Statistical Anisotropic Gaussian Simulations of the CMB Temperature Field
aa r X i v : . [ a s t r o - ph . C O ] A p r Statistical Anisotropic Gaussian Simulations of the CMBTemperature Field
Suvodip Mukherjee ∗ and Tarun Souradeep † IUCAA, Post Bag 4, Ganeshkhind, Pune-411007, IndiaApril 27, 2015
Abstract
Although theoretically expected to be Statistically Isotropic (SI), the observed Cosmic Mi-crowave Background (CMB) temperature & polarization field would exhibit SI violation dueto various inevitable effects like weak lensing, Doppler boost and practical limitations of ob-servations like non-circular beam, masking etc. However, presence of any SI violation beyondthese effects may lead to a discovery of inherent cosmic SI violation in the CMB temperature& polarization field. Recently, Planck presented strong evidence of SI violation as a dipolarpower asymmetry of the CMB temperature field in two hemispheres. Statistical studies of SIviolation effect require non-SI (nSI) Gaussian realizations of CMB temperature field. The nSIGaussian temperature field leads to non-zero 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 Non-Isotropic GaussianSky (CoNIGS) to generate nSI realizations of Gaussian CMB temperature field of Planck likeresolution with specific cases of SI violation. Realizations of nSI CMB temperature field areobtained for non-zero quadrupolar ( L = 2) BipoSH measurements by WMAP, dipolar asymme-try (resembles L = 1 BipoSH coefficients) with a scale dependent modulation field as measuredby Planck and for Doppler boosted CMB temperature field which also leads to L = 1 BipoSHspectra. Our method, CoNIGS can incorporate any kind of SI violation and can produce nSIrealizations efficiently. Cosmic Microwave Background (CMB) is a very powerful probe of our Universe. Several importantCMB experiments like COBE, WMAP, Planck, BOOMERanG, ACT, SPT etc., have opened anera of precision cosmology. Recent measurements from Planck [1] of the CMB temperature powerspectrum matches well with the minimal ΛCDM model at angular scales smaller than 2 degrees.However, at large angular scales, Planck [2] has revealed possible signature of Statistical Isotropy(SI) violation of CMB temperature field, which is beyond our present understanding. SI viola-tion can be measured by Bipolar Spherical Harmonics (BipoSH) coefficients, introduced in CMBtemperature measurements by Hajian & Souradeep [3, 4]. These are the linear combinations ofoff-diagonal terms in the covariance matrix and arise from the correlation between different CMBmultipoles l , in Spherical Harmonics (SH) space. ∗ [email protected] † [email protected] In our present understanding, temperature fluctuation of CMB originated from the quantum fluc-tuations of the ground state of the single inflation scalar field. This implies that the statistics ofthe CMB temperature fluctuation is expected to be Gaussian with zero mean. Recent results fromexperiments like Planck [23] also place fairly strong constraints on primordial non-Gaussianity tobe consistent with zero. In the next sections, we briefly review the statistics of CMB temperaturefield and its covariance matrix.
