11institutetext: University of Freiburg, Germany
11email<EMAIL_ADDRESS>22institutetext: TIB Leibniz Information
Centre for Science and Technology University Library, Germany 22email:
<EMAIL_ADDRESS>
# Toward Semantic Publishing in
Non-Invasive Brain Stimulation: A Comprehensive Analysis of rTMS
Studies††thanks: Supported by German BMBF project SCINEXT (ID 01lS22070), DFG
NFDI4DataScience (ID 460234259), and ERC ScienceGraph (ID 819536)

Swathi Anil 11 0009-0004-5976-1717 Jennifer D’Souza Both authors contributed
equally to this work.22 0000-0002-6616-9509

###### Abstract

Noninvasive brain stimulation (NIBS) encompasses transcranial stimulation
techniques that can influence brain excitability. These techniques have the
potential to treat conditions like depression, anxiety, and chronic pain, and
to provide insights into brain function. However, a lack of standardized
reporting practices limits its reproducibility and full clinical potential.
This paper aims to foster interinterdisciplinarity toward adopting Computer
Science Semantic reporting methods for the standardized documentation of
Neuroscience NIBS studies making them explicitly Findable, Accessible,
Interoperable, and Reusable (FAIR).

In a large-scale systematic review of 600 repetitive transcranial magnetic
stimulation (rTMS), a subarea of NIBS, dosages, we describe key properties
that allow for structured descriptions and comparisons of the studies. This
paper showcases the semantic publishing of NIBS in the ecosphere of knowledge-
graph-based next-generation scholarly digital libraries. Specifically, the
FAIR Semantic Web resource(s)-based publishing paradigm is implemented for the
600 reviewed rTMS studies in the Open Research Knowledge Graph.

###### Keywords:

Semantic Publishing Digital libraries Scholarly Publishing Infrastructure
Scholarly Publishing Workflows FAIR data principles Open Research Knowledge
Graph Noninvasive brain stimulation Repetitive Transcranial Magnetic
Stimulation.

##  1 Introduction

Noninvasive brain stimulation (NIBS) is a rapidly evolving field in
neuroscience that involves modulating neuronal activity in the brain without
surgical intervention, often employing electrical currents or magnetic fields
[24]. NIBS encompasses various subareas such as transcranial direct current
stimulation (tDCS), transcranial alternating current stimulation (tACS),
transcranial random noise stimulation (tRNS), repetitive transcranial magnetic
stimulation (rTMS), among others. These techniques harness the ability of
neural tissue to adapt to external stimulation by making structural,
functional and molecular changes, called neural plasticity. Over the past
decades, rTMS in particular has gained significant attention for its potential
to enhance cognitive abilities, including attention, memory, language, and
decision-making. It is also being explored as a treatment option for
neuropsychiatric disorders like pharmacoresistant major depressive disorder
(MDD), schizophrenia, and obsessive-compulsive disorder, as well as for stroke
rehabilitation, pain management, and investigation of brain function [7, 18,
9, 33, 13, 10, 2, 32, 20, 14]. The outcome of rTMS can be influenced by
various stimulation parameters, highlighting the importance of dosage
considerations. Stimulation parameters such as intensity, duration, frequency,
and pattern play a significant role in shaping the effects of rTMS on brain
activity and clinical outcomes.

An ongoing concern in the field of repetitive transcranial magnetic
stimulation (rTMS) is the lack of standardized reporting for dosing
parameters, which has implications for comprehensive understanding and
clinical application of these techniques [38]. Currently, the reporting of
rTMS dosage relies on textual descriptions, leading to varying levels of
detail and specificity. This variability hampers reproducibility and the
establishment of consistent protocols and effective stimulation parameters.
The absence of standardized reporting practices for rTMS dosage poses
challenges in comparing and replicating studies, limiting the accumulation of
robust evidence and the generalizability of findings across different
populations. This also poses a disadvantage in the ability to conduct meta-
analyses, hindering the translation of rTMS into clinical practice in an
effective manner. Therefore, efforts to establish standardized reporting
guidelines for rTMS studies are warranted, aiming to enhance transparency,
reproducibility, and collaboration within the field.

In this context, we recognize the opportunity for interdiciplinarity. In
Computer Science, the semantic web publishing method utilizes semantic
technologies and ontologies to structure and standardize data, enabling more
efficient and meaningful reporting, querying, and analysis of information
within the World Wide Web [5]. We thus posit the adoption of semantic web
methods from Computer Science as a solution to the problem of non-standardized
reporting of Neuroscience NIBS study doses. This is in keeping with the recent
growing general impetus driving a change in the status quo of discourse-based
scholarly communication toward Findable, Accessible, Interoperable, and
Resuable (FAIR) [37] publishing as structured scholarly knowledge graphs by
adopting the technologies of the semantic web [1, 19, 17, 4, 8, 34, 12, 3]. We
claim that the adoption of FAIR reporting paradigms facilitated by semantic
web technologies for NIBS dosing will mark a significant stride toward
overcoming the pressing challenge of reproducibility in this field. In turn,
it will facilitate tapping into the full clinical potential of NIBS.

Figure 1: Machine-actionable structured scholarly knowledge capture via
semantic publishing (in red) versus traditional discourse-based non-machine-
actionable publishing (in gray).

