Cosmic Microwave Background B-mode Research Articles

Identifying frequency de-correlated dust residuals in B-mode maps by exploiting the spectral capability of bolometric interferometry

Context. Astrophysical polarized foregrounds represent the most critical challenge in cosmic microwave background (CMB) B-mode experiments, requiring multifrequency observations to constrain astrophysical foregrounds and isolate the CMB signal. However, recent observations indicate that foreground emission may be more complex than anticipated. Not properly accounting for these complexities during component separation can lead to a bias in the recovered tensor-to-scalar ratio. Aims. In this paper we investigate how the increased spectral resolution provided by band-splitting in bolometric interferometry (BI) through a technique called spectral imaging can help control the foreground contamination in the case of an unaccounted-for Galactic dust frequency de-correlation along the line of sight (LOS). Methods. We focused on the next-generation ground-based CMB experiment CMB-S4 and compared its anticipated sensitivity, frequency, and sky coverage with a hypothetical version of the same experiment based on BI (CMB-S4/BI). We performed a Monte Carlo analysis based on parametric component separation methods (FGBuster and Commander) and computed the likelihood of the recovered tensor-to-scalar ratio, r. Results. The main result is that spectral imaging allows us to detect systematic uncertainties on r from frequency de-correlation when this effect is not accounted for in the component separation. Conversely, an imager such as CMB-S4 would detect a biased value of r and would be unable to spot the presence of a systematic effect. We find a similar result in the reconstruction of the dust spectral index, and we show that with BI we can more precisely measure the dust spectral index when frequency de-correlation is present and not accounted for in the component separation. Conclusions. The in-band frequency resolution provided by BI allows us to identify dust LOS frequency de-correlation residuals where an imager with a similar level of performance would fail. This creates the possibility of exploiting this potential in the context of future CMB polarization experiments that will be challenged by complex foregrounds in their quest for B-mode detection.

Design of a Fourier-transform spectrometer for detector characterization on CMB polarization telescope

We present the design of a Fourier-transform spectrometer (FTS) tailored for operation within the frequency range of 75 GHz to 300 GHz, offering a spectral resolution surpassing 2 GHz. This FTS can be used for characterizing detectors on a Cosmic Microwave Background (CMB) Polarization Telescope for the observation of CMB B-mode polarization. The FTS instrument is divided into the primary part, housing a Martin-Puplett interferometer (MPI) module, and the coupled part, which incorporates a beam expanding module and a beam steering module. An in-depth performance analysis focusing on the MPI module, involving considerations such as the orientation of the beam splitter, dihedral angle of the roof-top mirrors, and positioning uncertainty of the scanning mirror, has been conducted by using a simulation tool. Results indicate that the spectral characteristic remains unaffected by variations in the Reflection/Transmission ratio of the beam splitter, with any deviation from the 1:1 ratio uniformly decreasing intensity. The impact of dihedral angle errors of roof-top mirrors (below 0.02 degrees) and positioning uncertainty in the scanning mirror (lower than 3 × 10-3 mm) on the system performance is acceptable for the design of the FTS. The beam expanding module is capable of enlarging the output beam of the MPI module to roughly a 150 mm-radius. The beam steering module allows for directional adjustments within the range of +17 degrees to -17 degrees relative to the normal of the target plane. Preliminary testing on the MPI prototype reveals a spectral resolution of 1.5 GHz for the measured frequency.

Impact of beam far side-lobe knowledge in the presence of foregrounds for LiteBIRD

