2010), we have discovered that the WMAP spacecraft attitude data are asynchronous to the CMB differential data, thus existence of the timing error is apparently possible. This is further confirmed in Liu, Xiong & Li (2011) and Roukema (2010b).

Both the first and the second factors are called real pointing vector errors, because the antenna pointing vectors will be substantially in error due to them. However, besides the real pointing vector error, there is also equivalent pointing vector error caused by the third factor. In this case, the antenna pointing vectors might be accurate, but the recorded data are twisted, as if there were a pointing vector error.

Fig. 11. *Top Panels:* The expected temperature deviation caused by the equivalent pointing vector error due to the sidelobe uncertainty issue in the WMAP mission. *Bottom panels:* The quadrupole components of the top panels. From left to right: The equivalent pointing vector

Systematics in WMAP and Other CMB Missions 129

*<sup>c</sup>* **<sup>v</sup>** · (**ΔnA** <sup>−</sup> **<sup>Δ</sup>nB**) = *<sup>T</sup>*<sup>0</sup>

**Δn** is called the equivalent pointing vector error, because it isn't a real pointing error, but has the same effect as real pointing vector errors caused by direct reasons or the timing issue.

It's apparent that we are interested in the gain uncertainty Δ*G*, not the gain itself. It's very difficult to accurately determine the antenna sidelobe response, at least due to three reasons: 1, the sidelobe response is very weak; 2, the signal due to the sidelobe response is always mixed with other much stronger signals (some are even unknown); 3, the sidelobe response of the two antennas could overlap and it's almost impossible to exactly solve for both. Therefore, it's not strange that the WMAP sidelobe gain has very high uncertainty: as presented by Barnes et al. (2003), up to 30%; however, the provides 30% uncertainty contains only the average level uncertainty, with no inclusion of the pixel-to-pixel variance. Thus the

It's well known that the dot product of two vectors is invariant in coordinate transform, thus we can calculate Δ*dsidelobe* in any coordinates using Equation 14, and the result will be the same. Since **Δn** is determined by the gain and antenna pointing vectors, which are both constants in the spacecraft coordinate, the best coordinate to calculate Δ*dsidelobe* is certainly the spacecraft coordinate. However, even in the spacecraft coordinate, it's still impossible to exactly calculate the uncertainty due to **Δn** (otherwise it won't be called the "uncertainty"). In this case, a possible way out is to divide **Δn** into three components Δ*nx*, Δ*ny*, Δ*nz* along the *X*, *Y*, *Z* axes of the spacecraft coordinate. For each component, we can set any amplitude for it (we will explain why we can do like this below) and use the corresponding Δ*dsidelobe* instead of the differential data to obtain an output map Δ*Tx* Δ*Ty* or Δ*Tz* like we did for the real TOD

Both Fig. 10 and Fig. 11 illustrate the same thing: there could be significant systematical error in the WMAP CMB detection. Fig. 11 tell us more, that even if the antenna pointing vectors

*<sup>c</sup>* **<sup>v</sup>** · **<sup>Δ</sup><sup>n</sup>** . (14)

error is along the *x*, *y*, *z* axes in the spacecraft coordinate respectively.

<sup>Δ</sup>*dsidelobe* <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

overall uncertainty must be much higher than the claimed 30%.

in Sec. 2.1. Examples of Δ*Tx*, Δ*Ty*, Δ*Tz* are presented in Fig. 11.

and the uncertainty of *dsidelobe* can be expressed as

#### **4.5.2 Equivalent pointing vector error**

The third factor is a hidden factor that has never been noticed before. Generally speaking, radio antenna has response to all 4*π* solid angle, not merely along it's optical pointing. For the WMAP antennas, the 4*π* response is described by a normalized gain *G*: If there is only one beam *b* coming from a direction marked by pixel *i*, then the recorded signal should be *S* = *Gib*. For full sky signal like the CMB, the recorded signal is5:

$$S = \frac{1}{N\_{pix}} \sum\_{i=0}^{N\_{p\text{fix}}-1} G\_i T\_i \,. \tag{10}$$

