# Information-Theoretic Foundations for
Neural Scaling Laws

Hong Jun Jeon
Computer Science
Stanford University
Stanford, CA
<EMAIL_ADDRESS>
&Benjamin Van Roy
Stanford University
Stanford, CA
<EMAIL_ADDRESS>

###### Abstract

Neural scaling laws aim to characterize how out-of-sample error behaves as a
function of model and training dataset size. Such scaling laws guide
allocation of a computational resources between model and data processing to
minimize error. However, existing theoretical support for neural scaling laws
lacks rigor and clarity, entangling the roles of information and optimization.
In this work, we develop rigorous information-theoretic foundations for neural
scaling laws. This allows us to characterize scaling laws for data generated
by a two-layer neural network of infinite width. We observe that the optimal
relation between data and model size is linear, up to logarithmic factors,
corroborating large-scale empirical investigations. Concise yet general
results of the kind we establish may bring clarity to this topic and inform
future investigations.

_K_ eywords Information Theory $\cdot$ Neural Scaling Laws

##  1 Introduction

In recent years, foundation models have grown immensely, with some embodying
trillions of trainable parameters. While larger models have in general
produced better results, they also require much more compute to train. It has
become impractical to perform hyperparameter sweeps at the scale of these
modern models. This has required bypassing the practice of tuning
hyperparameters via extensive trial and error, as was previously common in
deep learning.

Among other things, hyperparameters control $1)$ the size, measured in terms
of the parameter count $p$, of the neural network model and $2)$ the number
$T$ of training tokens. If each parameter is adjusted in response to each
token then the computational requirements of training scale will the product
of these two quantities. For any compute budget $C$, one should carefully
balance between $p$ and $T$. Too few training tokens leads to model estimation
error, while too few parameters gives rise to mispecification error. As
evaluating performance across multiple choices of $p$ and $T$ becomes
computationally prohibitive at scale, alternative kinds of analysis are
required to guide allocation of computational resources.

Kaplan et al. [2020] and Hoffmann et al. [2022] have proposed the following
procedure for allocating a large compute budget: $1)$ Evaluate test errors of
models produced using various small compute budgets $C$ with many different
allocations to parameters $p$ versus training tokens $T$. $2)$ Extrapolate to
estimate the relation between $p$ and $T$ for large $C$. $3)$ Extrapolate to
estimate the relation between $p$ and $T$ for large $C$.

To give a sense of scales involved here, Hoffmann et al. [2022] evaluate test
errors across “small” models for which $p\times T$ ranges from around
$10^{18}$ to $10^{22}$ and extrapolates out to “large” models at around
$10^{24}$. Kaplan et al. [2020] and Hoffmann et al. [2022] each extrapolate
based on a hypothesized _scaffolding function_. Kaplan et al. [2020] guess a
scaffolding function based on results observed in small scale experiments.
Hoffmann et al. [2022] carry out an informal and somewhat speculative
mathematical analysis to guide their choice (see their Appendix D).

The analysis of Hoffmann et al. [2022] is somewhat generic rather than
specialized to the particular neural network architecture used in that paper.
In this paper, building on the work of Jeon and Van Roy [2022a, b], we develop
rigorous information-theoretic foundations and use them to derive similar
scaling laws. To keep things simple and concrete, we carry out the analysis
with a particular data generating process for which neural networks are well-
suited. The sorts of arguments developed by Hoffmann et al. [2022] are just as
relevant to this context as they are to language models.

Hoffmann et al. [2022] suggest that the compute optimal trade-off between
parameter count and number of training tokens is linear, though the authors
expressed some doubt and considered other possibilities that are near-linear
as well. We establish an upper bound on the minimal information-theoretically
achievable expected error as a function of $p$ and $T$ and derive the relation
required to minimize this bound for each compute budget. For large compute
budgets, this relation is linear, as suggested by Hoffmann et al. [2022].

Our main contributions include a first rigorous mathematical characterization
of the compute-optimal efficient frontier for a neural network model and
development of information-theoretic tools which enable that. A limitation of
our analysis is in its simplified treatment of computational complexity as the
product of the model and data set sizes; we do not assume any constraints on
computation beyond those imposed by choices of $p$ and $T$. In particular, we
analyze, algorithms which carry out perfect Bayesian inference with respect to
a model that is misspecificified due to its restricted size. While this
abstracts away the details of practical training algorithms, empirical
evidence suggests that our idealized framework leads to useful approximations
[Zhu et al., 2022]. In spite of these limitations, we hope our results set the
stage for further mathematical work to guide hyperparameter selection when
training large neural networks.

##  2 A Framework for Learning

###  2.1 Probabilistic Framework

We define all random variables with respect to a common probability space
$(\Omega,\mathbb{F},\mathbb{P})$. Recall that a random variable $F$ is simply
a measurable function $\Omega\mapsto\mathcal{F}$ from the sample space
$\Omega$ to an outcome set $\mathcal{F}$.

The probability measure $\mathbb{P}:\mathbb{F}\mapsto[0,1]$ assigns
probabilities to the events in the $\sigma$-algebra $\mathbb{F}$. For any
event $E\in\mathbb{F}$, $\mathbb{P}(E)$ to denotes the probability of the
event. For events $E,G\in\mathbb{F}$ for which $\mathbb{P}(G)>0$,
$\mathbb{P}(E|G)$ to denotes the probability of event $E$ conditioned on event
$G$.

For realization $z$ of a random variable $Z$, $\mathbb{P}(Z=z)$ is a function
of $z$. We denote its value evaluated at $Z$ by $\mathbb{P}(Z)$. Therefore,
$\mathbb{P}(Z)$ is a random variable (it takes realizations in $[0,1]$
depending on the value of $Z$). Likewise for realizations $(y,z)$ of random
variables $Y,Z$, $\mathbb{P}(Z=z|Y=y)$ is a function of $(y,z)$ and
$\mathbb{P}(Z|Y)$ is a random variable which denotes the value of this
function evaluated at $(Y,Z)$.

If random variable $Z:\Omega\mapsto\Re^{K}$ has density $p_{Z}$ w.r.t the
Lebesgue measure, the conditional probability $\mathbb{P}(E|Z=z)$ is well-
defined despite the fact that for all $z$, $\mathbb{P}(Z=z)=0$. If function
$f(z)=\mathbb{P}(E|Z=z)$ and $Y:\Omega\mapsto\Re^{K}$ is a random variable
whose range is a subset of $Z$’s, then we use the $\leftarrow$ symbol with
$\mathbb{P}(E|Z\leftarrow Y)$ to denote $f(Y)$. Note that this is different
from $\mathbb{P}(E|Z=Y)$ since this conditions on the event $Z=Y$ while
$\mathbb{P}(E|Z\leftarrow Y)$ indicates a change of measure.