Toward the semantic web publishing of NIBS dosages a few technical
considerations need to be in place. These include: defining the salient
entities, attributes, and relationships for the NIBS domain as standardized
resources accessible on the Web via uniform resource identifiers (URIs);
optionally, designing or reusing a suitable ontology that captures the
concepts, relationships, and constraints of interest; and publishing NIBS
doses using structured triples conforming to the RDF syntax recommended by the
World Wide Web Consortium (W3C).111https://www.w3.org/TR/rdf11-concepts/ It is
crucial to utilize standard ontologized vocabulary terms, incorporate
appropriate metadata, and ensure data publication on the web with unique URIs
for resource identification and interlinking. Fortunately, the latter required
FAIR semantic web publishing functionalities are readily available within
next-generation scholarly knowledge publishing digital libraries such as the
Open Research Knowledge Graph (ORKG) [3, 27]. The ORKG platform supports the
semantic publishing [26] of scholarly knowledge as structured, machine-
actionable data. Its specific functionalities include offering a robust and
reliable web namespace within which to specify the vocabulary of pertinent
scholarly knowledge via dereferenceable uniform resource identifiers (URIs);
intuitive frontend interfaces [22] to define scholarly
resources,222https://orkg.org/resources
properties,333https://orkg.org/properties classes,444https://orkg.org/classes
and templates555https://orkg.org/templates which via the platform’s backend
technologies automatically represent the growing scholarly knowledge graph in
RDF syntax;666https://orkg.org/api/rdf/dump given the machine-actionability
that semantic representations of knowledge generally entail, the ORKG supports
the creation of novel comparison views over comparably structured scholarly
knowledge in a similar way to comparisons of products on e-commerce websites
[23]; and finally, SPARQL queryable interfaces777https://orkg.org/triplestore,
https://orkg.org/sparql/ to further obtain customized snapshots or aggregated
views of the ORKG are also supported. Figure 1 juxtaposes the semantic
publishing of scholarly knowledge in the ORKG in terms of one of its workflows
compared with traditional discourse-based scholarly publishing paradigm.

Figure 2: Repetitive Transcranial Magnetic Stimulation (rTMS): A. Schematic
representation of an rTMS coil positioned over a brain. The illustration shows
the precise placement of the coil, adhering to the specific anatomical target,
alongside corresponding computational estimation of the electric field
(E-field) generated by the rTMS coil in the brain. The color gradient
indicates the intensity of the electric field, highlighting the spatial
distribution of the stimulation. B. Conventional rTMS device and experimental
setup. C. Parameters that can influence rTMS outcome include pulse width,
frequency, intensity and net protocol duration, among others. D. rTMS coil
shapes can influence the electrical field distribution of stimulation over the
brain, impacting the stimulation outcome. Courtesy: Biorender.com, SIMNIBS,
Magstim.

Returning then to the interdisciplinary vision laid out in this paper toward
adopting the semantic web publishing methodology for NIBS dosages, the only
remaining action items would then be to define the standardized vocabulary to
represent the domain. With this goal in mind, in this paper, we showcase in
practice the semantic publishing of repetitive transcranial magnetic
stimulation (rTMS) – a subarea of NIBS – as a use-case toward the semantic
publishing of NIBS studies, generally. rTMS is a non-invasive neuromodulation
technique that applies magnetic pulses to targeted brain regions to modulate
neural activity [36]. Extensive research in both humans and animals has
demonstrated the therapeutic potential of rTMS in neurological and psychiatric
disorders. Notably, rTMS has shown efficacy in conditions such as major
depressive disorder (MDD) and obsessive-compulsive disorder (OCD). Figure 2
offers an illustrative overview of the rTMS methdology in practice. The U.S.
Food and Drug Administration (FDA) has granted approval for the use of rTMS as
a therapeutic tool, specifically for the treatment of MDD and OCD. However,
due to the non-standardization of experimental protocols and data reporting of
rTMS, the evidence to inform clinical application is highly inconsistent [29,
11] and substantially based on trial and error. As a first step, is the
discovery of standardized vocabulary to describe rTMS. An early work in this
direction is the comprehensive review of rTMS studies between 1999 and 2020 by
Turi et al. [30] which introduced a set of salient properties based on which
over 600 rTMS studies were comparable. The standardized applicability of the
properties over the large number of studies indicates their potential to be
realized as a schema based on the philosophy of data-driven bottom-up ontology
construction [31]. Thus, in this paper we investigate the following main
research questions (RQs). RQ1: What are the properties for the structured
representation of rTMS studies? This is addressed in section 3. RQ2: How can
this semantic reporting of rTMS studies specifically, and NIBS studies
generally, be practically realized as a FAIR model? – addressed in subsection
3.1.

The primary goal of this work is to engage a broad and varied audience,
especially in the field of neuroscience. We aim to foster interdisciplinary
discussions to enhance understanding of the fast-evolving topic of semantic
scholarly knowledge publishing, both within this paper’s context and beyond.
The proposed semantics-based structured recording method is a concrete first
step that directly addresses the two main long-discussed problems of
transparency and reproducibility of NIBS studies to facilitate clinical
application [38]. Our concrete implementation of the semantic web-based
technology reporting of rTMS studies serves as a demonstration use-case and a
call-to-action to adopt and foster collaborations toward ORKG infrastructural
extensions as a platform to support publishing of NIBS studies.

##  2 Background on rTMS Dosage Data Management

The management of data in rTMS studies remains a challenging task due to the
varied practices in data collection, storage, and protection protocols [38].
Data is commonly collected directly from physiological signals and stored in
localized databases or documentation tools. While these practices fulfil basic
data storage needs, they raise significant issues concerning data
interoperability, reusability, and long-term accessibility, primarily due to
inconsistencies in data structures and metadata provision [38]. Moreover, data
protection presents a significant challenge in this context. The sensitivity
of personal data demands stringent measures to ensure security, which can
further complicate data accessibility. Ensuring secure access requires a fine
balance between protection and availability, often calling for robust
authentication protocols and persistent identifiers [37]. Adding to the
complexities is the variability in the selection of stimulation parameters, in
particular the stimulation intensity, across rTMS studies, as highlighted in
Turi et al.’s systematic review [30]. The lack of a standardized protocol for
this crucial parameter leads to substantial inconsistencies across studies,
impacting the reliability and comparability of research findings.

To refine this process, the integration of FAIR (Findable, Accessible,
Interoperable, and Reusable) principles into rTMS dosage data management
presents a potential solution [37]. With the application of these principles,
data could be catalogued accurately, improving findability. Accessibility
could be ensured with robust authentication protocols, allowing for a more
secure and open sharing of data. Interoperability, although often not
prioritized in traditional rTMS studies, could be significantly enhanced
through common data models and agreed-upon formats [6]. The proposed
adaptations in rTMS data handling, encompassing FAIR principles and
standardized intensity selection, can substantially elevate the quality of
research outputs, making them more robust, reliable, and universally
acceptable.