Since WMAP works in differential mode, the equation for WMAP should be:

$$D = \frac{1}{N\_{\rm pix}} \sum\_{i=0}^{N\_{\rm plx}-1} (G\_i^A - G\_i^B) T\_{i\prime} \tag{11}$$

where *Npix* is the number of pixels on the sky, *G<sup>A</sup> <sup>i</sup>* and *<sup>G</sup><sup>B</sup> <sup>i</sup>* are the normalized gains for the A-side and B-side antennas in the spacecraft coordinate<sup>6</sup> respectively, *Ti* is the CMB temperature. According to the WMAP convention, the normalization rule is <sup>∑</sup>*Npix*−<sup>1</sup> *<sup>i</sup>*=<sup>0</sup> *Gi* = *Npix*.

In Equation 11, only a few pixels stands for the main lobe, and most pixels belong to the sidelobe, but the sidelobe pixels have much lower gain amplitudes: the summation of all sidelobe normalized gains are less than 5%. The standard method to clean the sidelobe contamination is by deconvolution; however, this is very complex and slow, and the biggest disadvantage is missing of a clear physical picture. We discovered that the deconvolution can be greatly simplified for some special full sky signals, and we can get a very clear and simple physical scene for them. Fortunately, the strongest contamination in CMB experiments, the Doppler signal, is right such a kind of signal.

According to Equation 8 and Equation 11, we can calculate the 4*π* response to the Doppler signal by

$$d\_{\text{sidelob\text{\text\(\"\(\text{\(\}}\)\text{\(\text{\(\}}\)\text{\(\text{\(}}\)\text{\(\text{\(}}\)\text{\(}}\text{\(}\)\text{\(}}\text{\(}\text{)}\text{\(}\text{)}\text{\(}\text{)}]} = \frac{T\_0}{c}\mathbf{v} \cdot \sum\_{k} \left[\frac{(G\_k^A - G\_k^B)}{N}\mathbf{n}\_{\mathbf{k}}\right] / \tag{12}$$

where *k* stands for all sidelobe pixels.

Using Δ*G<sup>A</sup>* and Δ*G<sup>B</sup>* for the sidelobe gain uncertainties, and considering the fact that *G<sup>A</sup>* and *G<sup>B</sup>* are both constants in the spacecraft coordinate, we consequently obtain two constants in the spacecraft coordinate:

$$
\Delta \mathbf{n}\_{\mathbf{A}} = \sum\_{k} \frac{\Delta G\_{k}^{A} \mathbf{n}\_{\mathbf{k}}}{N}, \quad \Delta \mathbf{n}\_{\mathbf{B}} = \sum\_{k} \frac{\Delta G\_{k}^{B} \mathbf{n}\_{\mathbf{k}}}{N}, \tag{13}
$$

<sup>5</sup> The transmission factors are omitted here, e.g., the ratio between the antenna temperature and CMB temperature, and the ratio between temperature and electronic digital unit. That means, with all these simplifications, if the antenna response is a *δ* function which is none zero only at the optical direction, then the recorded signal is simply *S* = *b*

<sup>6</sup> Because the antennas are static in the spacecraft coordinate.

14 Will-be-set-by-IN-TECH

The third factor is a hidden factor that has never been noticed before. Generally speaking, radio antenna has response to all 4*π* solid angle, not merely along it's optical pointing. For the WMAP antennas, the 4*π* response is described by a normalized gain *G*: If there is only one beam *b* coming from a direction marked by pixel *i*, then the recorded signal should be

> *Npix*−1 ∑ *i*=0

> > (*G<sup>A</sup> <sup>i</sup>* <sup>−</sup> *<sup>G</sup><sup>B</sup>*

> > > *<sup>i</sup>* and *<sup>G</sup><sup>B</sup>*

*Npix*−1 ∑ *i*=0

the A-side and B-side antennas in the spacecraft coordinate<sup>6</sup> respectively, *Ti* is the CMB temperature. According to the WMAP convention, the normalization rule is <sup>∑</sup>*Npix*−<sup>1</sup>