###  2.2 Data

We consider a stochastic process which generates a sequence
$(X_{t},Y_{t+1}:t\in\mathbb{Z}_{+})$ of data pairs. For all $t$, we let
$H_{t}$ denote the history $(X_{0},Y_{1},\ldots,X_{t-1},Y_{t},X_{t})$ of
experience. We assume that there exists an underlying latent variable $F$ such
that $(X_{0},X_{1},\ldots)\perp F$ and $F$ prescribes a conditional
probability measure $F(\cdot|H_{t})$ to the next label $Y_{t+1}$. In the case
of an _iid_ data generating process, this conditional probability measure
would only depend on $H_{t}$ via $X_{t}$. Note that the current pre-training
objective of foundation models falls under this iid setting in which for all
$t$, $X_{t}$ is a random segment of the training corpus and $Y_{t+1}$ is the
subsequent token. As our framework is Bayesian, we represent our uncertainty
about $F$ by modeling it as a random variable with prior distribution
$\mathbb{P}(F\in\cdot)$.

###  2.3 A Learning Objective

We focus on a particular notion of error which facilitates analysis via
Shannon information theory and reflects the objective of modern foundation
models. For all $t\in\mathbb{Z}_{+}$, our algorithm is tasked with providing a
predictive distribution $P_{t}$ of $Y_{t+1}$ which may depend on the history
of data which it has already observed $H_{t}$. We express such an algorithm as
$\pi$ for which $P_{t}=\pi(H_{t})$. As aforementioned, an effective learning
system ought to leverage data as it becomes available and perform well across
all time. As a result, for any time horizon $T\in\mathbb{Z}_{+}$, we are
interested in quantifying the cumulative expected log-loss:

$\mathbb{L}_{T,\pi}=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}_{\pi}\left[-\ln
P_{t}(Y_{t+1})\right].$

Note that since we take all random variables to be defined with respect to a
common probability space, the expectation $\mathbb{E}$ integrates over all
random variables which we do not condition on. We use the subscript $\pi$ in
$\mathbb{E}_{\pi}$ to specify that all predictions $P_{t}$ for all $t$ are
produced by $\pi$. As $Y_{t+1}$ is the random variable which represents the
next label that is generated by the underlying stochastic process,
$P_{t}(Y_{t+1})$ denotes the probability that our algorithm’s prediction
$P_{t}$ assigns to label $Y_{t+1}$.

It is important to note that even for an _omniscient_ algorithm, the minimum
achievable log-loss is not $0$. Consider the _omniscient_ algorithm which
produces for all $t$ the prediction
$P^{*}_{t}=\mathbb{P}(Y_{t+1}\in\cdot|F,H_{t})$. Even this agent incurs a loss
of:

$\displaystyle\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}_{\pi}\left[-\ln\mathbb{P}(Y_{t+1}|F,H_{t})\right]$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{H}(Y_{t+1}|F,H_{t})$

where our point follows from the fact that the conditional entropy
$(\mathbb{H})$ of a discrete random variable $Y_{t+1}$ is non-negative. As a
result, we define the _reducible error_ as:

$\displaystyle\mathcal{L}_{T,\pi}$
$\displaystyle=\mathbb{L}_{T,\pi}-\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{H}(Y_{t+1}|F,H_{t})$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(P^{*}_{t}(\cdot)\|P_{t}(\cdot)\right)\right].$

reducible error represents the error which is reducible via observing
additional data and fitting a larger model. Therefore, we expect that this
error will consist of two terms which reflect $1)$ the error due to estimation
via finite data, $2)$ the error due to approximation with a finite parameter
model.

##  3 Error of Constrained Predictors

We introduce a general upper bound on the reducible error of a _constrained_
predictor. While the formulations remain abstract in this section, a useful
running example is the following: Assume that $F$ is an _infinite_ width
neural network which generates the data and $\tilde{F}$ is a _finite_ width
network.

###  3.1 A Constrained Predictor

$F$ may exhibit endless complexity, likely beyond what can be represented with
finite memory hardware. To represent the predictions made by a constrained
predictor, we first define a random variable $\tilde{F}$ whose range is a
subset of $F$’s. As aforementioned, this random variable can be a lossy
compression of $F$ i.e. if $F$ is represented by an infinite-width neural
network, $\tilde{F}$ could be a finite-width approximation. For all $t$, let
the constrained predictor be:

$\tilde{P}_{t}(\cdot)=\sum_{\tilde{f}}\mathbb{P}(\tilde{F}=\tilde{f}|H_{t})\cdot\mathbb{P}(Y_{t+1}\in\cdot|F=\tilde{f},X_{t}).$

The predictor performs inference on $\tilde{F}$ but performs predictions as if
$F=\tilde{F}$. We let

$\mathcal{L}_{T}(\tilde{F})=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(P^{*}_{t}(\cdot)\|P_{t}(\cdot)\right)\right].$

###  3.2 Error of Constrained Predictor

######  Theorem 3.1.

For all $T\in\mathbb{Z}_{++}$ and random variables
$F:\Omega\mapsto\mathcal{F},\tilde{F}:\Omega\mapsto\tilde{\mathcal{F}}$ for
which $\tilde{\mathcal{F}}\subseteq\mathcal{F}$, if
$((X_{t},Y_{t+1}):t\in\mathbb{Z}_{+})$ is iid conditioned on $F$, then

$\displaystyle\tilde{\mathcal{L}}_{T}(\tilde{F})$
$\displaystyle\leq\frac{\mathbb{I}(F;\tilde{F})}{T}+\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(P^{*}_{t}(\cdot)\|\hat{P}_{t}(\cdot)\right)\right],$

where
$\hat{P}_{t}(\cdot)=\mathbb{P}(Y_{t+1}\in\cdot|F\leftarrow\tilde{F},X_{t}).$

The first term denotes the _estimation error_ or the error which is reducible
via access to more data. This is evident by the fact it decreases linearly in
$T$ and the numerator reflects the _complexity_ of $\tilde{F}$. The more nats
of information that $\tilde{F}$ contains about the data stream, the more data
will be required to arrive at a good predictor.

The second term denotes the _misspecification error_ or the error which is
reducible via a larger learning model. The closer that $\tilde{F}$
approximates $F$, the smaller the KL divergence between $P^{*}_{t}$ and
$\hat{P}_{t}$ will be. In the following section, we will use Theorem 3.1 to
derive a concrete neural scaling law for an infinite-width neural network
example.

###  3.3 Scaling Law

For a FLOP constraint $C=p\cdot T$, it is clear that there is a tension
between $p$ and $T$ in minimizing the upper bound in Theorem 3.1. This can be
seen by first fixing a FLOP count $C$ and substituting $T=C/p$. The upper
bound becomes:

$\frac{p\cdot\mathbb{I}(F;\tilde{F})}{C}+\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(P^{*}_{t}(\cdot)\|\hat{P}_{t}(\cdot)\right)\right].$

Note that the first term is _increasing_ in $p$ whereas the second term is
_decreasing_ in $p$. Therefore, under a fixed FLOP budget, the designer ought
to select a value of $p$ which effectively balances the two sources of error.

##  4 An Illustrative Example

###  4.1 Data Generating Process

The generating process is described by a neural network with $d$ inputs, a
single asymptotically wide hidden layer of ReLU activation units, and a linear
output layer. We denote by $F$ the associated mapping from input to output.
Inputs and binary labels are generated according to
$X_{t}\overset{iid}{\sim}\mathcal{N}(0,I_{d})$ and
$\mathbb{P}(Y_{t+1}=1|F,X_{t})=\sigma(F(X_{t}))$ where $\sigma$ denotes the
sigmoid function.