We present a study of the impact of a beam far side-lobe lack of knowledge on the measurement of the Cosmic Microwave Background B-mode signal at large scale. Beam far side-lobes induce a mismatch in the transfer function of Galactic foregrounds between the dipole and higher multipoles which degrads the performances of component separation methods. This leads to foreground residuals in the CMB map. It is expected to be one of the main source of systematic effects in future CMB polarization observations. Thus, it becomes crucial for all-sky survey missions to take into account the interplays between beam systematic effects and all the data analysis steps. LiteBIRD is the ISAS/JAXA second strategic large-class satellite mission and is dedicated to target the measurement of CMB primordial B modes by reaching a sensitivity on the tensor-to-scalar ratio r of σ(r) ≤ 10-3 assuming r = 0. The primary goal of this paper is to provide the methodology and develop the framework to carry out the end-to-end study of beam far side-lobe effects for a space-borne CMB experiment. We introduce uncertainties in the beam model, and propagate the beam effects through all the steps of the analysis pipeline, most importantly including component separation, up to the cosmological results in the form of a bias δr. As a demonstration of our framework, we derive requirements on the calibration and modeling for the LiteBIRD's beams under given assumptions on design, simulation, component separation method and allocated error budget. In particular, we assume a parametric method of component separation with no mitigation of the far side-lobes effect at any stage of the analysis pipeline. We show that δr is mostly due to the integrated fractional power difference between the estimated beams and the true beams in the far side-lobes region, with little dependence on the actual shape of the beams, for low enough δr. Under our set of assumptions, in particular considering the specific foreground cleaning method we used, we find that the integrated fractional power in the far side-lobes should be known at the level of ∼ 10-4, to achieve the required limit on the bias δr < 1.9 × 10-5. The framework and tools developed for this study can be easily adapted to provide requirements under different design, data analysis frameworks and for other future space-borne experiments, such as PICO or CMB-Bharat. We further discuss the limitations of this framework and potential extensions to circumvent them.

Impact of half-wave plate systematics on the measurement of CMB B-mode polarization

Polarization of the cosmic microwave background (CMB) can help probe the fundamental physics behind cosmic inflation via the measurement of primordial B modes.As this requires exquisite control over instrumental systematics, some next-generation CMB experiments plan to use a rotating half-wave plate (HWP) as polarization modulator.However, the HWP non-idealities, if not properly treated in the analysis, can result in additional systematics.In this paper, we present a simple, semi-analytical end-to-end model to propagate the HWP non-idealities through the macro-steps that make up any CMB experiment (observation of multi-frequency maps, foreground cleaning, and power spectra estimation) and compute the HWP-induced bias on the estimated tensor-to-scalar ratio, r.We find that the effective polarization efficiency of the HWP suppresses the polarization signal, leading to an underestimation of r.Laboratory measurements of the properties of the HWP can be used to calibrate this effect, but we show how gain calibration of the CMB temperature can also be used to partially mitigate it.On the basis of our findings, we present a set of recommendations for the HWP design that can help maximize the benefits of gain calibration.

Effect of instrumental polarization with a half-wave plate on the B-mode signal: prediction and correction

We study the effect of incident unpolarized signal converted to polarized light produced by a realistic half-wave plate (HWP) and evaluate the impact of the effect in the measurement of Cosmic Microwave Background (CMB) B-mode polarization signal targeting to probe the tensor-to-scalar ratio r. The HWP is modeled with the Mueller formalism, and coefficients are decomposed for any incident angle into harmonics of the HWP rotation frequency due to azimuthal angle dependence. Although we use a general formalism, band-averaged matrix coefficients are calculated as an example for a 9-layer sapphire HWP using EM propagation simulations. We perform simulations of multi-detector observations in a band centered at 140 GHz using LiteBIRD instrumental configuration. We show both theoretically and with the simulations that most of the artefacts on Stokes parameter maps are produced by the dipole leakage on B-modes induced by the fourth harmonics MQI (4f) and MUI (4f). The resulting effect is strongly linked to the spin-2 focal plane scanning cross linking parameters. We develop a maximum likelihood-based method to correct the IP leakage by joint fitting of the Mueller matrix coefficients as well as the Stokes parameter maps. We show that the residual leakage after correction leads to an additional noise limited uncertainty on r of the order of 10-7, independently of the value of the Mueller matrix coefficients. We discuss the impact of the monopole signal and the potential coupling with other systematic effects such as gain variations and detector nonlinearities.

Anti-reflection coating with mullite and Duroid for large-diameter cryogenic sapphire and alumina optics.

We developed a broadband two-layer anti-reflection (AR) coating for use on a sapphire half-wave plate (HWP) and an alumina infrared (IR) filter for the cosmic microwave background (CMB) polarimetry. Measuring the faint CMB B-mode signals requires maximizing the number of photons reaching the detectors and minimizing spurious polarization due to reflection with an off-axis incident angle. Sapphire and alumina have high refractive indices of 3.1 and are highly reflective without an AR coating. This paper presents the design, fabrication, quality control, and measured performance of an AR coating using thermally sprayed mullite and Duroid 5880LZ. This technology enables large optical elements with diameters of 600mm. We also present a thermography-based nondestructive quality control technique, which is key to assuring good adhesion and preventing delamination when thermal cycling. We demonstrate the average reflectance of about 2.6% (0.9%) for two observing bands centered at 90/150 (220/280) GHz. At room temperature, the average transmittance of a 105mm square test sample at 220/280GHz is 83%, and it will increase to 90% at 100K, attributed to reduced absorption losses. Therefore, our developed layering technique has proved effective for 220/280GHz applications, particularly in addressing dielectric loss concerns. This AR coating technology has been deployed in the cryogenic HWP and IR filters of the Simons Array and the Simons observatory experiments and applies to future experiments such as CMB-S4.

