# CardioLab: Laboratory Values Estimation from Electrocardiogram Features - An
Exploratory Study

Juan Miguel Lopez Alcaraz1 Nils Strodthoff1

1 AI4Health Division  Carl von Ossietzky Universität Oldenburg  Oldenburg
Germany

###### Abstract

Introduction: Laboratory value represents a cornerstone of medical
diagnostics, but suffers from slow turnaround times, and high costs and only
provides information about a single point in time. The continuous estimation
of laboratory values from non-invasive data such as electrocardiogram (ECG)
would therefore mark a significant frontier in healthcare monitoring. Despite
its transformative potential, this domain remains relatively underexplored
within the medical community. Methods: In this preliminary study, we used a
publicly available dataset (MIMIC-IV-ECG) to investigate the feasibility of
inferring laboratory values from ECG features and patient demographics using
tree-based models (XGBoost). We define the prediction task as a binary
prediction problem of predicting whether the lab value falls into low or high
abnormalities. The model performance can then be assessed using AUROC.
Results: Our findings demonstrate promising results in the estimation of
laboratory values related to different organ systems based on a small yet
comprehensive set of features. While further research and validation are
warranted to fully assess the clinical utility and generalizability of ECG-
based estimation in healthcare monitoring, our findings lay the groundwork for
future investigations into approaches to laboratory value estimation using ECG
data. Such advancements hold promise for revolutionizing predictive healthcare
applications, offering faster, non-invasive, and more affordable means of
patient monitoring.

##  1 Introduction

Abnormal laboratory values play a crucial role in clinical settings as they
indicate underlying health conditions, sometimes posing severe and potentially
life-threatening risks, especially in cases of serious illnesses [1, 2]. These
abnormal values significantly increase the risk of mortality compared to
normal ranges [3], highlighting the need for vigilant monitoring and effective
management strategies. Continuous monitoring of patients often involves
inherent limitations such as invasive procedures like frequent blood sampling,
which are resource-intensive [4] and suffer from delays between sampling and
obtaining results [5] or potentially missing staffing of laboratory medicine
during the night, thereby limiting real-time monitoring capabilities.

While previous research has explored the correlation between abnormal
laboratory values and changes in electrocardiogram (ECG) readings [6, 7], the
full potential of utilizing ECG data for accurately estimating these
abnormalities remains largely untapped. Therefore, in this work, we propose a
first exploratory study in which we investigate the feasibility of estimating
laboratory values from ECG features with the addition of other data modalities
such as patient demographics.

##  2 Background

Significance of abnormal laboratory values In the United States, annual costs
for preoperative laboratory testing alone amount to $18 billion [4]. In
intensive care units (ICUs), these costs can rise to $14 billion, constituting
up to 25% of total ICU expenditures [8]. Identifying abnormal lab values is
crucial for patient care. In elderly surgical patients, preoperative tests
often show abnormal creatinine (12%), hemoglobin (10%), and glucose (7%)
levels [3]. In ICUs, these tests are essential for adjusting drug dosages and
procedures for critically ill patients, playing a key role in treatment
strategies [5].

Monitoring and management strategies Current guidelines for monitoring
patients with abnormal lab values recommend instruments and methods for early
detection, continuous monitoring, and effective interventions. They include
organ-specific tests, like liver function [9], and are adapted for settings
such as the emergency department (ED) [10]. While traditional phlebotomy is
common, non-invasive alternatives are emerging for biomarkers like glucose
[11] and hemoglobin [12].

Integration of ECG data for values prediction The use of ECG data for
laboratory values estimation has been briefly investigated, where previous
works demonstrated predictive capabilities in glucose [13], serum potassium
[14], and electrolyte imbalances [15, 16]. Lastly, other predictive modalities
such as vital signs from wearable sensors were investigated to predict
laboratory values [17].

##  3 Methods

Dataset The dataset comprises data sourced from the publicly available MIMIC-
IV [18, 19] and MIMIC-IV-ECG [20, 19] datasets. It includes a comprehensive
set of non-invasive features: demographics such as gender, age, and race
(Caucasian, African, Asian, Latino, or other); ECG features including RR
interval, P onset, P end, QRS onset, QRS end, and T end (all in milliseconds),
as well as P axis, QRS axis, and T axis (in degrees). In terms of target
abnormalities, we work on binary classification cases where we define a
positive case when the considered laboratory value is lower or higher than the
median low or high threshold values (provided as part of MIMIC-IV) across all
samples of the same laboratory value. For dataset splitting, we follow the
stratification based on patients, age, and gender from the previously proposed
MIMIC-IV-ECG-ICD [21] splits with a train, validation, and test ratio of
18:1:1 which ensures balanced diagnoses in each of the folds. Finally, for
this work, we consider final laboratory value cases where we obtain at least
10 positive and 10 negative cases per fold.

Models and performance evaluation We fit and train individual extreme gradient
boosting (XGBoost) tree models with a max depth of 1 per laboratory value in a
binary classification setting. We evaluate performances on the macro average
across all areas under the respective receiver operating curves (AUROC) (macro
AUROC), where to assess statistical uncertainty resulting from the finite size
and specific composition of the test set, we employ empirical bootstrap on the
test set with $n=1000$ iterations and report 95% confidence intervals.

##  4 Results

Table 1:  Performance results for laboratory values with AUROC > 0.70, presented by value name, abnormality setting, threshold defining abnormality by the number of units, (units), number of samples, class prevalence, and AUROC with 95% confidence intervals. All of these values represent blood fluid. Value | Threshold | Unit | Samples [Prev.] | AUROC (95% CI)
---|---|---|---|---
Urea Nitrogen | <6.0 | mg/dL | 240892 [2139] | 0.856 (0.855, 0.857)
NTproBNP | >353.0 | pg/mL | 23206 [18035] | 0.848 (0.848, 0.849)
Hematocrit | >51.0 | % | 238567 [1381] | 0.775 (0.773, 0.777)
Cortisol | >20.0 | ug/dL | 982 [486] | 0.771 (0.772, 0.781)
Osmolality | <275.0 | mOsm/kg | 2784 [730] | 0.765 (0.763, 0.768)
Hemoglobin | >17.5 | g/dL | 236802 [802] | 0.754 (0.753, 0.759)
Creatinine | <0.5 | mg/dL | 241968 [2499] | 0.742 (0.741, 0.744)
Urea Nitrogen | >20.0 | mg/dL | 240892 [100216] | 0.736 (0.736, 0.736)
Creatinine | >1.2 | mg/dL | 241968 [69012] | 0.735 (0.735, 0.736)
PT | <9.4 | sec | 137917 [380] | 0.732 (0.728, 0.734)
C-Reactive Protein | >5.0 | mg/L | 3520 [2107] | 0.730 (0.728, 0.734)
Anion Gap | >20.0 | mEq/L | 230666 [18299] | 0.726 (0.725, 0.727)
Cholesterol, HDL | <41.0 | mg/dL | 9023 [2103] | 0.725 (0.725, 0.729)
Lymphocytes | >42.0 | % | 171441 [9591] | 0.722 (0.721, 0.723)
RDW-SD | >46.3 | fL | 73001 [35887] | 0.720 (0.719, 0.721)
Bands | >5.0 | % | 16197 [2587] | 0.719 (0.718, 0.721)
Bicarbonate, Whole Blood | <21.0 | mEq/L | 8039 [1720] | 0.715 (0.712, 0.717)
Ferritin | <13.0 | ng/mL | 4643 [438] | 0.714 (0.714, 0.722)
Lymphocytes | <18.0 | % | 171441 [73323] | 0.713 (0.712, 0.713)
Monocytes | >11.0 | % | 9898/171442 | 0.713 (0.712, 0.714)
Bilirubin, Direct | > 0.3 | mg/dL | 3438 [1968] | 0.708 (0.704, 0.710)
Albumin | <3.5 | g/dL | 57040 [12834] | 0.708 (0.707, 0.708)
Sodium | >145.0 | mEq/L | 235578 [5361] | 0.706 (0.705, 0.707)
eAG | >123.0 | mg/dL | 9485 [4916] | 0.704 (0.702, 0.705)
Ferritin | >150.0 | ng/mL | 4643 [2013] | 0.704 (0.701, 0.705)
% Hemoglobin A1c | >5.9 | % | 10665 [5651] | 0.700 (0.699, 0.701)

Overall Predictive Performance Table 1 contains the performance results in
terms of AUROC for individual laboratory value abnormalities above 0.70. This
includes the threshold for their label definition, units, number of samples,
and prevalence. Overall, low urea nitrogen and NTproBNP, achieve an AUROC
above 0.80. The table lists 26 laboratory values with AUROC greater than 0.70,
indicating strong predictive performance. These values span various
physiological systems, demonstrating the model’s capability to predict
abnormalities across diverse domains beyond cardiovascular (NTproBNP,
hematocrit, hemoglobin), coagulation (PT), and electrolyte balance
(osmolality, anion gap, sodium). Notably, the model also encompasses renal
(urea nitrogen, creatinine), endocrine (cortisol, eAG), immune (bands,
monocytes, lymphocytes), inflammatory response (C-Reactive protein), metabolic
(cholesterol, bicarbonate), and respiratory (pO2) systems. This broad
applicability underscores the model’s extensive utility in clinical prediction
across a wide range of physiological parameters.