As alluded to by the asymptotic width, $F$ is a nonparametric model which we
will outline now. Let $\bar{\theta}$ be distributed according to a Dirichlet
process with base distribution $\mathrm{uniform}(\mathbb{S}^{d-1})$ and scale
parameter $K$. Realizations of this Dirichlet process are probability mass
functions on a countably infinite subset of $\mathbb{S}^{d-1}$. Let
$\mathcal{W}=\\{w\in\mathbb{S}^{d-1}:\bar{\theta}_{w}>0\\}$ denote this set.
For all $w\in\mathcal{W}$,

$\theta_{w}=\left\\{\begin{array}[]{ll}\overline{\theta}_{w}&\text{with
probability }1/2,\\\
-\overline{\theta}_{w}&\text{otherwise.}\end{array}\right.$

Finally, we have that

$F(X_{t})\ =\
\sqrt{K+1}\cdot\sum_{w\in\mathcal{W}}\theta_{w}\text{ReLU}\left(w^{\top}X_{t}\right).$

Since $\mathcal{W}$ has countably infinite cardinality, $F$ is characterized
by a neural network with infinite width. We let
$\theta=(\theta_{w}:w\in\mathcal{W})$ and $W=(w:w\in\mathcal{W})$ denote the
weights of such neural network and hence

$F(X_{t})\ =\ \sqrt{K+1}\cdot\theta^{\top}\text{ReLU}(WX_{t}).$

Note that the mean and variance structure satisfy

$\mathbb{E}[F(X)]=0,\qquad\mathbb{E}[F(X)^{2}]=1/2.$

Therefore, this model remains nontrivial as $d$ and $K$ grow as all of the
above quantities are invariant of $d$ and $K$.

###  4.2 Constrained Predictor

We will study the scaling law associated with a particular constrained
predictor characterized by a neural network of width $n$. Let
$\tilde{w}_{1},\tilde{w}_{2},\ldots,\tilde{w}_{n}$ be distributed iid ${\rm
Categorical}(\bar{\theta})$, where $\mathcal{W}$ are the classes. For any
$\epsilon>0$, let $\mathbb{S}^{d-1}_{\epsilon}$ be an $\epsilon$-cover w.r.t
$\|\cdot\|_{2}$ and for all $i\in[n]$, let

$\tilde{w}_{i,\epsilon}=\operatorname*{arg\,min}_{v\in\mathbb{S}^{d-1}_{\epsilon}}\|\tilde{w}_{i}-v\|^{2}_{2}.$

Finally, let

$\tilde{F}_{n,\epsilon}(X_{t})=\frac{\sqrt{K+1}}{n}\cdot\sum_{i=1}^{n}{\rm
sign}\left(\theta_{\tilde{w}_{i}}\right)\text{ReLU}\left(\tilde{w}_{i,\epsilon}^{\top}X_{t}\right).$

Let $\tilde{\theta}\in\Re^{n}$ is $({\rm
sign}(\theta_{\tilde{w}_{i}})/n:i\in[n])$ and
$\tilde{W}_{\epsilon}\in\Re^{n\times d}$ is
$(\tilde{w}_{i,\epsilon}:i\in[n])$. Therefore,

$\tilde{F}_{n,\epsilon}(X_{t})=\sqrt{K+1}\cdot\tilde{\theta}^{\top}\text{ReLU}\left(\tilde{W}_{\epsilon}X_{t}\right).$

We consider the performance of a constrained agent which for all $t$, produces
the prediction $\tilde{P}_{t}(\cdot)=$

$\sum_{\tilde{f}}\
\mathbb{P}(\tilde{F}_{n,\epsilon}=\tilde{f}|H_{t})\cdot\mathbb{P}(Y_{t+1}\in\cdot|F=\tilde{f},X_{t}).$

Note that this agent performs inference on the constrained model
$\tilde{F}_{n,\epsilon}$ and produces predictions about $Y_{t+1}$ as if
$\tilde{F}_{n,\epsilon}$ were the function $F$ which produced the data.

###  4.3 Error Bound

We will now study the error incurred by the constrained predictor described
above. We define

$\tilde{\mathcal{L}}_{T,n,\epsilon}=\frac{1}{T}\sum_{t=0}^{T-1}\
\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}\left(Y_{t+1}\in\cdot|\theta,X_{t}\right)\|\tilde{P}_{t}(\cdot)\right)\right]$

as the loss of interest.

######  Theorem 4.1.

For all $n,K,T\in\mathbb{Z}_{++}$ and $\epsilon\geq 0$, if for all
$t\in\\{0,1,2,\ldots,T-1\\}$, $(X_{t},Y_{t+1})$ is generated by $F$, then

$\displaystyle\tilde{\mathcal{L}}_{T,n,\epsilon}$
$\displaystyle\leq\underbrace{\frac{K\ln\left(1+\frac{n}{K}\right)\cdot\left(\ln(2n)+d\ln\left(\frac{3}{\epsilon}\right)\right)}{T}}_{\rm
estimation\ error}+\underbrace{\frac{3K(1+d\epsilon^{2})}{n}}_{\rm
misspecification\ error}.$

The estimation error represents the error which is incurred in the process of
learning $\tilde{F}$ from $H_{T}$. Notably, this error decays linearly in $T$,
but only depends _logarithmically_ in $n$. The misspecification error
represents the error which persists due to the fact that we approximate $F$
via $\tilde{F}_{n,\epsilon}$. As a result, this error decreases with greater
$n$ and smaller $\epsilon$, but is _independent_ of $T$. If we let
$\tilde{\mathcal{L}}_{T,n}=\inf_{\epsilon>0}\
\tilde{\mathcal{L}}_{T,n,\epsilon}$, then

######  Corollary 4.2.

For all $n\geq 3,K\geq 2,T\in\mathbb{Z}_{++}$, if for all
$t\in\\{0,1,2,\ldots,T-1\\}$, $(X_{t},Y_{t+1})$ is generated by $F$, then

$\displaystyle\tilde{\mathcal{L}}_{T,n}$
$\displaystyle\leq\frac{dK\ln\left(1+\frac{n}{K}\right)\left(\ln(e36TK)+\frac{2}{d}\ln(2n)\right)}{2T}+\frac{3K}{n}.$

Figure 1: Above (left), we depict the error bound from Corollary 4.2 for
$d=10,K=100$, and various FLOP counts $C$. Each curve consists of pairs
$(n,T)$ for which $d\cdot n\cdot T=C$. Therefore, each curve depicts the
possible error values attainable at a given FLOP count. Noticeably, an
_improper_ allocation of compute can lead to _higher_ error despite greater
resource investment. (Right) we depict the compute-optimal tradeoff between
parameter count and dataset size. The dashed line represents a line of slope
$1$. As a result, the relationship between optimal parameter count and dataset
size eventually looks linear (as suggested by Theorem 4.3).