Temperature anisotropy of CMB sky map ∆ T (ˆ n ) can be expanded in the orthonormal space ofSpherical Harmonics (SH) functions on the sphere,∆ T (ˆ n ) = X lm a lm Y lm (ˆ n ) , (1)where, a lm are the coefficients of Gaussian random field in SH space. The statistics of any Gaussiandistribution can be specified by its covariance matrix, G ij ≡ h a ∗ j a i i where, i = l ( l + 1) + m −
32s a single positive index representing an SH multipole ( l, m ). The probability distribution of thetemperature field is given by, P [ a j ] = 1(2 π ) N/ p | G | exp [ − ( a † i G − ij a j )2 ] , (2)where, we assume the Einstein summation rule. Given the largest multipole l max , the covariancematrix is of size N = l max ( l max + 1) + l max − . (3)Under the assumption of SI temperature field, the two-point correlation function on the sphere(sky) depends only upon angular separation between the two-points. This implies a diagonalcovariance matrix G , given by the angular power spectrum C l by, G ij ≡ h a ∗ j a i i = C l δ ij . (4)But for SI violated maps, covariance matrix is not diagonal. The non-zero off-diagonal elementsof non-SI (nSI) temperature field can be expressed by Bipolar Spherical Harmonics (BipoSH)coefficients introduced by Hajian & Souradeep [3]. In the next section we review the BipoSHcoefficients and its connection to the covariance matrix. As mentioned in the previous section, the CMB temperature fluctuations are Gaussian random fieldwith zero mean and we can express the statistics of this field by the two-point correlation function.In the full generality, the two-point correlation of SH coefficients of the CMB temperature anisotropy h a ∗ j a i i can be expanded in the tensor product basis of two SH space as, h a ∗ j a i i = X LM A LMll ′ ( − m ′ C LMlml ′ − m ′ , (5)where, A LMll ′ are called the BipoSH coefficients [3] and C LMlml ′ m ′ are the Clebsch-Gordan (CG) coef-ficients. These BipoSH coefficients with L = 0 , M = 0, are the diagonal elements related to theangular power spectrum by A ll ′ δ ll ′ = ( − l C l p (2 l + 1).For nSI case, BipoSH coefficients are non-zero for L = 0 , M = 0. We can relate eq.(5) to thecovariance matrix G by, G ij ≡ h a ∗ j a i i = X LM A LMll ′ ( − m ′ C LMlml ′ − m ′ ,G ij ≡ h a ∗ j a i i = C l δ ij + X LM ; L =0 A LMll ′ ( − m ′ C LMlml ′ − m ′ , (6)where i and j are related with l, m and l ′ , m ′ by i = l ( l + 1) + m − ,j = l ′ ( l ′ + 1) + m ′ − . (7)3sing the definition of BipoSH spectra from WMAP [9], eq.(6) becomes, G ij = C l δ ij + X LM ; L =0 ( − m ′ α LMll ′ Π ll ′ Π L C L l l ′ C LMlml ′ − m ′ , (8)where, BipoSH coefficients A LMll ′ , are related with BipoSH spectra, α LMll ′ , by A LMll ′ = α LMll ′ Π ll ′ Π L C L l l ′ , (9)and following the notation in [24]Π l l ...l n = p (2 l + 1)(2 l + 1) . . . (2 l n + 1) . (10)Any non-zero measurement of BipoSH coefficients indicates violation of SI of our Universe. Weaklensing, Doppler boost, masking and non-circular beam are some of important effects that lead toSI violation in the observed CMB sky even when the underlying CMB signal is SI. The central difference between the covariance matrix in SH space for SI and nSI CMB temperaturefield is the presence of off-diagonal terms in the SH space covariance matrix. This implies thatdifferent modes ( l, m ) of CMB are not independent in SH space for nSI temperature field. Thekey idea we implement is to make a change of basis from SH space to another space in which thiscovariance matrix is diagonal. A linear transformation does not change the Gaussian statisticsof the field. In the new space, different modes are no more correlated, but realizations of CMBtemperature in this basis are manifestly SI violated. On performing an inverse transformation ofCMB temperature map from the new space to SH space, gives us the nSI CMB temperature mapsin SH space.