Low urea nitrogen use-case: predictive performance for different patient
demographics By gender, males perform better than females despite having a
smaller prevalence rate. By race, the groups of Caucasians and others perform
above the overall population mean, Caucasians however, with the largest
prevalence, Asians have the lowest predictive performance, however, also the
ones with the fewest samples, to make a fair comparison, we down-sampled the
Caucasian training population to Asians population, and conclude that indeed
Caucasian perform better than (at least) Asians, even with the same number of
samples. By age, the 65 to 77 years age group is the most predictive across
all groups despite having a small prevalence.

Low urea nitrogen use-case: interpretability XGBoost enables explainable
methods like Shapley additive explanations (SHAP) [22]. Using SHAP, we
identify ECG features contributing most to abnormality predictions. Literature
indicates hemodialysis patients with reduced urea nitrogen levels often show
ECG abnormalities [23]. Our findings suggest: (1) High T end values (long T
wave) correlate with prolonged repolarization abnormalities, potentially
linked to observed ST segment changes (depression or elevation) and inverted T
waves in these patients. (2) Low T axis values correlate with T wave
inversions and other repolarization abnormalities. (3) High QRS axis values
indicate a shift in ventricular depolarization’s mean electrical axis,
possibly associated with ventricular hypertrophy or conduction abnormalities
in hemodialysis patients. Furthermore, high NTproBNP values, our second-best
performer, align with literature associating abnormal ECG features in a
specific cohort [24].

##  5 Discussion

Clinical significance This work enhances ECG predictive capabilities beyond
traditional applications, laying the groundwork for comprehensive patient
monitoring as in [25] for diagnostics. By analyzing ECG data alongside patient
demographics, we predict abnormalities in crucial laboratory markers such as
electrolyte levels (e.g., Anion gap, Sodium, Bicarbonate) and renal function
indicators like urea nitrogen and creatinine. This predictive approach
facilitates early detection of conditions such as arrhythmias due to
electrolyte imbalances or kidney dysfunction, enabling proactive interventions
that can optimize patient care by providing faster and low-cost results [26].

Limitations We identify several limitations in our current work, firstly, for
clinical consideration, one might need to externally validate these models on
a separate population cohort. Secondly, we aim to estimate the closest
absolute value within a horizon threshold, however, fixed intervals between
features and targets for all samples might mitigate variance. Lastly, A more
fine-grained setting can include drug-specific pharmacokinetic information to
account for drug effects in the target space. However, the complexity of it
would increase significantly as half-life, and mechanisms of action and
elimination are drug-specific.

Future work While this work investigates the predictive performance of
laboratory values by discriminating ECG features within a stratified
population mostly presenting diagnoses [21], it is also important to
understand how changes in ECG features contribute to predictions in a
population of only healthy individuals [27]. Additionally, previous research
demonstrates the superiority of using ECG signals over features for diagnosis
and decompensation prediction [28]. Extending this work to include waveforms
rather than relying solely on tabular data would be a valuable direction for
future research.

Data and code availability Code for dataset preprocessing and experimental
replications can be found in our dedicated repository [29].

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Nils Strodthoff
Carl von Ossietzky Universität Oldenburg Fakultät VI - Medizin und
Gesundheitswissenschaften Department für Versorgungsforschung Abteilung
AI4Health Ammerländer Heerstr. 114-118 26129 Oldenburg, Deutschland
<EMAIL_ADDRESS>
## Appendix

Table 2:  Performance results by demographic groups, presented by category, the number of samples, prevalence, and AUROC with 95% confidence intervals. Category | Samples [Prev.] | AUROC
---|---|---
Gender
Males | 122,125 [686] | 0.844 (0.842, 0.846)
Females | 118,767 [1453] | 0.829 (0.828, 0.831)
Race
Caucasians | 157,926 [1287] | 0.881 (0.880, 0.882)
Africans | 40,205 [432] | 0.784 (0.784, 0.790)
Asians | 7,295 [63] | 0.653 (0.652, 0.666)
Latinos | 13,842 [149] | 0.763 (0.765, 0.774)
Other | 17,306 [206] | 0.871 (0.869, 0.873)
Age by quantiles
(18-49yo.) | 54,830 [1181] | 0.723 (0.720, 0.723)
(50-64yo.) | 54,203 [617] | 0.702 (0.701, 0.708)
(65-77yo.) | 65,854 [227] | 0.808 (0.805, 0.815)
(>=78yo.) | 52,596 [114] | 0.709, (0.705, 0.708)


# Detection of Low-Energy Electrons with Transition-Edge Sensors

Carlo Pepe  Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91,
10135 Torino, Italy  Politecnico di Torino - Dipartimento di Elettronica e
Telecomunicazioni, Corso Duca degli Abruzzi 24, 10129 Torino, Italy  Benedetta
Corcione  Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Rome, Italy
Istituto Nazionale di Fisica Nucleare - Sezione di Roma, Piazzale Aldo Moro 2,
00185 Rome, Italy  Francesco Pandolfi<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare - Sezione di Roma, Piazzale Aldo Moro 2,
00185 Rome, Italy  Hobey Garrone  Istituto Nazionale di Ricerca Metrologica,
Strada delle Cacce 91, 10135 Torino, Italy  Politecnico di Torino -
Dipartimento di Elettronica e Telecomunicazioni, Corso Duca degli Abruzzi 24,
10129 Torino, Italy  Eugenio Monticone  Istituto Nazionale di Ricerca
Metrologica, Strada delle Cacce 91, 10135 Torino, Italy  Ilaria Rago  Istituto
Nazionale di Fisica Nucleare - Sezione di Roma, Piazzale Aldo Moro 2, 00185
Rome, Italy  Gianluca Cavoto  Sapienza Università di Roma, Piazzale Aldo Moro
2, 00185 Rome, Italy  Istituto Nazionale di Fisica Nucleare - Sezione di Roma,
Piazzale Aldo Moro 2, 00185 Rome, Italy  Alice Apponi  Dipartimento di Scienze
Università degli Studi Roma Tre, and Istituto Nazionale di Fisica Nucleare -
Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy  Alessandro
Ruocco  Dipartimento di Scienze Università degli Studi Roma Tre, and Istituto
Nazionale di Fisica Nucleare - Sezione di Roma Tre, Via della Vasca Navale 84,
00146 Rome, Italy  Federico Malnati  Dipartimento di Fisica, Università di
Torino, via Pietro Giuria 1, 10125, Torino, Italy  Danilo Serazio  Istituto
Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy
Mauro Rajteri  Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce
91, 10135 Torino, Italy

###### Abstract

We present the first detection of electrons with kinetic energy in the 100 eV
range with transition-edge sensors (TESs). This has been achieved with a
$(100\times 100)$ $\upmu$m2 Ti-Au bilayer TES, with a critical temperature of
about 84 mK. The electrons are produced directly in the cryostat by a cold-
cathode source based on field emission from vertically-aligned multiwall
carbon nanotubes. We obtain a Gaussian energy resolution between 0.8 and 1.8
eV for fully-absorbed electrons in the $(95-105)$ eV energy range, which is
found to be compatible with the resolution of this same device for photons in
the same energy range. This measurement opens new possibilities in the field
of electron detection.

Transition-edge sensors (TESs) are highly sensitive micro-calorimeters capable
of high-resolution single-photon counting across a wide energy spectrum Irwin
(1995); Cunningham  _et al._ (2002). The detection scheme is based on the
absorption of the incoming photons in a thin superconducting film, in which
their energy is transformed into heat. By operating a TES device at its
critical temperature $T_{\text{C}}$, even small variations in temperature lead
to measurable changes in its electrical resistance, owing to the steep
transition between the superconducting regime and the normal-conduction one.
TES devices have been capable of achieving single-photon Gaussian energy
resolutions below 50 meV for 0.8 eV photons Hattori  _et al._ (2022); Lolli
_et al._ (2013).