In Equation 11, only a few pixels stands for the main lobe, and most pixels belong to the sidelobe, but the sidelobe pixels have much lower gain amplitudes: the summation of all sidelobe normalized gains are less than 5%. The standard method to clean the sidelobe contamination is by deconvolution; however, this is very complex and slow, and the biggest disadvantage is missing of a clear physical picture. We discovered that the deconvolution can be greatly simplified for some special full sky signals, and we can get a very clear and simple physical scene for them. Fortunately, the strongest contamination in CMB experiments, the

According to Equation 8 and Equation 11, we can calculate the 4*π* response to the Doppler

Using Δ*G<sup>A</sup>* and Δ*G<sup>B</sup>* for the sidelobe gain uncertainties, and considering the fact that *G<sup>A</sup>* and *G<sup>B</sup>* are both constants in the spacecraft coordinate, we consequently obtain two constants in

*<sup>N</sup>* , **<sup>Δ</sup>nB** <sup>=</sup> ∑

<sup>5</sup> The transmission factors are omitted here, e.g., the ratio between the antenna temperature and CMB temperature, and the ratio between temperature and electronic digital unit. That means, with all these simplifications, if the antenna response is a *δ* function which is none zero only at the optical direction,

*k*

Δ*G<sup>B</sup> <sup>k</sup>* **nk**

*<sup>c</sup>* **<sup>v</sup>** · ∑ *k* [ (*G<sup>A</sup> <sup>k</sup>* <sup>−</sup> *<sup>G</sup><sup>B</sup> k* )

*GiTi* . (10)

*<sup>i</sup>* )*Ti* , (11)

*<sup>N</sup>* **nk**] , (12)

*<sup>N</sup>* , (13)

*<sup>i</sup>* are the normalized gains for

*<sup>i</sup>*=<sup>0</sup> *Gi* =

*S* = *Gib*. For full sky signal like the CMB, the recorded signal is5:

*<sup>S</sup>* <sup>=</sup> <sup>1</sup> *Npix*

Since WMAP works in differential mode, the equation for WMAP should be:

*<sup>D</sup>* <sup>=</sup> <sup>1</sup> *Npix*

*dsidelobe* <sup>=</sup> *<sup>T</sup>*<sup>0</sup>

Δ*G<sup>A</sup> <sup>k</sup>* **nk**

**ΔnA** = ∑ *k*

<sup>6</sup> Because the antennas are static in the spacecraft coordinate.

where *Npix* is the number of pixels on the sky, *G<sup>A</sup>*

Doppler signal, is right such a kind of signal.

where *k* stands for all sidelobe pixels.

then the recorded signal is simply *S* = *b*

the spacecraft coordinate:

*Npix*.

signal by

**4.5.2 Equivalent pointing vector error**

Fig. 11. *Top Panels:* The expected temperature deviation caused by the equivalent pointing vector error due to the sidelobe uncertainty issue in the WMAP mission. *Bottom panels:* The quadrupole components of the top panels. From left to right: The equivalent pointing vector error is along the *x*, *y*, *z* axes in the spacecraft coordinate respectively.

and the uncertainty of *dsidelobe* can be expressed as

$$
\Delta d\_{sidebo} = \frac{T\_0}{\mathcal{c}} \mathbf{v} \cdot (\Delta \mathbf{n}\_{\mathbf{A}} - \Delta \mathbf{n}\_{\mathbf{B}}) = \frac{T\_0}{\mathcal{c}} \mathbf{v} \cdot \Delta \mathbf{n} \,. \tag{14}
$$

**Δn** is called the equivalent pointing vector error, because it isn't a real pointing error, but has the same effect as real pointing vector errors caused by direct reasons or the timing issue.

