# Rapid building damage assessment workflow: An implementation for the 2023
Rolling Fork, Mississippi tornado event

Caleb Robinson1, Simone Fobi Nsutezo1, Anthony Ortiz1, Tina Sederholm1, Rahul
Dodhia1,
Cameron Birge2, Kasie Richards3, Kris Pitcher3, Paulo Duarte3, Juan M. Lavista
Ferres1

1Microsoft AI for Good Research Lab, 2Microsoft Philanthropies,
3American Red Cross Corresponding author<EMAIL_ADDRESS>
###### Abstract

Rapid and accurate building damage assessments from high-resolution satellite
imagery following a natural disaster is essential to inform and optimize first
responder efforts. However, performing such building damage assessments in an
automated manner is non-trivial due to the challenges posed by variations in
disaster-specific damage, diversity in satellite imagery, and the dearth of
extensive, labeled datasets. To circumvent these issues, this paper introduces
a human-in-the-loop workflow for rapidly training building damage assessment
models after a natural disaster. This article details a case study using this
workflow, executed in partnership with the American Red Cross during a tornado
event in Rolling Fork, Mississippi in March, 2023. The output from our human-
in-the-loop modeling process achieved a precision of 0.86 and recall of 0.80
for damaged buildings when compared to ground truth data collected post-
disaster. This workflow was implemented end-to-end in under 2 hours per
satellite imagery scene, highlighting its potential for real-time deployment.

Figure 1: (Left) Post disaster Planet SkySat imagery over Rolling Fork,
Mississippi from March 25th; (Right) Building level model damage estimates
(color gradient indicates damage from no damage colored in white to destroyed
in red).

##  1 Introduction

Rapid building damage assessment in the wake of a natural disaster can inform
first responders about the types and magnitudes of resources needed on the
ground to support affected populations. Advancements in automated building
damage assessment have been supported by the availability of high resolution
imagery pre and post a disaster paired with corresponding damage labels in
datasets such as xBD [5] and xFBD [8]. Variants of semantic segmentation
methods have been applied to pre and post-disaster imagery to perform the dual
tasks of outlining building footprints and classify damaged buildings ([6, 7,
1, 4]). Despite these advancements, building damage assessment models suffer
from significant degradation when evaluated on out-of-distribution datasets
[2, 4] due to the breadth of variations in the type of disaster, image
modality, capture conditions, weather, and data size, among others. These
issues limit the usability of existing disaster assessment models in real
world scenarios where quick and accurate results are needed for a specific
area using any available imagery.

We present a rapid response damage assessment workflow that utilizes a small
number of labels and only post-disaster imagery. We demonstrate a fast
mechanism by which a model can be tuned to perform well under a new disaster
using only a small number of training examples thereby eliminating the need
for generalization. In addition, this method reduces the dependence on pre-
post image pairs, resulting in a more streamlined damage assessment pipeline.
This workflow was deployed immediately after the occurrence of a natural
disaster and validated in collaboration with the American Red Cross (ARC),
showing that this approach can support rapid disaster response.

##  2 Application Context

On Friday, March 24, 2023, an EF4 tornado formed in Mississippi. The tornado
lasted for 71 minutes and caused 16 fatalities and 165 injuries. It first
touched down in Issaquena County, and tracked northeastward through Sharkey
and Humphreys Counties, causing extensive damage in Rolling Fork, Midnight,
and Silver City. The most severe damage was observed in Rolling Fork, where
numerous structures were destroyed or severely damaged, including homes,
businesses, a hospital, schools, and a water tower. Throughout its path, the
tornado produced varying degrees of damage, from minor tree and power pole
damage to catastrophic destruction of buildings and infrastructure. The
tornado eventually dissipated after having traveled 59.4 miles (95.6 km) in
total.

On March 25th, our partners at the ARC suggested that we use the event to test
our damage assessment workflow, to provide them with satellite imagery-based
damage analysis to inform ground operations that would occur over the
following week. The same day, a Planet Labs SkySat satellite captured a clear
post-disaster image of the town of Rolling Fork, which provided us with the
necessary data to implement our workflow (described in Section 3).

On March 26th, we executed our damage assessment workflow, and provided the
ARC with a set of 1347 building footprints (from the Global Microsoft Building
ML Footprints dataset [9]), and an estimate of how damaged each building was
on a scale of 0 to 100.

Over the following weeks, the ARC deployed more than 800 trained disaster
workers to respond to the event. These disaster workers provided sheltering to
those displaced by the disaster as well as supported health, mental health
services, and recovery support. ARC disaster workers provided shelf-stable
meals in the community, where other resources were not available and provided
relief items including comfort kits to people in need. The disaster response
team trained in damage assessment conducted more than 3500 assessments of
residential damage across the state of Mississippi. The ground truth damage
assessment included over 909 addresses in Rolling Fork where they categorized
the building at each address according to the FEMA building damage scale: no
visible damage, affected, minor damage, major damage, or destroyed [3].

In the following sections we outline our damage assessment workflow, describe
our modeled output over Rolling Fork (as shown in Figure 1), and perform a
post-hoc analysis of the model output using the ground truth data.

##  3 Damage Assessment Workflow

Given a new disaster area of interest (AOI), we outline our building damage
assessment workflow:

#### Satellite image and building footprint acquisition.

We acquire post-disaster high-resolution satellite imagery scenes overlapping
the AOI. A scene is defined as an image captured from the same pass by a
satellite111Large AOIs are often made up of multiple scenes from potentially
different satellite passes at different times of day, with different imaging
angles, cloud coverage, illumination, and on-the-ground conditions.. We obtain
corresponding building footprints (polygons) for the same scene from
OpenStreetMap [11] or the Microsoft Building Footprint dataset [9].

#### Satellite image-based label acquisition.

We set up an instance of the “satellite imagery labeling tool” [10] with post-
disaster imagery. We manually label a random selection of building footprints
in the AOI as “damaged” or “not damaged” based on a visual inspection of the
post-disaster imagery and use them for validation of the modeling output.

#### Model training.

First, we manually label a total of $\approx 100$ example polygons of
“background”, “building”, and “damaged building” classes using the labeling
tool instance. Next, we fine-tune an pre-trained semantic segmentation model
222We used a U-Net with an ImageNet pre-trained ResNet-50 backbone as our
semantic segmentation architecture, but practitioners can choose alternative
architectures with pre-trained models better suited to the task. with the
cumulative labeled examples. We run the fine-tuned model on the entire scene
to classify each pixel as: “background”, “building”, or “damaged building”. We
compute, for each building footprint in the AOI, the percentage area of the
footprint that is classified as “damaged building”, $\hat{y}_{i}$. These
damage proportions are used to represent the likelihood that a building is
damaged. This process is repeated, per scene, until the performance of the
modeled output is acceptable (with respect to the validation labels).

Formally, the output of this building damage assessment process is a set of
$N$ building footprints (polygons) and the associated damage estimates,
$\\{\hat{y}_{i}\\}_{i=1}^{N}$ where $\hat{y}_{i}$ is the estimated percent
damage of the $i^{\text{th}}$ building, i.e.
$\hat{y}_{i}\in\left[0,100\right]$.

#### Validation.

Given building level percent damage estimates obtained from a model, we
determine the performance of the model by computing a set of validation
metrics with respect to labeled building level data. Here, labeled building
level data can be obtained either from a ground truth assessment by experts
(e.g. a categorization of each building according to the FEMA building damage
criteria) or, less desirably, by an independent visual interpretation of the
post-disaster imagery. Formally, we assume that we have $N$ labels over $C$
damage classes, $\\{y_{i}\\}_{i=1}^{N}$, where $y_{i}\in\\{1,\cdots,C\\}$ and
the damage classes represent categories such as “no visible damage”,
“affected”, “minor damage”, “major damage”, and “destroyed”.

First, we aggregate the $C$ damage classes into two classes, i.e. converting
the problem into a binary classification problem. For example, if our damage
labels follow the FEMA categorization, we may consider “major damage” and
“destroyed” as a single “damaged” class with the remaining categories grouped
into a “not damaged” class.

Now, given a percent damage estimate, $\hat{y}_{i}$, and binary damage label,
$y_{i}$, for each building, we compute precision and recall for the damaged
class under an any damage threshold (i.e. where $\hat{y}_{i}>0$), as well as a
precision-recall curve and average precision metric. More specifically, given
a threshold value, $\theta$, we compute the following:

$\text{tp}_{\theta}$

true positives, the number of buildings in which $\hat{y}_{i}>\theta$ and
$y_{i}=1$, i.e. the number of buildings where our model estimated a damage
percentage greater than $\theta$ and the building was “damaged”.

$\text{fp}_{\theta}$

false positives, the number of buildings in which $\hat{y}_{i}>\theta$ and
$y_{i}=0$, i.e. the number of buildings where our model estimated a damage
percentage greater than $\theta$ and the building was “not damaged”.

$\text{fn}_{\theta}$

false negatives, the number of buildings in which $\hat{y}_{i}\leq\theta$ and
$y_{i}=1$, i.e. the number of buildings where our model estimated a damage
percentage less than or equal to $\theta$ and the building was “damaged”.

Precision and recall under the any damage threshold ($\theta$ = 0) are then:

$\displaystyle\text{{Precision}}_{0}$
$\displaystyle=\frac{\text{tp}_{0}}{\text{tp}_{0}+\text{fp}_{0}}$ (1)
$\displaystyle\text{{Recall}}_{0}$
$\displaystyle=\frac{\text{tp}_{0}}{\text{tp}_{0}+\text{fn}_{0}}$ (2)

Intuitively, precision is the fraction of buildings that the model estimates
to be damaged, that are, in fact, damaged. E.g. if precision is 0.8, then 1
out of every 5 buildings the model predicts to be damaged, will not be
damaged. Similarly, recall is the fraction of damaged buildings that the model
correctly estimates to be damaged. E.g. if recall is 0.8 then the model
identified 80% of the damaged buildings in the scene.

Figure 2: Precision-recall curve for our building damage estimates compared to
image interpretation based labels.

Setting $\theta$ to all possible values between 0 and 1 and plotting the
resulting precision and recall values gives us a “precision-recall curve” that
shows the trade off between precision and recall. As the percent damage
threshold with which we use to classify a building as damaged increases, the
precision of the model will increase while the recall decreases. Intuitively,
the precision-recall curve serves as proxy for damage localization, where
“major damage” and “destroyed” buildings should have large percent damage
estimates.