In principle, this detection scheme should also be sensitive to low-energy
electrons absorbed in the superconducting film, as seems to be confirmed by
simulations Patel  _et al._ (2021). However, there is currently very limited
research on the use of TES devices for electron detection, except for a recent
unpublished result on keV electrons Patel  _et al._ (2024). Low-energy
electrons have been detected with other detection schemes, such as micro-
channel plates Apponi  _et al._ (2022a), which have only very poor energy
resolution, and also feature geometrical inefficiencies due to their non-
unitary fill factor; and silicon detectors, such as avalanche photo-diodes
Apponi  _et al._ (2020) and silicon drift detectors Gugiatti  _et al._ (2020),
which are characterized by the presence of a dead layer at the detector
entrance, in which low-energy electrons are absorbed before generating a
signal. On the other hand TES devices offer, in principle, a detection scheme
without dead layers and with unitary fill factor, therefore with the potential
for high efficiency and excellent intrinsic energy resolution on low-energy
electrons. This would be of great interest for a wide range of experiments,
such as, for example, the PTOLEMY project Betti  _et al._ (2019a, b); Apponi
_et al._ (2022b), which plans to search for the cosmic neutrino background by
analyzing the endpoint of the beta decay of tritium with unprecedented
electron energy resolution.

Figure 1: TES layout and experimental setup. (a): Schematic view of the TES
device and its shield layer (y-axis to scale, x-axis not to scale). (b): 3D
model of the TES chip. (c): Schematic view of the setup used for electron
counting: the carbon nanotubes (CNTs) are hosted on the top copper plate and
oriented with their tips pointing towards the TES.

In this work, electrons are produced by field emission from vertically-
aligned, multi-wall carbon nanotubes (CNTs). This ‘cold-cathode source’
solution overcomes the issues in interfacing standard hot-filament-based
electron sources with the TES working at cryogenic temperature, thus allowing
to place the source directly inside the cryostat close to the TES. The results
presented here were obtained in the Innovative Cryogenic Detectors Laboratory
of Istituto Nazionale di Ricerca Metrologica (INRiM) in Torino (Italy) Pepe
(2023). The TES detectors operate on the 30 mK stage of an adiabatic
demagnetization refrigerator cryostat. Since electrons in motion are affected
by magnetic fields, and TESs and DC-SQUIDs are also sensitive to magnetic
fields, a cryogenic magnetic shield was installed within the cryostat around
the working area to suppress any magnetic interference that could degrade the
sensor performance and alter electron trajectories.

Figure 2: Results of the TES optical characterization with
$\lambda_{\gamma}=406$ nm photons. The histogram of the pulse amplitudes
report peaks corresponding to a number of photons $N_{\gamma}=31$, 32, 33, and
34 which are clearly distinguishable, and fitted with a sum of Gaussian
functions (red line).

A schematic view of the experimental setup is shown in Figure 1: a cross
section of the layers which compose the TES and the shield is shown in panel
(a), while panel (b) shows a schematic 3D model of it. The TES device used in
this work was fabricated at QR Lab, INRiM, has an area of $(100\times 100)$
$\mathrm{\upmu m}^{2}$, and is a Ti-Au bilayer device, composed of a 15 nm
layer of titanium covered by a 30 nm layer of gold, deposited by thermal
evaporation on a 500 nm silicon nitride substrate Monticone  _et al._ (2021).
It is wired with 50 nm superconducting Nb strips, which were deposited on the
substrate via sputtering. The TES has a critical temperature
$T_{\mathrm{C}}=84$ mK and was characterized in the $(0-140)$ eV energy range
with 406 nm photons from a pulsed laser. The result of this characterization,
in the energy range of interest for this work, is shown in Figure 2: the peaks
corresponding to a number of photons $N_{\gamma}=31$, 32, 33, and 34 are
clearly distinguishable, and fitted with a sum of Gaussian functions.

This TES design was adapted for electron detection by adding a shield layer,
which is needed as the electron source has a significantly larger area
(approximately $3\times 3$ mm2) compared to the TES active area ($100\times
100$ $\upmu$m2). The shield leaves the TES active area exposed, but covers the
area surrounding it, specifically the Nb wiring and the substrate, as direct
electron hits on the wiring would induce electrical noise, while on the
insulating substrate would lead to charge build-up. The shield layer is
produced by thermal evaporation, depositing an insulating layer consisting of
300 nm of amorphous silicon oxide (SiOx) Monticone  _et al._ (2000), followed
by a thin (5 nm) layer of titanium, and finally a 50 nm layer of gold. The
thin titanium layer is necessary for best adhesion of gold to the SiOx.

The electron source consists of a sample of CNTs which were synthesized in the
INFN laboratory ‘TITAN’ at Sapienza University of Rome Schifano  _et al._
(2023); Yadav  _et al._ (2024); Sarasini  _et al._ (2022); Tayyab  _et al._
(2024). The nanotubes were grown through chemical vapor deposition on a 500
$\upmu$m silicon substrate, and are approximately 200 $\upmu$m in length,
while covering a surface of roughly $(3\times 3)$ mm2. Thanks to the high
geometrical field enhancement factor of their tips, nanotubes are capable of
emitting electrons through quantum tunneling (field emission) without the
necessity of applying very high voltages de Heer  _et al._ (1995); Lahiri and
Choi (2011); Semet  _et al._ (2002); Lin  _et al._ (2015). Furthermore, as
this emission is quantic in nature, and not thermal, it does not generate heat
and can therefore be used in a cryostat. Moreover, within the energy range
under consideration, we can assume that this source is monochromatic: the
energy spread of the emitted electrons, in fact, is expected to be of the
order of $kT$, which at cryogenic temperatures is only a few $\upmu$eV.

The TES and the nanotubes were placed on two copper plates, facing each other,
separated by 0.5 mm sapphire spacers, which ensure electrical insulation while
guaranteeing a good degree of thermal conductance. The top copper plate, where
the nanotubes are hosted, was connected to a power supply, and was provided
negative voltage $V_{\mathrm{cnt}}$ in order to produce field-emission
electrons; the bottom plate was put in thermal contact with the cryostat, and
grounded electrically through it. The distance between the tips of the
nanotubes and the surface of the TES, in this setup, was $d=0.5$ mm. A
schematic view of the setup is shown in panel (c) of Figure 1.

The electron emission from the CNTs was read-out in two different ways. In the
‘anode’ configuration the TES was short-circuited to the metallic shield
layer, and the two were used as a large-area metallic plate sensitive to the
full electron current $I_{\mathrm{cnt}}$ emitted by the CNTs by connecting it
to a Keithley 6487 picoammeter. In the ‘counting’ configuration the TES output
was sensitive to single-particle absorption. This was achieved by voltage
biasing the TES to its critical temperature $T_{\text{C}}$ via electrothermal
feedback Irwin (1995). The TES working point was chosen to be
$R_{\mathrm{0}}=0.35\cdot R_{N}$, where $R_{N}$ is the resistance of the TES
in normal state. The TES was read-out with a DC-SQUID transimpedence amplifier
Drung  _et al._ (2007).

Figure 3: Current $I_{\mathrm{cnt}}$ emitted by the nanotube source (black
curve, left vertical scale), as a function of the negative voltage
$V_{\mathrm{cnt}}$ provided to it, when reading the TES and the shield in
anode configuration, compared to the rate of counts (red markers, right
vertical scale) recorded by the TES in counting configuration. The inset shows
a typical TES pulse shape for $V_{\text{cnt}}=100$ V with a recovery time of
approximately 10 $\upmu$s.

The results in the anode configuration are summarized by the black curve of
Figure 3, where the current $I_{\mathrm{cnt}}$ emitted by the nanotubes is
shown as a function of $V_{\mathrm{cnt}}$. As can be seen, $I_{\mathrm{cnt}}$
exhibits an exponential rise, compatible with the Fowler-Nordheim theory on
field emission Fowler and Nordheim (1928). Superimposed on the same plot with
red markers is the rate of counts recorded by the TES in counting mode. As can
be seen the increase in counting rate as a function of $V_{\mathrm{cnt}}$
follows the same exponential rise as $I_{\mathrm{cnt}}$, therefore proving
that the signals recorded by the TES are due to electrons. The inset in Figure
3 shows a typical TES signal for $V_{\mathrm{cnt}}=100$ V, characterized by a
very fast rise time $\tau_{+}\approx 200$ ns, and a recovery time
$\tau_{\mathrm{eff}}\approx 10$ $\upmu$s.