Table 1: rTMS key properties and associated resources Property |  Resources
---|---
Type of rTMS |  Conventional rTMS (rTMS), Intermittent theta burst stimulation (iTBS), Continuous theta burst stimulation (cTBS), Quadripulse stimulation (QPS)
Intrabust Frequency |  -
Stimulation Intensity Selection Approach |  Active motor threshold (AMT), Resting motor threshold (RMT), Unspecified motor threshold (MT), Functional lesion (FL), Phosphene threshold (PT), Fixed intensity (FXD), Electric field (EF)
Threshold-estimation strategies |  Method of limit (ML), 5 step procedure (5STEP), Threshold hunting (TH), Maximum likelihood based threshold hunting (MLTH), Parameter estimation by sequential testing (PEST), TMS Motor Threshold Assessment Tool (MTAT)
Threshold Measurement |  Electrode (E), Visual (V)
Amplitude of the Motor Evoked Potential (mV) |  -
Threshold Ratio |  -
Percentage or the Amplitude of the Motor Threshold Contraction |  -
Percent of Stimulation Intensity |  Min value, Max value
Maximum Stimulator Output |  -
Stimulator Company |  Cad, MedDan, MagSti, NeoNet, NeuNet, MagVen, NexSti, MagMor, Yir, BraSwa, DeyDia, YunTec, NeuSof
Stimulator Model |  HS, MP, MES10, R, SR, SR2, NP, 16E05, 200, 200 2, MLR25, 200 BI, QP500, HF, MP30, MPX100, 2100CRS, MP100, R2, MPR30, NBS, PM100, CCYI, CCYIA, DMXT, NS, Sys4.3, R2P1, N-MS/D, MPC, MS/D
Coil Shape |  F8, R, F8-D, D
Coil Size |  -
Coil Model |  MC125, MC-125, MC-B70, MCF-B70, MCF-B-65, MCF-B65, WC, AC, DC, PN9925, 992500, C-B60, FC, FC-B70, HP, Cool B65, cool-B65, Cool-DB80, Cool B56, H-ADD, H, H1, AF, DB-80, B65, MMC-140, 70BF-Cool

##  3 Supporting Semantic Technology and Semantification of rTMS Studied Doses

As a first step, for the semantic publishing of rTMS dosage, we need to define
the rTMS-domain-specific resources and properties. Using the ORKG semantic
scholarly knowledge publishing platform, we decide to define the rTMS
resources and predicates as resources on the web with deferenceable URIs in
the ORKG namespace. The use of the ORKG namespace offers: 1\. Standardization
and Interoperability. It provides a framework for representing scholarly
knowledge, ensuring seamless integration and data sharing across platforms.
2\. Community Collaboration. Researchers and practitioners can actively
contribute to the growth and improvement of the ORKG ecosystem, refining
ontology and shaping scholarly resources. 3\. Enhanced Discoverability.
Adoption of the ORKG namespace improves the visibility and indexing of
scholarly resources, increasing their chances of being accessed and cited. 4\.
Semantic Enrichment. The comprehensive ontology of the ORKG namespace enriches
scholarly resources with detailed metadata, facilitating advanced queries and
analysis. 5\. Community Trust and Credibility. Utilizing the ORKG namespace
adds credibility to scholarly resources through community validation,
fostering trust within the scholarly community. 6\. Long-term Sustainability.
The ORKG namespace, supported by an active community, ensures the maintenance
and longevity of semantic web resources, giving researchers confidence in
their continuity. Overall, the use of the ORKG namespace provides researchers
and organizations with a standardized, collaborative, and credible framework
for creating scholarly resources and predicates in the semantic web.

rTMS doses encompass multiple adjustable parameters that shape the
characteristics of the stimulation [16, 15, 25, 35, 21, 28]. The longitudinal
review of rTMS studies between 1999 and 2020 by Turi et al. [30] characterized
the rTMS dose in terms of 15 salient properties. Based on the philosophy of
bottom-up discovery of standardized domain vocabulary [31], these 15
properties and their resources are defined as URIs in the ORKG namespace and
then applied to document the individual rTMS doses reviewed. The 15-properties
strong rTMS recording resources, detailed in Table 1, are: 1. type of rTMS
with four resource candidates, viz. conventional rTMS (rTMS), intermittent
theta burst stimulation (iTBS), continuous theta burst stimulation (cTBS), and
quadripulse stimulation (QPS). 2. intrabust frequency. 3. stimulation
intensity selection approach with seven resource candidates, viz. active motor
threshold (AMT), resting motor threshold (RMT), unspecified motor threshold
(MT), functional lesion (FL), phosphene threshold (PT), fixed intensity (FXD),
and electric field (EF). 4. threshold-estimation strategies with six resource
candidates, viz. method of limit (ML), 5 step procedure (5STEP), threshold
hunting (TH), maximum likelihood based threshold hunting (MLTH), parameter
estimation by sequential testing (PEST), and TMS Motor Threshold Assessment
Tool (MTAT). 5. threshold measurement with two resource candidates, viz.
electrode (E) and visual (V). 6. amplitude of the motor evoked potential in
microvolts. 7. threshold ratio. 8. percentage or the amplitude of the motor
threshold contraction. 9. percent of stimulation intensity with sub-properties
9.a. percent of stimulation intensity (min value) and 9.b. percent of
stimulation intensity (max value). 10. maximum stimulator output. 11.
stimulator company with 13 and counting resource candidates: Cad, MedDan,
MagSti, NeoNet, NeuNet, MagVen, NexSti, MagMor, Yir, BraSwa, DeyDia, YunTec,
and NeuSof. 12. stimulator model with 31 resources: HS, MP, MES10, R, SR, SR2,
NP, 16E05, 200, 200_2, MLR25, 200_BI, QP500, HF, MP30, MPX100, 2100CRS, MP100,
R2, MPR30, NBS, PM100, CCYI, CCYIA, DMXT, NS, Sys4.3, R2P1, N-MS/D, MPC, and
MS/D. 13. coil shape with 4 resources: F8, R, F8-D, and D. 14. coil size. 15.
coil model with 27 resources: MC125, MC-125, MC-B70, MCF-B70, MCF-B-65,
MCF-B65, WC, AC, DC, PN9925, 992500, C-B60, FC, FC-B70, HP, Cool B65,
cool-B65, Cool-DB80, Cool B56, H-ADD, H, H1, AF, DB-80, B65, MMC-140, and
70BF-Cool. These properties are implemented as a template
https://orkg.org/template/R211955 which can be extended and reused for the
explicit and standardized recording of rTMS doses. As a demonstration use-
case, the 600 reviewed studies [30] were semantically published resulting in
an rTMS-KG as a subgraph of the ORKG. Comparisons over the structured rTMS
data were computed and published in six parts, i.e. 1, 2, 3, 4, 5, & 6, with
$\sim$100 studies compared in each part.