###  4.4 Resulting Scaling Law

Corollary 4.2 provides an upper bound on loss. We conjecture that this upper
bound is tight to within logarithmic factors. As such, we consider its use as
an approximation of loss to guide allocation of compute resources in the
following result:

######  Theorem 4.3.

(compute-optimal parameter count) For all $d,K\in\mathbb{Z}_{++}$ and FLOP
counts $C\in\mathbb{Z}_{++}$, if $K\geq 2,d\geq 3$, and $n^{*}$ minimizes the
upper bound of Corollary 4.2 subject to $d\cdot n\cdot T\leq C$, then

$d\cdot n^{*}=\tilde{\Theta}\left(\sqrt{C}\right).$

A proof of Theorem 4.3 can be found in Appendix A.2. Recall that the FLOP
count $C$ is the product of the parameter count $(n\cdot d)$ and the dataset
size $T$. The above result states that for any FLOP budget $C$, we can
minimize the loss upper bound from Corollary 4.2 by allocating
$\tilde{\Theta}(\sqrt{C})$ to the _parameter count_. This would dictate that
the optimal dataset size would _also_ be $\tilde{\Theta}(\sqrt{C})$. This
result is consistent with the insights of Hoffmann et al. [2022] that, up to
logarithmic factors, the optimal parameter count grows linearly with the
training dataset size.

##  5 Conclusion

Our results provide a first step in developing rigorous mathematics for the
purposes of analyzing scaling laws for foundation models. We hope that this
will inspire further theoretical research on the subject. Our analysis is
based on an error upper bound and furthermore, our analysis restricts
attention to single-hidden-layer feedforward neural networks. Generalizing the
results to treat state-of-the-art architectures remains an open issue.
Furthermore, we have only considered allocation of pretraining compute. State-
of-the-art performance in modern application domains relies on subsequent
fine-tuning (see, e.g., [Ziegler et al., 2019]) through reinforcement learning
from human feedback. How best to allocate resources between pretraining and
fine-tuning is another area that deserves attention. An information-theoretic
framework that treats pretraining, fine-tuning, and decision making in a
unified and coherent manner, perhaps in the vein of [Lu et al., 2023], might
facilitate theoretical developments on this front.

## Acknowledgments

Financial support from the NSF GRFP fellowship and the Army Research Office
(ARO) Grant W911NF2010055 is gratefully acknowledged.

## References

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##  Appendix A Proofs of Theoretical Results

See 3.1

###### Proof.

$\displaystyle\tilde{\mathcal{L}}_{T}(\tilde{F})$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T}\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(P^{*}_{t}(\cdot)\bigg{\|}\sum_{\tilde{f}\in\tilde{\mathcal{F}}}\mathbb{P}(Y_{t+1}\in\cdot|F=\tilde{f},H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|H_{t})\right)\right]$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{I}(X_{t+1};F|H_{t})+\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y_{t+1}\in\cdot|H_{t})\bigg{\|}\sum_{\tilde{f}\in\tilde{\mathcal{F}}}\mathbb{P}(Y_{t+1}\in\cdot|\mathcal{F}=\tilde{f},H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|H_{t})\right)\right]$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{I}(X_{t+1};F|H_{t})$
$\displaystyle\
+\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\sum_{\tilde{f}\in\tilde{\mathcal{F}}}\sum_{y\in\mathcal{Y}}\mathbb{P}(Y_{t+1}=y|H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|Y_{t+1}=y,H_{t})\ln\frac{\mathbb{P}(Y_{t+1}=y|H_{t})}{\sum_{\tilde{f}\in\tilde{\mathcal{F}}}\mathbb{P}(Y_{t+1}=y|F=\tilde{f},H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|H_{t})}\right]$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{I}(X_{t+1};F|H_{t})$
$\displaystyle\
+\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\sum_{\tilde{f}\in\tilde{\mathcal{F}}}\sum_{y\in\mathcal{Y}}\mathbb{P}(Y_{t+1}=y|H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|Y_{t+1}=y,H_{t})\ln\frac{\sum_{\tilde{f}\in\tilde{F}}\mathbb{P}(Y_{t+1}=y|H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|Y_{t+1}=y,H_{t})}{\sum_{\tilde{f}\in\tilde{\mathcal{F}}}\mathbb{P}(Y_{t+1}=y|F=\tilde{f},H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|H_{t})}\right]$
$\displaystyle\overset{(a)}{\leq}\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{I}(X_{t+1};F|H_{t})$
$\displaystyle\
+\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\sum_{y\in\mathcal{Y}}\sum_{\tilde{f}\in\tilde{\mathcal{F}}}\mathbb{P}(Y_{t+1}=y|H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|Y_{t+1}=y,H_{t})\ln\frac{\mathbb{P}(Y_{t+1}=y|H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|Y_{t+1}=y,H_{t})}{\mathbb{P}(Y_{t+1}=y|F=\tilde{f},H_{t})\cdot\mathbb{P}(\tilde{F}=\tilde{f}|H_{t})}\right]$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{I}(X_{t+1};F|H_{t})$
$\displaystyle\
+\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(\tilde{F}\in\cdot|Y_{t+1},H_{t})\|\mathbb{P}(\tilde{F}\in\cdot|H_{t})\right)\right]+\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y_{t+1}\in\cdot|H_{t})\|\mathbb{P}(Y_{t+1}\in\cdot|F\leftarrow\tilde{F},H_{t})\right)\right]$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y_{t+1}\in\cdot|F,H_{t})\|\mathbb{P}(Y_{t+1}\in\cdot|H_{t})\right)\right]$
$\displaystyle+\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{I}(Y_{t+1};\tilde{F}|H_{t})+\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y_{t+1}\in\cdot|H_{t})\|\mathbb{P}(Y_{t+1}\in\cdot|F\leftarrow\tilde{F},H_{t})\right)\right]$
$\displaystyle=\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{I}(Y_{t+1};\tilde{F}|H_{t})+\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y_{t+1}\in\cdot|F,H_{t})\|\mathbb{P}(Y_{t+1}\in\cdot|F\leftarrow\tilde{F},H_{t})\right)\right]$
$\displaystyle=\frac{\mathbb{I}(H_{T};\tilde{F})}{T}+\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y_{t+1}\in\cdot|F,H_{t})\|\mathbb{P}(Y_{t+1}\in\cdot|F\leftarrow\tilde{F},H_{t})\right)\right],$

where $(a)$ follows from the log-sum inequality. ∎

###  A.1 Proof of Dirichlet Process Results

######  Lemma A.1.

(squared error upper bounds KL) For all real-valued random variables $G$ and
$\tilde{G}$, if $Y$ is a binary random variable for which
$\mathbb{P}(Y=1|G)=\frac{1}{1+e^{-G}}$, then

$\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}(\mathbb{P}(Y\in\cdot|G)\|\mathbb{P}(Y\in\cdot|G\leftarrow\tilde{G}))\right]\leq\mathbb{E}\left[\left(G-\tilde{G}\right)^{2}\right].$