Figure 4: Power $P_{\mathrm{J}}$ needed to bring the TES to its working point
(black markers, left vertical scale) and local temperature
$T_{\mathrm{local}}$ around the TES device (red markers, right vertical scale)
for different values of $V_{\mathrm{cnt}}$. The bath temperature
$T_{\mathrm{bath}}$ is indicated by the red dashed line.

A feature of Fowler-Nordheim emission is that the current of emitted electrons
by the nanotubes depends on the electric field $|\vec{E}|$ in proximity of
their tips, which in our planar configuration corresponds, in first
approximation, to $|\vec{E}|=V_{\mathrm{cnt}}/d$. At the same time, if we
assume that electrons are emitted by the nanotubes with zero initial kinetic
energy, $V_{\mathrm{cnt}}$ also determines the kinetic energy of the electrons
when entering the TES. Therefore, in this setup, the signal rate and electron
energy are not independent parameters, as they both depend on
$V_{\mathrm{cnt}}$.

The electrons are emitted from a relatively large area compared to the TES,
and the additional heating created by the electrons hitting its nearby
environment must be considered. When the CNTs are off ($V_{\mathrm{cnt}}=0$)
or at low $V_{\mathrm{cnt}}$, the electron rate and the energy deposited are
minimal, keeping the temperature of the area around the TES at the cryostat
bath temperature ($T_{\text{bath}}\sim 41$ mK). In this condition, the Joule
power required to bring the TES to its critical temperature $T_{\text{C}}$ is
given by
$P_{\text{J}}(T_{\text{bath}})=I_{\text{TES}}^{2}R_{0}=\kappa(T_{\text{C}}^{n}-T_{\text{bath}}^{n})$.
For this device, the exponential parameter was found to be $n=4.78\pm 0.07$
and the coupling constant $\kappa=(10\pm 1)\cdot 10^{-7}$ W$\text{K}^{-n}$.

However, as $V_{\mathrm{cnt}}$ increases, the rate of electrons and the
associated energy deposition in the vicinity of the TES increase, raising the
local temperature above the bath temperature. As a result, the local
temperature $T_{\text{local}}$ becomes higher than $T_{\text{bath}}$ and
closer to $T_{\text{C}}$. Consequently, less $P_{J}$ is needed to bring the
TES to its working point at the critical temperature. This effect is shown in
Figure 4, where the black markers represent the power needed to bring the TES
to its nominal working point, and it can be seen that this power decreases as
$V_{\mathrm{cnt}}$ increases. The reduction of the dissipated power is
interpreted as an increase in the local temperature of the area in direct
contact with the TES, shown with red markers in Figure 4. When operating the
electron source at $V_{\mathrm{cnt}}=105$ V, the local temperature is already
higher than 60 mK, compared to the initial temperature of about 41 mK before
electron emission. This implies that results at different $V_{\mathrm{cnt}}$
values are not rigorously comparable, as the TES operates under slightly
different conditions.

Figure 5: Typical spectrum of TES signal amplitudes. This spectrum was
obtained with $V_{\mathrm{cnt}}=100$ V.

For each value of $V_{\mathrm{cnt}}$, the amplitude of the signals produced by
the TES were analyzed. A typical spectrum, obtained for $V_{\mathrm{cnt}}=100$
V is shown in Figure 5: as can be seen the distribution presents a high-
amplitude peak, corresponding to the full absorption of the electron energy in
the sensitive layer of the TES; a marked tail to the left of the peak, due to
partial absorption of electrons, and is most likely due to non-containment in
the thickness of the device, as the TES bilayer is very thin (45 nm); and a
low-amplitude peak, truncated by the trigger threshold of 166 mV, compatible
with electrons back-scattered out of the TES after exciting an internal mode
in the Au layer.

Figure 6: Example fits, for four different values of $V_{\mathrm{cnt}}$, of
the high-amplitude peak with the asymmetric Gaussian function.

We fit the high-amplitude peaks of these distributions with an asymmetric
Gaussian function, described by its peak position $\mu$ and its left
($\sigma_{\mathrm{L}}$) and right ($\sigma_{\mathrm{R}}$) tails. Some example
fits, for $V_{\mathrm{cnt}}=97$, 101, 103, and 105 V, are shown in Figure 6.

Figure 7:  TES electron energy resolution for photons (black square markers)
and electrons (red circular markers) as a function of the particle energy.

Figure 7 shows the TES Gaussian energy resolution for electrons (red circular
markers) and photons (black square markers) as a function of the incoming
particle energy. The electron energy resolution is defined as
$\sigma_{\mathrm{e}}=(\sigma_{\mathrm{R}}/\mu)\cdot E_{\mathrm{e}}$, where
$E_{\mathrm{e}}=V_{\mathrm{cnt}}\cdot\frac{C}{e}$, where $C$ is the Coulomb
charge and $e$ is the charge of the electron. The parameter
$\sigma_{\mathrm{R}}$ describes the broadness of the high-energy tail of the
absorption peak in the TES amplitude distribution. While the left tail
$\sigma_{\mathrm{L}}$ is dominated by electron non-containment effects, the
right tail $\sigma_{\mathrm{R}}$ is dominated by the energy resolution of the
device, plus negligible effects due to the non-monochromaticity of the source.
The photon energy resolution is instead obtained from the characterization
shown in Figure 2. As can be seen, there is no significant difference in TES
energy resolution between electrons and photons in this energy range. The
electron Gaussian resolution, in particular, is measured to be
$0.8<\sigma_{\mathrm{e}}<1.8$ eV for electrons in the $95\leq
E_{\mathrm{e}}\leq 105$ eV energy range.

To conclude, this work reports the detection of low-energy electrons with a
transition-edge sensor device. This has been achieved with the use of a cold
electron source, based on field emission from vertically-aligned carbon
nanotubes. We achieved a Gaussian energy resolution between 0.8 and 1.8 eV for
fully-contained electrons in the $(95-105)$ eV energy range. Notably, this
resolution is comparable to the energy resolution of the same TES device for
photons in the same energy range, demonstrating for the first time a direct
comparison between photon and electron energy resolutions. This work, together
with Patel  _et al._ (2024), open new possibilities in the detection of
electrons, as they represent the first detection of such particles with
intrinsic energy resolution without using electrostatic analyzers (as
currently done in electron spectroscopy Boumsellek  _et al._ (1990)), and with
no form of particle multiplication (such as the case of ionization-based
devices). TES devices, furthermore, have no dead layers and are therefore
capable of excellent energy resolution, as proved by this work.

##  I Acknowledgements

The authors are grateful to Elio Bertacco, Martina Marzano, Matteo Fretto and
Ivan De Carlo for technical support, as well as the PTOLEMY collaboration for
useful discussions. This research was partially funded by the contribution of
grant 62313 from the John Templeton Foundation, by PRIN grant ‘ANDROMeDa’
(PRIN_2020Y2JMP5) of Ministero dell’Università e della Ricerca, and from the
EC project ATTRACT (Grant Agreement No. 777222).

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# Coma cluster $\gamma$-ray and radio emission is consistent with a secondary
electron origin for the radio halo

Doron Kushnir1, Uri Keshet2 and Eli Waxman1
1Dept. of Particle Phys. & Astrophys., Weizmann Institute of Science, Rehovot
76100, Israel
2Physics Department, Ben-Gurion University of the Negev, POB 653, Be’er-Sheva
84105, Israel
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Observations of diffuse, non-thermal radio emission spanning several
megaparsecs have been documented in over 100 galaxy clusters. This emission,
classified as giant radio halos (GHs), mini halos, and radio relics based
mainly on their location and morphology, is interpreted as synchrotron
radiation and implies the presence of relativistic electrons and magnetic
fields in the intra-cluster medium (ICM). GHs were initially thought to be
generated by secondary electrons resulting from inelastic $p+p\rightarrow
X+\pi^{\pm}$ collisions. However, recent literature has leaned towards
primary-electron turbulent (re)acceleration models, partly due to claimed
upper limits on the $\gamma$-ray emission from $\pi^{0}$ decay. We demonstrate
that the observed GH and $\gamma$-ray flux in the Coma cluster are consistent
with a secondary origin for the GH across a broad range of magnetic field
values. Although the constraints on magnetic field configuration are not
stringent, they align well with previous estimates for Coma. Within this
magnetic field range, the energy density of cosmic-ray protons (CRp)
constitutes a few percent to tens of percent of the ICM energy density, as
predicted and observed for a sample of radio-emitting galaxy clusters.
Notably, we detect a rise in the ratio of CRp to ICM energy densities towards
the outer regions of the cluster. This phenomenon was anticipated to arise
from either adiabatic compression of CRp accelerated by accretion shocks or,
more likely, from strong CRp diffusion.