It's apparent that we are interested in the gain uncertainty Δ*G*, not the gain itself. It's very difficult to accurately determine the antenna sidelobe response, at least due to three reasons: 1, the sidelobe response is very weak; 2, the signal due to the sidelobe response is always mixed with other much stronger signals (some are even unknown); 3, the sidelobe response of the two antennas could overlap and it's almost impossible to exactly solve for both. Therefore, it's not strange that the WMAP sidelobe gain has very high uncertainty: as presented by Barnes et al. (2003), up to 30%; however, the provides 30% uncertainty contains only the average level uncertainty, with no inclusion of the pixel-to-pixel variance. Thus the overall uncertainty must be much higher than the claimed 30%.

It's well known that the dot product of two vectors is invariant in coordinate transform, thus we can calculate Δ*dsidelobe* in any coordinates using Equation 14, and the result will be the same. Since **Δn** is determined by the gain and antenna pointing vectors, which are both constants in the spacecraft coordinate, the best coordinate to calculate Δ*dsidelobe* is certainly the spacecraft coordinate. However, even in the spacecraft coordinate, it's still impossible to exactly calculate the uncertainty due to **Δn** (otherwise it won't be called the "uncertainty"). In this case, a possible way out is to divide **Δn** into three components Δ*nx*, Δ*ny*, Δ*nz* along the *X*, *Y*, *Z* axes of the spacecraft coordinate. For each component, we can set any amplitude for it (we will explain why we can do like this below) and use the corresponding Δ*dsidelobe* instead of the differential data to obtain an output map Δ*Tx* Δ*Ty* or Δ*Tz* like we did for the real TOD in Sec. 2.1. Examples of Δ*Tx*, Δ*Ty*, Δ*Tz* are presented in Fig. 11.

Both Fig. 10 and Fig. 11 illustrate the same thing: there could be significant systematical error in the WMAP CMB detection. Fig. 11 tell us more, that even if the antenna pointing vectors

Fig. 12. *Top Panels:* The expected temperature deviation caused by the equivalent pointing vector error due to the sidelobe uncertainty issue in the Planck mission. *Bottom panels:* The quadrupole components of the top panels. From left to right: The equivalent pointing vector

Systematics in WMAP and Other CMB Missions 131

Planck scan strategy and possible resulting CMB deviation due to real or equivalent pointing vector error (Liu & Li, 2011), and found that similar CMB deviation may also occur in the Planck mission, as shown in Fig. 12. If in the final data release of Planck, the CMB result after least-square fitting removal is consistent to the WMAP result with the similar treatment, then

Understanding systematic effects in experiments and observations is a difficult task. It is not strange for a newly explored waveband in astronomy that early observation results contain unaware systematics. As an example, COS B is the second mission for gamma-ray band. The COS B group published the 2CG catalog of high energy gamma-ray sources detected with the selection criteria for no more than one spurious detection above the adopted threshold from background fluctuation could be expected (Swanenburg et al., 1981). However, Li & Wolfendale (1982) found that about half of the released 25 sources are pseudo-ones produced by Galactic gamma-ray background fluctuation. The mistake came from a simplified model of the diffuse Galactic gamma-rays used by the COS B group, leading to systematically overestimate significance of source detection. After many years debating, the pseudo-sources have been finally deleted from the 2CG catalog, and instead of the simplified background model, more reliable structured models for diffuse Galactic gamma-rays have been adopted by all following gamma-ray missions. For microwave band, WMAP is just the second mission. Although the WMAP team made huge effort to study systematic effects, there still notably exist systematic errors in their released results. It seems that the foreground issue has been carefully and effectively treated, but the CMB dipole effect has not yet. The Doppler-dipole moment dominates CMB anisotropies, various errors in CMB experiments can produce pseudo-dipole signals and then artificial anisotropies in resultant CMB maps, seriously affecting cosmological studies. Beside the foreground emissions, the pseudo-dipole-induced anisotropy is another key systematic problem common for all CMB missions. Without carefully and effectively treating the dipole issue, CMB maps from COBE,

error is along the *x*, *y*, *z* axes in the spacecraft coordinate respectively.

the corresponding result would certainly be more robust and import.

WMAP, and Planck are not reliable for studying the CMB anisotropy.