Finally, average precision (AP) is a summary of precision and recall across
multiple thresholds. Specifically, AP is a weighted mean of precisions over
the set of possible $\theta$ values, where the weighting is obtained from the
change in recall between thresholds [12]:

$\text{{AP}}=\sum_{n}{(\text{Recall}_{\theta_{n}}-\text{Recall}_{\theta_{n-1}})*\text{Precision}_{\theta_{n}}}$
(3)

##  4 Rolling Fork, Mississippi Results

Figure 3: Precision-recall curves for our building damage estimates compared
to ground truth labels under two definitions of “damaged” buildings. The
(left) panel shows the result when “major damage” and “destroyed” points are
considered “damaged” while the (right) panel shows the result when only the
“destroyed” points are considered “damaged”.

For our Rolling Fork building damage assessment, we produced percent damage
estimates for 1,347 buildings over the city of Rolling Fork, Mississippi using
Planet SkySat imagery from March 25th, 2023333Scene ID:
20230325_195842_ssc11_u0001.. We annotated 153 examples of “background” (34
examples), “undamaged building” (71 examples), and “damaged building” (48
examples), then used these to train a semantic segmentation model that can
make predictions over the entire image.

We perform two validation efforts – using labeled data collected through a
visual interpretation of the satellite imagery and using labeled data
collected through a ground truth analysis. We detail the results of these
efforts below.

###  4.1 Validation with image-based labels

Our initial validation effort involves annotating a set of 250 randomly
selected building footprints as “damaged” or “not damaged” based on an
interpretation of the post-disaster imagery that was independent of the
annotation effort used to train the damage assessment model. Our building
level damage estimates have an average precision of 0.96 with a precision-
recall curve shown in Figure 2. The precision and recall under the “any
damage” threshold are 0.96 and 0.74.

###  4.2 Validation with ground truth based labels

We received 909 labels from the ARC based on a ground truth analysis that they
performed during their disaster response. These labels were assigned to
addresses and categorized according to the FEMA building damage assessment
scale. We first geocoded the addresses to obtain latitude, longitude point
locations, and matched them to the set of building footprints that we
performed the damage assessment over. Due to noise in the geocoding process
and the fact that not all buildings were analyzed by the ARC, sometimes the
ground truth points are not contained in a building footprint444Examples are
i) the point occurs along a road, ii) several points occur on a single
building or iii) a building has no matching points.. Thus, we associate each
ground truth point with the nearest building footprint in a 20 meter radius
(if any).

We compute the validation metrics over the 830 ground truth points that are
associated with a building footprint.

If we consider ground truth points that are categorized as “major damage” or
“destroyed” to be the “damaged” class, and the remaining categories to be an
“undamaged” class, precision and recall under the “any damage” threshold are
0.862 and 0.801. When only the “destroyed” class is considered to be
“damaged”, then precision and recall are 0.84 and 0.90. Figure 3 shows the
localization performance of the model where average precision increases from
0.81 (“major” and “destroyed”) to 0.86 (only “destroyed”).

##  5 Conclusion

Our work demonstrates that a human-in-the-loop workflow can provide a rapid
and effective method of building damage assessment post-natural disasters. Our
method has demonstrated its practical usefulness by quickly training with
minimal data, effectively bypassing the requirement for extensive labeled
datasets and the challenges of training models that account for diverse
disaster scenarios and satellite imagery variations. A key advantage of the
workflow is its capability to be implemented in under 2 hours per satellite
imagery scene by any user familiar with annotating satellite imagery, which
enables its potential use in production scenarios where immediate action and
response are of paramount importance. This was evident in the case of the
tornado event in Rolling Fork, Mississippi, where the model, having been
trained and validated collaboratively with the ARC, resulted in a precision of
0.86 and a recall of 0.80 when evaluated against ground truth data. Future
work will focus on refining this workflow, exploring scalability across
disaster types, and strengthening collaboration between AI and human
expertise.

## Acknowledgments

This work builds off of previous work done between Microsoft Philanthropies
and the Netherlands Red Cross 510 data team. We would like to thank the
anonymous reviewers for their feedback.

## References

  * [1] Yanbing Bai, Junjie Hu, Jinhua Su, Xing Liu, Haoyu Liu, Xianwen He, Shengwang Meng, Erick Mas, and Shunichi Koshimura.  Pyramid pooling module-based semi-siamese network: A benchmark model for assessing building damage from xBD satellite imagery datasets.  Remote Sensing, 12(24):4055, 2020.
  * [2] Vitus Benson and Alexander Ecker.  Assessing out-of-domain generalization for robust building damage detection.  arXiv preprint arXiv:2011.10328, 2020.
  * [3] FEMA.  FEMA Preliminary Damage Assessment Guide, August 2021.  https://www.fema.gov/sites/default/files/documents/fema_2021-pda-guide.pdf, 2021\.
  * [4] Shahrzad Gholami, Caleb Robinson, Anthony Ortiz, Siyu Yang, Jacopo Margutti, Cameron Birge, Rahul Dodhia, and Juan Lavista Ferres.  On the deployment of post-disaster building damage assessment tools using satellite imagery: A deep learning approach.  In 2022 IEEE International Conference on Data Mining Workshops (ICDMW), pages 1029–1036. IEEE, 2022.
  * [5] Ritwik Gupta, Richard Hosfelt, Sandra Sajeev, Nirav Patel, Bryce Goodman, Jigar Doshi, Eric Heim, Howie Choset, and Matthew Gaston.  xbd: A dataset for assessing building damage from satellite imagery.  arXiv preprint arXiv:1911.09296, 2019.
  * [6] Rohit Gupta and Mubarak Shah.  Rescuenet: Joint building segmentation and damage assessment from satellite imagery.  In 2020 25th International Conference on Pattern Recognition (ICPR), pages 4405–4411. IEEE, 2021.
  * [7] Eugene Khvedchenya and Tatiana Gabruseva.  Fully convolutional siamese neural networks for buildings damage assessment from satellite images.  arXiv preprint arXiv:2111.00508, 2021.
  * [8] Dennis Melamed, Cameron Johnson, Chen Zhao, Russell Blue, Philip Morrone, Anthony Hoogs, and Brian Clipp.  xFBD: Focused building damage dataset and analysis.  arXiv preprint arXiv:2212.13876, 2022.
  * [9] Microsoft.  Global ML Building Footprints.  https://github.com/microsoft/GlobalMLBuildingFootprints, 2023.
  * [10] Microsoft.  Satellite Imagery Labeling Tool.  https://github.com/microsoft/satellite-imagery-labeling-tool, 2023\.
  * [11] OpenStreetMap contributors.  Planet dump retrieved from https://planet.osm.org .  https://www.openstreetmap.org, 2017.
  * [12] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay.  Scikit-learn: Machine learning in Python.  Journal of Machine Learning Research, 12:2825–2830, 2011.


# Improved Constraints on Cosmic Birefringence from the WMAP and Planck Cosmic
Microwave Background Polarization Data

Johannes R. Eskilt<EMAIL_ADDRESS>Institute of Theoretical
Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway
Eiichiro Komatsu<EMAIL_ADDRESS>Max Planck Institute for
Astrophysics, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany  Kavli
Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI),
Todai Institutes for Advanced Study, The University of Tokyo, Kashiwa
277-8583, Japan

###### Abstract

The observed pattern of linear polarization of the cosmic microwave background
(CMB) photons is a sensitive probe of physics violating parity symmetry under
inversion of spatial coordinates. A new parity-violating interaction might
have rotated the plane of linear polarization by an angle $\beta$ as the CMB
photons have been traveling for more than 13 billion years. This effect is
known as “cosmic birefringence.” In this paper, we present new measurements of
cosmic birefringence from a joint analysis of polarization data from two space
missions, Planck and WMAP. This dataset covers a wide range of frequencies
from 23 to 353 GHz. We measure
$\beta=0.342^{\circ\,+0.094^{\circ}}_{\phantom{\circ\,}-0.091^{\circ}}$ (68%
C.L.) for nearly full-sky data, which excludes $\beta=0$ at 99.987% C.L. This
corresponds to the statistical significance of $3.6\sigma$. There is no
evidence for frequency dependence of $\beta$. We find a similar result, albeit
with a larger uncertainty, when removing the Galactic plane from the analysis.

##  I Introduction

Photons of the cosmic microwave background (CMB), the afterglow of the
primordial fireball Universe [1], are linearly polarized [2]. One can
decompose the observed pattern of linear polarization into eigenstates of
parity called $E$ and $B$ modes, which transform differently under inversion
of spatial coordinates [3, 4]. This property can be used to probe new physics
beyond the standard model of elementary particles and fields that violates
parity symmetry [5].

A pseudoscalar “axionlike” field, $\phi$, that couples to electromagnetism is
an example of such new physics. Consider a Lagrangian density given by [6, 7]

${\mathcal{L}}=-\frac{1}{2}(\partial\phi)^{2}-V(\phi)-\frac{1}{4}F^{2}-\frac{\alpha}{4f}\phi
F\tilde{F}\,,$ (1)

where $(\partial\phi)^{2}\equiv
g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ with the metric tensor
$g^{\mu\nu}$, $V(\phi)$ is $\phi$’s potential, $F^{2}\equiv
g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}F_{\mu\nu}F_{\mu^{\prime}\nu^{\prime}}$,
$F\tilde{F}\equiv
F_{\mu\nu}\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}F_{\mu^{\prime}\nu^{\prime}}/(2\sqrt{-g})$,
and $\alpha$ and $f$ are a dimensionless coupling constant and the so-called
axion decay constant, respectively. Here,
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
antisymmetric electromagnetic tensor with the vector potential $A_{\mu}$,
$\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}$ is a totally antisymmetric symbol
with $\epsilon^{0123}=1$, and $g$ is the determinant of the metric tensor.

The last term in Eq. (1) is familiar in particle physics, as it appears in the
low-energy effective action of quantum chromodynamics [8]. A pion is a
pseudoscalar [9] and a neutral pion decays into two photons via this term with
$\alpha$ and $f$ determined precisely by experiments. In cosmology, $\phi$ is
a new pseudoscalar and a candidate for dark matter and dark energy [10, 11]
with $\alpha/f$ being a free parameter.

When $\phi$ depends on spacetime, the plane of linear polarization of photons
rotates [12, 13, 14] by an angle
$\beta(\hat{\mathbf{n}})=\frac{\alpha}{2f}\left[\phi(\eta_{\textrm{o}})-\phi(\eta_{\textrm{e}},r\hat{\mathbf{n}})\right]$,
where $\hat{\mathbf{n}}$ is the direction of an observer’s line of sight,
$\eta_{\textrm{o}}$ and $\eta_{\textrm{e}}$ are the conformal times of
observation and emission of photons, respectively, and
$r\equiv\eta_{\textrm{o}}-\eta_{\textrm{e}}$ is the conformal distance to the
emitter. We take the speed of light to be unity throughout this paper.