In this section, we discuss the method of diagonalization of the covariance matrix by CholeskyDecomposition (CD) algorithm. In CMB analysis, CD has been implemented by Gorski [25] fordiagonalization of coupling matrix for SH on the cut sky. When SI violation is captured in a limitedset of non-zero BipoSH coefficients (like L = 1 captures Doppler boosted sky, scale dependentdipole modulation, L = 2 captures quadrupolar anisotropy and any other pattern), it is possibleto implement an efficient code that scale by N . implying ∼ l . max ( l max determines the angularresolution of the map), instead of the well-known N / G of a Gaussian distribution is always positive definite which satisfiesthe following two conditions [26],1. For any vector y ∈ R , y T Gy >
0. This condition implies a strong constrain on diagonal terms,i.e. G ii > | G | ≥
0. Since the inverseof the covariance matrix ( G − ) must exist, this makes | G | > C LMlml ′ m ′ = 0; iff | l − l ′ | < L < l + l ′ ; m + m ′ = M ; C L l l ′ = 0; iff l + l ′ + L = 2 n ; n is an integer . (11)Due to the conditions mentioned in eq.(11), the covariance matrix is sparsely populated, whichmakes the diagonalization of the covariance matrix efficient. Cholesky decomposition leads todecomposition of covariance matrix into a lower triangular matrix L and its conjugate transpose L † , G = LL † , (12)where elements of L are related to the elements of G by [26], L ii = vuut ( G ii − i − X k =1 L ik ) , L ji = ( G ij − P i − k =1 L ik L jk ) L ii ; j = i + 1 , .....n. (13)On performing CD on covariance matrix G , the modification to the probability distribution functioneq.(2) can be written as, P [ a ] = 1(2 π ) N/ p | G | exp [ − ( a † ( LL † ) − a )2 ] , = 1(2 π ) N/ p | G | exp [ − ( a † ( L † ) − L − a )2 ] , = 1(2 π ) N/ p | G | exp [ − ( L − a ) † ( L − a )2 ] , = 1(2 π ) N/ p | G | exp [ − x † Ix , (14)where, we define x j = L − ji a i . x is the Gaussian CMB temperature map in the new space withunit variance. On performing inverse of this transformation, we get, a i = L ij x j . This map a i , ismanifestly SI violated in SH space and average over random realizations of a i should match theinput BipoSH coefficients. In the previous section we discussed the key idea of making nSI simulations of CMB temperaturefield. In this section, we discuss the logical steps, we use to produce these nSI maps.To efficiently diagonalize the covariance matrix of size N ≈ l max , we use the properties of CGcoefficients mentioned in eq.(11). For a given value of angular power spectra C l and BipoSHspectra α LMll ′ , we develop a numerical code to diagonalize the covariance matrix G , into lowertriangular matrix L and L † . L is also a sparsely populated matrix when incorporated physicaleffects like scale dependent modulation, Doppler boost and quadrupolar anisotropies. The numberof non-zero elements in L depends upon the number of elements which satisfy the properties of CG5oefficients given in eq.(11). For a dipole statistical anisotropy ( L = 1 non-zero BipoSH coefficients)and a quadrupolar statistical anisotropy ( L = 2 non-zero BipoSH coefficients), only correlated off-diagonal terms are ( l, l + 1) and ( l, l + 2) respectively. This makes the covariance matrix sparselypopulated. By the choice of coordinate system such that only M = 0 BipoSH coefficients are non-zero, we can make the matrix very sparse with only one off-diagonal term present in the covariancematrix along with the diagonal term. After CD, nSI maps can be rotated to any other coordinatesystem, by using the rotate alm subroutine of HEALPix [27].In Fig. 1(a), we plot the computational time requires for the diagonalization of the covariancematrix for Doppler boost, dipole modulation and quadrupolar anisotropies using CD. Using theproperties of CG coefficients, CoNIGS scales with size of the matrix N by N . implying ∼ l . max for diagonalization. For a denser covariance matrix with more BipoSH multipoles ( L max ), thescaling of the computational time increases. In Fig. 1(b), we also plot the scaling of CD codewith l max for different values of L max . The computational time for performing the multiplication L ij x j , to produce real and imaginary part of nSI maps in SH space are plotted in Fig. 2. For agiven angular power spectra C l and BipoSH spectra α LMll ′ , we need to generate the lower triangularmatrix L only once, and different realizations can be obtained by multiplying L with different unitvariance Gaussian realizations x i .The temperature realizations a i in SH space are complex numbers with both real and imaginarypart, i.e. a i = c i + i d i , (15)where, c i and d i are the real Gaussian random variables. But for the azimuthal symmetric modes( m = 0), imaginary part d i vanishes. Hence, the map x i in the new basis should be transformedboth to the real part c i and imaginary part d i of the temperature map a i . We define x Ri and x Ii , astwo parts of the map x i , which contribute respectively to the real and imaginary part of the map a i , in SH space. Then, the variance of x i should satisfies, σ x R + σ x I ≡ σ x = 1; for m = 0 modes σ x R ≡ σ x = 1; for m = 0 modes . (16)Following are the steps incorporated to produce nSI realizations of CMB temperature field usingCoNIGS.1. Compute the covariance matrix using CG subroutine and implement CD on the covariancematrix G , to produce lower triangular matrix L and its conjugate transpose L † . This stepscales as l . max , plotted in Fig. 1(a), and takes CPU times ≈