###  3.1 rTMS-KG - Findable, Accessible, Interoperable, and Reusable

rTMS-KG is an open-source machine-actionable knowledge graph of 600 rTMS doses
that satisfies the FAIR publishing guidelines [37]. Each published comparison
of studies in the rTMS-KG can be persistently identified with a Digital Object
Identifier (DOI) (‘Findable’). This resolves to a landing page, providing
access to metadata, respective rTMS study data, and versioning, all of which
is indexed and searchable (‘Findable’, ‘Accessible’, and ‘Reusable’). rTMS-KG
can be downloaded over the Web from the ORKG in RDF (‘Interoperable’). The
rTMS vocabulary can be mapped to existing relevant ontologies via same-as
links (‘Interoperable’). Finally, the ORKG provides public search interfaces
to build custom views of rTMS-KG (‘Accessible’).

##  4 Conclusion

Standardization of experimental protocols and data reporting is pivotal for
the advancement of the rTMS field in particular, and NIBS more broadly.
Adopting consensus reporting guidelines and protocols would foster uniformity
and comparability across studies, thereby facilitating a more precise
evaluation of NIBS dose efficacy. Transparent reporting practices, coupled
with data sharing, can bolster reproducibility, reduce publication bias, and
encourage collaboration. This enhancement can be directly achieved through the
semantic web publishing of scholarly knowledge, as exemplified by the ORKG. By
defining a standardized vocabulary of properties and resources, and
documenting upcoming studies within this predefined semantic framework, the
issues of transparency and reproducibility in NIBS studies are addressed. As a
result, the NIBS field can continue its trajectory of progress, delivering
effective, evidence-based neuromodulation interventions for neurological and
psychiatric disorders [38].

This study, for the first time, showcases the practical application of the
semantic web resource-based reporting methodology for NIBS studies at large,
and rTMS studies in particular. It offers transformative value by: (1)
promoting standardization in the recording of NIBS enhances transparency
through the explicit logging of key study properties, which in turn bolsters
reproducibility, and (2) promoting the adoption of such machine-actionable
documentation for NIBS studies, as evidenced by the demonstrated rTMS subarea
use-case.

## Acknowledgements

We thank the anonymous reviewers for their detailed and insightful comments on
an earlier draft of the paper. This work was jointly supported by the German
BMBF project SCINEXT (ID 01lS22070), DFG NFDI4DataScience (ID 460234259), and
ERC ScienceGraph (ID 819536).

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# Isotropic cosmic birefringence from early dark energy

Kai Murai  ICRR, The University of Tokyo, Kashiwa, 277-8582, Japan  Kavli IPMU
(WPI), UTIAS, The University of Tokyo, Kashiwa, 277-8583, Japan  Fumihiro
Naokawa  Department of Physics, Graduate School of Science, The University of
Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  Research Center for the Early
Universe, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  Toshiya
Namikawa  Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, 277-8583,
Japan  Eiichiro Komatsu  Max Planck Institute for Astrophysics, Karl-
Schwarzschild-Str. 1, 85748 Garching, Germany  Kavli IPMU (WPI), UTIAS, The
University of Tokyo, Kashiwa, 277-8583, Japan

###### Abstract

A tantalizing hint of isotropic cosmic birefringence has been found in the
$EB$ cross-power spectrum of the cosmic microwave background (CMB)
polarization data with a statistical significance of $3\sigma$. A pseudoscalar
field coupled to the CMB photons via the Chern-Simons term can explain this
observation. The same field may also be responsible for early dark energy
(EDE), which alleviates the so-called Hubble tension. Since the EDE field
evolves significantly during the recombination epoch, the conventional formula
that relates $EB$ to the difference between the $E$\- and $B$-mode auto-power
spectra is no longer valid. Solving the Boltzmann equation for polarized
photons and the dynamics of the EDE field consistently, we find that currently
favored parameter space of the EDE model yields a variety of shapes of the
$EB$ spectrum, which can be tested by CMB experiments.

##  I Introduction

Observations and interpretation of the cosmic microwave background (CMB)
temperature and polarization anisotropies have established the standard
cosmological model, the $\Lambda$ Cold Dark Matter ($\Lambda$CDM) model [1, 2,
3, 4, 5, 6, 7]. There are, however, several observational hints for new
physics beyond the $\Lambda$CDM model and the standard model of elementary
particles and fields [8]. In this paper we connect two such hints: the so-
called “cosmic birefringence” [9] and “Hubble tension” [10].

Cosmic birefringence is a rotation of the plane of linear polarization of
photons [11, 12, 13], which induces the cross-correlation between parity-even
$E$ and parity-odd $B$ modes in the CMB polarization field [14]. A tantalizing
hint of cosmic birefringence has been found by the recent analyses of the WMAP
and Planck data [15, 16, 17, 18]. The reported signal is isotropic and
consistent with a rotation angle of
$\beta=0.342^{\circ\,+0.094^{\circ}}_{\phantom{\circ\,}-0.091^{\circ}}$ (68%
C.L.), independent of the photon frequency [18].