This effect is often called “cosmic birefringence,” and is best probed by
analyzing the oldest polarized light in the Universe, that is, the CMB (see
Ref. [15] for a review). Discovery of a non-zero value of $\beta$ would have
profound implications for the fundamental physics behind dark energy [16, 17],
dark matter [18, 19], and quantum gravity [20, 21].

When analyzing CMB data, one decomposes the Stokes parameters for linear
polarization as [3, 4]

$Q(\hat{\mathbf{n}})\pm
iU(\hat{\mathbf{n}})=-\sum_{\ell=2}^{\ell_{\mathrm{max}}}\sum_{m=-\ell}^{\ell}\left(E_{\ell
m}\pm iB_{\ell m}\right){}_{\pm 2}Y_{\ell}^{m}(\hat{\mathbf{n}})\,,$ (2)

where $E_{\ell m}$ and $B_{\ell m}$ are the spherical harmonics coefficients
of the $E$ and $B$ modes, respectively, ${}_{\pm
2}Y_{\ell}^{m}(\hat{\mathbf{n}})$ are the spin-2 spherical harmonics, and
$\ell_{\mathrm{max}}$ is the maximum multipole used for the analysis.

The coefficients transform under inversion of spatial coordinates,
$\hat{\mathbf{n}}\to-\hat{\mathbf{n}}$, as $E_{\ell m}\to(-1)^{\ell}E_{\ell
m}$ and $B_{\ell m}\to(-1)^{\ell+1}B_{\ell m}$. The cross-power spectrum of
$E$ and $B$ modes,
$C_{\ell}^{EB}\equiv(2\ell+1)^{-1}\sum_{m}\operatorname{Re}(E_{\ell m}B_{\ell
m}^{*})$, has odd parity and is sensitive to $\beta$ [5]. When the plane of
linear polarization rotates uniformly on the sky by an angle $\beta$, the
observed $E$ and $B$ modes become $E_{\ell m}^{\textrm{o}}=E_{\ell
m}\cos(2\beta)-B_{\ell m}\sin(2\beta)$ and $B_{\ell m}^{\textrm{o}}=E_{\ell
m}\sin(2\beta)+B_{\ell m}\cos(2\beta)$, respectively. One thus finds that [22,
23, 24, 25, 26]

$C_{\ell}^{EB,{\textrm{o}}}=\frac{\tan(4\beta)}{2}\left(C^{EE,{\textrm{o}}}_{\ell}-C^{BB,{\textrm{o}}}_{\ell}\right)+\frac{C_{\ell}^{EB}}{\cos(4\beta)}\,,$
(3)

where the last term is the intrinsic $EB$ correlation at the time of emission,
and $C_{\ell}^{EE}$ and $C_{\ell}^{BB}$ are the auto-power spectra of $E$ and
$B$ modes, respectively. We adopt the sign convention such that $\beta>0$ is a
clockwise rotation of the plane of linear polarization on the sky.

However, miscalibration angles will arise if one does not know precisely
enough how the polarization-sensitive orientations of detectors on the focal
plane of a telescope are related to the sky coordinates and how polarization
of the incoming light is rotated by optical components [27, 28, 29, 30]. These
angles will be degenerate with the cosmic birefringence angle $\beta$. We will
from now on denote miscalibration angles by $\alpha$, which should not be
confused with the coupling constant in Eq. (1). Hence, the observed $EB$ power
spectrum measured by an instrument gets an extra rotation contribution from
its miscalibration angle $\alpha$, inducing a total rotation of
$\alpha+\beta$. Without knowledge of $\alpha$, one can only determine the sum
of the two angles, $\alpha+\beta$.

The sky contains not only the CMB, but also the polarized emission of
interstellar gas, called the Galactic foreground emission. Photons of the
foreground emission do not travel for a long distance, receiving only a
negligible amount of $\beta$ when $\phi$ varies slowly in spacetime; thus, the
foreground polarization is rotated only by the miscalibration angle $\alpha$
[26]. The foreground emission might possess a non-vanishing intrinsic $EB$
correlation, which needs to be taken into account.

These considerations lead to

$\displaystyle\begin{bmatrix}E^{\textrm{o}}_{\ell m}\\\ B^{\textrm{o}}_{\ell
m}\end{bmatrix}$ $\displaystyle=\begin{bmatrix}\cos(2\alpha)&-\sin(2\alpha)\\\
\sin(2\alpha)&\phantom{-}\cos(2\alpha)\end{bmatrix}\begin{bmatrix}E^{\textrm{fg}}_{\ell
m}\\\ B^{\textrm{fg}}_{\ell m}\end{bmatrix}$
$\displaystyle+\begin{bmatrix}\cos(2\alpha+2\beta)&-\sin(2\alpha+2\beta)\\\
\sin(2\alpha+2\beta)&\phantom{-}\cos(2\alpha+2\beta)\end{bmatrix}\begin{bmatrix}E^{\textrm{CMB}}_{\ell
m}\\\ B^{\textrm{CMB}}_{\ell m}\end{bmatrix}\,,$ (4)

for a single channel. Here, “fg” and “CMB” denote the foreground and CMB,
respectively. One finds that [26]

$\displaystyle C_{\ell}^{EB,{\textrm{o}}}$
$\displaystyle=\frac{\tan(4\alpha)}{2}\left(C_{\ell}^{EE,{\textrm{o}}}-C_{\ell}^{BB,{\textrm{o}}}\right)+\frac{C_{\ell}^{EB,\text{fg}}}{\cos(4\alpha)}$
$\displaystyle+\frac{\sin(4\beta)}{2\cos(4\alpha)}\left(C_{\ell}^{EE,\text{CMB}}-C_{\ell}^{BB,\text{CMB}}\right)$
$\displaystyle+\frac{\cos(4\beta)}{\cos(4\alpha)}C_{\ell}^{EB,\text{CMB}}\,.$
(5)

This equation allows us to determine $\alpha$ and $\beta$ simultaneously,
independent of the $E$\- and $B$-mode auto-power spectra of the foreground,
$C^{EE,\textrm{fg}}_{\ell}$ and $C^{BB,\textrm{fg}}_{\ell}$.

Eq. (5) still requires knowledge of $C_{\ell}^{EB,\mathrm{fg}}$ and
$C_{\ell}^{EB,\mathrm{CMB}}$. Discovery of the latter would be similarly
revolutionary in cosmology; however, we ignore this term in this paper because
the current data are not yet sensitive enough to detect it.

We take into account the effect of $C_{\ell}^{EB,\mathrm{fg}}$. As the
foreground helps us constrain $\alpha$, neglecting any non-zero
$C_{\ell}^{EB,\mathrm{fg}}$ in the analysis can bias the measurement of
$\alpha$. A biased $\alpha$ will necessarily bias $\beta$ since the CMB highly
constrains the sum, $\alpha+\beta$.

After being verified on simulations with multiple frequency channels [31], the
method was applied to the high frequency instrument (HFI [32]) maps of the
Planck Public Release 3 (PR3) in Ref. [33]. The authors measured
$\beta=0.35^{\circ}\pm 0.14^{\circ}$ for nearly full-sky data. We quote the
68% confidence level (C.L.) intervals throughout this paper. The method was
then applied to the Planck PR4 HFI data [34], yielding $\beta=0.30^{\circ}\pm
0.11^{\circ}$ [35]. The low frequency instrument (LFI [36]) maps of the Planck
PR4 were included in the analysis of Ref. [37], which found
$\beta=0.33^{\circ}\pm 0.10^{\circ}$. The statistical significance exceeds
$3\sigma$.

Ref. [37] also measured the frequency dependence of the signal by fitting a
power-law model, $\beta(\nu)\propto\nu^{n}$, finding
$n=-0.35^{+0.48}_{-0.47}$. This is consistent with a frequency-independent
signal predicted by the axionlike field, Eq. (1).

These authors [33, 35, 37] initially ignored $C_{\ell}^{EB,\mathrm{fg}}$ in
the analysis. As the foreground is dominated by polarized thermal dust
emission at the HFI frequencies [38], they noted that the positive dust $TB$
and $TE$ correlations found from Planck [38] would suggest
$C_{\ell}^{EB,\mathrm{fg}}\simeq C_{\ell}^{EB,\mathrm{dust}}>0$, which would
make the measured value of $\beta$ a lower bound. This expectation was
confirmed by Ref. [35], which found a larger value, $\beta=0.36^{\circ}\pm
0.11^{\circ}$, from nearly full-sky data of the Planck HFI PR4 assuming
$C_{\ell}^{EB,\mathrm{dust}}/C_{\ell}^{EE,\mathrm{dust}}\propto
C_{\ell}^{TB,\mathrm{dust}}/C_{\ell}^{TE,\mathrm{dust}}$ [39].

When $C_{\ell}^{EB,\mathrm{dust}}$ is ignored, the inferred value of $\beta$
decreases as the Galactic plane is masked and removed from the analysis via
the impact of $C_{\ell}^{EB,\mathrm{dust}}$ on the determination of $\alpha$
[35]. Although one finds similar values of $\beta$ from the Planck data
regardless of the sky fraction used for the analysis when $\alpha=0$, that is,
the $EB$ signal of $\alpha+\beta\simeq 0.3^{\circ}$ is isotropic in the sky,
the inferred value of $\alpha$ depends on the sky fraction via
$C_{\ell}^{EB,\mathrm{dust}}$. This was foreseen in the work by Ref. [39],
which argued that there should be a mostly positive
$C_{\ell}^{EB,\mathrm{dust}}$ for smaller sky fractions (larger Galactic
masks). This would bias a measurement of $\beta$ towards a lower value
whenever $C_{\ell}^{EB,\mathrm{dust}}$ is ignored.

The authors of Ref. [35] suggested an ansatz to model
$C_{\ell}^{EB,\mathrm{dust}}$ based on the results of Ref. [40, 39]. Including
their ansatz to the equations confirmed that $C_{\ell}^{EB,\mathrm{dust}}$ was
the cause of the decline of $\beta$ for lower sky fractions. Including
$C_{\ell}^{EB,\mathrm{dust}}$ in the inference gave robust positive
measurements of $\beta$ at all sky fractions.

Therefore, the results so far indicate the presence of an isotropic and
frequency-independent signal of cosmic birefringence in the Planck data with a
statistical significance of $3\sigma$. The detailed study [35, 41] using the
simulations of PR4 [34] shows that the impact of the known systematics of the
Planck HFI on $\beta$ is negligible compared with the statistical uncertainty.

