Implications of scattering for CMB foreground emission modelling

Since the discovery of the Cosmic Microwave Background (CMB) by Penzias and Wilson (Penzias & Wilson 1965), the CMB has emerged as a powerful cosmological probe, enabling deeper understanding of the history and evolution of our universe, as well as rigorous testing of cosmological models.

The WMAP (Bennett et al. 2013) and Planck (Planck Collaboration I 2020) space missions have achieved remarkable precision in measuring the full-sky temperature anisotropies of the CMB, while also providing invaluable survey data concerning CMB polarization modes. Still, much information can yet be extracted from more sensitive and accurate observations of the CMB. This motivates robust ongoing ground-based, balloon-borne, and space-borne observational programs. Observations with experiments such as the POLARBEAR (Arnold et al. 2010), SPT (Carlstrom et al. 2011), BICEP-Keck experiment (BICEP2 Collaboration 2014), ACTPol (Naess et al. 2014), and CLASS (Dahal et al. 2020), among others, have gradually improved the accuracy and sensitivity of polarization measurements over patches of the southern sky, paving the way for even more ambitious ground-based observations with AliCPT (Salatino et al. 2020), the Simons Observatory (Ade et al. 2019) and with CMB-S4 (Abazajian et al. 2019), as well as with next-generation space missions such as LiteBIRD (LiteBIRD Collaboration et al. 2023), PICO (Hanany et al. 2019), CORE (Delabrouille et al. 2018), PIXIE (Kogut et al. 2011) and PRISM (André et al. 2014).

One of the key objectives of these experiments is the detection, in CMB polarization patterns, of the tiny signature of primordial gravitational waves produced during a phase of cosmic inflation in the extremely early universe. These so-called primordial B-modes of CMB polarization are expected to be orders of magnitude fainter than foreground polarized emission from the Galactic interstellar medium (ISM), which has motivated numerous studies of the detectability of CMB B-modes in the presence of foreground contamination (Tucci et al. 2005; Betoule et al. 2009; Dunkley et al. 2009; Stompor et al. 2016; Remazeilles et al. 2018; Thorne et al. 2019; Aurlien et al. 2023; Zhang et al. 2024). However, the conclusion of those studies typically depends on the reliability of the assumptions being made on the properties of the foreground emission.

It is generally assumed that two main components in the Galactic ISM contribute to the foreground polarization emission for CMB observations. Synchrotron emission, generated by relativistic electrons spiralling in the Galactic magnetic field, is strongly polarized, and dominates at low frequencies, below $\nu\simeq 80$ GHz, while thermal dust emission from elongated dust grains, which dominates above $\nu\simeq 80$ GHz, is polarized by reason of grain alignment perpendicular to the Galactic magnetic field. Free-free emission from warm ionized medium, dust emission from spinning dust grains, and line emissions from atoms and molecules such as carbon monoxide, which significantly contribute to foreground emission in intensity, are usually assumed to be unpolarized, or very little polarized, and thus to negligibly impact the measurement of CMB polarization in low-foreground emission regions of the sky.

With ever more precise CMB observations, it becomes crucial to thoroughly examine the possible interactions of these components and their impact on the detailed properties of polarized foreground emission. In general, free electrons and Galactic dust grains can absorb or scatter incoming radiation, potentially generating distortions in the model electromagnetic spectra by second order effects.

For example, while primordial thermal free-free emission is unpolarized due to the isotropy of the random coulombian collisions between particles, Ichiki (2014) pointed out that Thomson scattering by free electrons which generate the free-free emission will produce polarized photons. In another instance, fee electrons responsible for free-free emission can also absorb synchrotron radiation at low frequencies, around 408 MHz, which may produce bias in the modelling of synchrotron radiation when extrapolated to higher frequency (Sun et al. 2008; Delabrouille et al. 2013). And also, dust particles absorb CMB photons, generating a “CMB shadow” that generates an effective (negative) component of polarized emission by interstellar dust (Nashimoto et al. 2020).