A pseudoscalar “axionlike” field can explain the observed isotropic cosmic
birefringence signal [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33]. In general, a pseudoscalar field can couple to photons via the Chern-
Simons term. The known example in the standard model of elementary particles
and fields is a neutral pion [34]. The temporal and spatial difference in the
pseudoscalar field values induces cosmic birefringence independent of the
photon frequency. The evolution of a homogeneous pseudoscalar field induces
the isotropic signal.

The Hubble tension refers to the discrepancy between the local and early-
universe measurements of the Hubble constant, $H_{0}$ [10, 35]. Among the
proposed models to alleviate this tension (see Refs. [36, 37] for reviews) are
early dark energy (EDE) models [38, 39, 40, 41, 42], which utilize an
additional energy component to modify the value of $H_{0}$ inferred from CMB
observations. The EDE field behaves like dark energy in the early universe and
starts to oscillate in the pre-recombination epoch. In this paper we focus on
an EDE model with a potential of $V(\phi)\propto[1-\cos(\phi/f)]^{n}$ with
$n>1$ [43, 40]. Such a potential may be generated for an axionlike field by
higher-order instanton corrections [44].

Several authors estimated the EDE parameters such as the maximum energy-
density fraction of EDE, $f_{\mathrm{EDE}}$, from the current cosmological
datasets and discussed the viability of EDE [42, 45, 46, 47, 48, 49, 50, 51].
The authors of Ref. [52] analyzed the EDE model with $n=3$ using a profile
likelihood to avoid the volume effect on marginalization, and obtained
$f_{\mathrm{EDE}}=0.072\pm 0.036$ at 68% C.L.

In this paper we consider the case where the EDE field has a Chern-Simons
coupling to photons. Whereas the previous study [53] was focused on
fluctuations in cosmic birefringence from EDE, we focus on isotropic cosmic
birefringence. To this end we extend the linear Boltzmann solver developed in
Ref. [54] and evaluate the $EB$ power spectrum for fixed values of
$f_{\mathrm{EDE}}$ with the best-fitting parameters of the EDE model given in
Ref. [52].

The rest of this paper is organized as follows. We review the physics of
cosmic birefringence induced by an axionlike field in Sec. II. We explain the
EDE model in Sec. III and show the $EB$ power spectra in Sec. IV. We summarize
our findings and conclude in Sec. V.

##  II Isotropic cosmic birefringence

We consider an axionlike field, $\phi$, coupled to photons through the Chern-
Simons term. The Lagrangian density is written as

$\displaystyle\mathcal{L}=-\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}g\phi
F_{\mu\nu}\tilde{F}^{\mu\nu}\,,$ (1)

where $V(\phi)$ is the potential which is specified in Sec. III, $F_{\mu\nu}$
is the field strength tensor of the photon field, $\tilde{F}^{\mu\nu}$ is its
dual, and $g$ is the Chern-Simons coupling constant of mass dimension $-1$.

When $\phi$ varies slowly compared to the photon frequency, the dispersion
relations of photons are modified as [11, 12, 13]

$\displaystyle\omega_{\pm}\simeq
k\mp\frac{g}{2}\left(\frac{\partial\phi}{\partial
t}+\frac{\bm{k}}{k}\cdot\bm{\nabla}\phi\right)=k\mp\frac{g}{2}\frac{\mathrm{d}\phi}{\mathrm{d}t}\,,$
(2)

where $\mathrm{d}/\mathrm{d}t$ denotes a total derivative along the photon
trajectory, and $+$ and $-$ correspond to the right- and left-handed circular
polarization states of photons, respectively, in right-handed coordinates with
the $z$ axis taken in the direction of propagation of photons.

In this paper we focus on isotropic cosmic birefringence and ignore its
anisotropy, which is not found yet [55, 56, 57, 58]. See Refs. [59, 60, 61,
62, 63, 64, 65, 66, 67] for study on anisotropic cosmic birefringence.

The helicity-dependent dispersion relation given in Eq. (2) induces a rotation
of the plane of linear polarization. The rotation angle from a given time $t$
to the present time $t_{0}$ is given by [11, 12, 13]

$\displaystyle\beta(t)=-\frac{1}{2}\int_{t}^{t_{0}}\mathrm{d}\tilde{t}\,(\omega_{+}-\omega_{-})=\frac{g}{2}\left[\phi(t_{0})-\phi(t)\right]\,.$
(3)

Here, we use the CMB convention for the position angle of linear polarization,
i.e., $\beta>0$ represents a clockwise rotation of linear polarization in the
sky. The rotation angle depends only on the difference in the axionlike field
values between the emission and observation and does not depend on the
evolution history [13].

Next, we discuss the effect of cosmic birefringence on the CMB polarization
field (see Ref. [9] for a review). In this paper we focus on scalar-mode
perturbations and ignore tensor modes. The evolution of the CMB polarization
field is described by the Boltzmann equation [68]. We define the Fourier
transform of the Stokes parameters of linear polarization, $Q\pm iU$, by
${}_{\pm 2}\Delta_{P}(\eta,q,\mu)$. Here, $\eta$ is the conformal time,
$\bm{q}$ is a wave vector in the Fourier space, and
$\mu\equiv\bm{q}\cdot\bm{k}/(qk)$ parameterizes the angle between $\bm{q}$ and
the photon momentum $\bm{k}$. The Boltzmann equation for ${}_{\pm
2}\Delta_{P}(\eta,q,\mu)$ is given by [20, 69, 70]