In this paper, we continue to search for the isotropic $\beta$ by including
the polarization data of the Wilkinson Microwave Anisotropy Probe (WMAP)
9-year observations [42] in a joint analysis with the polarized Planck
channels. The polarization data of WMAP have a lower signal-to-noise ratio
than those of the Planck HFI, but the inclusion of the WMAP channels gives
rise to many cross-power spectra with both itself and the Planck LFI and HFI
channels. This allows us to increase the precision of $\beta$. As there is no
evidence [43, 44, 45, 46] for fluctuations in $\beta(\hat{\mathbf{n}})$, we
focus on the isotropic $\beta$.

The rest of the paper is organized as follows. We describe the data and the
analysis method in Sec. II. We present the results in Sec. III and conclude in
Sec. IV.

##  II Data and Analysis method

We use the WMAP 9-year maps for each differencing assembly, which contain 1,
1, 2, 2, and 4 maps at $\nu=23$, 33, 41, 61, and 94 GHz, respectively [42]. As
in Refs. [35, 37] we also use the Planck PR4 (often called NPIPE-reprocessed
data [34]) maps at $\nu=30$, 44, 70, 100, 143, 217, and 353 GHz. The NPIPE
pipeline divided detectors of a given frequency band into two groups, hence
creating 2 detector split maps for each band. However, as there were not
enough detectors to make 2 maps for each of 30 and 44 GHz bands, we use the
so-called half-mission maps for these bands. This means that the 30 and 44 GHz
bands have 1 miscalibration angle each, whereas the others have 1
miscalibration angle per detector split map.

The analysis method we use in this paper is similar to those presented in
Refs. [33, 35, 37], which we summarize here. The multi-channel generalization
of Eq. (5) is

$\displaystyle
C_{\ell}^{E_{i}B_{j},\textrm{o}}=R^{T}(\alpha_{i},\alpha_{j})\textbf{R}^{-1}(\alpha_{i},\alpha_{j})\begin{bmatrix}C_{\ell}^{E_{i}E_{j},\textrm{o}}\\\
C_{\ell}^{B_{i}B_{j},\textrm{o}}\end{bmatrix}+$
$\displaystyle\bigg{[}R^{T}(\alpha_{i}+\beta_{i},\alpha_{j}+\beta_{j})-R^{T}(\alpha_{i},\alpha_{j})\textbf{R}^{-1}(\alpha_{i},\alpha_{j})$
$\displaystyle\cdot\textbf{R}(\alpha_{i}+\beta_{i},\alpha_{j}+\beta_{j})\bigg{]}\begin{bmatrix}C_{\ell}^{E_{i}E_{j},\text{CMB}}\\\
C_{\ell}^{B_{i}B_{j},\text{CMB}}\end{bmatrix}\,,$ (6)

where

$\displaystyle\mathbf{R}(\theta_{i},\theta_{j})$
$\displaystyle=\begin{bmatrix}\cos(2\theta_{i})\cos(2\theta_{j})&\sin(2\theta_{i})\sin(2\theta_{j})\\\
\sin(2\theta_{i})\sin(2\theta_{j})&\cos(2\theta_{i})\cos(2\theta_{j})\end{bmatrix}\,,$
(7) $\displaystyle R(\theta_{i},\theta_{j})$
$\displaystyle=\begin{bmatrix}\phantom{-}\cos(2\theta_{i})\sin(2\theta_{j})\\\
-\sin(2\theta_{i})\cos(2\theta_{j})\end{bmatrix}\,.$ (8)

Here, $\alpha_{i}$ is the miscalibration angle for a given frequency band and
data split, $i$. As we also allow $\beta$ to depend on frequency, $\beta_{i}$
denotes the value of $\beta(\nu)$ at $\nu=\nu_{i}$. We group the cross-power
spectra of different combinations of maps into the observed power spectra
vector
$\vec{C}^{\textrm{o}}_{\ell}=\begin{bmatrix}C^{E_{i}E_{j},\textrm{o}}_{\ell},C^{B_{i}B_{j},\textrm{o}}_{\ell},C^{E_{i}B_{j},\textrm{o}}_{\ell}\end{bmatrix}^{T}$.
The CMB power spectra, $C_{\ell}^{E_{i}E_{j},\text{CMB}}$ and
$C_{\ell}^{B_{i}B_{j},\text{CMB}}$, are computed from the Boltzmann solver
CAMB[47]111https://github.com/cmbant/CAMB with the best-fitting cosmological
parameters given in Ref. [48]. We beam-smooth them using the beam transfer
functions, $b^{X}_{\ell}$, and pixel window functions, $w^{i}_{\textrm{pix}}$,

$\vec{C}^{\Lambda\text{CDM}}_{\ell}=\begin{bmatrix}C^{EE,\text{CAMB}}_{\ell}b^{E_{i}}_{\ell}b^{E_{j}}_{\ell}w^{i}_{\text{pix},\ell}w^{j}_{\text{pix},\ell},\\\
C^{BB,\text{CAMB}}_{\ell}b^{B_{i}}_{\ell}b^{B_{j}}_{\ell}w^{i}_{\text{pix},\ell}w^{j}_{\text{pix},\ell}\end{bmatrix},$
(9)

where $C^{EE,\text{CAMB}}_{\ell}$ and $C^{BB,\text{CAMB}}_{\ell}$ are the
power spectra from CAMB. As not all the official transfer functions of WMAP
reach up to $\ell_{\textrm{max}}=1490$ used in our analysis, we set
$b^{X}_{\ell}=0$ for $\ell$ in which transfer functions are not available.

Following Refs. [40, 39, 35], we model $C_{\ell}^{EB,\mathrm{dust}}$ by
assuming that the dust $EB$ is proportional to the observed dust $TB$, that
is, $C_{\ell}^{EB,\mathrm{dust}}/C_{\ell}^{EE,\mathrm{dust}}\propto
C_{\ell}^{TB,\mathrm{dust}}/C_{\ell}^{TE,\mathrm{dust}}$. Specifically, we use

$C^{EB,\text{dust}}_{\ell}=A_{\ell}C^{EE,\text{dust}}_{\ell}\sin\left(4\psi_{\ell}\right)\,,$
(10)

where $A_{\ell}\geq 0$ is a free amplitude parameter and

$\psi_{\ell}=\frac{1}{2}\arctan\left(\frac{C^{TB,\text{dust}}_{\ell}}{C^{TE,\text{dust}}_{\ell}}\right)\,.$
(11)

We calculate $\psi_{\ell}$ from smoothing each of the symmetrized $TB$ and
$TE$ spectra of the 353 GHz A and B split maps using a one-dimensional
Gaussian filter. The parameter $A_{\ell}$ is expected to depend on $\ell$
weakly. Following Refs. [35, 37], we sample $A_{\ell}$ for 4 ranges in
$51\leq\ell\leq 130$, $131\leq\ell\leq 210$, $211\leq\ell\leq 510$, and
$511\leq\ell\leq 1490$ with flat positive priors.

Unlike Refs. [35, 37], we do not relate Eq. (10) to the effective angle of the
foreground $EB$, $\gamma_{\ell}$ (cf. Eq. (2) of Ref. [35]). We instead use
Eq. (10) directly in our analysis. This procedure is more reliable because it
does not require a small-angle approximation or multiplying a noisy factor
${C^{EE,\textrm{dust}}_{\ell}/(C^{EE,\textrm{dust}}_{\ell}-C^{BB,\textrm{dust}}_{\ell})}$
taken from the 353 GHz channels.

We define matrices

$\displaystyle\mathbf{A}_{\ell,ij}=$
$\displaystyle\left[-\Lambda_{\ell}^{T}(\alpha_{i},\alpha_{j})\mathbf{\Lambda}_{\ell}^{-1}(\alpha_{i},\alpha_{j}),\,1\right],$
(12) $\displaystyle\textbf{B}_{\ell,ij}=$
$\displaystyle\bigg{[}R^{T}(\alpha_{i}+\beta_{i},\alpha_{j}+\beta_{j})-\Lambda_{\ell}^{T}(\alpha_{i},\alpha_{j})\mathbf{\Lambda}_{\ell}^{-1}(\alpha_{i},\alpha_{j})$
$\displaystyle\cdot\textbf{R}(\alpha_{i}+\beta_{i},\alpha_{j}+\beta_{j})\bigg{]}\,,$
(13)

where

$\displaystyle\mathbf{\Lambda}_{\ell}(\alpha_{i},\alpha_{j})$
$\displaystyle=\mathbf{R}(\alpha_{i},\alpha_{j})+\mathbf{D}(\alpha_{i},\alpha_{j})\mathbf{F}_{\ell},$
(14) $\displaystyle\Lambda_{\ell}^{T}(\alpha_{i},\alpha_{j})$
$\displaystyle=R^{T}(\alpha_{i},\alpha_{j})+D^{T}(\alpha_{i},\alpha_{j})\mathbf{F}_{\ell},$
(15) $\displaystyle\mathbf{D}(\theta_{i},\theta_{j})$
$\displaystyle=\begin{bmatrix}-\cos(2\theta_{i})\sin(2\theta_{j})&-\sin(2\theta_{i})\cos(2\theta_{j})\\\
\phantom{-}\sin(2\theta_{i})\cos(2\theta_{j})&\phantom{-}\cos(2\theta_{i})\sin(2\theta_{j})\end{bmatrix},$
(16) $\displaystyle D(\theta_{i},\theta_{j})$
$\displaystyle=\begin{bmatrix}\phantom{-}\cos(2\theta_{i})\cos(2\theta_{j})\\\
-\sin(2\theta_{i})\sin(2\theta_{j})\end{bmatrix}\,,$ (17)
$\displaystyle\mathbf{F}_{\ell}$
$\displaystyle=A_{\ell}\sin(4\psi_{\ell})\begin{bmatrix}1&0\\\
1&0\end{bmatrix}\,.$ (18)

Here, the matrix $\mathbf{F}$ is defined differently from that given in Refs.
[35, 37] to simplify the expression.

We bin the observed power spectra with a bin size $\Delta\ell=20$, and limit
the range of multipoles to
$\ell_{\textrm{min}}\leq\ell\leq\ell_{\textrm{max}}$ with
$\ell_{\textrm{min}}=51$ and $\ell_{\textrm{max}}=1490$. We thus use
$N_{\textrm{bin}}=72$ bins. The bin size and multipole range are the same as
in the previous work [49, 33, 35, 37]. The results are robust against changes
in $\ell_{\textrm{min}}$ and $\ell_{\textrm{max}}$.