###### Proof.

$\displaystyle\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}(\mathbb{P}(Y\in\cdot|G)\|\mathbb{P}(Y\in\cdot|G\leftarrow\tilde{G}))\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{1+e^{G}}\ln\left(\frac{1+e^{\tilde{G}}}{1+e^{G}}\right)\right]$
$\displaystyle\quad+\mathbb{E}\left[\frac{1}{1+e^{-G}}\ln\left(\frac{1+e^{-\tilde{G}}}{1+e^{-G}}\right)\right]$
$\displaystyle\overset{(a)}{\leq}\mathbb{E}\left[\left(G-\tilde{G}\right)^{2}\right]$

where $(a)$ follows from the fact that for all $x,y\in\Re$,
$\frac{1}{1+e^{x}}\ln\left(\frac{1+e^{y}}{1+e^{x}}\right)+\frac{1}{1+e^{-x}}\ln\left(\frac{1+e^{-y}}{1+e^{-x}}\right)\leq(x-y)^{2}$.
∎

######  Lemma A.2.

For all $d,n,N\in\mathbb{Z}_{++}$, if $X\sim\mathcal{N}(0,I_{d})$, then

$\mathbb{E}\left[\left(F(X)-\tilde{F}_{n,0}(X)\right)^{2}\right]\leq\frac{K+1}{n}.$

###### Proof.

$\displaystyle\mathbb{E}\left[\left(\theta^{\top}\text{ReLU}(WX)-\tilde{\theta}^{\top}\text{ReLU}(\tilde{W}X)\right)^{2}\right]$
$\displaystyle\overset{(a)}{\leq}\frac{K+1}{n^{2}}\cdot\mathbb{E}\left[\text{ReLU}(\tilde{W}X)^{\top}\left(\tilde{\theta}\tilde{\theta}^{\top}\right)\text{ReLU}(\tilde{W}X)\right]$
$\displaystyle=\frac{K+1}{n^{2}}\cdot\mathbb{E}\left[\text{ReLU}(\tilde{W}X)^{\top}I_{n}\text{ReLU}(\tilde{W}X)\right]$
$\displaystyle\leq\mathbb{E}\left[\frac{K+1}{n^{2}}\cdot\sum_{i=1}^{n}(\tilde{w}_{i,0}^{\top}X)^{2}\right]$
$\displaystyle=\frac{K+1}{n}.$

where $(a)$ follows from the fact that the two functions have equal
conditional expectation conditioned on $\theta$. ∎

######  Lemma A.3.

For all $d,n,K\in\mathbb{Z}_{++}$,

$\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y\in\cdot|F,X)\|\mathbb{P}(Y\in\cdot|F\leftarrow\tilde{F}_{n,0},X)\right)\right]\
\leq\ \frac{K+1}{n}.$

###### Proof.

$\displaystyle\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}(\mathbb{P}(Y\in\cdot|F,X)\|\mathbb{P}(Y\in\cdot|F\leftarrow\tilde{F}_{n,0},X))\right]$
$\displaystyle\overset{(a)}{=}\mathbb{E}\left[\left(F(X)-\tilde{F}_{n,0}(X)\right)^{2}\right]$
$\displaystyle\overset{(b)}{\leq}\frac{K+1}{n},$

where $(a)$ follows from Lemma A.1, $(c)$ follows from the fact that the
distribution of $\theta$ is the limiting distribution
$\lim_{N\rightarrow\infty}$ of a Dirichlet $[K/N,\ldots,K/N]$ random variable,
$(d)$ follows from the dominated convergence theorem, and $(e)$ follows from
Lemma A.2. ∎

######  Lemma A.4.

For all $d,n,K\in\mathbb{Z}_{++}$,

$\mathbb{E}\left[\left(F_{n,0}(X)-F_{n,\epsilon}(X)\right)^{2}\right]\ \leq\
\frac{d(K+1)\epsilon^{2}}{n}.$

###### Proof.

$\displaystyle\mathbb{E}\left[\left(F_{n,0}(X)-F_{n,\epsilon}(X)\right)^{2}\right]$
$\displaystyle=\mathbb{E}\left[(K+1)\cdot\left\|\sum_{i=1}^{n}\tilde{\theta}_{i}\left(\text{ReLU}(\tilde{w}_{i,0}^{\top}X)-\text{ReLU}(\tilde{w}_{i,\epsilon}^{\top}X)\right)\right\|^{2}\right]$
$\displaystyle=\mathbb{E}\left[(K+1)\cdot\sum_{i=1}^{n}\tilde{\theta}_{i}^{2}\cdot\left\|\left(\text{ReLU}(\tilde{w}_{i,0}^{\top}X)-\text{ReLU}(\tilde{w}_{i,\epsilon}^{\top}X)\right)\right\|^{2}\right]$
$\displaystyle\leq\frac{d(K+1)\epsilon^{2}}{n}$

∎

######  Lemma A.5.

For all $d,n,K\in\mathbb{Z}_{++}$ and $\epsilon\geq 0$,

$\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y\in\cdot|F,X)\|\mathbb{P}(Y\in\cdot|F\leftarrow\tilde{F}_{n,\epsilon},X)\right)\right]\
\leq\ \frac{3K(1+d\epsilon^{2})}{n}.$

###### Proof.

$\displaystyle\mathbb{E}\left[\mathbf{d}_{\mathrm{KL}}\left(\mathbb{P}(Y\in\cdot|F,X)\|\mathbb{P}(Y\in\cdot|F\leftarrow\tilde{F}_{n,\epsilon},X)\right)\right]$
$\displaystyle\overset{(a)}{\leq}\mathbb{E}\left[\left(F(X)-\tilde{F}_{n,\epsilon}(X)\right)^{2}\right]$
$\displaystyle=\mathbb{E}\left[2\left(F(X)-F_{n,0}(X)\right)^{2}+2\left(F_{n,0}(X)-F_{n,\epsilon}(X)\right)^{2}\right]$
$\displaystyle\overset{(b)}{\leq}\frac{2(K+1)}{n}+\frac{2d(K+1)\epsilon^{2}}{n}$
$\displaystyle\leq\frac{3K(1+d\epsilon^{2})}{n},$

where $(a)$ follows from Lemma A.1 and $(b)$ follows from Lemmas A.4 and A.5.
∎

######  Lemma A.6.

(entropy upper bound) For all $d,n,K\in\mathbb{Z}_{++}$ and $\epsilon>0$,

$\mathbb{H}(\tilde{F}_{n,\epsilon})\leq
K\ln\left(1+\frac{n}{K}\right)\cdot\left(\ln(2n)+d\ln\left(\frac{3}{\epsilon}\right)\right).$

###### Proof.

$\displaystyle\mathbb{H}(\tilde{F}_{n,\epsilon})$
$\displaystyle\overset{(a)}{\leq}\mathbb{E}\left[\sum_{w\in\mathcal{W}}\mathbbm{1}_{|\tilde{\theta}_{w}|>0}\cdot\left(\ln(2n)+d\ln\left(\frac{3}{\delta}\right)\right)\right]$
$\displaystyle\overset{(b)}{\leq}K\ln\left(1+\frac{n}{K}\right)\left(\ln(2n)+d\ln\left(\frac{3}{\delta}\right)\right)$

where $(a)$ follows from the fact that the output weight can take on at most
$2n$ different values
$(-\frac{n\sqrt{K+1}}{n},-\frac{(n-1)\sqrt{K+1}}{n},\ldots,-\frac{\sqrt{K+1}}{n},\frac{\sqrt{K+1}}{n},\ldots\frac{n\sqrt{K+1}}{n})$
and the fact that $|\mathbb{S}^{d-1}_{\epsilon}|\leq(3/\delta)^{d}$ and $(b)$
follows as a commonly known fact about the number of unique classes of a
dirichlet-multinomial distribution. ∎