${}_{\pm 2}\Delta_{P}^{\prime}+iq\mu\,_{\pm 2}\Delta_{P}=$
$\displaystyle\tau^{\prime}\left[-_{\pm
2}\Delta_{P}+\sqrt{\frac{6\pi}{5}}\,_{\pm 2}Y_{2}^{0}(\mu)\Pi(\eta,q)\right]$
$\displaystyle\pm 2i\beta^{\prime}\,_{\pm 2}\Delta_{P}\,,$ (4)

where ${}_{\pm 2}Y_{l}^{m}$ is the spin-2 spherical harmonics, $\Pi(\eta,q)$
is the polarization source term [71], and $\tau^{\prime}\equiv
a(\eta)n_{e}(\eta)\sigma_{T}$ is the differential optical depth with the
Thomson scattering cross section $\sigma_{T}$ and the number density of
electrons $n_{e}$. Here, the prime denotes the derivative with respect to
$\eta$. It is convenient to expand ${}_{\pm 2}\Delta_{P}$ with the spin-2
spherical harmonics as

${}_{\pm 2}\Delta_{P}(\eta,q,\mu)\equiv\sum_{l}i^{-l}\sqrt{4\pi(2l+1)}\,_{\pm
2}\Delta_{P,l}(\eta,q)\,_{\pm 2}Y_{l}^{0}(\mu)\,.$ (5)

Then, we obtain the formal solution to the Boltzmann equation as

${}_{\pm
2}\Delta_{P,l}(\eta_{0},q)=-\frac{3}{4}\sqrt{\frac{(l+2)!}{(l-2)!}}\int_{0}^{\eta_{0}}\mathrm{d}\eta\,$
$\displaystyle\tau^{\prime}e^{-\tau(\eta)}\Pi(\eta,q)$ $\displaystyle\times$
$\displaystyle\frac{j_{l}(x)}{x^{2}}e^{\pm 2i\beta(\eta)}\,,$ (6)

where $j_{l}(x)$ is the spherical Bessel function with $x\equiv
q(\eta_{0}-\eta)$ and
$\tau(\eta)\equiv\int^{\eta_{0}}_{\eta}\mathrm{d}\eta_{1}\,\tau^{\prime}(\eta_{1})$.

To discuss the parity violation, we work with parity eigenstates of the CMB
polarization, $E$ and $B$ modes [71, 72]. In this basis, parity violation is
imprinted in the $EB$ correlation [14]. We write the coefficients of $E$ and
$B$ modes as

$\displaystyle\Delta_{E,l}(q)\pm i\Delta_{B,l}(q)\equiv-\,_{\pm
2}\Delta_{P,l}(\eta_{0},q)\,.$ (7)

Cosmic birefringence, $\beta\neq 0$, induces the imaginary part of ${}_{\pm
2}\Delta_{P,l}$, or equivalently $B$ modes. Using these coefficients, we
obtain the polarization power spectra as

$\displaystyle C_{l}^{XY}=4\pi\int\mathrm{d}(\ln
q)\,\mathcal{P}_{s}(q)\Delta_{X,l}(q)\Delta_{Y,l}(q)\,,$ (8)

where $\mathcal{P}_{s}(q)$ is the primordial scalar curvature power spectrum,
and $X,Y=E$ or $B$.

If $\beta$ is independent of $\eta$, the coefficients of $E$ and $B$ modes are
given by

$\displaystyle\Delta_{E,l}\pm i\Delta_{B,l}=e^{\pm
2i\beta}\left(\tilde{\Delta}_{E,l}\pm i\tilde{\Delta}_{B,l}\right)\,,$ (9)

where the tildes denote quantities without cosmic birefringence. Thus, cosmic
birefringence transforms the polarization power spectra as [69, 73]

$\displaystyle C_{l}^{EE}$
$\displaystyle=\cos^{2}(2\beta)\tilde{C}_{l}^{EE}+\sin^{2}(2\beta)\tilde{C}_{l}^{BB}\,,$
(10) $\displaystyle C_{l}^{BB}$
$\displaystyle=\cos^{2}(2\beta)\tilde{C}_{l}^{BB}+\sin^{2}(2\beta)\tilde{C}_{l}^{EE}\,,$
(11) $\displaystyle C_{l}^{EB}$
$\displaystyle=\frac{1}{2}\sin(4\beta)\left(\tilde{C}_{l}^{EE}-\tilde{C}_{l}^{BB}\right)\,,$
(12)

which can be combined to give
$C_{l}^{EB}=\frac{1}{2}\tan(4\beta)(C_{l}^{EE}-C_{l}^{BB})$ [74]. When the
primordial $B$ modes are negligible compared to the $E$ modes, we obtain
$C_{l}^{EB}\simeq\tan(2\beta)C_{l}^{EE}$.

In general, $\beta$ depends on $\eta$ through the evolution of $\phi$. Roughly
speaking, there are two main contributions to the CMB polarization power
spectra. One is the reionization epoch contributing to low multipoles,
$l\lesssim 10$ [75], and the other is the recombination epoch contributing to
higher $l$ [76]. Thus, the $EB$ power spectrum at low and high $l$ is largely
determined by the evolution of $\phi$ after the reionization and recombination
epochs, respectively [69, 77, 78, 54].

If $\phi$ starts to oscillate after the reionization epoch, $\beta$ is
approximately constant and we obtain $C_{l}^{EB}\propto C_{l}^{EE}$. If $\phi$
starts to oscillate before the reionization epoch, the reionization bump in
the $EB$ power spectrum at low $l$ is suppressed [78, 54]. If $\phi$
oscillates during the recombination epoch, the shape of the $EB$ power
spectrum can be different from that of the $EE$ power spectrum at high $l$
[20, 54], which is relevant to our study. The information in the shape of the
$EB$ power spectrum thus allows for “tomographic approach,” with which we can
constrain the mechanism inducing cosmic birefringence or axion parameters [78,
54].

##  III Early dark energy

EDE is an additional energy component that behaves like dark energy at early
times and contributes to the energy density budget around the recombination
epoch [38]. This contribution increases the Hubble expansion rate around the
recombination epoch, which, in turn, reduces the size of the sound horizon
[10]. As a result, EDE increases the inferred value of $H_{0}$ from CMB
observations and alleviates the Hubble tension.