Disentangling patchy reionization signatures from primordial gravitational waves using CMB E-mode and B-mode polarization

ABSTRACT The detection of large angular scale B mode in the cosmic microwave background (CMB) polarization signal will open a direct window into not only the primary CMB anisotropies caused by the primordial gravitational waves (PGW) originating in the epoch of inflation, but also the secondary anisotropies imprinted during the epoch of cosmic reionization. The existence of patchiness in the electron density during reionization produces a unique distortion in the CMB B-mode polarization, which can be distinguished from the PGW signal with the aid of spatial frequency modes. In this work, we employ an EB estimator by combining E-mode and B-mode polarization for the τ power spectrum signal generated in a photon-conserving seminumerical reionization model called SCRIPT. We developed a Bayesian framework for the joint detection of the PGW and reionization signal from CMB observations and show the efficacy of this technique for upcoming CMB experiments. We find that, for our model, the τ power spectrum signal effectively tracks the inhomogeneous electron density field, allowing for robust constraints on the patchy B-mode signal. Further, our results indicate that employing the EB estimator for the τ signal will facilitate ground-based CMB-S4 to detect the patchy B-mode signal at approximately ≥2σ confidence level, while observations with space-based PICO will improve this detection to ≥3σ going as high as ≥7σ for extreme reionization models. These findings not only highlight the future potential of these experiments to provide an improved picture of the reionization process but also have important implications towards an unbiased measurement of r.

New Probe of Inflationary Gravitational Waves: Cross-Correlations of Lensed Primary CMB B-Modes with Large-Scale Structure.

We propose a new probe of inflationary gravitational waves (IGWs): the cross-correlation of the lensing of inflationary B-mode polarization with a large-scale structure (LSS) tracer, which can also be a cosmic microwave background (CMB) lensing map. This is equivalent to measuring a three-point function of two CMB B-modes and an LSS tracer. We forecast expected 1σ constraints on the tensor-to-scalar ratio r, albeit with a simplistic foreground treatment, and find constraints of σ_{r}≃7×10^{-3} from the correlation of CMB-S4-Deep B-mode lensing and LSST galaxies, σ_{r}≃5×10^{-3} from the correlation of CMB-S4-Deep B-mode lensing and CMB-S4-Deep CMB lensing, and σ_{r}≃10^{-2} from the correlation of LiteBIRD B-mode lensing and CMB-S4-Wide lensing. Because this probe is inherently non-Gaussian, simple Gaussian foregrounds will not produce any biases to the measurement of r. While a detailed investigation of non-Gaussian foreground contamination for different cross-correlations will be essential, this observable has the potential to be a useful probe of IGWs, which, due to different sensitivity to many potential sources of systematic errors, can be complementary to standard methods for constraining r.

BEYONDPLANCK

We presentPlanckLow Frequency Instrument (LFI) frequency sky maps derived within the BEYONDPLANCKframework. This framework draws samples from a global posterior distribution that includes instrumental, astrophysical, and cosmological parameters, and the main product is an entire ensemble of frequency sky map samples, each of which corresponds to one possible realization of the various modeled instrumental systematic corrections, including correlated noise, time-variable gain, as well as far sidelobe and bandpass corrections. This ensemble allows for computationally convenient end-to-end propagation of low-level instrumental uncertainties into higher-level science products, including astrophysical component maps, angular power spectra, and cosmological parameters. We show that the two dominant sources of LFI instrumental systematic uncertainties are correlated noise and gain fluctuations, and the products presented here support – for the first time – full Bayesian error propagation for these effects at full angular resolution. We compared our posterior mean maps with traditional frequency maps delivered by the Planck Collaboration, and find generally good agreement. The most important quality improvement is due to significantly lower calibration uncertainties in the new processing, as we find a fractional absolute calibration uncertainty at 70 GHz of Δg0/g0 = 5 × 10−5, which is nominally 40 times smaller than that reported byPlanck2018. However, we also note that the originalPlanck2018 estimate has a nontrivial statistical interpretation, and this further illustrates the advantage of the new framework in terms of producing self-consistent and well-defined error estimates of all involved quantities without the need of ad hoc uncertainty contributions. We describe how low-resolution data products, including dense pixel-pixel covariance matrices, may be produced from the posterior samples directly, without the need for computationally expensive analytic calculations or simulations. We conclude that posterior-based frequency map sampling provides unique capabilities in terms of low-level systematics modeling and error propagation, and may play an important role for future Cosmic Microwave Background (CMB)B-mode experiments aiming at nanokelvin precision.