In this paper, we investigate two more of these second-order effects, and estimate their magnitudes and impacts for modelling foreground emission. Of specific interest are the scattering of dust and synchrotron emission by a distribution of thermal free electrons, as well as the scattering of synchrotron and free-free emission by dust particles. Indeed, these second-order effects would generate apparent emission processes with frequency scaling different from that of the intrinsic emission of those scatterers.

The rest of this paper is organised as follows: In Section 2 we derive mathematical formulae and approximation for the scattering of Galactic emission by free electrons, and derive corresponding numerical estimates of the scattering of incoming radiation for a recent model of free-electron distribution. Section 3 checks the amplitude of Rayleigh scattering caused by dust grains, and we conclude in Section 4.

Scattering of anisotropic radiation by free electrons generates polarized emission originating from the scatterers. This is the exact process that generates most of CMB polarization at the surface of last scattering. In our own Galaxy, free electrons are potential scatterers of incoming emission originating from the Milky Way. This may generate a component of polarized emission associated with the warm ionized medium in the galactic ISM.

We now estimate the amplitude of scattering emission off dust grains in the ISM.

The particle size distributions within ISM can be approximated by power-law functions, as outlined by the model presented in (Mathis et al. 1977),

For graphite particles, $0.005\thinspace\mathrm{\mu m}<a<1\thinspace\mathrm{\mu m}$; and for other materials, $0.025\thinspace\mathrm{\mu m}<a<0.25\thinspace\mathrm{\mu m}$. Specific constants $\mathcal{A}_{\mathrm{x}}$ are used for different materials: $\mathcal{A}_{\mathrm{Si}}=10^{-25.10}\mathrm{cm}^{2.5}$ and $\mathcal{A}_{\mathrm{C}}=10^{-25.13}\mathrm{cm}^{2.5}$ (Draine & Lee 1984). Here, $N_{\mathrm{H}}$ represents the number density of hydrogen nuclei (encompassing both atomic and molecular forms) (Weingartner & Draine 2001) and accordingly, $\differential N_{\mathrm{x}}$ is the number density of dust grains with material “x”, and size between $a$ and $a+\differential a$. However, it should be noted that, determining the precise number of large and small particles encounters challenges currently: the former, characterised by a size $a>\lambda$ ($\lambda$ is for visible and infrared band), extinguish stellar light with negligible frequency dependence, a phenomenon known as ”grey extinction”. Since most methods for measuring dust extinction are related to reddening rather than direct extinction, they exhibit insensitivity to grey extinction (Gorbikov & Brosch 2010). In the latter case, the dust particles reside in the Rayleigh limit of the visible light band, resulting in an extinction cross-section that is proportional to the volume of the particles. Consequently, the extinction cross-section per unit mass remains size-independent, making it difficult to determine the size distribution of these smallest particles through extinction measurements (Mathis et al. 1977).

CMB observations mostly occur in the millimetre to centimetre wavelength range (30 GHz to 300 GHz), for which the size of dust grains $a$ smaller than $1/10\lambda$. This leads to Rayleigh scattering dominating the scattering interaction of CMB foreground emission with ISM dust. Though dust particles within ISM usually exhibit aspherical shapes (Lazarian 2007), it is reasonable to perform an order of magnitude estimate by modelling them as uniform, isotropic dielectric spheres with radius $a$ and relative permittivity $\epsilon_{r}=\Re{\epsilon_{r}}+\mathrm{i}\Im{\epsilon_{r}}$, where the imaginary component of the permittivity contributes to the dissipation of electromagnetic waves in dielectric materials. For naturally occurring dissipative media, $\Im{\epsilon_{r}}<0$.⁷⁷7 The sign of the imaginary part of permittivity depends on the Fourier transform convention. In this paper we adopt the following one: $$f_{\omega}=\frac{1}{2\uppi}\int_{-\infty}^{+\infty}f(t)\mathrm{e}^{-\mathrm{i}\omega t}\mathrm{d}t.$$ (13) The absorption cross-section for a single uniform isotropic dielectric sphere with volume $V$ is expressed as (Nordebo et al. 2019):