**6. Discussion**

are accurate, it's still impossible to conclude that the CMB results are hence reliable. Not to mention the fact confirmed by Roukema (2010b) and Liu, Xiong & Li (2011) that there are more potential uncertainties in the data calibration. Thus it's really too early to be so sure about the CMB detection or a final CMB model right now.

#### **4.6 How to remove the artificial components?**

We have seen possible CMB temperature deviations due to real or equivalent antenna pointing errors in Fig 10, 11, but they don't 100% ensure that the final CMB result by WMAP is wrong. However, we can still do something to get a more reasonable CMB estimation. We know that if there is extra uncertainty due to any reason in CMB detection, the corresponding deviation map, if available, should be removed from the final CMB temperature map to ensure a clean and reliable result. Now we have successfully obtained the final temperature deviation pattern due to the possible pointing error (Fig. 10 or Fig. 11, no matter real or equivalent), then we surely need to remove it from the obtained CMB results. The last hamper is that we don't know the exact amplitude of Δ*nx*, Δ*ny*, Δ*nz*; however, there are already methods for that, which has already been adopted by the WMAP team in removing the foreground emission.

When we try to remove the foreground emission, we are facing exactly the same difficulty: The foreground emission can be modeled by astrophysical emission mechanism including free-free, synchrotron, and dust emissions, however, the exact amplitude of each emission mechanism can't be predicted by the model. Moreover, all emission mechanism take effect together in CMB detection, and they are also combined with the CMB signal, thus the only way to determine the amplitudes of each emission mechanism is by model fitting. In this process, three a priori emission maps are presented as *Tf f* , *Tsync*, and *Tdust* (Bennett et al., 2003; Finkbeiner et al., 1999; Finkbeiner, 2003; Gold et al., 2009), and the clean temperature *T* is supposed to be

$$T = T^\* - \mathcal{c}\_{ff} T\_{ff} - \mathcal{c}\_{sync} T\_{sync} - \mathcal{c}\_{dust} T\_{dust} \tag{15}$$

where the amplitudes *c f f* , *csync*, *cdust* are calculated by least-square fitting. The same technic can be used here: The real CMB temperature should be

$$T = T^\* - \mathfrak{c}\_{\mathfrak{X}} \Delta T\_{\mathfrak{X}} - \mathfrak{c}\_{\mathfrak{Y}} \Delta T\_{\mathfrak{Y}} - \mathfrak{c}\_{\mathfrak{Z}} \Delta T\_{\mathfrak{Z}} \tag{16}$$

and like above, the coefficients *cx*, *cy*, *cz* can be determined by least-square fitting. It's important to notice that, in least square fitting, the amplitudes of the templates Δ*Tx*, Δ*Ty*, Δ*Tz* are not important, that's why we said above that we can use any amplitude for Δ*nx*, Δ*ny*, and Δ*nz*. As shown by Liu & Li (2010), the result is well consistent to our previous work (Liu & Li, 2009b): the CMB quadrupole after template fitting removal is about 10 ∼ 20 *μ*K, much lower than the WMAP release. In other words, without supposing an a priori pointing error, we have self-consistently confirmed it posteriorly.

#### **5. Planck and future CMB detection missions**

Although the Planck spacecraft is significantly improved compared to WMAP, the basic detecting method are similar: antenna that suffers from 4*π* sidelobe response is used; the spacecraft works on the same L2 spot; the scan pattern is similar: a three-axis rotation consists of rotation around the Sun, spin around it's symmetry axis, and cycloidal procession around the Sun-to-spacecraft axis. Thus the three factors we have listed: direct pointing error, timing error, and sidelobe uncertainty can all take effect in the Planck mission. We have simulated the 16 Will-be-set-by-IN-TECH

are accurate, it's still impossible to conclude that the CMB results are hence reliable. Not to mention the fact confirmed by Roukema (2010b) and Liu, Xiong & Li (2011) that there are more potential uncertainties in the data calibration. Thus it's really too early to be so sure about the