###### keywords:

XXX – YYY

††pubyear: 2024††pagerange: Coma cluster $\gamma$-ray and radio emission is
consistent with a secondary electron origin for the radio halo–Coma cluster
$\gamma$-ray and radio emission is consistent with a secondary electron origin
for the radio halo

##  1 introduction

Diffuse, non-thermal radio emission on scales as large as a few Mpc has been
observed in more than $100$ clusters of galaxies (see van Weeren et al., 2019,
for a recent review). The radio emission is interpreted as synchrotron
radiation, suggesting that relativistic electrons and magnetic fields exist in
the intra-cluster medium (ICM). The emission is classified as giant radio
halos (GHs), mini halos (MHs), and radio relics, depending mainly on their
location and morphology. Several models for synchrotron emission in halos have
been presented in the literature. These models differ in the assumptions
regarding the origin of the emitting electrons. In some models, the emitting
electrons are secondary electrons and positrons that were generated by $p-p$
interactions of a cosmic-ray proton (CRp) population with the ICM (e.g.
Dennison, 1980; Blasi & Colafrancesco, 1999) while in others, the emitting
electrons are reaccelerated by turbulence from a preexisting population of
nonthermal seeds in the ICM (secondary or otherwise, e.g. Brunetti et al.,
2001; Gitti et al., 2002).

Secondary (i.e. hadronic) models explain the diffuse radio emission from the
ICM more naturally than primary (leptonic) models, not only in GHs (Kushnir,
Katz, & Waxman, 2009), MHs (Keshet & Loeb, 2010), and even relics (Keshet,
2010), but also in the transitional, hybrid stages of these sources (Keshet,
2010, 2023), which in primary models would require fine-tuning different
acceleration mechanisms to match each other. Fundamentally, while the presence
of extended CR ion populations in clusters is thought to be both unavoidable
and at the level needed to produce the observed secondary-electron radio
sources, primary models invoke non-standard processes of electron
(re)acceleration in turbulence or weak shocks, which are neither well
established theoretically, nor confirmed observationally elsewhere, nor
plausible in the presence of the strong diffusion inferred in the ICM (Keshet,
2023).

The secondary nature of these diffuse sources is affirmed by accumulated
observations, including morphological, energetic, and spectral evidence tying
GHs, MHs, relics, and transients such as halo–relic bridges as different
manifestations of the same underlying mechanism; the nearly universal,
$-1.2\lesssim\alpha\lesssim-1.0$ spectral index of integrated sources (except
at transitional stages), including weak-shock relics (which in primary models
should be softer); sufficient acceleration of CRp in virial shocks established
by $\gamma$-ray to radio (Keshet et al., 2017; Reiss & Keshet, 2018; Hou et
al., 2023) emission detected from their CR electron counterparts; and
radio–X-ray (Kushnir, Katz, & Waxman, 2009) and other correlations between
coincident signals (see Keshet, 2023, and references therein).

Arguments raised against GHs as secondary sources target their spatial and
spectral properties (_e.g.,_ Brunetti & Jones, 2014), but such criticisms tend
to invoke oversimplified hadronic models. Some GHs, such as in A521, show a
very soft spectrum but are likely transient sources associated with a very
recent merger, in which rapid magnetic growth (Keshet, 2010) combined with
strong CR diffusion (Keshet, 2023) softens the spectrum. Even evolved GHs such
as in Coma, with a flat integrated spectrum, can show regions of very soft
spectra in particular in their peripheries, but such softening is naturally
explained by the strong CR electron diffusion inferred observationally
(Keshet, 2023).

The allegedly strongest and most decisive argument invoked against the
secondary model in GHs is that the CRp population required to produce the
radio emission from the Coma cluster emits a $\gamma$-ray flux through $p-p$
collisions that is larger than observed or constrained by Fermi-LAT (Brunetti,
Zimmer, & Zandanel, 2017; van Weeren et al., 2019; Adam et al., 2021).

In this paper, we show that the GH and $\gamma$-ray flux of the Coma cluster
are fully consistent with secondaries from the same CRp population. The
relevant radio and $\gamma$-ray observations for our analysis are described in
Section 2. We focus on the $144\,\rm{MHz}$ LOFAR observations (Bonafede et
al., 2022, hereafter B22) and on Fermi-LAT observations of the Coma cluster.
We present in Section 3 an order of magnitude assessment utilizing the ratio
of radio to $\gamma$-ray flux, demonstrating the consistency of the
observations with a secondary source for the GH. A straightforward model for
the CRp population and magnetic field is presented in Section 4. In Section 5,
we calculate the CRp population required to reproduce the observed GH emission
for a given magnetic field. We then calculate the $\gamma$-ray flux from this
population and compare it to the Fermi-LAT observations. We find that the
secondary origin of the Coma GH is consistent with observations over a large
range of magnetic field values. Over this range, the CRp energy density is a
few percent to tens of percent of the ICM energy density. We find that the
ratio between the CRp and the ICM energy densities increases towards the
cluster’s outskirts. This observation was anticipated to arise from either
adiabatic compression of accretion-shock accelerated CRp (Kushnir & Waxman,
2009) or, more favorably, as the result of strong CRp diffusion (Keshet, 2010,
2023), with unconstrained accelerators. In Section 6, we discuss the
implications of our findings and show, in particular, that an exceedingly high
magnetic field, regarded as necessary for the hadronic model in prior
investigations, stems from the steep CRp spectrum assumed in these studies.

##  2 Observations

In this section, we describe the relevant radio (Section 2.1) and $\gamma$-ray
(Section 2.2) observations used in our analysis.

###  2.1 Radio

The B22 $144\,\rm{MHz}$ observations of the Coma cluster include a radio halo,
a radio relic, and a bridge connecting the two. B22 provides the radial radio
brightness profiles of the GH in three sectors (excluding the SW sector, which
harbors the bridge region); see their figure 12. The profiles are computed in
elliptical annuli, with a major-to-minor axis ratio of $\mathord{\sim}1.3$. In
what follows, we treat the radial profiles as circular and average them, which
may lead at most to a few tens of percent error, not impacting our main
conclusions. Moreover, we note that for a slightly higher frequency,
$342\,\rm{MHz}$, the major-to-minor axis ratio decreases to
$\mathord{\sim}1.1$, approaching spherical symmetry. We defer the construction
of a more detailed model to reflect the deviations of the radio surface
brightness from a circular shape to future work. The mean radial profile is
presented with black symbols in the top panel of Figure 1. The error bars
reflect the scatter between the three sectors. The spectral index computed
between $144\,\rm{MHz}$ and $342\,\rm{MHz}$ is consistent with $\alpha=-1$, up
to $\theta_{b}\approx 2000^{\prime\prime}$ (red dashed line), within which
$>95\%$ of the surface brightness resides. The spectral index becomes steeper
at larger radii.

Figure 1: Results of the fitting as a function of the distance from the
cluster center for several example values of $B_{0}$ and $\eta$. Upper panel:
Radio surface brightness at $144\,\rm{MHz}$. Black symbols represent the mean
radial profile of B22 , with error bars denoting the scatter between the three
sectors averaged for the mean profile. Solid lines and a dashed black line
depict the results for cases 1-4 (refer to the text for details). Dashed
vertical lines denote $\theta_{b}$, $\theta_{500}$, and $\theta_{200}$ (as
described in the text), highlighted in red, blue, and green, respectively.
Lower panel: The assumed magnetic field profile (illustrated with dotted
lines) and the calibrated $\beta$ profiles (displayed with solid lines and a
dashed black line) for each case (using the same colors and line types as in
the upper panel). Circles represent the extrapolated post-shock thermal plasma
energy deposited in CRp ($\eta_{p}$), with open circles corresponding to cases
1-3 (with matching colors) and the filled circle corresponding to case 4.
Squares and a triangle denote the ratio between the total energy of the CRp
component (integrated up to $\theta_{200}$) and the thermal energy of the gas,
$E_{nt}/E_{th}$. Squares correspond to cases 1-3 (with matching colors), while
the triangle corresponds to case 4. A green line illustrates the assumed
electron density of the ICM.