And the total scattering cross-section across all directions is described by (Jackson 1999, Eq. (10.11)):

Similarly to the Thomson scattering case discussed in the previous section, with the distinction that the Rayleigh scattering cross-section depends on the incident wave’s frequency, the polarization tensor of scattered light off single dust particle can be written as:

Thus the total scattered light by all dust particles along one sight-line is:

The dust responsible for interstellar extinction appears to be relatively well-mixed with the gas in the Milky Way (Draine 2010). A well-established value exists for the mean ratio of total neutral hydrogen $\Sigma_{\rm{H}}$ to colour excess (Bohlin et al. 1978), given by:

Lines of sight traversing diffuse gas in the Milky Way tend to have an average value of $R_{V}\equiv\dfrac{A_{V}}{E(B-V)}\simeq 3.1$ (Draine 2010). For $R_{V}=3.1$, a connection can be established between the column density of hydrogen atoms and dust extinction at $V$ band, $A_{V}$:

Consequently, Eq. (17) can be related to the dust extinction data at $V$ band, yielding:

where

With $\nu=100\thinspace\mathrm{GHz}$ and $A_{V}\simeq 10\thinspace\mathrm{mag}$, the ratio is approximately $10^{-12}$. This implies that only about $10^{-12}$ of the incident foreground emission will be scattered by ISM dust. Fig. 5 shows the amplitude of the scattered foreground by interstellar dust at 143 GHz. It is reasonable to conclude that the scattering effect from ISM dust can be disregarded in the analysis of the CMB foreground emission.

This paper addresses the computation of second-order effects in Galactic ISM emission arising from electric dipole scattering within the interstellar medium on the CMB foreground signals, which encompass both Rayleigh scattering by dust grains and Thomson scattering by thermal free electrons within ionized regions.

The impact of scattering by interstellar dust particles on foreground signals is a relative correction of the order of $\sim 10^{-12}$. Consequently, the contribution of Rayleigh scattering can be safely disregarded in the analysis of CMB foreground contamination.

Thomson scattering by free electrons, while still faint, is not obviously completely negligible for the most sensitive upcoming CMB experiments. This is due to the higher number density of interstellar free electrons as scatterers, the extreme sensitivity required for the detection of CMB polarization signals corresponding to the primordial tensor perturbations, and the high degree of polarization of scattered radiation. It might be worth to refine our estimate in the future and possibly take the scattering of foreground emission by free electrons in the Milky Way into account in the next generation of CMB telescope programs aimed at searching for the trace of primordial gravitational waves.

The spectral energy distribution of the scattered Galactic foreground emission by free electrons is identical to that of the incident emission, and its spatial distribution is highly correlated with free-free emission. Another key point is that the foreground emission scattered by free electrons is polarized, but free-free emission itself is not. One may try to detect the scattered emission by searching Q-type polarization with a mean ISM emission spectrum in mid to high Galactic latitude regions of ionized ISM.

We point out that the uncertainty of the work involved in this paper mainly comes from the spatial distribution of free electrons within the Milky Way, which is not very well measured as of now. However, based on the DM map from Hutschenreuter et al. (2023), we could get a first-order estimate of the magnitude of the scattering effect, which may be refined in the future.

Our assumption that electrons as scatterers are all much closer to Earth than sources of incident foreground signals is another source of uncertainty. Theoretically, a full three-dimensional model of the distribution of matter within the Milky Way would be necessary to refine our calculations.

We thank Brandon Hensley, Chang Feng, Václav Vavryčuk, Ya-Qi Zhao, and Rui Shi for helpful discussions and suggestions. This work is supported in part by National Key R&D Program of China (2021YFC2203100), by NSFC (12261131497), by CAS young interdisciplinary innovation team (JCTD-2022-20), by 111 Project (B23042), by Fundamental Research Funds for Central Universities, by CSC Innovation Talent Funds, by USTC Fellowship for International Cooperation, by USTC Research Funds of the Double First-Class Initiative. Kavli IPMU is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. All numerical calculations were operated on the computer clusters LINDA & JUDY in the particle cosmology group at USTC.