We have seen possible CMB temperature deviations due to real or equivalent antenna pointing errors in Fig 10, 11, but they don't 100% ensure that the final CMB result by WMAP is wrong. However, we can still do something to get a more reasonable CMB estimation. We know that if there is extra uncertainty due to any reason in CMB detection, the corresponding deviation map, if available, should be removed from the final CMB temperature map to ensure a clean and reliable result. Now we have successfully obtained the final temperature deviation pattern due to the possible pointing error (Fig. 10 or Fig. 11, no matter real or equivalent), then we surely need to remove it from the obtained CMB results. The last hamper is that we don't know the exact amplitude of Δ*nx*, Δ*ny*, Δ*nz*; however, there are already methods for that, which has already been adopted by the WMAP team in removing the foreground emission. When we try to remove the foreground emission, we are facing exactly the same difficulty: The foreground emission can be modeled by astrophysical emission mechanism including free-free, synchrotron, and dust emissions, however, the exact amplitude of each emission mechanism can't be predicted by the model. Moreover, all emission mechanism take effect together in CMB detection, and they are also combined with the CMB signal, thus the only way to determine the amplitudes of each emission mechanism is by model fitting. In this process, three a priori emission maps are presented as *Tf f* , *Tsync*, and *Tdust* (Bennett et al., 2003; Finkbeiner et al., 1999; Finkbeiner, 2003; Gold et al., 2009), and the clean temperature *T*

where the amplitudes *c f f* , *csync*, *cdust* are calculated by least-square fitting. The same technic

and like above, the coefficients *cx*, *cy*, *cz* can be determined by least-square fitting. It's important to notice that, in least square fitting, the amplitudes of the templates Δ*Tx*, Δ*Ty*, Δ*Tz* are not important, that's why we said above that we can use any amplitude for Δ*nx*, Δ*ny*, and Δ*nz*. As shown by Liu & Li (2010), the result is well consistent to our previous work (Liu & Li, 2009b): the CMB quadrupole after template fitting removal is about 10 ∼ 20 *μ*K, much lower than the WMAP release. In other words, without supposing an a priori pointing

Although the Planck spacecraft is significantly improved compared to WMAP, the basic detecting method are similar: antenna that suffers from 4*π* sidelobe response is used; the spacecraft works on the same L2 spot; the scan pattern is similar: a three-axis rotation consists of rotation around the Sun, spin around it's symmetry axis, and cycloidal procession around the Sun-to-spacecraft axis. Thus the three factors we have listed: direct pointing error, timing error, and sidelobe uncertainty can all take effect in the Planck mission. We have simulated the

*T* = *T*<sup>∗</sup> − *c f f Tf f* − *csyncTsync* − *cdustTdust* , (15)

*T* = *T*<sup>∗</sup> − *cx*Δ*Tx* − *cy*Δ*Ty* − *cz*Δ*Tz* , (16)

CMB detection or a final CMB model right now.

**4.6 How to remove the artificial components?**

can be used here: The real CMB temperature should be

error, we have self-consistently confirmed it posteriorly.

**5. Planck and future CMB detection missions**

is supposed to be

Fig. 12. *Top Panels:* The expected temperature deviation caused by the equivalent pointing vector error due to the sidelobe uncertainty issue in the Planck mission. *Bottom panels:* The quadrupole components of the top panels. From left to right: The equivalent pointing vector error is along the *x*, *y*, *z* axes in the spacecraft coordinate respectively.

Planck scan strategy and possible resulting CMB deviation due to real or equivalent pointing vector error (Liu & Li, 2011), and found that similar CMB deviation may also occur in the Planck mission, as shown in Fig. 12. If in the final data release of Planck, the CMB result after least-square fitting removal is consistent to the WMAP result with the similar treatment, then the corresponding result would certainly be more robust and import.