See 4.1

###### Proof.

The result follows from Theorem 3.1 and Lemmas A.6 and A.5 ∎

See 4.2

###### Proof.

The result holds from Theorem 4.1 by setting
$\epsilon^{2}=\frac{nK\ln(1+\frac{n}{K})}{4T(K+1)}$ and the fact that for
$n\geq 3,K\geq 2,$ $\frac{36T(K+1)}{nK\ln(1+n/K)}\leq 36KT$. ∎

###  A.2 Optimal Width

See 4.3

###### Proof.

$\displaystyle n^{*}$
$\displaystyle=\operatorname*{arg\,min}_{n\in\left[\frac{C}{d}\right]}\frac{3K}{n}+\frac{K\ln\left(1+\frac{n}{K}\right)\cdot\ln\left(2n\right)}{t}+\frac{dK\ln\left(1+\frac{n}{K}\right)\left(1+\frac{1}{2}\ln\left(36KT\right)\right)}{t};\
\text{ s.t. }n\cdot d\cdot t\leq C$
$\displaystyle=\operatorname*{arg\,min}_{n\in\left[\frac{C}{d}\right]}\frac{3}{n}+\frac{\ln\left(1+\frac{n}{K}\right)\cdot\ln\left(2n\right)}{t}+\frac{d\ln\left(1+\frac{n}{K}\right)\left(1+\frac{1}{2}\ln\left(36KT\right)\right)}{t};\
\text{ s.t. }n\cdot d\cdot t\leq C$
$\displaystyle=\operatorname*{arg\,min}_{n\in\left[\frac{C}{d}\right]}\frac{3}{n}+\frac{nd\ln\left(1+\frac{n}{K}\right)\cdot\ln\left(2n\right)}{C}+\frac{nd^{2}\ln\left(1+\frac{n}{K}\right)\left(1+\frac{1}{2}\ln\left(\frac{36KC}{nd}\right)\right)}{C}$
$\displaystyle\overset{(a)}{=}n\text{ s.t.
}\frac{3}{n^{2}}=\frac{d\left(\ln\left(1+\frac{n}{K}\right)+\ln\left(1+\frac{n}{K}\right)ln(2n)+\frac{n}{K+n}\ln\left(2n\right)\right)}{C}$
$\displaystyle\qquad+\frac{d^{2}\ln\left(1+\frac{n}{K}\right)\ln\left(\frac{36KC}{nd}\right)}{2C}+\frac{d^{2}n}{C(n+K)}+\frac{nd^{2}\ln\left(\frac{36KC}{nd}\right)}{2C\left(n+K\right)}$
$\displaystyle=n\text{ s.t.
}C=\frac{dn^{2}\left(\ln\left(1+\frac{n}{K}\right)+\ln\left(1+\frac{n}{K}\right)ln(2n)+\frac{n}{K+n}\ln\left(2n\right)\right)}{3}$
$\displaystyle\qquad+\frac{d^{2}n^{2}\ln\left(1+\frac{n}{K}\right)\ln\left(\frac{36KC}{nd}\right)}{6}+\frac{d^{2}n^{3}}{3(n+K)}+\frac{n^{3}d^{2}\ln\left(\frac{36KC}{nd}\right)}{6\left(n+K\right)}.$

where $(a)$ follows from $1$st order optimality conditions

Due to monotonicity, we can drive upper and lower bounds for the value of $n$
via lower and upper bounds of the above RHS respectively.

We begin with the upper bound for $n^{*}$:

$\displaystyle n^{*}$ $\displaystyle\overset{(a)}{\leq}n\text{ s.t.
}C=\frac{d^{2}n^{2}}{3}$ $\displaystyle=\frac{\sqrt{3C}}{d}.$

where $(a)$ follows from the fact that $d^{2}n^{2}/3$ is a lower bound of the
above RHS.

We now derive the lower bound for $n^{*}$:

$\displaystyle n^{*}$ $\displaystyle\overset{(a)}{\geq}n\text{ s.t.
}C=dn^{2}\ln(2n)\ln(36KC)$
$\displaystyle=\tilde{\Omega}\left(\frac{\sqrt{C}}{d}\right).$

where $(a)$ follows from the fact that $dn^{2}\ln(2n)\ln(36KC)$ is an upper
bound of the above RHS. The result follows. ∎


# Constraints on Electron Acceleration in Gamma-Ray Bursts Afterglows from
Radio Peaks

Ruby A. Duncan,1 Alexander J. van der Horst1, Paz Beniamini2,3
1Department of Physics, The George Washington University, 725 21st Street NW,
Washington, DC 20052, USA
2Department of Natural Science, The Open University of Israel, P.O. Box 808,
Ra’anana 4353701, Israel
3Astrophysics Research Center of the Open University (ARCO), The Open
University of Israel, P.O. Box 808, Ra’anana 4353701, Israel E-mail:
<EMAIL_ADDRESS>
(1 November 2022)

###### Abstract

Studies of gamma-ray bursts (GRBs) and their multi-wavelength afterglows have
led to insights in electron acceleration and emission properties from
relativistic, high-energy astrophysical sources. Broadband modeling across the
electromagnetic spectrum has been the primary means of investigating the
physics behind these sources, although independent diagnostic tools have been
developed to inform and corroborate assumptions made in particle acceleration
simulations and broadband studies. We present a methodology to constrain three
physical parameters related to electron acceleration in GRB blast waves: the
fraction of shock energy in electrons, $\epsilon_{e}$; the fraction of
electrons that gets accelerated into a power-law distribution of energies,
$\xi_{e}$; and the minimum Lorentz factor of the accelerated electrons,
$\gamma_{m}$. These parameters are constrained by observations of the peaks in
radio afterglow light curves and spectral energy distributions. From a sample
of 49 radio afterglows, we are able to find narrow distributions for these
parameters, hinting at possible universality of the blast wave microphysics,
although observational bias could play a role in this. Using radio peaks and
considerations related to the prompt gamma-ray emission efficiency, we
constrain the allowed parameter ranges for both $\epsilon_{e}$ and $\xi_{e}$
to within about one order of magnitude, $0.01\lesssim\epsilon_{e}\lesssim 0.2$
and $0.1\lesssim\xi_{e}\lesssim 1$. Such stringent constraints are
inaccessible for $\xi_{e}$ from broadband studies due to model degeneracies.