Mitigating Cosmic Microwave Background Shadow Degradation of Tensor-to-scalar Ratio Measurements through Map-based Studies

It has been pointed out that the spurious cosmic microwave background (CMB) B-mode polarization signals caused by the absorption of the CMB monopole component due to the Galactic interstellar matter, called the CMB shadow, degrade the accuracy of detecting the CMB B-mode polarization signals imprinted by primordial gravitational waves. We have made a realistic estimation using simulated sky maps of how the CMB shadow affects forthcoming high-precision CMB B-mode experiments for the first time. The delta-map method, an internal template method taking into account the first-order spatial variation of foregrounds’ spectral parameters, is applied as a foreground-removal method. We show that if the CMB shadow effects are not taken into account in the foreground-removal process, future observations would lead to the false detection of the CMB B-mode polarization signals originating from primordial gravitational waves. We also show that the effect of the CMB shadow can be mitigated by our revised delta-map method to target the CMB B-mode polarization signals at the level of tensor-to-scalar ratio r = 0.001.

A framework to mitigate patchy reionization contamination on the primordial gravitational wave signal

ABSTRACT One of the major goals of future cosmic microwave background (CMB) B-mode polarization experiments is the detection of primordial gravitational waves through an unbiased measurement of the tensor-to-scalar ratio r. Robust detection of this signal will require mitigating all possible contamination to the B-mode polarization from astrophysical origins. One such extragalactic contamination arises from the patchiness in the electron density during the reionization epoch. Along with the signature on CMB polarization, the patchy reionization can source secondary anisotropies on the CMB temperature through the kinetic Sunyaev–Zeldovich (kSZ) effect. In order to study the impact of this foreground for the upcoming CMB missions, we present a self-consistent framework to compute the CMB anisotropies based on a physically motivated model of reionization. We show that the value of r can bias towards a higher value if the secondary contribution from reionization is neglected. However, combining small-scale kSZ signal, large-scale E-mode polarization, and B-mode polarization measurements, we can put constraints on the patchiness in electron density during reionization and can mitigate its impact on the value of r. CMB missions such as CMB-S4 and PICO may experience a bias of &amp;gt;0.17σ which can go as high as ∼0.73σ for extreme reionization models allowed by the Planck and SPT CMB measurements. As future experiments target to measure r at 5σ, this is likely to affect the measurement significance and hence possibly affect the claim of detection of r, if not mitigated properly by using joint estimations of different reionization observables.

Optimization of a Microwave Polarimeter for Astronomy with Optical Correlation and Detection

Cosmic Microwave Background (CMB) B-modes detection is the main focus of future CMB experiments because of the valuable information it contains, particularly to probe the physics of the very early universe. For this reason, we have developed an optimized polarimeter demonstrator sensitive to the 10-20 GHz band in which the signal received by each antenna is modulated into a Near Infrared (NIR) laser by a Mach-Zehnder modulator. Then, these modulated signals are optically correlated and detected using photonic back-end modules consisting of voltage-controlled phase shifters, a 90-degree optical hybrid, a pair of lenses, and an NIR camera. During laboratory tests, a 1/f-like noise signal related to the low phase stability of the demonstrator has been found experimentally. To solve this issue, we have developed a calibration method that allows us to remove this noise in an actual experiment, until obtaining the required accuracy level in the measurement of polarization.

Dark energy with a triplet of classical U(1) fields

We present a new mechanism for cosmic acceleration consisting of a scalar field coupled to a triplet of classical U(1) gauge fields. The gauge fields are arranged in a homogeneous, isotropic configuration, with both electric- and magnetic-like vacuum expectation values. The gauge fields provide a mass-like term via a Chern-Simons interaction that suspends the scalar away from its potential minimum, thereby enabling potential-dominated evolution. We show this mechanism can drive a brief period of acceleration, such as dark energy, without the need for fine tunings. We obtain simple analytic results for the dark energy equation of state and dependence on model parameters. In this model, the presence of the gauge field generically leads to a suppression of long-wavelength gravitational waves, with implications for the experimental search for cosmic microwave background B-modes and direct detection of a stochastic gravitational wave background.