It is worth noting for future reference that the total radio flux at
$144\,\rm{MHz}$ emitted by the cluster within an angular radius of
$\theta_{500}\approx 0.75^{\circ}$ (indicated by the blue dashed line in the
upper panel of Figure 1) amounts to $f_{r}\equiv 2\pi\int\nu I_{\nu}rdr\approx
1.7\times 10^{-14}\,\rm{erg\,s^{-1}\,cm^{-2}}$. This estimation encompasses a
slight extrapolation of around $5\%$ from the last radio data point to
$\theta_{500}$, and it remains unaffected by the spatial modeling of the
brightness profile.

###  2.2 $\gamma$-rays

The $\gamma$-ray flux from the Coma cluster has been analyzed in a few works.
Xi et al. (2018) claimed a detection of $\mathord{\sim}2.5\times
10^{-9}\,\rm{ph}\,\rm{s}^{-1}\,\rm{cm}^{-2}$ diffuse emission within
$1.23^{\circ}\approx\theta_{200}$ (green dashed line in the upper panel of
Figure 1) in the energy range $[200\,\rm{MeV}-300\,\rm{GeV}]$ (see, e.g.,
their disc+p1 model). Adam et al. (2021) claimed a detection of
$\mathord{\sim}10^{-9}\,\rm{ph}\,\rm{s}^{-1}\,\rm{cm}^{-2}$ within
$\theta_{500}$ in the same energy range. The exact flux depends on the
underlying models for the background and the signal angular distribution. We
include this by taking a $\mathord{\sim}3$ uncertainty factor on the measured
flux (see, e.g., table 1 of Adam et al., 2021). Baghmanyan et al. (2022)
reported a [$200$ MeV–$300$ GeV] flux of $(2.41\pm 0.45)\times
10^{-9}\rm{\,ph\,s}^{-1}\rm{cm}^{-2}$ for a uniform disc model, probably
adopting their $0.82_{-0.05}^{+0.10}$ degree $68\%$ containment radius. The
spectral index of the emission is consistent with a flat spectrum, although
difficult to measure and model-dependent: Xi et al. (2018) obtained
$2.2\lesssim\Gamma\lesssim 3.2$, Adam et al. (2021) obtained $\Gamma=2.45\pm
0.19$, and Baghmanyan et al. (2022) obtained $\Gamma=2.23\pm 0.11$. Skeptical
readers may use $3\times 10^{-9}\,\rm{ph}\,\rm{s}^{-1}\,\rm{cm}^{-2}$ as an
upper limit for the $\gamma$-ray flux from the cluster.

##  3 The consistency of the observations with a secondary origin for the GH

Prior to employing a straightforward model in Section 4 to demonstrate the
consistency of the observations with a secondary source for the GH, it proves
beneficial to conduct an order of magnitude assessment utilizing the ratio of
radio to $\gamma$-ray flux. We adopt a flat CRp spectrum
($dn/d\varepsilon\propto\varepsilon^{-2}$; denoted as $\delta_{p}=-2$), which
generates $\gamma$-rays and secondary particles (electrons and positrons)
through $p-p$ collisions with the nuclei of the ICM. Our choice of the CRp
spectral index, where the energy density per logarithmic energy interval
remains constant with energy as expected for particles accelerated at high
Mach number shocks (Blandford & Eichler, 1987), aligns with the observed
spectral indices of both the radio halo and $\gamma$-ray emissions (refer
below). This flat spectrum is also consistent with the spectrum observed from
virial shocks in stacked $\gamma$-ray (Reiss & Keshet, 2018) and radio (Hou et
al., 2023) clusters and in the Coma cluster (Keshet & Reiss, 2018), as well as
with the aforementioned universal $-1.2\lesssim\alpha\lesssim-1.0$ of
integrated GHs, MHs, and relics. We assume a scenario where the distribution
of radio-emitting secondaries is in a steady state, wherein within the
relevant energy ranges, the generated secondaries lose all their energy to
synchrotron radiation and inverse-Compton scattering of Cosmic Microwave
Background (CMB) photons.

In the context of a flat CRp spectrum, the energy density per logarithmic
energy interval remains invariant with energy, hence the $p-p$ $\gamma$-ray
luminosity per logarithmic photon energy bin exhibits an approximately energy-
independent behavior, $\Gamma\equiv 1-d\ln
f_{\varepsilon}/d\ln{\varepsilon}\approx-\delta_{p}=2$, where
$f_{\varepsilon}$ is the energy flux per unit energy, consistent with the
observations (note that constraining the $\gamma$-ray spectrum in an
individual cluster is challenging, as discussed in Section 2.2). Likewise, the
synchrotron emissivity per logarithmic frequency interval is directly
proportional to the energy production rate of secondaries per logarithmic
secondary energy interval, also maintaining an energy-independent spectrum,
resulting in $\alpha\approx\delta_{p}/2=-1$, consistent with the radio
observations within the hundreds of MHz frequency range (refer to the right
panel of B22 Figure 9).

The ratio of the radio to the $\gamma$-ray flux is given to a good
approximation by (Katz & Waxman, 2008; Kushnir & Waxman, 2009)

$\frac{\nu I_{\nu}}{\varepsilon
f_{\varepsilon}}\approx\frac{1}{4}\frac{B^{2}}{B^{2}+B^{2}_{\rm{CMB}}}\approx
0.1,$ (1)

where $B_{\rm{CMB}}\approx 3\mu\textrm{G}$ represents the equivalent CMB
energy density magnetic field, and we employed $B=3\,\mu\textrm{G}$ for the
last equality. Utilizing $f_{r}\approx 1.7\times
10^{-14}\,\rm{erg\,s^{-1}\,cm^{-2}}$ from Section 2.1 and the $\gamma$-ray
flux $\mathord{\sim}3\times 10^{-10}-3\times
10^{-9}\,\rm{ph}\,\rm{s}^{-1}\,\rm{cm}^{-2}$ from Section 2.2 (within
$\theta_{500}$), yields (with $\varepsilon=200\,\rm{MeV}$):

$\frac{\nu I_{\nu}}{\varepsilon f_{\varepsilon}}\approx 0.02-0.2,$ (2)

consistent with Equation (1). This consistency persists with the more detailed
model outlined in Section 4.

The large magnetic field, regarded as necessary for the hadronic model in
prior investigations, stems from the steep CRp spectrum assumed in these
studies (refer to Section 6 for a detailed discussion). For now, note that the
frequency at which an electron emits most of its synchrotron power is dictated
by $\nu=\nu_{0}\gamma^{2}$, where $\nu_{0}=3eB/(4\pi m_{e}c)$ and $\gamma$
represents the Lorentz factor of the electron. Consequently:

$\gamma\approx 6\times
10^{3}\left(\frac{\nu}{144\,\rm{MHz}}\right)^{1/2}B_{-6}^{-1/2},$ (3)

indicating an electron energy of approximately $\mathord{\sim}3\,$GeV. Since
the energy of secondary electrons from a $p-p$ collision is roughly half of
the $\gamma$-ray energy (for a given CRp energy), the parent CRp of the
$\mathord{\sim}3\,$GeV electrons produce $\mathord{\sim}6\,$GeV $\gamma$-rays,
surpassing the $200\,\rm{MeV}$ energy of the $\gamma$-rays which contribute
most to the flux. This discrepancy signifies that the parent CRp of the
electrons carries more than $10$ times the energy of the parent CRp of the
$\gamma$-rays, with implications discussed in Section 6. Finally, note that
the cooling time of these electrons, ranging from
$\mathord{\sim}0.1-0.4\,\rm{Gyr}$ for $B_{-6}=0.1-10$, is shorter than the
dynamical time of the cluster, allowing the application of the steady-state
assumption.