12 seconds for l max = 2048 ona single processor with 2 . scale dependent modulationand quadrupolar anisotropy cases. This step needs to be performed only once for a givencovariance matrix.2. Generate Gaussian random variables x Ri and x Ii with variance satisfying eq.(16). This is thetemperature map x i , in the new vector space.3. Multiply x i with the lower triangular matrix L , to obtain temperature map, a i in SH space.This step takes ≈
25 seconds of CPU time for producing one SI violated realization in SH spacewhich can incorporate Doppler boost, scale dependent modulation and quadrupolar effect for l max = 2048. The scaling of the computational time with l max is plotted in Fig. 2. This stepneeds to be implemented on every unit variance map x i to produce many realizations.6. Using the alm map subroutine in HEALPix [27], we generate the temperature map from a i .The scaling of this part is the usual scaling for alm map subroutine.5. Using the HEALPix subroutine rotate alm , we rotate the nSI maps to change the co-ordinatesystem. Log N10 -1 L o g T , [ s e c o n d s ] Computational time for Cholesky decompositionN
Fit (a) Log N10 L o g T , [ s e c o n d s ] Computational time for L max =1N
FitComputational time for L max =2N
FitComputational time for L max =3N
FitComputational time for L max =10N
Fit (b)
Figure 1: ( a ) Scaling of computational time for Cholesky Decomposition (CD) with L = 1 , M = 0or L = 2 , M = 0. Blue circles are the time taken by the code for different N = l max values. For l max = 2048, computational time requires on a single processor of clock speed 2 .
60 GHz is ∼ N . ∼ l . max for CD. ( b ) We plot CD fordifferent set of L max = 1 ( yellow ) , blue ) , red ) ,
10 ( green ) and M = [ − L max , L max ] values withthe dimension of the SH space covariance matrix, N . The scaling of each cases are also shown bya fit in solid line. 7 Log N10 -1 L o g T , [ s e c o n d s ] Computational time for generating nSI map in spherical harmonics basisN
Fit
Figure 2: Scaling of computational time on a single processor of clock speed 2 .
60 GHz for generatinga single SH space map, a i as a function of the dimension of SH space covariance matrix, N ≈ l max .Blue circles are the time taken by the code for different values of l max . For l max = 2048, withspecific BipoSH coefficients, L = 1 or L = 2 have a computational time of ∼
25 seconds on a singleprocessor of clock speed 2 .
60 GHz for generating the maps. Green line is fit to these circles with N . ∼ l . max for generating nSI maps in SH space. We study three different cases of nSI maps with the BipoSH spectra that have come under discus-sions and study due to results obtained by recent experiments like WMAP and Planck.1. The quadrupolar ( L = 2) BipoSH spectra as measured by WMAP [9].2. BipoSH spectra for L = 1 with a scale dependent dipole modulation strength which resultsin dipolar asymmetry as detected by Planck [2].3. Doppler boosted CMB temperature map with non-zero BipoSH spectra for L = 1 as measuredby Planck [6]. L = 2 . The measurement of BipoSH spectra for L = 2 , M = 0 by WMAP [9] is a signature of SI violation.With angular power spectrum C l and only non-zero BipoSH spectra α ll and α ll +2 , we obtainthe covariance matrix G using eq.(8). Using the numerical algorithm CoNIGS, we obtain the nSIrealization of CMB temperature field for l max = 2048, given in Fig. 3. To show the visual effect ofthe non-zero BipoSH coefficients, we take the difference of SI and nSI realization (with the same seedvalue) and is plotted in Fig. 4. The range of the variation of difference in temperature fluctuationis [ − . µ K , . µ K], which is ∼