###### keywords:

gamma-ray burst: general - acceleration of particles - relativistic processes
- shock waves - radiation mechanisms: non-thermal

††pubyear: 2022††pagerange: Constraints on Electron Acceleration in Gamma-Ray
Bursts Afterglows from Radio Peaks–References

##  1 Introduction

The multi-wavelength afterglow of gamma-ray bursts (GRBs; van Paradijs et al.,
1997; Costa et al., 1997; Frail et al., 1997) allows for detailed studies of
the physics behind these highly energetic cosmic explosions. Observations
across the electromagnetic spectrum can be used to model their extreme,
collimated outflows. From the afterglow observations one can derive physical
parameters characterizing the energetics of the explosion, the circumburst
medium, and the microphysics of the observed radiation (Sari et al., 1998;
Wijers & Galama, 1999). The latter includes parameters describing the
acceleration of the emitting particles, predominantly electrons, and the
magnetic fields in which these electrons are accelerated and radiate.

From GRB afterglow observations, a spectral energy distribution (SED) can be
constructed over a broad frequency range, from radio wavelengths to gamma-ray
energies (e.g., Perley et al., 2014; MAGIC Collaboration et al., 2019). The
dominant radiation process is synchrotron emission, with inverse Compton
effects important at X-ray and higher energies (Sari & Esin, 2001). The
rapidly evolving SED results in varying light curves at all observing
frequencies, from which one can study the different stages of the afterglow
evolution. Observations in the radio regime are particularly useful in
constructing the SED, as they can determine the evolution of the spectral peak
and the synchrotron self-absorption frequency (Frail et al., 1997; Wijers &
Galama, 1999), two observables necessary to constrain the physical parameters
of the GRB jet and its environment.

Multi-wavelength studies of afterglows are predominantly performed within the
relativistic blast wave or fireball model (Rees & Meszaros, 1992). In this
framework, a relativistic shock at the front of the jet accelerates electrons
that produce the observed radiation. Broadband modeling of the afterglow
observations have been carried out with (semi-)analytical approximations of
the blast wave evolution (e.g., Chevalier & Li, 1999; Granot & Sari, 2002;
Panaitescu & Kumar, 2002; Yost et al., 2003), but more recently this has also
been done by fitting hydrodynamical simulations directly to the data (van
Eerten et al., 2012). Broadband modeling of a well-sampled afterglow can
result in tight constraints on all the physical parameters. While these
methods have proven to be powerful for studying the physics of the outflow and
its environment, the results reported show wide ranges for each of the
parameter values (Cenko et al., 2011), as well as conflicting results across
studies. Even in the cases of modeling a singular GRB afterglow, different
methods and modeling codes can produce widely varying values for the physical
parameters (Granot & van der Horst, 2014). This makes it, for instance,
difficult to establish the widths of distributions for these parameters.
Several studies have suggested the possibility of universal values for some of
the parameters (e.g., Kirk et al., 2000; Achterberg et al., 2001), especially
the micro-physical ones, but the current broadband modeling studies do not
allow for pinning this down.

To support the estimation of parameters through broadband modeling, there have
been a few studies focused on developing alternative means for parameter
estimation, in particular using only a few observables to constrain parameter
values independently. Nava et al. (2014) derived narrow distributions for two
of the physical parameters, the fraction of shock energy in electrons and the
efficiency of the prompt radiation mechanism, through normalized light curves
at high-energy gamma rays. Beniamini et al. (2015) used a combination of the
high-energy gamma-ray and X-ray emission to find a narrow distribution for the
efficiency of the prompt emission as well.

The work presented in this paper focuses on another diagnostic tool: using the
peaks in radio afterglows to constrain physical parameters that describe the
physics of electrons accelerated by the blast wave. Beniamini & van der Horst
(2017) used the peaks in radio light curves to constrain the fraction of shock
energy in electrons, $\epsilon_{e}$. They concluded that $\epsilon_{e}$ is
relatively narrowly distributed around $0.13-0.15$, with a width in log space
of $0.26-0.31$. We expand upon this work by increasing the sample size and
taking into account the peaks in broadband radio SEDs as well, thus further
constraining $\epsilon_{e}$, and placing constraints on two more physical
parameters: the minimum Lorentz factor of the accelerated electrons,
$\gamma_{m}$, and the fraction of electrons accelerated by the shock to a
power-law distribution, $\xi_{e}$.

Our sample of radio light curves and SEDs is outlined in Section 2, and the
methodology we follow is described in Section 3. Our results are presented in
Section 4, and in Section 5 we discuss our findings, with our conclusions
given in Section 6. We adopt the cosmological parameters $H_{0}$ = 69.6
km/s/Mpc, $\Omega_{\rm{M}}$ = 0.286, and $\Omega_{\Lambda}$ = 0.714.

##  2 Radio Sample

The sample used in this work consists of GRBs with radio afterglow
observations that provide well-sampled light curves or SEDs needed to
determine a peak. Beniamini & van der Horst (2017) used a sample of 36
afterglows compiled from the Chandra & Frail (2012) catalog of GRB radio
observations from 1997 to 2011, with a majority of the data taken with the
Very Large Array (VLA) at 8.5 GHz. For the new sample, we use bursts observed
from 2010 to 2019, from different radio observatories: the upgraded VLA,
Westerbork Synthesis Radio Telescope (WSRT), Arcminute Microkelvin Imager
Large Array (AMI-LA), Combined Array for Research in Millimeter-wave Astronomy
(CARMA), and Karoo Array Telescope (MeerKAT).

The sample has GRBs for which radio peaks could be determined in either light
curves or SEDs, with measurements ranging from from 1.3 to 15.7 GHz and 4 to
31 days. All GRBs in our sample have a spectroscopic redshift and a well-
determined isotropic equivalent gamma-ray energy,
$\text{E}_{\gamma,\text{iso}}$. This results in a sample size of 13 afterglows
(hereafter referred to as Sample 2), which we add to the Beniamini & van der
Horst (2017) data set (Sample 1), totaling 49 GRBs. For well-sampled
afterglows, we were able to identify peaks in more than one light curve or
SED. In those cases, we used the best-constrained peak in our analysis.
Results for the other peaks are included in Table 1 and Table 2, denoted in
italics. They have not been included in the statistical analysis on the
physical parameters derived from the radio peaks, nor in the results discussed
in Section 4.1.

The 36 afterglows from Sample 1 have a peak flux and time from the Chandra &
Frail (2012) catalog. In order to obtain the peak parameters for Sample 2, we
fit a smoothly broken power-law model to each well-defined peak, with a fairly
sharp smoothness parameter ($s=5$). The free parameters in the fitting routine
are the peak flux $F_{\nu_{p}}$, the peak time $t_{p}$ (for light curves) or
peak frequency $\nu_{p}$ (for SEDs), and the pre- and post-break power-law
slopes. The fit results for every GRB are listed in Table 1.

For Sample 1, 21 of the GRBs had a jet-break time derived from achromatic
breaks in optical and/or X-ray light curves, obtained from Chandra & Frail
(2012). 10 of the 13 GRBs in Sample 2 have jet break times reported in the
literature, which we give in Table 1.