The EDE energy density should decay faster than the matter density
contribution after recombination, so as not to affect the late-time cosmic
expansion. The EDE models that satisfy this property are based on a
pseudoscalar field potential given by [40]

$\displaystyle
V(\phi)=m^{2}f^{2}\left[1-\cos\left(\frac{\phi}{f}\right)\right]^{n}\,,$ (13)

where $f$ is the breaking scale of the symmetry related to $\phi$ and $m$
determines the potential scale. Around $\phi=0$, this potential is
approximated by $V(\phi)\propto\phi^{2n}$. Thus, once $\phi$ starts to
oscillate, its energy density decreases as $\rho_{\phi}\propto a^{-6n/(n+1)}$
[79], which means that $\rho_{\phi}$ decreases faster than the matter density
for $n>1$. In the following, we choose $n=3$ as in the previous work [52].

EDE is usually characterized by three parameters,
$(f_{\mathrm{EDE}},z_{c},\theta_{i})$, where $f_{\mathrm{EDE}}$ is the maximum
energy density fraction of EDE at the critical redshift $z_{c}$, and
$\theta_{i}$ is the initial value of $\theta\equiv\phi/f$. The previous work
[52] has performed a profile likelihood analysis to avoid volume effects in
the Markov chain Monte Carlo analysis and derived a robust constraint of
$f_{\mathrm{EDE}}=0.072\pm 0.036$ at 68% C.L. They have also obtained the
best-fitting values of the 6 parameters of the base-$\Lambda$CDM model and
$(z_{c},\theta_{i})$ for fixed values of $f_{\mathrm{EDE}}$. In this paper we
adopt their best-fitting values for $f_{\mathrm{EDE}}=0.01$, $0.07$, and
$0.14$, which are summarized in Table 1.

Table 1: Best-fitting values of the base-$\Lambda$CDM and EDE model parameters for $f_{\mathrm{EDE}}=0.01,0.07$, and $0.14$.  $f_{\mathrm{EDE}}$ | 0.01 | 0.07 | 0.14
---|---|---|---
100 $\omega_{b}$ | 2.248 | 2.259 | 2.276
$\omega_{\mathrm{cdm}}$ | 0.1200 | 0.1260 | 0.1341
$100\,\theta_{s}$ | 1.042 | 1.042 | 1.041
$\ln(10^{10}A_{s})$ | 3.046 | 3.056 | 3.070
$n_{s}$ | 0.9704 | 0.9794 | 0.9927
$\tau_{\mathrm{reio}}$ | 0.05492 | 0.05492 | 0.05534
$\log_{10}z_{c}$ | 3.551 | 3.548 | 3.560
$|\theta_{i}|$ | 2.752 | 2.758 | 2.769

We show the time evolution of $\phi$ in Fig. 1. The orange and blue regions
represent the recombination and reionization epochs, respectively.

Figure 1:  Time evolution of $\phi$ for the best-fitting parameters with
$f_{\mathrm{EDE}}=0.01,0.07$, and $0.14$ given in Table 1. The dashed lines
represent $-\phi$. The orange and blue regions represent the recombination and
reionization epochs, respectively.

If the EDE has a Chern-Simons coupling to photons, the time evolution of
$\phi$ induces isotropic cosmic birefringence. The difference in the EDE
parameters affects the amplitude and phase of oscillations. While the
amplitude is proportional to $\beta$ (that is, the amplitude of $C_{l}^{EB}$),
the phase affects the shape of $C_{l}^{EB}$.

##  IV Cosmic birefringence from EDE

The authors of Ref. [54] included the dynamics of $\phi$ in the Boltzmann
equation, assuming that the energy density of $\phi$ is negligible in the
Friedmann equation and ignoring the gravitational lensing effect of the CMB.
In this paper we account for both effects.

Specifically, we modify the publicly-available CLASS_EDE code [45], which is
based on the linear Boltzmann solver CLASS [80, 81] and solves the equation of
motion for $\phi$ with the potential given in Eq. (13), to include
$\beta(\eta)$ in the line-of-sight integral solution [Eq. (6)] and output the
$EB$ power spectrum using Eq. (8). We include the lensing effect as explained
in Ref. [82].

In Fig. 2 we show the $EB$ power spectrum induced by the EDE field shown in
Fig. 1.

Figure 2:  $EB$ power spectrum in the EDE model for the best-fitting
parameters given in Table 1 and $g=M_{\mathrm{Pl}}^{-1}$. The colored lines
represent the $EB$ power spectra for $f_{\mathrm{EDE}}=0.01$, $0.07$, and
$0.14$, whereas the black line represents the $EE$ power spectrum multiplied
by a constant factor for comparison. The solid and dashed lines show positive
and negative values, respectively.

Here, we fix $g=M_{\mathrm{Pl}}^{-1}$ by the reduced Planck constant
$M_{\mathrm{Pl}}\equiv(8\pi G)^{-1/2}$. One can rescale our results for
arbitrary values of $g$ by
$C_{l}^{EB}=gM_{\mathrm{Pl}}C_{l}^{EB}(g=M_{\mathrm{Pl}}^{-1})$. As the sign
of $C_{l}^{EB}$ depends on those of $g$ and $\theta_{i}$, we assume $g>0$ and
$\theta_{i}<0$ so that $C_{l}^{EB}>0$ at $l\gtrsim 500$. For comparison, the
black line represents $C_{l}^{EE}$ multiplied by a constant.

We find that $C_{l}^{EB}$ has a wide variety of shapes, which are different
from $C_{l}^{EE}$ commonly assumed in the data analysis [15, 16, 18, 17]. One
can understand the shape of $C_{l}^{EB}$ from the time evolution of $\phi$.
Since the CMB photons contribute to the power spectrum in a different way
depending on the time of last scattering, the time-dependent $\beta$ leads to
a modulation of the shape of $C_{l}^{EB}$ with respect to that of $C_{l}^{EE}$
[54]. In our case, $\phi$ flips the sign during the recombination epoch as
seen in Fig. 1, and thus $C_{l}^{EB}$ can also flip the sign at $10\lesssim
l\lesssim 500$. This effect also shifts the peak positions of $C_{l}^{EB}$ at
high $l$ as seen in Fig. 3. Since the positive contribution to $C_{l}^{EB}$
comes from the earlier stage of the recombination epoch, the peak shifts to
higher $l$.