100 times smaller than the range of temperature fluctuation ofnSI map.The average two-point correlation function estimated from these maps should match the input8ovariance matrix G . To test the consistency of the two-point correlation for these nSI maps withthe input covariance matrix, we obtain 1000 realizations of nSI maps and then using the BipoSHestimation code [28], we obtain the two-point correlation from the simulated temperature maps.The comparison between input and output values of angular power spectra and BipoSH spectra areplotted in Fig. 5 and Fig. 6 respectively. The comparison between input and output power spectraof temperature field ensures that the ensemble average of nSI realizations generated by our methodrecover the input angular power spectra and BipoSH spectra. This provides an efficient means forcreating Monte Carlo ensembles of CMB maps useful in studying different models of SI violationand to study their statistical properties.Figure 3: nSI realization for CMB temperature field with L = 2 BipoSH spectra produced usingCoNIGS in ecliptic coordinates.Figure 4: Difference between the nSI map with L = 2 BipoSH spectra and SI map for the sameseed of random realization in ecliptic coordinates. The difference in temperature is in the range[ − . µK, . µK ] to be compared to the range of nSI map [ − µK, µK ] given in Fig. 3.9
500 1000 1500 2000 2500CMB Multipole, l01000200030004000500060007000 D l [ ( µ K ) ] Output from 1000 nSI mapsInput
Figure 5: Comparison of input and output values of D l = l ( l + 1) C l / π obtained from 1000 nSIrealizations. D ll [ ( µ K ) ] Output from 1000 nSI mapsInput (a) D ll + [ ( µ K ) ] Output from 1000 nSI mapsInput (b)
Figure 6: Comparison of BipoSH spectra, (a) D ll = l ( l + 1) α ll / π , (b) D ll +2 = l ( l + 1) α ll +2 / π obtained from 1000 realizations with the input value of D ll and D ll +2 . Here the BipoSH spectraare binned with ∆ l = 50. 10 .2 Dipolar asymmetry: Scale dependent non-zero dipolar ( L = 1) BipoSHspectra from Planck measurement.
The nSI estimator on Planck CMB map measured a 3 . σ detection of BipoSH spectra for L = 1,modeled as a dipolar modulation of SI temperature field with a scale dependent modulation strength[2]. So, it is interesting to make realizations of nSI maps that incorporate this feature. CoNIGS is anefficient, fast method to produce nSI realization for any kind of scale dependent modulation field .In CoNIGS, we encapsulate scale dependent modulation in terms of L = 1 BipoSH coefficients,which are the complete representation for any SI violation. In terms of BipoSH spectra, scaledependent modulation strength can be incorporated by the expression, α ll +1 = m l (cid:20) C T Tl + C T Tl +1 (cid:21) , (17)where, we used Planck best fit ΛCDM lensed C l [1, 29]. m l / π Figure 7: Modulation strength m l with a scale dependence as measured by Planck [2] used forproducing nSI maps.Using CoNIGS, we make realizations of such nSI CMB sky with the BipoSH spectra for L = 1with a scale dependent modulation strength, m l , given in Fig. 7. This complicated modulationstrength is taken to imitate the modulation strength as measured by Planck [2]. In Fig. 8, we plota nSI realization and in Fig. 9 we plot the corresponding difference between SI and nSI realization(with the same seed value) for scale dependent dipole modulation.Using CoNIGS, we also make realizations with scale independent modulation strength with m =0 . scale independent modulated nSI map is plotted inFig. 10. On comparing the difference maps plotted in Fig. 9 and Fig. 10, we can conclude thatthe scale independent modulation strength results in fluctuation at all scales, in contrast to the scale dependent modulation, where more fluctuation are at larger angular scale. The comparisonbetween input and output values of angular power spectra and BipoSH spectra from 1000 scaledependent dipole modulated realizations are plotted in Fig. 5 and Fig. 11 respectively. nSI realization with scale dependent modulation can also be made by modulating each temperature multipoleas, T (ˆ n ) = P l (1 + w l (ˆ n )) P m a lm Y lm (ˆ n ). But the BipoSH spectra for them are slightly different from the BipoSHspectra given in eq.(17). L = 1 non-zero 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 scaledependent modulation strength (Fig. 7) as measured by Planck [2]. The difference in temperatureis in the range [ − . µK, . µK ] about ∼
500 times smaller than nSI map given in Fig. 8.12igure 10: Difference between the SI and nSI realization with L = 1 BipoSH spectra for scaleindependent modulation strength m = 0 . − . µK, . µK ] about ∼
250 times smaller than the nSI realization. D ll + [ ( µ K ) ] Output from 1000 nSI mapsInputZero BipoSH value
Figure 11: Comparison of BipoSH spectra, D ll +1 = l ( l + 1) α ll +1 / π obtained from 1000 realizationsof scale dependent dipolar asymmetric nSI maps with the input value of D ll +1 . Here, the BipoSHspectra are binned with ∆ l = 64. The error bar for the 1 st bin is 168. The blue line plots zeroBipoSH spectra. Using minimum variance estimator [30, 31], a significant detection of BipoSHspectra is made by Planck [2]. ( L = 1) BipoSH spectra from our localmotion.
