A comprehensive study of modulation effects on CMB polarization

Abstract

The Cosmic Microwave Background is characterized by temperature and linear polarization fields. Dipole modulation in the temperature field has been extensively studied in the context of hemispherical power asymmetry. In this article, we show that a dipole modulation, and in general, any kind of modulation isn’t allowed in the E and B modes. This is the main result of this paper. This result explains why no evidence of modulation in E mode has been found in the literature. On the contrary, the linear polarization fields Q and U have no such restrictions. We show that modulation under certain situations can be thought of as local U(1) gauge transformations on the surface of a sphere. As far as the modulation function is concerned, we show that physical considerations enforce it to be (i) a spin 0 field and (ii) a scalar under parity. As masking is a specific type of modulation, our study suggests that a direct masking of E mode isn’t also possible. Masking in E map can only be applied through Q and U fields. This means that in principle, leaking of E and B mode powers into each other is unavoidable.

Spherical harmonic coefficients of the modulating function

Fig. 2

Plot of the modulating function f in different cases. For all these cases, we take $A_0=0$. From left to right, a a linear combination of dipole and quadrupole $\rightarrow$ $A_{i}=0.2\delta _{1i}+0.1\delta _{2i}$, ${\varvec{\uplambda }}_{i}=(0,0,1)\delta _{1i}+(0,1,1)\delta _{2i}$ b a pure quadrupole $\rightarrow$ $A_{i}=0.2\delta _{2i}$, ${\varvec{\uplambda }}_{i}=(0,0,1)\delta _{2i}$ and (c) a linear combination of quadrupole and hexadecapole $\rightarrow$ $A_{i}=0.09\delta _{2i}+0.1\delta _{4i}$, ${\varvec{\uplambda }}_{i}=(-1,-1,0)\delta _{2i}+(0,0,1)\delta _{4i}$

Our analysis till this point restricts the function f to only have spin 0 and being scalar under parity. But in principle it can take any form. In this section, we study specific forms of the modulating function f. Our choice is motivated by the dipole modulation model that has been employed to study hemispherical power asymmetry in the T field of CMB (Eq. 3.1). A similar kind of dipole modulation has been used for $Q\pm iU$ fields [24, 33, 34, 44]. This modulation has only one amplitude A and a direction ${\varvec{\uplambda }}$.

In general, we can have different alignments of dipolar, quadrupolar, octupolar, etc., modulations along different directions ${\varvec{\uplambda }}_i$ and with different amplitudes $A_i$. These would be proportional to different exponents of ${\varvec{\uplambda }}_i\cdot {\mathbf {n}}$. This motivates the following modulating function,

$$\begin{aligned} f({\mathbf {n}})=1+A_{1}({\varvec{\uplambda }}_1\cdot {\mathbf {n}})+A_{2}({\varvec{\uplambda }}_2\cdot {\mathbf {n}})^{2}+\ldots =\sum _{i=0}^{\infty }A_{i}(\cos \gamma _{i})^{i},\ \ A_0=1,\ \ A_i\in {\mathbb {C}}. \nonumber \\ \end{aligned}$$

(A.1)

In the above equation, we have defined $\cos \gamma _i={\varvec{{\uplambda }}}_i\cdot {\mathbf {n}}$. Notice that we have written the modulating function as a linear combination of pure dipole, quadrupole, etc., terms. In Fig. 2, we have shown the plots of the modulating function f with various possibilities.

In order to calculate the corresponding modulated coefficients, our objective is to find out the spherical harmonic coefficients $f_{\ell m}$ of the modulating function f. For that, we notice that any power of $\cos \gamma _{i}$ can be written as a linear combination of the Legendre’s polynomials ${\mathcal {P}}_{\ell }(\cos \gamma _{i})$ with appropriate coefficients. So we write (no sum over i on either sides)

$$\begin{aligned} (\cos \gamma _{i})^{i}=\sum _{\ell \ge 0}\alpha _{i,\ell }{\mathcal {P}}_{\ell }(\cos \gamma _{i}). \end{aligned}$$

(A.2)

The ‘base change’ coefficients $\alpha _{i,\ell }$ can be easily found using any table on Legendre’s polynomials. Using addition theorem of spherical harmonics, we can express the Legendre’s polynomials in terms of spherical harmonics

$$\begin{aligned} {\mathcal {P}}_{\ell }(\cos \gamma _{i})=\frac{4\pi }{2\ell +1}\sum _{m=-\ell }^{\ell }Y_{\ell m}({\mathbf {n}})Y_{\ell m}^{*}({\varvec{\uplambda }}_{i}). \end{aligned}$$

(A.3)

Finally, using Eqs. (A.3) and (A.2) in (A.1), the spherical harmonic coefficients $f_{\ell m}$ are found to be

$$\begin{aligned} f_{\ell m} = \frac{4\pi }{2\ell +1}\sum _{i=0}^\infty A_i\,\alpha _{i,\ell }\,Y_{\ell m}^*({\varvec{\uplambda }}_i) \end{aligned}$$

(A.4)

These harmonic coefficients for some special cases of pure monopole, dipole, etc., modulations are given in Table 1.

Table 1 Spherical harmonic coefficients $f_{\ell m}$ of the modulating function f in some specific cases