#### **6. Discussion**

Understanding systematic effects in experiments and observations is a difficult task. It is not strange for a newly explored waveband in astronomy that early observation results contain unaware systematics. As an example, COS B is the second mission for gamma-ray band. The COS B group published the 2CG catalog of high energy gamma-ray sources detected with the selection criteria for no more than one spurious detection above the adopted threshold from background fluctuation could be expected (Swanenburg et al., 1981). However, Li & Wolfendale (1982) found that about half of the released 25 sources are pseudo-ones produced by Galactic gamma-ray background fluctuation. The mistake came from a simplified model of the diffuse Galactic gamma-rays used by the COS B group, leading to systematically overestimate significance of source detection. After many years debating, the pseudo-sources have been finally deleted from the 2CG catalog, and instead of the simplified background model, more reliable structured models for diffuse Galactic gamma-rays have been adopted by all following gamma-ray missions. For microwave band, WMAP is just the second mission. Although the WMAP team made huge effort to study systematic effects, there still notably exist systematic errors in their released results. It seems that the foreground issue has been carefully and effectively treated, but the CMB dipole effect has not yet. The Doppler-dipole moment dominates CMB anisotropies, various errors in CMB experiments can produce pseudo-dipole signals and then artificial anisotropies in resultant CMB maps, seriously affecting cosmological studies. Beside the foreground emissions, the pseudo-dipole-induced anisotropy is another key systematic problem common for all CMB missions. Without carefully and effectively treating the dipole issue, CMB maps from COBE, WMAP, and Planck are not reliable for studying the CMB anisotropy.

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#### **7. Acknowledgments**

This work is Supported by the National Natural Science Foundation of China (Grant No. 11033003). The data analysis made use of the WMAP data archive and the HEALPix software package.

#### **8. References**


18 Will-be-set-by-IN-TECH

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package.

**8. References**


**1. Introduction**

high power laser technology and plasma physics [25].

Historically, the modifications to standard electrodynamics were introduced for preventing the appearance of infinite physical quantities in theoretical analysis involving electromagnetic interactions. For instance, Born-Infeld [1] proposed a model in which the infinite self energy of point particles (typical of linear electrodynamics) are removed by introducing an upper limit on the electric field strength and by considering the electron an electrically charged particle of finite radius. Along this line, other Lagrangians for nonlinear electrodynamics were proposed by Plebanski, who also showed that Born-Infeld model satisfy physically acceptable requirements [20], due to its feature of being inspired on the special relativity principles. Applications and consequences of nonlinear electrodynamics have been extensively studied in literature, ranging from cosmological and astrophysical models [22] to nonlinear optics,

**Nonlinear Electrodynamics Effects on**

**the Cosmic Microwave Background:**

Herman J. Mosquera Cuesta1,2,3,4 and Gaetano Lambiase5,6 *1Departamento de Física, Centro de Ciências Exatas e Tecnológicas (CCET),* 

*4International Institute for Theoretical Physics and High Mathematics* 

*3International Center for Relativistic Astrophysics Network (ICRANet), Pescara,* 

*2Instituto de Cosmologia, Relatividade e Astrofísica (ICRA-BR), Centro Brasileiro de Pesquisas Físicas, Urca Rio de Janeiro, RJ,* 

*Universidade Estadual Vale do Acaraú, Sobral, Ceará,* 

**Circular Polarization** 

*5Dipartimento di Fisica "E. R. Caianiello", Universitá di Salerno, Fisciano (Sa),* 

*Einstein-Galilei, PRATO,* 

*6INFN, Sezione di Napoli,* 

*1,2Brazil 3,4,5,6Italy* 

**8**

In this paper we investigate the polarization of CMB photons if electrodynamics is inherently nonlinear. We compute the polarization angle of CMB photons propagating in an expanding Universe, by considering in particular cosmological models with planar symmetry. It is shown that the polarization does depend on the parameter characterizing the nonlinearity of electrodynamics, which is thus constrained by making use of the recent data from WMAP and BOOMERanG (for other models see [26]). It is worth to point out that the effect we are investigating, i.e. the rotation of the polarization angle as radiation propagates in a planar geometry, is strictly related to Skrotskii effect [27]. The latter is analogous to Faraday effect

