Statistically anisotropic Gaussian simulations of the CMB temperature field

Although theoretically expected to be statistically isotropic (SI), the observed cosmic microwave background (CMB) temperature and polarization field would exhibit SI violation due to various inevitable effects like weak lensing, Doppler boost, and practical limitations of observations like noncircular beam, masking, etc. However, the presence of any SI violation beyond these effects may lead to a discovery of inherent cosmic SI violation in the CMB temperature and polarization field. Recently, Planck presented strong evidence of SI violation as a dipolar power asymmetry of the CMB temperature field in two hemispheres. Statistical studies of SI violation effect require non-SI (nSI) Gaussian realizations of the CMB temperature field. The nSI Gaussian temperature field leads to nonzero off-diagonal terms in the spherical harmonics (SH) space covariance matrix encoded in the coefficients of the bipolar spherical harmonics (BipoSH) representation. We discuss an effective numerical algorithm, code for nonisotropic Gaussian sky (CoNIGS) to generate nSI realizations of the Gaussian CMB temperature field of Planck-like resolution with specific cases of SI violation. Realizations of nSI CMB temperature field are obtained for nonzero quadrupolar ($L=2$) BipoSH measurements by WMAP, dipolar asymmetry (resembles $L=1$ BipoSH coefficients) with a scale-dependent modulation field as measured by Planck and for Doppler-boosted CMB temperature field which also leads to $L=1$ BipoSH spectra. Our method, CoNIGS can incorporate any kind of SI violation and can produce nSI realizations efficiently.

Figure 1

(a) Scaling of computational time for CD with $L=1$, $M=0$ or $L=2$, $M=0$. Blue circles are the time taken by the code for different $N=l_{\max }^2$ values. For $l_{\max }=2048$, computational time required on a single processor of clock speed 2.60 GHz is $\sim 12$ seconds for CD. The green line is fit to these circles with $N^{0.853}\sim l_{\max }^{1.70}$ for CD. (b) We plot CD for a different set of $L_{\max }=1(\text{yellow})$, 2 (blue), 3 (red), 10 (green), and $M=[-L_{\max },L_{\max }]$ values with the dimension of the SH space covariance matrix, $N$. The scaling of each case is also shown by a fit in solid line.

Figure 2

Scaling of computational time on a single processor of clock speed 2.60 GHz for generating a single SH space map $a_i$ as a function of the dimension of SH space covariance matrix, $N\approx l_{\max }^2$. Blue circles are the time taken by the code for different values of $l_{\max }$. For $l_{\max }=2048$, with specific BipoSH coefficients, $L=1$ or $L=2$ have a computational time of $\sim 25$ seconds on a single processor of clock speed 2.60 GHz for generating the maps. The green line is fit to these circles with $N^{0.77}\sim l_{\max }^{1.54}$ for generating nSI maps in SH space.

Figure 3

nSI realization for the CMB temperature field with $L=2$ BipoSH spectra produced using CoNIGS in ecliptic coordinates.

Figure 4

Difference between the nSI map with $L=2$ BipoSH spectra and SI map for the same seed of random realization in ecliptic coordinates. The difference in temperature is in the range $[-5.56\mu K,5.74\mu K]$ to be compared to the range of the nSI map $[-534\mu K,505\mu K]$ given in Fig. 3.

Figure 5

Comparison of input and output values of $D_l^{00}=l(l+1)C_l/2\pi$ obtained from 1000 nSI realizations.

Figure 6

Comparison of BipoSH spectra, (a) $D_{ll}^{20}=l(l+1){\alpha }_{ll}^{20}/2\pi$, (b) $D_{ll+2}^{20}=l(l+1){\alpha }_{ll+2}^{20}/2\pi$ obtained from 1000 realizations with the input value of $D_{ll}^{20}$ and $D_{ll+2}^{20}$. Here the BipoSH spectra are binned with $\mathrm{\Delta }l=50$.

Figure 7

Modulation strength $m_l^1$ with a scale dependence as measured by Planck [2] used for producing nSI maps.

Figure 8

nSI realization with $L=1$ nonzero BipoSH spectra due to dipolar asymmetry with a scale-dependent modulation strength (Fig. 7) as measured by Planck [2].

Figure 9

Difference between the SI and nSI realization with $L=1$ BipoSH spectra for scale-dependent modulation strength (Fig. 7) as measured by Planck [2]. The difference in temperature is in the range $[-0.972\mu K,0.916\mu K]$, about $\sim 500$ times smaller than nSI map given in Fig. 8.

Figure 10

Difference between the SI and nSI realization with $L=1$ BipoSH spectra for scale-independent modulation strength $m_1=0.008$. The difference in temperature is in the range $[-2.34\mu K,2.26\mu K]$, about $\sim 250$ times smaller than the nSI realization.

Figure 11

Comparison of BipoSH spectra, $D_{ll+1}^{10}=l(l+1){\alpha }_{ll+1}^{10}/2\pi$ obtained from 1000 realizations of scale-dependent dipolar asymmetric nSI maps with the input value of $D_{ll+1}^{10}$. Here, the BipoSH spectra are binned with $\mathrm{\Delta }l=64$. The error bar for the 1st bin is 168. The blue line plots zero BipoSH spectra. Using a minimum variance estimator [30, 31], a significant detection of BipoSH spectra is made by Planck [2].

Figure 12

nSI realization with $L=1$ nonzero BipoSH spectra due to Doppler boost for $\beta =1.23\times {10}^{-3}$ along $z$-axis.

Figure 13

Difference between SI and Doppler boosted nSI realization. The difference in temperature is in the range $[-0.862\mu K,0.895\mu K]$ which is $\sim 500$ times smaller than the range of fluctuation in the corresponding nSI map given in Fig. 12.

Figure 14

Comparison of BipoSH spectra, $D_{ll+1}^{10}=l(l+1){\alpha }_{ll+1}^{10}/2\pi$ obtained from Doppler boosted 1000 realizations of the CMB temperature sky with the input value of $D_{ll+1}^{10}$. Here, the BipoSH spectra are binned with $\mathrm{\Delta }l=128$. The error bar for the 1st bin is 160. The blue line plots zero BipoSH spectra. The deviation in Doppler boosted BipoSH spectra from the blue line indicates the possibility for detection of $\beta$ from the small angular scales of the CMB temperature field. Using quadratic estimators a significant detection of $\beta$ is made by Planck [6].