##  4 Model

In our simple model, the ICM contains a population of CRp with a power law
distribution $\varepsilon^{2}dn/d\varepsilon=\beta 3nT/2$, where $n$ and $T$
are the number density and (constant) temperature of the ICM, respectively. We
assume that $\beta$, the ratio of the CRp energy density (within a logarithmic
CRp energy interval) and the thermal energy of the ICM, scales with the ICM
number density as $\beta=\beta_{0}(n/n_{0})^{\delta}$, where subscript $0$
denotes values at the center of the cluster. Since CRp produced by the
accretion shock are later affected mainly by adiabatic expansion and
compression, the difference between the adiabatic indices of the relativistic
CRp ($4/3$) and nonrelativistic thermal plasma ($5/3$) implies $\delta=-1/3$
(Kushnir & Waxman, 2009). More detailed modeling suggests that typically
$\delta<-1/3$ because of the effect of weak merger shocks (Kushnir & Waxman,
2009). If CRp diffusion is sufficiently strong, we expect a constant CRp
density, $\delta=-1$, regardless of the CRp acceleration sites (as long as the
acceleration is at high Mach number shocks to produce a flat spectrum),

The electron density of the ICM is assumed to follow a $\beta_{m}$-model
(Cavaliere & Fusco-Femiano, 1978), $n=n_{0}(1+(r/r_{c})^{2})^{-3\beta_{m}/2}$,
and we adopt the values $n_{0}=3.36\times 10^{-3}\,\rm{cm}^{-3}$,
$r_{c}=310\,\rm{kpc}$, $\beta_{m}=0.75$, used by Adam et al. (2021) (see green
line in the lower panel of Figure 1). We further assume a constant
temperature, $T=8.25\,\textrm{keV}$, for the ICM (Reiprich & Böhringer, 2002).

The magnetic field strength profile of the Coma cluster was inferred by
Bonafede et al. (2010, hereafter B10) using Faraday rotation measures to be
$B(n)=B_{0}(n/n_{0})^{\eta}$ with the best-fit values of $B_{0,-6}=4.7$
($B_{-6}=B/\mu\rm{G}$) and $\eta=0.5$. Here we adopt the parametric form of
B10, but due to possible biases in the analysis of B10 (see, e.g., discussion
in Brunetti, Zimmer, & Zandanel, 2017), we allow a large range of $B_{0}$ and
$\eta$ values.

We use the parametrization of Kamae et al. (2006) for the secondary spectrum
of $p-p$ interactions, and the exact formula for synchrotron emission given by
Blumenthal & Gould (1970) to calculate the $144\,\rm{MHz}$ surface brightness
and the $[200\,\rm{MeV}-300\,\rm{GeV}]$ $\gamma$-ray flux.

##  5 Results

For each combination of $B_{0}$ and $\eta$, we find the values of $\beta_{0}$
and $\delta$ that best fit the observed $144\,\rm{MHz}$ surface brightness.
For example, our results for $B_{0,-6}=4.7$ and $\eta=0.5$ (dotted black line
at the lower panel of Figure 1) are presented in the top panel of Figure 1
(black solid line; Case 1). We find $\beta_{0,-4}=1.6$ (defining
$\beta_{0,-4}=\beta_{0}/10^{-4}$) and $\delta=-1.2$, which nicely agree with
the observed surface brightness. The value of $\beta_{0}$ is close to the
value $2\times 10^{-4}$ calibrated to Coma by Kushnir & Waxman (2009), to a
sample of radio-emitting galaxy clusters by Kushnir, Katz, & Waxman (2009,
based on the correlation between the radio luminosity and the thermal X-ray
luminosity), and to a sample of GHs and MHs by Keshet & Loeb (2010). The value
of the calibrated $\delta$ satisfies the expectation that $\delta<-1/3$ (the
radial profile of $\beta$ is presented as a solid black line at the lower
panel of Figure 1). Interestingly, the value of $\delta$ is close to the
strong diffusion expectation ($-1$), suggesting this limit is achieved,
regardless of the CRp acceleration sites. The obtained $\gamma$-ray flux for
the calibrated $\beta_{0}$ and $\delta$ values is
$\mathord{\approx}0.85(1.45)\times
10^{-9}\,\rm{ph}\,\rm{s}^{-1}\,\rm{cm}^{-2}$ within
$\theta_{500}(\theta_{200})$, in an agreement with the observations of Adam et
al. (2021) (Xi et al., 2018) and slightly below the observation of Baghmanyan
et al. (2022). A significant fraction of the $\gamma$-ray flux,
$\mathord{\approx}77(45)\%$, is contained within the angular distance for
which the radio observations directly constrain $\beta$, such that the
uncertainty due to the extrapolation to $\theta_{500}(\theta_{200})$ is not
large.

We may also extrapolate $\beta$ to $\theta_{200}\approx 1.3^{\circ}$ (dashed
green line in the top panel of Figure 1), where the accretion shock is
predicted to reside (but see discussion in Section 6). We can then multiply
the extrapolated $\beta$ by the number of logarithmic intervals that span the
CRp distribution ($\mathord{\approx}20$; see, e.g., Kushnir & Waxman, 2009) to
find the fraction $\eta_{p}\approx 3.7\times 10^{-2}$ of the post-shock
thermal plasma energy deposited in CRp (the black disc in the bottom panel of
Figure 1). The ratio between the total energy of the CRp component (integrated
up to $\theta_{200}$) and the thermal energy of the gas is
$\mathord{\approx}0.005$ per logarithmic CRp energy interval, leading to a
ratio of $E_{nt}/E_{th}\approx 0.07-0.1$ between these two components,
depending on the number of logarithmic intervals that the CRp distribution
spans ($\mathord{\approx}14(20)$ for maximal energy of $10^{6(9)}\,\rm{GeV}$
relevant for supernova remnants (galaxy cluster accretion shocks)). This
estimate is plotted with a black square in the bottom panel of Figure 1.

We next allow $B_{0,-6}$ to vary in the range $1-10$ and $\eta$ to vary in the
range $0-1.1$. We repeat our analysis for each combination of $B_{0}$ and
$\eta$, and the results are presented in Figure 2. As can be seen in the top-
left (top-right) panel of Figure 2, the $\gamma$-ray observations of Adam et
al. (2021) and Baghmanyan et al. (2022) (Xi et al., 2018) are consistent with
a wide range of $B_{0}$ and $\eta$ values (all combinations with $\gamma$-ray
flux within the two black lines; we plot the 2$\sigma$ constraints of
Baghmanyan et al. (2022)). Specifically, the constraints for $B_{0}$ and
$\eta$ of B10 are consistent with the observations. Two extreme cases of
$B_{0}$ and $\eta$ values, $B_{0,-6}=1$, $\eta=0$ and $B_{0,-6}=10$,
$\eta=0.9$ (Case 2 and Case 3, respectively), which are consistent with the
observations, are presented in Figure 1 (with red and blue colors,
respectively). While the agreement with the radio surface brightness at the
largest radial bins is slightly worse than the agreement in Case 1, these
cases cannot be ruled out based on the radio observations. The ranges of the
calibrated $\beta_{0,-4}$ and $\delta$ is $[0.94,13.5]$ and $[-1.7,-0.56]$
(middle-right and middle-left panels of Figure 2, respectively) for all models
with consistent $\gamma$-ray flux. For the same models, the obtained range of
$\eta_{p}$ is $[0.02,0.25]$ (for $20$ logarithmic intervals of the CRp energy
distribution) and of $E_{nt}/E_{th}$ is $[0.02,0.6]$ (for $14$ logarithmic
intervals of the CRp energy distribution). The values within the ranges of the
calibrated parameters are all reasonable, not allowing stronger constraints on
combinations of $B_{0}$ and $\eta$.

Figure 2: The outcomes of the analysis for each pairing of $B_{0}$ and $\eta$
within the surveyed range. The enclosed dashed contours delineate the best-fit
values of B10 (same $1-3\sigma$ contours in all panels). Top-left (top-right)
panels: The obtained $[200\,\rm{MeV}-300\,\rm{GeV}]$ $\gamma$-ray flux within
angular diameter of $0.75^{\circ}\approx\theta_{500}$
($1.23^{\circ}\approx\theta_{200}$) as inspected by Adam et al. (2021) and
Baghmanyan et al. (2022) (Xi et al., 2018). The constrained $\gamma$-ray flux
of Adam et al. (2021) and Baghmanyan et al. (2022) lies between the two solid
and dotted black lines, respectively (we plot the 2$\sigma$ constraint of
Baghmanyan et al., 2022). The constrained $\gamma$-ray flux of Xi et al.
(2018) lies between the dot-dashed black lines. Subsequent panels depict
colormaps of the calibrated $B_{0}$, $\log_{10}(\beta_{0})$,
$\log_{10}(\eta_{p})$ at $R_{200}$, and $\log_{10}(E_{nt}/E_{th})$. The black
lines correspond to the lines in the upper panels ("Adam" for Adam et al.
(2021), "Xi" for Xi et al. (2018), and "Bag." for Baghmanyan et al. (2022)).
The values within the calibrated parameter ranges are all reasonable,
precluding stronger constraints on combinations of $B_{0}$ and $\eta$.