Doppler boost of CMB temperature and polarization field due to our local motion with velocity( β ≡ | v | /c = 1 . × − ) induces non-zero dipolar ( L = 1) BipoSH coefficients as shown byMukherjee et al. [7]. Recent result from Planck [6] estimated β from the off-diagonal terms of theSH space covariance matrix which are related to the L = 1 BipoSH coefficients. Doppler boosting13f CMB temperature field results into two kinds of effect, modulation and aberration. Our methodis an efficient way of simulating high resolution maps as mentioned in Fig. 1(a), 2 and also canincorporate aberration and modulation effect in terms of BipoSH spectra up to any order in β . Thelinear order effect of Doppler boost on CMB temperature field leads to non-zero BipoSH spectragiven by [6, 7] α ll +1 = β (cid:20) ( l + b ν ) C T Tl − ( l + 2 − b ν ) C T Tl +1 (cid:21) , (18)where, β is taken along z -axis and b ν is the frequency dependent effect on Doppler boost given by[6], b ν = νν coth (cid:18) ν ν (cid:19) − , (19)with ν = 57 GHz. We estimate the BipoSH spectra with b ν ≈ ν = 217 GHz.We plot nSI realizations of CMB temperature in Fig. 12 with induced Doppler boost for the valueof β = 1 . × − . The difference between SI and nSI realization (with the same seed) is plottedin Fig. 13. The realizations produced by CoNIGS are manifestly nSI and have non-zero valueof BipoSH spectra for L = 1, which we recover back from the BipoSH estimation code [28]. InFig. 14, we compare the consistency of input BipoSH spectra with the BipoSH spectra from 1000realizations. Power spectra, C l from the 1000 simulations also matches the given input C l as plottedin Fig. 5.Figure 12: nSI realization with L = 1 non-zero BipoSH spectra due to Doppler boost for β =1 . × − along z -axis. 14igure 13: Difference between SI and Doppler boosted nSI realization. The difference in temper-ature is in the range [ − . µK, . µK ] which is ∼
500 times smaller than range of fluctuationin the corresponding nSI map given in Fig. 12. D ll + [ ( µ K ) ] Output from 1000 nSI mapsInputZero BipoSH value
Figure 14: Comparison of BipoSH spectra, D ll +1 = l ( l + 1) α ll +1 / π obtained from Doppler boosted1000 realizations of CMB temperature sky with the input value of D ll +1 . Here, the BipoSH spectraare binned with ∆ l = 128. The error bar for the 1 st bin is 160. The blue line plots zero BipoSHspectra. The deviation in Doppler boosted BipoSH spectra from the blue line indicates the possi-bility for detection of β from the small angular scales of CMB temperature field. Using quadraticestimators a significant detection of β is made by Planck [6]. Statistical Isotropy (SI) violation of CMB temperature and polarization maps are an unavoidableconsequence of weak lensing by large scale structures, Doppler boost due to local motion, non-trivial cosmic topology etc. Many observational systematics like masking, non-circularity of beam15tc., can also lead to SI violation of CMB temperature field. Recent experiments like WMAPand Planck measured statistically significant non-zero BipoSH coefficients. A likely explanationof non-zero quadrupolar ( L = 2) BipoSH detection by WMAP as arising from uncorrected non-circular beam effect is mentioned by several authors [9, 10, 11, 12, 13]. Doppler boost due to ourlocal motion leads to L = 1 BipoSH spectra, which is also measured by Planck [6]. Planck alsomade a 3 . σ detection of dipolar ( L = 1) BipoSH spectra apart from Doppler boost, known as thedipolar asymmetry [2], which is beyond the understanding of present SI Cosmological models. Thismeasurement indicates, CMB temperature field to be manifestly SI violated. To study these effectsand to understand the statistics of Non-SI (nSI) CMB sky, it is important to make nSI realizationsof CMB temperature and polarization field.In this paper, we have described an efficient numerical algorithm, Code for Non-Isotropic GaussianSky (CoNIGS) developed for generating nSI Gaussian realizations of CMB temperature field. nSICMB temperature field leads to non-zero off-diagonal (BipoSH coefficients) terms in the SH spacecovariance matrix, in contrast to the SI temperature field that has only non-zero diagonal (angularpower spectra) terms. The central idea of the technique is to diagonalize once the covariance matrixusing Cholesky Decomposition (CD) into lower triangular matrix L and L † as mentioned in eq.