To find the posterior distributions of the sampled parameters ($\alpha_{i}$,
$\beta_{i}$, $A_{\ell}$), we use a Metropolis Markov Chain Monte Carlo sampler
to evaluate

$\ln
L=-\frac{1}{2}\sum_{b=1}^{N_{\text{bin}}}\left(\vec{v}_{b}^{T}\textbf{M}_{b}^{-1}\vec{v}_{b}+\ln|\textbf{M}_{b}|\right)\,,$
(19)

where $b$ is the bin number, $\textbf{M}_{b}$ is the binned covariance matrix,
and
$\vec{v}^{T}_{b}\equiv\textbf{A}\vec{C}^{\textrm{o}}_{b}-\textbf{B}\vec{C}^{\Lambda\text{CDM}}_{b}$.
The unbinned covariance matrix is given by
$\textbf{M}_{\ell}=\textbf{A}\text{Cov}(\vec{C}^{\textrm{o}}_{\ell},\vec{C}^{\textrm{o}}_{\ell})\textbf{A}^{T}$.
We bin $\text{Cov}(\vec{C}^{\textrm{o}}_{\ell},\vec{C}^{\textrm{o}}_{\ell})$
as [31]

$\text{Cov}(C^{XY}_{b},C^{ZW}_{b})=\frac{1}{\Delta\ell^{2}}\sum_{\ell\in
b}\text{Cov}(C^{XY}_{\ell},C^{ZW}_{\ell})\,,$ (20)

where we use an approximate covariance for each $\ell$,

$\text{Cov}(C^{XY}_{\ell},C^{ZW}_{\ell})=\frac{C^{XZ,\textrm{o}}_{\ell}C^{YW,\textrm{o}}_{\ell}+C^{XW,\textrm{o}}_{\ell}C^{YZ,\textrm{o}}_{\ell}}{(2\ell+1)f_{\textrm{sky}}}\,,$
(21)

with $f_{\textrm{sky}}$ being the fraction of sky used for the analysis.

We avoid using $EB$ terms in the right-hand side of Eq. (21) due to the
statistical fluctuations of $C_{\ell}^{EB,\textrm{o}}$. We thus set
$\text{Cov}(C^{E_{i}B_{j}}_{\ell},C^{E_{p}B_{q}}_{\ell})=C^{E_{i}E_{p}}_{\ell}C^{B_{j}B_{q}}_{\ell}/[(2\ell+1)f_{\textrm{sky}}]$
where $i,j,p,q$ denote different maps. We also do not use off-diagonal
elements in the covariance matrix of Eq. (21).

We use PolSpice222http://www2.iap.fr/users/hivon/software/PolSpice/ [50] to
get the observed power spectra in $\vec{C}^{\textrm{o}}_{\ell}$. We use 14
Planck and 10 WMAP maps, which give $24\cdot 23=552$ cross-power spectra,
whereas auto-power spectra are excluded.

Our baseline result is based on the largest, nearly full-sky coverage mask
used in Ref. [37]. This mask excludes pixels in which the intensity of a
carbon-monoxide (CO) line is stronger than
$45\,\textrm{K}_{\textrm{RJ}}\,\textrm{km}\,\textrm{s}^{-1}$. The CO emission
is not polarized, but it could induce intensity-to-polarization leakage.
Although the CO line exists only in some of the HFI maps, we apply the same CO
mask to all the maps to simplify the analysis. The mask also excludes the
locations of known polarized point sources. Specifically we use the union of
the point-source masks of all the polarized Planck maps.

We calculate the sky fraction using [51, 52]

$f_{\textrm{sky}}=\frac{1}{N_{\text{pix}}}\frac{\left(\sum^{N_{\text{pix}}}_{i=1}w_{i}^{2}\right)^{2}}{\sum^{N_{\text{pix}}}_{i=1}w_{i}^{4}},$
(22)

where $N_{\text{pix}}$ is the number of pixels and $w_{i}$ is the weight of
the apodized mask at the $i$th pixel. We find $f_{\textrm{sky}}=0.92$ for the
baseline CO and point-source mask.

To explore the dependence of $\alpha_{i}$ and $C_{\ell}^{EB,\mathrm{dust}}$ on
the mask, we also use a 30% Galactic mask in union with the CO and point-
source masks. The sky fraction is $f_{\textrm{sky}}=0.62$. These two masks
correspond to the largest and smallest $f_{\textrm{sky}}$ used in Ref. [37].

The code to reproduce the results of this paper is publicly
available333https://github.com/LilleJohs/Cosmic_Birefringence.

##  III Results

| $f_{\textrm{sky}}=0.92$ | $f_{\textrm{sky}}=0.62$
---|---|---
With the filamentary dust $EB$ model
Planck HFI | $\alpha_{i}\neq 0$ | $0.36^{\circ}\pm 0.11^{\circ}$ | $0.29^{\circ}\pm 0.28^{\circ}$
Planck HFI+LFI | $\alpha_{i}\neq 0$ | $\mathbf{0.342^{\circ\,+0.094^{\circ}}_{\phantom{\circ\,}-0.091^{\circ}}}$ | $0.37^{\circ}\pm 0.14^{\circ}$
+WMAP
Without the filamentary dust $EB$ model
Planck HFI | $\alpha_{i}=0$ | $0.288^{\circ}\pm 0.032^{\circ}$ | $\phantom{-}0.330^{\circ}\pm 0.035^{\circ}$
$\alpha_{i}\neq 0$ | $0.30^{\circ}\pm 0.11^{\circ}$ | $-0.25^{\circ}\pm 0.23^{\circ}$
Planck HFI+LFI | $\alpha_{i}=0$ | $0.298^{\circ}\pm 0.032^{\circ}$ | $0.343^{\circ}\pm 0.035^{\circ}$
+WMAP | $\alpha_{i}\neq 0$ | $0.288^{\circ}\pm 0.091^{\circ}$ | $0.18^{\circ}\pm 0.14^{\circ}$
Table 1: Measurements of the cosmic birefringence angle, $\beta$. The baseline
result is shown in bold face. For “$\alpha_{i}\neq 0$” we jointly sample
$\beta$, the miscalibration angles, $\alpha_{i}$, and the dust $EB$ amplitude,
$A_{\ell}$ [Eq. (10)], whereas for “$\alpha_{i}=0$” we only sample $\beta$.
The numbers in the third and seventh row (Planck HFI, $\alpha_{i}\neq 0$) are
taken from Ref. [35].

First, we assume $\alpha_{i}=0$ and measure $\beta=0.288^{\circ}\pm
0.032^{\circ}$ and $0.330^{\circ}\pm 0.035^{\circ}$ from the Planck HFI data
for $f_{\mathrm{sky}}=0.92$ and 0.62, respectively (see the 6th row in Table
1). We find similar results for other $f_{\mathrm{sky}}$. As the CMB can only
determine the sum of $\beta$ and miscalibration angles, this result shows that
the $EB$ signal is isotropic in the sky, and we robustly measure
$\bar{\alpha}+\beta\simeq 0.3^{\circ}$ regardless of $f_{\mathrm{sky}}$, where
$\bar{\alpha}$ is some suitable average value of $\alpha_{i}$ for the HFI.
This result is not affected by $C_{\ell}^{EB,\mathrm{fg}}$ and is consistent
with the Planck team’s result performed on foreground-cleaned maps [49]. The
foreground emission is not responsible for this signal.

Using all the WMAP and Planck polarization data and still assuming
$\alpha_{i}=0$, we measure $\beta=0.298^{\circ}\pm 0.032^{\circ}$ and
$0.343^{\circ}\pm 0.035^{\circ}$ for $f_{\mathrm{sky}}=0.92$ and 0.62,
respectively (second-last row). The statistical power of the data is
sufficient to make a significant detection of $\beta$, provided that we know
$\alpha_{i}$.

As both the WMAP and Planck have flown and ended the missions already, we
cannot precisely measure the miscalibration angles of their detectors anymore.
We thus rely on the foreground to measure $\alpha_{i}$ as done in Refs. [33,
35, 37]. When $C_{\ell}^{EB,\mathrm{fg}}$ is ignored in the analysis, the WMAP
and Planck data yield $\beta=0.288^{\circ}\pm 0.091^{\circ}$ for
$f_{\mathrm{sky}}=0.92$. This is more precise than the Planck HFI result,
$\beta=0.30^{\circ}\pm 0.11^{\circ}$ [35], showing the additional information
gained from the LFI and WMAP data.

Still ignoring $C_{\ell}^{EB,\mathrm{fg}}$, we find that the Galactic mask
reduces the value of $\beta$ to $0.18^{\circ}\pm 0.14^{\circ}$ for
$f_{\mathrm{sky}}=0.62$, but not as much as to $-0.25^{\circ}\pm 0.23^{\circ}$
found for the Planck HFI-only result [35]. The inclusion of the cross-power
spectra with low frequency bands, in which the intensity of polarized dust
emission is much weaker, reduces the impact of $C_{\ell}^{EB,\mathrm{dust}}$
and significantly increases the measured value of $\beta$. The Planck LFI+HFI
analysis reported in Ref. [37] gave $\beta=0.14^{\circ}\pm 0.17^{\circ}$ for
$f_{\textrm{sky}}=0.62$. The inclusion of the WMAP channels further increases
the mean value and tightens the uncertainty of $\beta$.

Figure 1:  Posterior distributions of the cosmic birefringence angle, $\beta$,
and the dust $EB$ amplitude, $A_{\ell}$ [Eq. (10)], in 4 bins for nearly full-
sky data ($f_{\mathrm{sky}}=0.92$; the 4th row in Table 1). The miscalibration
angles, $\alpha_{i}$, are jointly sampled with $\beta$ and $A_{\ell}$ but not
shown here. See Fig. 3 for the 1-d marginalized posterior distribution of
$\alpha_{i}$.  Figure 2: Same as Fig. 1 but for $f_{\mathrm{sky}}=0.62$.

We now account for $C_{\ell}^{EB,\mathrm{fg}}$. The foreground emission in the
Planck LFI bands and similar frequency bands of WMAP is dominated by
synchrotron rather than by dust. Unlike for dust, there is no evidence for the
intrinsic $EB$ correlation of synchrotron emission [53]. We thus ignore
$C_{\ell}^{EB,\mathrm{synch}}$, but use $C_{\ell}^{EB,\mathrm{dust}}$
presented in Eq. (10) to model the dust $EB$ correlation and apply it only to
the HFI maps of Planck and the 94 GHz maps of WMAP.

Figure 3: Marginalized posterior distributions of 22 miscalibration angles,
$\alpha_{i}$, for the baseline result. The distribution of $\beta$ (thick
black line) is the same as in the top-left corner of Fig. 1.

Jointly sampling a frequency-independent $\beta$, 22 miscalibration angles
$\alpha_{i}$, and the dust $EB$ amplitudes $A_{\ell}$ in 4 ranges of $\ell$,
we measure
$\beta=0.342^{\circ\,+0.094^{\circ}}_{\phantom{\circ\,}-0.091^{\circ}}$ and
$0.37^{\circ}\pm 0.14^{\circ}$ for $f_{\textrm{sky}}=0.92$ and 0.62,
respectively (the 4th row in Table 1). The former is our baseline result,
which excludes $\beta=0$ at 99.987% C.L. The latter agrees with the former
within $1\sigma$ and excludes $\beta=0$ at 99.5% C.L.