Constraining the polarisation flux density and angle of point sources by training a convolutional neural network

Context. Constraining the polarisation properties of extragalactic point sources is a relevant task not only because they are one of the main contaminants for primordial cosmic microwave background B-mode detection if the tensor-to-scalar ratio is lower than r = 0.001, but also for a better understanding of the properties of radio-loud active galactic nuclei. Aims. We develop and train a machine learning model based on a convolutional neural network to learn how to estimate the polarisation flux density and angle of point sources embedded in cosmic microwave background images knowing only their positions. Methods. To train the neural network, we used realistic simulations of patches of 32 × 32 pixels in area at the 217 GHz Planck channel with injected point sources at their centres. The patches also contain a realistic background composed of the cosmic microwave background signal, the Galactic thermal dust, and instrumental noise. We split our analysis into three parts: firstly, we studied the comparison between true and estimated polarisation flux densities for P, Q, and U simulations. Secondly, we analysed the comparison between true and estimated polarisation angles. Finally, we studied the performance of our model with the 217 GHz Planck map and compared our results against the detected sources of the Second Planck Catalogue of Compact Sources (PCCS2). Results. We find that our model can be used to reliably constrain the polarisation flux density of sources above the 80 mJy level. For this limit, we obtain relative errors of lower than 30% in most of the flux density levels. Training the same network with Q and U maps, the reliability limit is above ±250 mJy when determining the polarisation angle of both Q and U sources. Above that cut, the network can constrain angles with a 1σ uncertainty of ±29° and ±32° for Q and U sources, respectively. We test this neural network against real data from the 217 GHz Planck channel, obtaining similar results to the PCCS2 for some sources; although we also find discrepancies in the 300–400mJy flux density range with respect to the Planck catalogue. Conclusions. Based on these results, our model appears to be a promising tool for estimating the polarisation flux densities and angles of point sources above 80 mJy in any catalogue with very small computational time requirements.

Constraining cosmic inflation with observations: Prospects for 2030

ABSTRACT The ability to test and constrain theories of cosmic inflation will advance substantially over the next decade. Key data sources include cosmic microwave background (CMB) measurements and observations of the distribution of matter at low-redshift from optical, near-infrared, and 21-cm intensity surveys. A positive detection of a CMB B-mode consistent with a primordial stochastic gravitational wave background (SGWB) is widely viewed as a smoking gun for an inflationary phase. Still, a null result does not exclude inflation. However, in a significant class of inflationary scenarios, a low SGWB amplitude is correlated with a more significant running, αs, in the primordial density perturbations than is seen with the simplest inflationary potentials. With this motivation, we forecast the precision with which the spectral index ns and αs can be constrained by currently envisaged observations, including CMB (Simons Observatory, CMB-S4 and LiteBIRD), optical/near infra-red (DESI and SPHEREx), and 21-cm intensity mapping (Tianlai and CHIME) surveys. We identify optimal combinations of data sets for constraining the running and show that they may yield additional and informative constraints on the overall inflationary parameter space if the SGWB remains undetected.

Toward inflation with ns=1 in light of the Hubble tension and implications for primordial gravitational waves

Recently, it has been found that complete resolution of the Hubble tension might point to a scale-invariant Harrison-Zeldovich spectrum of primordial scalar perturbation, i.e., ${n}_{s}=1$ for ${H}_{0}\ensuremath{\sim}73\text{ }\text{ }\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. We show that, for well-known slow-roll models, if inflation ends by a waterfall instability with respect to another field in the field space while inflaton is still at a deep slow-roll region, then ${n}_{s}$ can be lifted to ${n}_{s}=1$. A surprise of our result is that with prerecombination early dark energy, chaotic ${\ensuremath{\phi}}^{2}$ inflation, strongly disfavored by $\mathrm{Planck}+\mathrm{BICEP}/\mathrm{Keck}$ in standard $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$, can be revived, which is now well within testable region of upcoming cosmic microwave background B-mode experiments.