Figure 3:  $EB$ power spectrum in the EDE model at $800\leq l\leq 1200$. The
lines are the same as in Fig. 2.

We can also understand the diverse behaviors of $C_{l}^{EB}$ at $l\lesssim 10$
from the variation of the oscillation phases during the reionization epoch.

Let us relate our $C_{l}^{EB}$ to the observed value of $\beta$. The
observational determination of $\beta$ typically assumes a constant rotation
angle multiplying $C_{l}^{EE}-C_{l}^{BB}$ [15, 16, 18, 17]. Although our
$C_{l}^{EB}$ is not proportional to $C_{l}^{EE}-C_{l}^{BB}$, one may find an
approximate proportionality factor at $l\gtrsim 500$ and translate our
$C_{l}^{EB}$ into $\beta$ by comparing the maximum values of
$C_{l}^{EE}-C_{l}^{BB}$ and $C_{l}^{EB}$. Considering Eq. (12), we define the
effective value of $\beta$, $\beta_{\mathrm{eff}}$, by

$\displaystyle\mathrm{max}\left[C_{l}^{EB}\right]\equiv\frac{1}{2}\tan(4\beta_{\mathrm{eff}})\,\mathrm{max}\left[C_{l}^{EE}-C_{l}^{BB}\right].$
(14)

By requiring $\beta_{\mathrm{eff}}=0.34^{\circ}$, we estimate $g$ as

$\displaystyle\frac{g}{M_{\mathrm{Pl}}^{-1}}=\left(1.2,0.42,0.47\right)\,,$
(15)

for $f_{\mathrm{EDE}}=0.01$, $0.07$, and $0.14$, respectively. Since the decay
constant, $f$, is typically of $\mathcal{O}(0.1M_{\mathrm{Pl}})$ for the best-
fitting parameters, this estimation implies
$g=\mathcal{O}(0.01\,\text{--}\,0.1)f^{-1}$.

Finally, we forecast the testability of the EDE model by measuring the $EB$
power spectrum in future CMB experiments. We consider to fit the $EB$ power
spectrum predicted by the EDE model to that measured by a future CMB
experiment. We assume that the fiducial cosmic birefringence signal is
described by a constant rotation of $\beta=0.34^{\circ}$. Here, we consider
two experiments; the Simons Observatory (SO) [83] and CMB-S4 [84]. We choose
the same experimental setup for CMB-S4 as described in Ref. [54], while for SO
we use the noise curves provided by the SO
collaboration111https://github.com/simonsobs/so_noise_models. For the current
best-fitting EDE model with $f_{\rm EDE}=0.07$, we minimize the chi-squared by
varying $g$. We find that the $p$-value of the minimum chi-squared is close to
unity for SO, implying that we cannot distinguish the EDE model from a
constant rotation. For CMB-S4, however, the $p$-value becomes close to $0$,
and the EDE model will be distinguished from a constant rotation.

##  V Summary

Polarization of the CMB is a powerful probe of new physics beyond the standard
model of elementary particles and fields [9]. In this paper we connected two
hints of new physics from the current cosmological datasets: isotropic cosmic
birefringence and EDE. We included the dynamics of the EDE field, $\phi$, and
its coupling to photons in the Boltzmann equation for the CMB polarization and
evaluated the $EB$ power spectrum, $C_{l}^{EB}$, induced by isotropic cosmic
birefringence. The shape of $C_{l}^{EB}$ (Fig. 2) is different from that of
$C_{l}^{EE}$ and depends sensitively on the EDE parameters given in Table 1.
We understood this result in terms of the time evolution of $\phi$ shown in
Fig. 1.

We roughly translated the obtained $C_{l}^{EB}$ into an effective rotation
angle, $\beta_{\mathrm{eff}}$. By requiring that $\beta_{\mathrm{eff}}$ is
equal to the observed rotation angle, we found
$g=\mathcal{O}(0.01\,\text{--}\,0.1)f^{-1}$. We also discussed the testability
of the EDE model and showed that the cosmic birefringence predicted by the EDE
model can be distinguished from that with a constant rotation angle by
measuring $C_{l}^{EB}$ in future CMB experiments such as CMB-S4.

Our analysis can also be applied to other EDE models. Since the shape of
$C_{l}^{EB}$ is sensitive to the oscillation phase of the EDE field during the
recombination and reionization epoch, the shape of $C_{l}^{EB}$ can be
different in other models. Although we have focused on the effect of isotropic
cosmic birefringence on $C_{l}^{EB}$, the perturbations of the EDE field will
induce the anisotropic cosmic birefringence [53], which may provide a distinct
signature in $C_{l}^{EB}$. We leave this to future work.

###### Acknowledgements.

We thank E. G. M. Ferreira and L. Herold for their help with the EDE
parameters and CLASS_EDE code, and the participants of the workshop
YITP-T-21-08 on “Upcoming CMB observations and Cosmology” for useful
discussion. This work was supported in part by the Program of Excellence in
Photon Science (K.M.), the Forefront Physics and Mathematics Program to Drive
Transformation (FoPM), a World-leading Innovative Graduate Study (WINGS)
Program, the University of Tokyo (F.N.), and JSPS KAKENHI Grant No. JP20J20248
(K.M.), No. JP20H05850 (E.K.), No. JP20H05859 (E.K. and T.N.), and No.
JP22K03682 (T.N.). Part of this work uses resources of the National Energy
Research Scientific Computing Center (NERSC). The Kavli IPMU is supported by
World Premier International Research Center Initiative (WPI Initiative), MEXT,
Japan.

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