In Figs. 1 and 2 we show the posterior distributions of $\beta$ and $A_{\ell}$
in 4 bins for $f_{\textrm{sky}}=0.92$ and 0.62, respectively. The dust $EB$
amplitudes are consistent with zero except for the first multipole bin
($51\leq\ell\leq 130$) for $f_{\mathrm{sky}}=0.62$.

In Fig. 3 we show the 1-dimensional marginalized posterior distributions of 22
miscalibration angles, $\alpha_{i}$, for the baseline result. The
distributions for the WMAP and Planck LFI are broader than those of the Planck
HFI, as expected from the signal-to-noise ratio. The measured $\alpha_{i}$ are
in agreement with the reported calibration uncertainties, $1.5^{\circ}$ for
WMAP [29] and $1^{\circ}$ for Planck [54].

We now introduce a frequency-dependent cosmic birefringence signal as
$\beta(\nu)=\beta_{0}(\nu/150~{}\mathrm{GHz})^{n}$ [37]. Using the WMAP and
Planck data and accounting for $C_{\ell}^{EB,\mathrm{dust}}$, we find
$\beta_{0}=0.30^{\circ}\pm 0.10^{\circ}$ and $n=-0.20^{+0.41}_{-0.39}$. Our
measurements are thus consistent with a frequency-independent cosmic
birefringence signal predicted by the axionlike field.

Figure 4:  Stacked observed $EB$ power spectrum (upper) and residuals with
respect to the best-fitting model (lower) for nearly full-sky data
($f_{\textrm{sky}}=0.92$). The beam transfer and pixel window functions are
not deconvolved. The blue and red shaded areas show the $1\sigma$ bands for
the cosmic birefringence and miscalibration contributions, respectively. The
green solid line shows the total best-fitting model.

These results support the following hypothesis. Synchrotron emission has
little to no intrinsic $EB$ correlations that could bias the measurement of
$\beta$, whereas dust contains $EB$ correlations which bias the measurement
when not accounted for, especially for smaller $f_{\mathrm{sky}}$. Although no
synchrotron $EB$ has been found so far [53], more evidence is needed to rule
out synchrotron $EB$ as one of the culprits. The C-BASS and QUIJOTE
experiments will give us a better understanding of synchrotron in the near
future [55, 56].

Figure 5:  Same as Fig. 4 but for $f_{\textrm{sky}}=0.62$.

Finally, we show the average of the observed $EB$ power spectra (“stacked
observed $EB$ power spectrum”) with the uncertainty in Fig. 4 and Fig. 5 for
$f_{\textrm{sky}}=0.92$ and 0.62, respectively. We calculate the inverse-
variance weighted average of the observed $EB$ power spectra from

$\textrm{E}\left(C^{EB,\textrm{o}}_{b}\right)=\frac{\vec{1}\cdot\textbf{M}^{-1}_{b}\cdot\vec{v}_{b}}{\vec{1}\cdot\textbf{M}^{-1}_{b}\cdot\vec{1}}\,,$
(23)

and the variance from

$\textrm{Var}\left(C^{EB,\textrm{o}}_{b}\right)=\frac{1}{\vec{1}\cdot\textbf{M}^{-1}_{b}\cdot\vec{1}}\,.$
(24)

We use $\alpha_{i}=0$ and $\beta=0$ in $\vec{v}_{b}$ to get the stacked $EB$
power spectrum shown in the black points with error bars in the upper panels
of Fig. 4 and Fig. 5. We find similar stacked $EB$ power spectra for both sky
fractions, which are expected because similar values of $\beta$ are found when
assuming $\alpha_{i}=0$ (see the 4th row of Table 1).

The blue $1\sigma$ bands show the $EB$ power spectra as predicted by the
baseline cosmic birefringence angle,
$\beta=0.342^{\circ\,+0.094^{\circ}}_{\phantom{\circ\,}-0.091^{\circ}}$ for
$f_{\textrm{sky}}=0.92$, and $\beta=0.37^{\circ}\pm 0.14^{\circ}$ for
$f_{\textrm{sky}}=0.62$. We fix $\alpha_{i}=0$ in the covariance matrix,
$\textbf{M}_{b}$, for this case. The red $1\sigma$ bands show the
contributions from $\alpha_{i}$ where $\alpha_{i}$ is included in
$\textbf{M}_{b}$. The smaller sky fraction increases the uncertainty on the
contribution from $\alpha_{i}$, but still yields consistent results.

In the lower panels we show the residual with respect to the best-fitting
$A_{\ell}$, $\alpha_{i}$ and $\beta$ model. The $\chi^{2}$ for the degree of
freedom of 72 is 65.3 and 65.8 for $f_{\textrm{sky}}=0.92$ and 0.62,
respectively. We thus conclude that the residuals are consistent with null.

##  IV Conclusions

Datasets | $\beta$ | Dust $EB$ model
---|---|---
Planck PR3 HFI [33] | $0.34^{\circ}\pm 0.14^{\circ}$ | No
Planck PR4 HFI [35] | $0.30^{\circ}\pm 0.11^{\circ}$ | No
Planck PR4 HFI [35] | $0.36^{\circ}\pm 0.11^{\circ}$ | Yes
Planck PR4 HFI + LFI [37] | $0.33^{\circ}\pm 0.10^{\circ}$ | No
Planck PR4 + WMAP | $0.342^{\circ\,+0.094^{\circ}}_{\phantom{\circ\,}-0.091^{\circ}}$ | Yes
Table 2: Previous measurements of cosmic birefringence at nearly full-sky that
adopts the method of Refs. [26, 31]. The right column indicates if the
filamentary dust model of $C^{EB,\textrm{dust}}_{\ell}$ [Eq. (10)] was applied
to the high-frequency maps where dust is the dominating foreground
contribution. The last row displays this work.

We have presented new constraints on the cosmic birefringence angle, $\beta$,
from a joint analysis of the $EB$ power spectra of the WMAP 9-year maps [42]
and the Planck PR4 LFI and HFI maps [34], which cover a wide range of
frequencies from $\nu=23$ to 353 GHz. We used the method based on Refs. [40,
39, 35] to account for the potential impact of the intrinsic $EB$ correlation
of polarized dust emission on the determination of instrumental miscalibration
angles, $\alpha_{i}$, for $\nu\geq 94$ GHz. Marginalizing over $\alpha_{i}$
and the dust $EB$ amplitudes, we measure
$\beta=0.342^{\circ\,+0.094^{\circ}}_{\phantom{\circ\,}-0.091^{\circ}}$ (68%
C.L.) for nearly full-sky data, excluding zero at 99.987% C.L. This is
consistent with, and more precise than, the previous results from the Planck
data [33, 35, 37], and corresponds to the statistical significance of
$3.6\sigma$.

The consistent results of the joint analysis reinforce the conclusion of the
Planck HFI analysis [35, 41] that errors due to instrumental systematics are
smaller than statistical errors. In Table 2 we show previous measurements of
cosmic birefringence where there is a clear trend of increased statistical
significance when new datasets are added or a filamentary dust model for $EB$
is included.

If we remove the Galactic plane from the analysis, we find
$\beta=0.37^{\circ}\pm 0.14^{\circ}$ for $f_{\mathrm{sky}}=0.62$, excluding
zero at 99.5% C.L. We thus find consistent signals of cosmic birefringence for
both sky fractions. We also find that adding the cross-power spectra with
lower frequency data from WMAP and Planck LFI reduces the impact of polarized
dust emission on $\alpha_{i}$ compared to the HFI-only analysis [37]. We have
not accounted for the intrinsic $EB$ correlation of synchrotron emission which
dominates at low frequencies because there is no evidence for it [53, 57].

We find no evidence for frequency dependence of $\beta$. For
$\beta(\nu)\propto\nu^{n}$, we measure $n=-0.20^{+0.41}_{-0.39}$ when
accounting for $C^{EB,\textrm{dust}}_{\ell}$. This is consistent with $n=0$
predicted by the axionlike field [Eq. (1)] but disfavors, for example, $n=-2$
predicted by the Faraday rotation effect from the intergalactic (including
primordial) or interstellar magnetic field.

A better understanding of the foreground emission is needed to completely rule
out the foreground $EB$ as the culprit. However, the best way forward is to
improve upon the calibration work rather than relying on the foreground
emission to determine $\alpha_{i}$. If the calibration accuracy reaches $\pm
0.06^{\circ}$, $\beta$ can be reliably detected with a statistical
significance of $>5\sigma$. We also need confirmation from more independent
datasets to completely rule out the (unknown) systematics of WMAP and Planck.

To this end, both on-going and future ground-based [58, 59, 60, 61, 62, 63,
64, 65, 66], balloon-borne [67, 68], and space-borne [69, 70] experiments are
expected to lead to a convincing discovery (or otherwise) of cosmic
birefringence. If proven to be a cosmological signal, isotropic cosmic
birefringence would have a profound impact on cosmology, particle physics, and
quantum gravity [71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81].

###### Acknowledgements.

We thank Patricia Diego-Palazuelos, Hans Kristian Eriksen, Kris M. Górski,
Yuto Minami, Matthieu Tristram, Duncan Watts, and Ingunn Wehus for useful
discussions and comments on the paper. This work was supported in part by the
European Research Council (ERC) under the Horizon 2020 Research and Innovation
Programme (Grant agreement No. 819478), JSPS KAKENHI Grants No. JP20H05850 and
No. JP20H05859, and the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311. This
work has also received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No.
101007633. The Kavli IPMU is supported by World Premier International Research
Center Initiative (WPI), MEXT, Japan. We acknowledge the use of the Legacy
Archive for Microwave Background Data Analysis (LAMBDA), part of the High
Energy Astrophysics Science Archive Center (HEASARC). HEASARC/LAMBDA is a
service of the Astrophysics Science Division at the NASA Goddard Space Flight
Center. Planck is a project of the European Space Agency (ESA) with
instruments provided by two scientific consortia funded by ESA member states
and led by Principal Investigators from France and Italy, telescope reflectors
provided through a collaboration between ESA and a scientific consortium led
and funded by Denmark, and additional contributions from NASA (USA). Some of
the results in this paper have been derived using the HEALPix package [82].