Constraints on the Optical Depth to Reionization from Balloon-borne Cosmic Microwave Background Measurements

We assess the uncertainty with which a balloon-borne experiment, nominally called Tau Surveyor (τS), can measure the optical depth to reionization σ(τ) with given realistic constraints of instrument noise and foreground emissions. Using a τS fiducial design with six frequency bands between 150 and 380 GHz, with white and uniform map noise of 7 μK arcmin, achievable with a single midlatitude flight, and including Planck's 30 and 44 GHz data, we assess the error σ(τ) obtained with three foreground models and as a function of sky fraction f sky between 40% and 54%. We carry out the analysis using both parametric and blind foreground separation techniques. We compare the σ(τ) values to those obtained with low-frequency and high-frequency versions of the experiment called τS-lf and τS-hf, which have only four and up to eight frequency bands with narrower and wider frequency coverage, respectively. We find that with τS, the lowest constraint is σ(τ) = 0.0034, obtained for one of the foreground models with f sky = 54%. σ(τ) is larger, in some cases by more than a factor of 2, for smaller sky fractions, with τS-lf, or as a function of foreground model. The τS-hf configuration does not lead to significantly tighter constraints. The exclusion of the 30 and 44 GHz data, which give information about synchrotron emission, leads to significant τ misestimates. Decreasing noise by an ambitious factor of 10, while keeping f sky = 40%, gives σ(τ) = 0.0031. The combination of σ(τ) = 0.0034, baryon acoustic oscillation data from DESI, and future cosmic microwave background B-mode lensing data from the CMB-S3/CMB-S4 experiments could give σ(∑m ν ) = 17 meV.

Broadband Multi-layer Anti-reflection Coatings with Mullite and Duroid for Half-wave Plates and Alumina Filters for CMB Polarimetry

A broadband two-layer antireflection (AR) coating was developed for use on a sapphire half-wave plate (HWP) and an alumina infrared (IR) filter for cosmic microwave background (CMB) polarimetry. Measuring tiny CMB B-mode signals requires maximizing the number of photons reaching the detectors and minimizing spurious polarization due to reflection with an off-axis incident angle. However, a sapphire HWP and an alumina IR filter have high refractive indices of \(\simeq\) 3.1, and an AR coating must be applied to them. Thermally sprayed mullite and Duroid 5880LZ were selected in terms of index and coefficient of thermal expansion for use at cryogenic temperatures. With these materials, the reflectivity was reduced to about 2% at 90/150 GHz and <1% at 220/280 GHz. The design, fabrication, and optical performance evaluation of the AR coatings are described. The coatings were used in a current ground-based CMB experiment called the Simons Array. Also, they could be applied to next-generation CMB experiments, such as the Simons Observatory.

Improved constraints on primordial gravitational waves in light of the H0 tension and BICEP/Keck data

The Hubble tension that the standard $\Lambda$CDM model is suffering from can be resolved with pre-recombination early dark energy. We present the first constraint on the tensor-to-scalar ratio $r$ in corresponding Hubble-tension-free cosmologies using the most recent BICEP/Keck cosmic microwave background (CMB) B-mode polarization data. We find, combining BICEP/Keck with Planck18 CMB and baryon acoustic oscillation data, that the models with larger Hubble constant $H_0$ will have tighter upper bound on $r$, and resolution $H_0\sim73$ km/s/Mpc of the Hubble tension tightens the upper bound to $r<0.028\ (95\%\text{C.L.})$, $25\%$ tighter than the $\Lambda$CDM constraint $r<0.036$. We clarify the origin of this tightening bound.

Mitigating Bias in CMB B-modes from Foreground Cleaning Using a Moment Expansion

One of the primary challenges facing upcoming cosmic microwave background (CMB) polarization experiments aiming to measure the inflationary B-mode signal is the removal of polarized foregrounds. The thermal dust foreground is often modeled as a single modified blackbody; however, overly simplistic foreground models can bias measurements of the tensor-to-scalar ratio r. As CMB polarization experiments become increasingly sensitive, thermal dust emission models must account for greater complexity in the dust foreground while making minimal assumptions about the underlying distribution of dust properties within a beam. We use Planck dust temperature data to estimate the typical variation in dust properties along the line of sight and examine the impact of these variations on the bias in r if a single modified blackbody model is assumed. We then assess the ability of the moment method to capture the effects of spatial averaging and to reduce bias in the tensor-to-scalar ratio for different possible toy models of dust emission. We find that the expected bias due to temperature variations along the line of sight is significant compared to the target sensitivities of future CMB experiments, and that the use of the moment method could reduce bias as well as shed light into the distribution of dust physical parameters.