(12)and then producing random nSI realization a i , by multiplying the lower triangular matrix L witha unit variance Gaussian realization x j , as discussed in Sec. 3.1. For L = 1, L = 2 SI violationcases, the non-zero elements of the covariance matrix are related to the non-zero Clebsch-Gordan(CG) coefficients as mentioned in eq.(8), which ensures that the covariance matrix is sparselypopulated. So, we can diagonalize the covariance matrix faster than the usual CD. The time takenfor diagonalization using CoNIGS depends on the dimension of matrix, N by N . implying a ∼ l . max as given in Fig. 1(a), in contrast to the usual CD, which depends upon N by N /
6. Ontaking non-zero BipoSH coefficients, A LMll ′ , with larger L values, covariance matrix becomes denser,which results into more computational time as showed in Fig. 1(b). As mentioned in Sec. 3.1,nSI realization in SH space are obtained by multiplying lower triangular matrix L with the unitvariance Gaussian realization x j , this step scales with dimension of covariance matrix by N . asplotted in Fig. 2.Using this method we have generated nSI realizations for temperature field, plotted in Fig. 3, with L = 2 BipoSH spectra as measured by WMAP. The difference between the SI and nSI realization(with the same seed) is plotted in Fig. 4. The comparison between input and output values ofangular power spectra and BipoSH spectra from 1000 realizations are plotted in Fig. 5 and Fig. 6respectively.Planck [2] measurement of BipoSH coefficients for L = 1 dipolar asymmetry, shows a scale dependent modulation as plotted in Fig. 7. Incorporating SI violation in terms of BipoSH coefficients, we canmake nSI realizations for any scale dependent modulation strength using CoNIGS. For the scaledependent modulation strength, plotted in Fig. 7, nSI map and the difference between the SIand nSI map for the same seed value of the random realization are plotted in Fig. 8 and Fig. 9respectively. We have also made nSI realizations for scale independent modulation strength andplotted the difference between SI realization and nSI realization for a scale independent modulationstrength, m = 0 .
008 in Fig. 10. Fig. 9 and Fig 10 shows the effect of a scale dependent and scale independent modulation strength respectively on a SI CMB temperature field. Presence ofprominent fluctuations at large angular scales and absence of fluctuations at smaller angular scalesin the difference map in Fig. 9, are the signature of scale dependent modulation strength given inFig. 7. Whereas, for constant modulation strength, Fig. 10 shows fluctuations at all angular scales.The comparison between the input and output angular power spectra is plotted in Fig. 5. The plotfor comparison between input and output BipoSH power spectra from 1000 realizations is given inFig. 11. 16e also made realizations for Doppler boosted CMB temperature map which gives rise to L = 1BipoSH spectra. The realization for nSI temperature sky and the difference between SI and nSIrealization with the same seed value of the random realization are given in Fig. 12 and Fig. 13respectively. The comparison between input and output values of angular power spectra andBipoSH spectra from 1000 realizations are plotted in Fig. 5 and Fig. 14 respectively. CoNIGS is afast and efficient method for producing nSI realizations of CMB temperature field. This is a veryimportant tool to understand the statistics and also to estimate any signal in the nSI temperaturefield. Acknowledgement:
We have used the HPC facility at IUCAA. We also used the HEALPix [27] package to make themaps. SM acknowledges Council for Science and Industrial Research (CSIR), India, for the financialsupport as Senior Research Fellow. SM also thanks Santanu Das and Aditya Rotti for many usefuldiscussions. TS acknowledges support from Swarnajayanti fellowship, DST, India.