## References

  * Peebles  _et al._ [2009] P. J. E. Peebles, L. A. Page, and R. B. Partridge,  _Finding the Big Bang_ (Cambridge University Press, 2009).
  * Kosowsky [1996] A. Kosowsky, Cosmic microwave background polarization, Annals Phys. 246, 49 (1996), arXiv:astro-ph/9501045 .
  * Zaldarriaga and Seljak [1997] M. Zaldarriaga and U. Seljak, An all sky analysis of polarization in the microwave background, Phys. Rev. D 55, 1830 (1997), arXiv:astro-ph/9609170 .
  * Kamionkowski  _et al._ [1997] M. Kamionkowski, A. Kosowsky, and A. Stebbins, Statistics of cosmic microwave background polarization, Phys. Rev. D 55, 7368 (1997), arXiv:astro-ph/9611125 .
  * Lue  _et al._ [1999] A. Lue, L.-M. Wang, and M. Kamionkowski, Cosmological signature of new parity violating interactions, Phys. Rev. Lett. 83, 1506 (1999), arXiv:astro-ph/9812088 .
  * Ni [1977] W.-T. Ni, Equivalence Principles and Electromagnetism, Phys. Rev. Lett. 38, 301 (1977).
  * Turner and Widrow [1988] M. S. Turner and L. M. Widrow, Inflation Produced, Large Scale Magnetic Fields, Phys. Rev. D 37, 2743 (1988).
  * Weinberg [1996] S. Weinberg,  _The quantum theory of fields. Vol. 2: Modern applications_ (Cambridge University Press, 1996).
  * Chinowsky and Steinberger [1954] W. Chinowsky and J. Steinberger, Absorption of Negative Pions in Deuterium: Parity of the Pion, Phys. Rev. 95, 1561 (1954).
  * Marsh [2016] D. J. E. Marsh, Axion Cosmology, Phys. Rept. 643, 1 (2016), arXiv:1510.07633 [astro-ph.CO] .
  * Ferreira [2021] E. G. M. Ferreira, Ultra-light dark matter, Astron. Astrophys. Rev. 29, 7 (2021), arXiv:2005.03254 [astro-ph.CO] .
  * Carroll  _et al._ [1990] S. M. Carroll, G. B. Field, and R. Jackiw, Limits on a Lorentz and Parity Violating Modification of Electrodynamics, Phys. Rev. D 41, 1231 (1990).
  * Carroll and Field [1991] S. M. Carroll and G. B. Field, The Einstein equivalence principle and the polarization of radio galaxies, Phys. Rev. D 43, 3789 (1991).
  * Harari and Sikivie [1992] D. Harari and P. Sikivie, Effects of a Nambu-Goldstone boson on the polarization of radio galaxies and the cosmic microwave background, Phys. Lett. B 289, 67 (1992).
  * Komatsu [2022] E. Komatsu, New physics from the polarized light of the cosmic microwave background, Nature Rev. Phys. 4, 10.1038/s42254-022-00452-4 (2022), arXiv:2202.13919 [astro-ph.CO] .
  * Carroll [1998] S. M. Carroll, Quintessence and the Rest of the World: Suppressing Long-Range Interactions, Phys. Rev. Lett. 81, 3067 (1998), arXiv:astro-ph/9806099 [astro-ph] .
  * Panda  _et al._ [2011] S. Panda, Y. Sumitomo, and S. P. Trivedi, Axions as Quintessence in String Theory, Phys. Rev. D 83, 083506 (2011), arXiv:1011.5877 [hep-th] .
  * Finelli and Galaverni [2009] F. Finelli and M. Galaverni, Rotation of Linear Polarization Plane and Circular Polarization from Cosmological Pseudo-Scalar Fields, Phys. Rev. D 79, 063002 (2009), arXiv:0802.4210 [astro-ph] .
  * Fedderke  _et al._ [2019] M. A. Fedderke, P. W. Graham, and S. Rajendran, Axion Dark Matter Detection with CMB Polarization, Phys. Rev. D 100, 015040 (2019), arXiv:1903.02666 [astro-ph.CO] .
  * Myers and Pospelov [2003] R. C. Myers and M. Pospelov, Ultraviolet modifications of dispersion relations in effective field theory, Phys. Rev. Lett. 90, 211601 (2003), arXiv:hep-ph/0301124 .
  * Arvanitaki  _et al._ [2010] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, String Axiverse, Phys. Rev. D 81, 123530 (2010), arXiv:0905.4720 [hep-th] .
  * Liu  _et al._ [2006] G.-C. Liu, S. Lee, and K.-W. Ng, Effect on cosmic microwave background polarization of coupling of quintessence to pseudoscalar formed from the electromagnetic field and its dual, Phys. Rev. Lett. 97, 161303 (2006), arXiv:astro-ph/0606248 [astro-ph] .
  * Feng  _et al._ [2005] B. Feng, H. Li, M. Li, and X. Zhang, Gravitational leptogenesis and its signatures in CMB, Phys. Lett. B620, 27 (2005), arXiv:hep-ph/0406269 [hep-ph] .
  * Zhao  _et al._ [2015] G.-B. Zhao, Y. Wang, J.-Q. Xia, M. Li, and X. Zhang, An efficient probe of the cosmological CPT violation, J. Cosmol. Astropart. Phys. 07 (2015), 032, arXiv:1504.04507 [astro-ph.CO] .
  * Gruppuso  _et al._ [2016] A. Gruppuso, G. Maggio, D. Molinari, and P. Natoli, A note on the birefringence angle estimation in CMB data analysis, J. Cosmol. Astropart. Phys. 05 (2016), 020, arXiv:1604.05202 [astro-ph.CO] .
  * Minami  _et al._ [2019] Y. Minami, H. Ochi, K. Ichiki, N. Katayama, E. Komatsu, and T. Matsumura, Simultaneous determination of the cosmic birefringence and miscalibrated polarization angles from CMB experiments, Progress of Theoretical and Experimental Physics 2019, 083E02 (2019), arXiv:1904.12440 [astro-ph.CO] .
  * Wu  _et al._ [2009] E. Y. S. Wu _et al._ (QUaD Collaboration), Parity Violation Constraints Using Cosmic Microwave Background Polarization Spectra from 2006 and 2007 Observations by the QUaD Polarimeter, Phys. Rev. Lett. 102, 161302 (2009), arXiv:0811.0618 [astro-ph] .
  * Miller  _et al._ [2009] N. J. Miller, M. Shimon, and B. G. Keating, CMB Polarization Systematics Due to Beam Asymmetry: Impact on Cosmological Birefringence, Phys. Rev. D 79, 103002 (2009), arXiv:0903.1116 [astro-ph.CO] .
  * Komatsu  _et al._ [2011] E. Komatsu _et al._ (WMAP Collaboration), Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, Astrophys. J. Suppl. 192, 18 (2011), arXiv:1001.4538 [astro-ph.CO] .
  * Krachmalnicoff  _et al._ [2022] N. Krachmalnicoff _et al._ (LiteBIRD), In-flight polarization angle calibration for LiteBIRD: blind challenge and cosmological implications, J. Cosmol. Astropart. Phys. 01 (2022), 039, arXiv:2111.09140 [astro-ph.CO] .
  * Minami and Komatsu [2020a] Y. Minami and E. Komatsu, Simultaneous determination of the cosmic birefringence and miscalibrated polarization angles II: Including cross frequency spectra, Progress of Theoretical and Experimental Physics 2020, 103E02 (2020a), arXiv:2006.15982 [astro-ph.CO] .
  * Planck Collaboration III [2020] Planck Collaboration III, Planck 2018 results. III. High Frequency Instrument data processing, Astron. Astrophys. 641, A3 (2020), arXiv:1807.06207 .
  * Minami and Komatsu [2020b] Y. Minami and E. Komatsu, New Extraction of the Cosmic Birefringence from the Planck 2018 Polarization Data, Phys. Rev. Lett. 125, 221301 (2020b), arXiv:2011.11254 [astro-ph.CO] .
  * Planck Collaboration Int. LVII [2020] Planck Collaboration Int. LVII, Planck intermediate results. LVII. NPIPE: Joint Planck LFI and HFI data processing, Astron. Astrophys. 643, A42 (2020), arXiv:2007.04997 .
  * Diego-Palazuelos  _et al._ [2022a] P. Diego-Palazuelos _et al._ , Cosmic Birefringence from the Planck Data Release 4, Phys. Rev. Lett. 128, 091302 (2022a), arXiv:2201.07682 [astro-ph.CO] .
  * Planck Collaboration II [2020] Planck Collaboration II, Planck 2018 results. II. Low Frequency Instrument data processing, Astron. Astrophys. 641, A2 (2020), arXiv:1807.06206 .
  * Eskilt [2022] J. R. Eskilt, Frequency-Dependent Constraints on Cosmic Birefringence from the LFI and HFI Planck Data Release 4, Astron. Astrophys. 662, A10 (2022), arXiv:2201.13347 [astro-ph.CO] .
  * Planck Collaboration XI [2020] Planck Collaboration XI, Planck 2018 results. XI. Polarized dust foregrounds, Astron. Astrophys. 641, A11 (2020), arXiv:1801.04945 .
  * Clark  _et al._ [2021] S. E. Clark, C.-G. Kim, J. C. Hill, and B. S. Hensley, The Origin of Parity Violation in Polarized Dust Emission and Implications for Cosmic Birefringence, Astrophys. J. 919, 53 (2021), arXiv:2105.00120 [astro-ph.GA] .
  * Huffenberger  _et al._ [2020] K. M. Huffenberger, A. Rotti, and D. C. Collins, The Power Spectra of Polarized, Dusty Filaments, Astrophys. J. 899, 31 (2020), arXiv:1906.10052 [astro-ph.CO] .
  * Diego-Palazuelos  _et al._ [2022b] P. Diego-Palazuelos _et al._ , Cosmic Birefringence from Planck Public Release 4, in  _56th Rencontres de Moriond on Cosmology_ (2022) arXiv:2203.04830 [astro-ph.CO] .
  * Bennett  _et al._ [2013] C. L. Bennett _et al._ (WMAP), Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results, Astrophys. J. Suppl. 208, 20 (2013), arXiv:1212.5225 [astro-ph.CO] .
  * Contreras  _et al._ [2017] D. Contreras, P. Boubel, and D. Scott, Constraints on direction-dependent cosmic birefringence from Planck polarization data, J. Cosmol. Astropart. Phys. 12 (2017), 046, arXiv:1705.06387 [astro-ph.CO] .
  * Bianchini  _et al._ [2020] F. Bianchini _et al._ (SPT Collaboration), Searching for Anisotropic Cosmic Birefringence with Polarization Data from SPTpol, Phys. Rev. D 102, 083504 (2020), arXiv:2006.08061 [astro-ph.CO] .
  * Namikawa  _et al._ [2020] T. Namikawa _et al._ (ACT Collaboration), Atacama Cosmology Telescope: Constraints on cosmic birefringence, Phys. Rev. D 101, 083527 (2020), arXiv:2001.10465 [astro-ph.CO] .