The data allows us to rule out the case of density-independent $\beta$ (i.e.
$\delta=0$). Forcing $\delta=0$ for $B_{0,-6}=4.7$ and $\eta=0.5$ leads to a
radial surface brightness profile that is too concentrated (dashed black line
in the top panel of Figure 1, Case 4). The same argument holds for all
$\delta>0$ values.

##  6 Discussion

We showed that the GH and $\gamma$-ray flux of the Coma cluster are fully
consistent with a secondary origin for the GH over a large range of magnetic
field values. Although the constraints on the magnetic field configuration are
not stringent, they align well with previous estimates for Coma (B10). Within
this magnetic field range, the energy density of CRp constitutes a few percent
to tens of percent of the ICM energy density, as predicted and observed for a
sample of radio-emitting galaxy clusters (Kushnir, Katz, & Waxman, 2009). We
find that the ratio between the CRp and the ICM energy densities increases
towards the cluster’s outskirts. This observation was anticipated to arise
from either adiabatic compression of accretion-shock accelerated CRp (Kushnir
& Waxman, 2009) or, more favorably, as the result of strong CRp diffusion
(Keshet, 2023), allowing also for other possibilities for the acceleration
sites.

Our analysis assumed a spherical symmetry for the cluster, probably
introducing at most a few tens of percent error. We defer the construction of
a more detailed model to reflect the deviations of the radio surface
brightness from a circular shape to future work.

The recent radio and $\gamma$-ray observations of the Coma cluster are found
to be entirely consistent with a hadronic origin, contrasting with previous
claims (Brunetti, Zimmer, & Zandanel, 2017; van Weeren et al., 2019; Adam et
al., 2021). In these earlier studies, a substantially large magnetic field was
deemed necessary for the hadronic model, implying that a significant portion
of the Coma cluster’s energy was attributed to the magnetic field. However,
our analysis reveals a different scenario, where the best-fit values of B10
align well with our hadronic model. The discrepancy arises from the differing
assumptions regarding the CRp spectrum. Previous works assumed a steep CRp
spectrum ($\delta_{p}<-2$), while we adopted a flat CRp spectrum
($\delta_{p}=-2$). The steep spectrum in previous studies was motivated by the
purported steep spectrum of the GH (note that the $\gamma$-ray spectrum is
difficult to constrain, see Section 2.2). For instance, Brunetti, Zimmer, &
Zandanel (2017) claimed $\alpha=-1.22\pm 0.04$, resulting in
$\delta_{p}\approx 2\alpha=-2.44$. However, the GH spectral index was
determined by fitting a broad range of radio frequencies, including high
frequencies where the parent CRp’s energy is too high to contribute to the
observed $\gamma$-ray flux (see below). When limiting the radio frequency
range to hundreds of MHz, a slope of $\alpha\simeq-1$ emerges (see the right
panel of B22 Figure 9; the observed spectral softening at the edge of the GH
is expected for secondary electrons diffusing outward (see Keshet, 2023)),
rendering the index assumed by Brunetti, Zimmer, & Zandanel (2017) untenable.
Other analyses adopted even softer CRp spectra, such as $\delta_{p}<-2.7$ in
Adam et al. (2021), largely based on the poorly determined $\gamma$-ray
spectrum.

Assuming a steep spectrum for the CRp led those studies to artificial
inflation of the energy density of low-energy CRp, resulting in an
overproduction of $\gamma$-rays and effectively increasing the required
magnetic field. This effect is illustrated in Figure 3, where we depict the
ratio of the radio to $\gamma$-ray flux as:

$\frac{\nu
I_{\nu}}{200\,\textrm{MeV}f_{[200\,\textrm{MeV},200\,\textrm{GeV}]}},$ (4)

plotted against the magnetic field. This ratio is observed to fall within the
range $0.02-0.2$ (within $\theta_{500}$, see Section 3). The figure displays
the obtained ratio for calculations assuming different indexes for the CRp
spectra, utilizing the parametrization of Kamae et al. (2006) for the
secondary spectrum of $p-p$ interactions and the exact formula for synchrotron
emission by Blumenthal & Gould (1970) to compute the radio and $\gamma$-ray
fluxes. Additionally, the simple estimate from Equation (1) pertinent for
$\delta_{p}\approx-2$ is included, exhibiting a deviation by less than a
factor of $2$ from the more precise calculation. As depicted in the figure,
the calculation for $\delta_{p}=-2$ aligns with observations for
$0.5<B_{-6}<3$, consistent with earlier sections. However, given that the
parent CRp of the electrons carries more than $10$ times the energy of the
parent CRp of the $\gamma$-rays (refer to Section 3), a steep spectrum
augments the energy density of the parent CRp of the electrons compared to the
energy density of the parent CRp of the $\gamma$-rays, necessitating larger
magnetic fields to uphold a fixed ratio of radio to $\gamma$-ray flux, as
evidenced in Figure 3 for $\delta_{p}=-2.45,-2.7$. As demonstrated here, for
$\delta_{p}=-2$, which is fully consistent with the GH spectral index of B22,
the magnetic field constraints align perfectly with the best-fit values of
B10. It’s worth noting that a slight deviation from $\delta_{p}=-2$ has a
negligible impact on our findings, as the energy difference of the parent CRp
is only about one order of magnitude.

Figure 3: The relationship between the radio to $\gamma$-ray flux ratio and
the magnetic field. Solid lines represent the calculated ratios for different
indexes of the CRp spectra ($\delta_{p}=-2,-2.45,-2.7$ in black, red, and
blue, respectively). The dotted line shows the simple estimate from Equation
(1) pertinent for $\delta_{p}\approx-2$, exhibiting a deviation by less than a
factor of $2$ from the more precise calculation. The observed ratio falls
within the two dashed lines (within $\theta_{500}$, see Section 3). The
calculation for $\delta_{p}=-2$ is consistent with observations for
$0.5<B_{-6}<3$. However, considering that the parent CRp of electrons carries
more than $10$ times the energy of the parent CRp of $\gamma$-rays (see
Section 3), a steep spectrum increases the energy density of the parent CRp of
electrons relative to that of $\gamma$-rays, necessitating larger magnetic
fields to maintain a fixed ratio of radio to $\gamma$-ray flux.

It should be emphasized that previous studies that argued for inconsistency
between hadronic halo interpretations and $\gamma$-ray observations could not
show a substantial discrepancy exceeding the underlying systematic
uncertainties, as evident from the variability in the results obtained by
different authors and the arguments sketched above. For instance, Adam et al.
(2021) argued that the secondary counterpart to their $\gamma$-ray model is
too weak to explain the halo, necessitating some unknown primary electron
acceleration. However, the discrepancy they report is only a factor of $\sim
4$, so the putative primary electron emission component should be comparable,
i.e., only a factor $\sim 3$ stronger than the secondary component. Given the
very different origins of secondary and putative primary electrons, such a
coincidence is unlikely. On the contrary: given the substantial systematics
and the challenges facing primary models, a radio–$\gamma$-ray discrepancy by
a factor of a few in the hadronic picture should be considered a qualitative
success of the model.

Recent evidence indicates that cluster virial shocks extend well beyond
$\theta_{200}$, with strong stacked signatures found in the narrow range
$[2.2,2.5]\theta_{500}$, or equivalently$[1.4,1.6]\theta_{200}$, and weak
signatures of larger semi-major radii (Reiss & Keshet, 2018; Hou et al., 2023;
Ilani et al., 2024a, b). We avoid extrapolating the $\beta$ profile beyond
$\theta_{200}$, which would incur substantial systematic errors and may be
unwarranted for the semi-minor axis in Coma (Keshet et al., 2017; Keshet &
Reiss, 2018). Consequently, our extrapolated CRp fraction at the virial shock
probably underestimates its true post-shock value, reaching tens of percent at
$\gtrsim 1.4\theta_{200}$. Hence, our results are consistent with strong
cosmic-ray diffusion not only in terms of the $\delta\sim-1$ profile, but
also, given the order-unity energy fraction typically anticipated in post-
shock CRp, in our normalization of the CRp distribution; strong diffusion also
accounts for the spectral softening at the edge of the GH (as secondary
electrons diffuse outward; see Keshet, 2023), seen for example in the top
panel of Figure 1, and for additional observations such as the GH–relic bridge
(Keshet, 2010, 2023).

## Acknowledgements

We thank Boaz Katz for useful discussions. DK is supported by a research grant
from The Abramson Family Center for Young Scientists and by the Minerva
Stiftung. UK is supported by ISF grant No. 2126/22.

## Data availability

All data used in this study is available through other publications.

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