  * Gruppuso  _et al._ [2020] A. Gruppuso, D. Molinari, P. Natoli, and L. Pagano, Planck 2018 constraints on anisotropic birefringence and its cross-correlation with CMB anisotropy, J. Cosmol. Astropart. Phys. 11 (2020), 066, arXiv:2008.10334 [astro-ph.CO] .
  * Lewis  _et al._ [2000] A. Lewis, A. Challinor, and A. Lasenby, Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models, Astrophys. J. 538, 473 (2000), astro-ph/9911177 .
  * Planck Collaboration VI [2020] Planck Collaboration VI, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), arXiv:1807.06209 .
  * Planck Collaboration Int. XLIX [2016] Planck Collaboration Int. XLIX, Planck intermediate results. XLIX. Parity-violation constraints from polarization data, Astron. Astrophys. 596, A110 (2016), arXiv:1605.08633 .
  * Chon  _et al._ [2004] G. Chon, A. Challinor, S. Prunet, E. Hivon, and I. Szapudi, Fast estimation of polarization power spectra using correlation functions, Mon. Not. Roy. Astron. Soc. 350, 914 (2004), arXiv:astro-ph/0303414 .
  * Hivon  _et al._ [2002] E. Hivon, K. M. Gorski, C. B. Netterfield, B. P. Crill, S. Prunet, and F. Hansen, MASTER of the cosmic microwave background anisotropy power spectrum: A fast method for statistical analysis of large and complex cosmic microwave background data sets, The Astrophysical Journal 567, 2 (2002).
  * Challinor and Chon [2005] A. Challinor and G. Chon, Error analysis of quadratic power spectrum estimates for CMB polarization: Sampling covariance, Mon. Not. Roy. Astron. Soc. 360, 509 (2005), arXiv:astro-ph/0410097 .
  * Martire  _et al._ [2022a] F. A. Martire, R. B. Barreiro, and E. Martínez-González, Characterization of the polarized synchrotron emission from Planck and WMAP data, J. Cosmol. Astropart. Phys. 04 (2022), 003, arXiv:2110.12803 [astro-ph.CO] .
  * Rosset  _et al._ [2010] C. Rosset _et al._ , Planck pre-launch status: High Frequency Instrument polarization calibration, Astron. Astrophys. 520, A13 (2010), arXiv:1004.2595 [astro-ph.CO] .
  * Dickinson  _et al._ [2019] C. Dickinson _et al._ , The C-Band All-Sky Survey (C-BASS): Constraining diffuse Galactic radio emission in the North Celestial Pole region, Mon. Not. Roy. Astron. Soc. 485, 2844 (2019), arXiv:1810.11681 [astro-ph.GA] .
  * Poidevin  _et al._ [2018] F. Poidevin _et al._ (QUIJOTE), The QUIJOTE Experiment: Prospects for CMB B-MODE polarization detection and foregrounds characterization, in  _13th Rencontres du Vietnam: Cosmology 2017_ (2018) arXiv:1802.04594 [astro-ph.CO] .
  * Martire  _et al._ [2022b] F. A. Martire, R. B. Barreiro, and E. Martínez-González, Characterization of the polarized synchrotron emission from Planck and WMAP data, in  _56th Rencontres de Moriond on Cosmology_ (2022) arXiv:2203.06724 [astro-ph.CO] .
  * Adachi  _et al._ [2020] S. Adachi _et al._ (Polarbear), A Measurement of the CMB $E$-mode Angular Power Spectrum at Subdegree Scales from670 Square Degrees of POLARBEAR Data, Astrophys. J. 904, 65 (2020), arXiv:2005.06168 [astro-ph.CO] .
  * Choi  _et al._ [2020] S. K. Choi _et al._ (ACT Collaboration), The Atacama Cosmology Telescope: a measurement of the Cosmic Microwave Background power spectra at 98 and 150 GHz, J. Cosmol. Astropart. Phys. 12 (2020), 045, arXiv:2007.07289 [astro-ph.CO] .
  * Dutcher  _et al._ [2021] D. Dutcher _et al._ (SPT-3G Collaboration), Measurements of the E-mode polarization and temperature-E-mode correlation of the CMB from SPT-3G 2018 data, Phys. Rev. D 104, 022003 (2021), arXiv:2101.01684 [astro-ph.CO] .
  * Ade  _et al._ [2021] P. A. R. Ade _et al._ (BICEP, Keck), Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett. 127, 151301 (2021), arXiv:2110.00483 [astro-ph.CO] .
  * Dahal  _et al._ [2022] S. Dahal _et al._ , Four-year Cosmology Large Angular Scale Surveyor (CLASS) Observations: On-sky Receiver Performance at 40, 90, 150, and 220 GHz Frequency Bands, Astrophys. J. 926, 33 (2022), arXiv:2107.08022 [astro-ph.IM] .
  * Hamilton  _et al._ [2022] J. C. Hamilton _et al._ (QUBIC), QUBIC I: Overview and science program, J. Cosmol. Astropart. Phys. 04 (2022), 034, arXiv:2011.02213 [astro-ph.IM] .
  * Ade  _et al._ [2019] P. Ade _et al._ (Simons Observatory), The Simons Observatory: Science goals and forecasts, J. Cosmol. Astropart. Phys. 02 (2019), 056, arXiv:1808.07445 [astro-ph.CO] .
  * Moncelsi  _et al._ [2020] L. Moncelsi _et al._ , Receiver development for BICEP Array, a next-generation CMB polarimeter at the South Pole, Proc. SPIE Int. Soc. Opt. Eng. 11453, 1145314 (2020), arXiv:2012.04047 [astro-ph.IM] .
  * Abazajian  _et al._ [2019] K. Abazajian _et al._ , CMB-S4 Science Case, Reference Design, and Project Plan, arXiv preprint (2019), arXiv:1907.04473 [astro-ph.IM] .
  * Ade  _et al._ [2022] P. A. R. Ade _et al._ (SPIDER), A Constraint on Primordial B-modes from the First Flight of the Spider Balloon-borne Telescope, Astrophys. J. 927, 174 (2022), arXiv:2103.13334 [astro-ph.CO] .
  * Addamo  _et al._ [2021] G. Addamo _et al._ (LSPE), The large scale polarization explorer (LSPE) for CMB measurements: performance forecast, J. Cosmol. Astropart. Phys. 08 (2021), 008, arXiv:2008.11049 [astro-ph.IM] .
  * LiteBIRD Collaboration [2022] LiteBIRD Collaboration, Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey, arXiv preprint (2022), arXiv:2202.02773 [astro-ph.CO] .
  * Hanany  _et al._ [2019] S. Hanany _et al._ (NASA PICO), PICO: Probe of Inflation and Cosmic Origins, arXiv preprint (2019), arXiv:1902.10541 [astro-ph.IM] .
  * Fujita  _et al._ [2021a] T. Fujita, Y. Minami, K. Murai, and H. Nakatsuka, Probing axionlike particles via cosmic microwave background polarization, Phys. Rev. D 103, 063508 (2021a), arXiv:2008.02473 [astro-ph.CO] .
  * Fujita  _et al._ [2021b] T. Fujita, K. Murai, H. Nakatsuka, and S. Tsujikawa, Detection of isotropic cosmic birefringence and its implications for axionlike particles including dark energy, Phys. Rev. D 103, 043509 (2021b), arXiv:2011.11894 [astro-ph.CO] .
  * Takahashi and Yin [2021] F. Takahashi and W. Yin, Kilobyte Cosmic Birefringence from ALP Domain Walls, J. Cosmol. Astropart. Phys. 04 (2021), 007, arXiv:2012.11576 [hep-ph] .
  * Mehta  _et al._ [2021] V. M. Mehta, M. Demirtas, C. Long, D. J. E. Marsh, L. McAllister, and M. J. Stott, Superradiance in string theory, J. Cosmol. Astropart. Phys. 07 (2021), 033, arXiv:2103.06812 [hep-th] .
  * Nakagawa  _et al._ [2021] S. Nakagawa, F. Takahashi, and M. Yamada, Cosmic Birefringence Triggered by Dark Matter Domination, Phys. Rev. Lett. 127, 181103 (2021), arXiv:2103.08153 [hep-ph] .
  * Alvey and Escudero Abenza [2021] J. Alvey and M. Escudero Abenza, Constraints on global symmetry breaking in quantum gravity from cosmic birefringence measurements, Phys. Lett. B 823, 136752 (2021), arXiv:2106.04226 [hep-th] .
  * Choi  _et al._ [2021] G. Choi, W. Lin, L. Visinelli, and T. T. Yanagida, Cosmic birefringence and electroweak axion dark energy, Phys. Rev. D 104, L101302 (2021), arXiv:2106.12602 [hep-ph] .
  * Obata [2021] I. Obata, Implications of the Cosmic Birefringence Measurement for the Axion Dark Matter Search, arXiv preprint (2021), arXiv:2108.02150 [astro-ph.CO] .
  * Gasparotto and Obata [2022] S. Gasparotto and I. Obata, Cosmic birefringence from monodromic axion dark energy, JCAP 08 (08), 025, arXiv:2203.09409 [astro-ph.CO] .
  * Nakatsuka  _et al._ [2022] H. Nakatsuka, T. Namikawa, and E. Komatsu, Is cosmic birefringence due to dark energy or dark matter? A tomographic approach, Phys. Rev. D 105, 123509 (2022), arXiv:2203.08560 [astro-ph.CO] .
  * Kitajima  _et al._ [2022] N. Kitajima, F. Kozai, F. Takahashi, and W. Yin, Power spectrum of domain-wall network and its implications for isotropic and anisotropic cosmic birefringence, arXiv preprint (2022), arXiv:2205.05083 [astro-ph.CO] .
  * Górski  _et al._ [2005] K. M. Górski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, and M. Bartelman, HEALPix - A Framework for high resolution discretization, and fast analysis of data distributed on the sphere, Astrophys. J. 622, 759 (2005), arXiv:astro-ph/0409513 .


<concept_desc>Software and its engineering General programming languages</concept_desc>
<concept_desc>Social and professional topics History of programming languages</concept_desc>

[500]Software and its engineering General programming languages
[300]Social and professional topics History of programming languages

CycleQ: an efficient basis  for cyclic equational reasoning]CycleQ: an efficient basis
for cyclic equational reasoning

Eddie Jones

Department of Computer Science
University of Bristol
United Kingdom

C.-H. Luke Ong

Department of Computer Science
University of Oxford
United Kingdom

Steven Ramsay

Department of Computer Science
University of Bristol
United Kingdom