Table 1: Radio peak and other relevant parameters for Sample 2, from light curve and SED modeling of observations from the Very Large Array (VLA), Westerbork Synthesis Radio Telescope (WSRT), Arcminute Microkelvin Imager Large Array (AMI-LA), Combined Array for Research in Millimeter-wave Astronomy (CARMA), and Karoo Array Telescope (MeerKAT). [1] Laskar et al. (2015); [2] Antonelli et al. (2010); [3] Marshall et al. (2011); [4] de Ugarte Postigo et al. (2018); [5] Moin et al. (2013); [6] Chornock et al. (2010); [7] van der Horst et al. (2015); [8] Zauderer et al. (2013); [9] Tello et al. (2012); [10] Laskar et al. (2018a); [11] Tanvir et al. (2014); [12] Higgins et al. (2019); [13] Anderson et al. (2018); [14] Cucchiara et al. (2015); [15] Perley et al. (2014); [16] de Ugarte Postigo & Tomasella (2015); [17] Golenetskii et al. (2015); [18] Laskar et al. (2016); [19] Tanvir et al. (2016a); [20] Alexander et al. (2017); [21] Xu et al. (2016); [22] Zhang et al. (2018); [23] Laskar et al. (2018b); [24] Tanvir et al. (2016b); [25] Alexander et al. (2019); [26] Kankare et al. (2017); [28] Valeev et al. (2019); [29] Rhodes et al. (2020) GRB | $F_{\nu_{p}}$ | $\nu_{p}$ | $t$ | $d_{28}$ | $z$ | $E_{\gamma,\text{iso}}$ | $t_{j}$ | Telescope | Ref.
---|---|---|---|---|---|---|---|---|---
| (mJy) | (Ghz) | (d) | $(10^{28}\text{cm})$ |  | $(10^{53}\text{ergs})$ | (d) |  |
_100418A_ | _0.58 $\pm$ 0.06_ | _4.8_ | _38 $\pm$ 21_ | _1.15_ | _0.624_ | _0.0099_ | _17_ | _VLA, WSRT_ | _1,2,3,4,5_
100418A | 1.42 $\pm$ 0.08 | 8.46 | 52.8 $\pm$ 6.1 | 1.15 | 0.624 | 0.0099 | 17 | VLA | 1,2,3,5
100901A | 0.32 $\pm$ 0.02 | 4.8 | 9.4 $\pm$ 1.7 | 3.16 | 1.41 | 0.8 | 0.96 | VLA, WSRT | 1,6
111215A | 0.84 $\pm$ 0.13 | 4.8 | 24.3 $\pm$ 6.5 | 5.06 | 2.01 | 0.45 | $>31$ | VLA, WSRT | 7,8
120326A | 0.83 $\pm$ 0.08 | 28.7 $\pm$ 7.3 | 15.4 | 4.28 | 1.80 | 0.32 | 1.5 | VLA, CARMA | 1,9
_120326A_ | _0.79 $\pm$ 0.10_ | _10.3 $\pm$ 1.6_ | _31.3_ | _4.28_ | _1.80_ | _0.32_ | _1.5_ | _VLA_ | _1,9_
140311A | 0.42 $\pm$ 0.06 | 23.0 $\pm$ 4.0 | 4.5 | 13.3 | 4.59 | 2.7 | 0.6 | VLA | 10,11
140713A | 1.67 $\pm$ 0.14 | 15.7 | 11.1 $\pm$ 0.8 | 1.90 | 0.935 | 0.017 | 30 | AMI-LA | 12,13
141121A | 0.22 $\pm$ 0.03 | 15.7 | 16.4 $\pm$ 3.4 | 3.34 | 1.47 | 0.80 | 3 | AMI-LA, VLA | 13,14,15
150413A | 0.25 $\pm$ 0.06 | 15.7 | 4.7 $\pm$ 0.4 | 8.45 | 3.14 | 6.5 | … | AMI-LA | 13,16,17
160509A | 1.39 $\pm$ 0.09 | 8.1 $\pm$ 0.6 | 4.1 | 2.51 | 1.17 | 5.76 | 6 | VLA | 18,19
_160509A_ | _0.59 $\pm$ 0.02_ | _5.0_ | _2.3 $\pm$ 0.2_ | _2.51_ | _1.17_ | _5.76_ | _6_ | _VLA_ | _18,19_
_160625B_ | _0.61 $\pm$ 0.19_ | _2.1 $\pm$ 0.8_ | _12.5_ | _3.15_ | _1.41_ | _30_ | _25_ | _VLA_ | _20,21,22_
160625B | 0.59 $\pm$ 0.07 | 1.9 $\pm$ 0.4 | 22.5 | 3.15 | 1.41 | 30 | 25 | VLA | 20,21,22
161219B | 3.90 $\pm$ 0.28 | 22.8 $\pm$ 1.1 | 1.52 | 0.13 | 0.148 | 0.0018 | 32 | VLA | 23,24,25
171010A | 1.35 $\pm$ 0.02 | 14.2 $\pm$ 0.8 | 9.66 | 0.53 | 0.33 | 2.2 | … | VLA | 26,27
190829A | 1.67 $\pm$ 0.33 | 1.3 | 12.9 $\pm$ 3.8 | 0.011 | 0.0785 | 0.0030 | … | MeerKAT | 28,29

##  3 Methodology

The afterglow of GRBs can be described by an expanding, relativistic shell or
blast wave propagating outward through an external medium (Rees & Meszaros,
1992). The broadband synchrotron radiation we observe is emitted by electrons
accelerated by the shock interacting with the ambient medium, producing the
afterglow that can be seen from X-ray to radio wavelengths on timescales of
seconds to years (e.g., Sari et al., 1998; Wijers & Galama, 1999).

The dynamics of the blast wave is characterized by its isotropic equivalent
kinetic energy, $E_{iso}$, which is driving the shock outward through the
ambient medium, where the density of the medium, $n$, also plays a role in the
blast wave evolution. The ambient medium is typically assumed to be
homogeneous or structured as a stellar wind, with the density dropping off as
one over the distance from the central engine squared (Chevalier & Li, 1999).
The electrons behind the shock are accelerated to a power-law distribution of
energies, with power-law index $p$; and with $\epsilon_{e}$ describing the
ratio of shock energy in electrons to the total energy density. Similarly,
$\epsilon_{B}$ describes the ratio of energy density in the magnetic field to
the total energy.

These micro- and macrophysical parameters can be determined from several
observables that describe the synchrotron emission spectrum. The latter is
typically characterized as a series of connected power-laws, with three
characteristic frequencies connecting each regime: the peak frequency
$\nu_{m}$, the cooling frequency, $\nu_{c}$, and the synchrotron self-
absorption frequency, $\nu_{a}$. These observable parameters, as well as the
peak flux, evolve with time and determine the shape of the spectrum at a given
time (e.g., Granot & Sari, 2002). $\nu_{c}$ is typically found in the X-ray or
optical bands, while $\nu_{m}$ is found to evolve from the optical to the
radio regime (as a power law in time with slope -3/2 or even faster than
that), and $\nu_{a}$ is usually detected in the radio from the onset of a GRB.
As these three characteristic frequencies evolve over time, they will cross
through observing frequencies, creating light curve breaks or peaks. The
latter can also occur across the electromagnetic spectrum at the same time,
which is caused by dynamical rather than spectral effects. Examples of this
are the jet-break time $t_{j}$, which is related to the opening angle of the
jet, or the time at which the blast wave becomes sub- or non-relativistic.

In this work, we assume that the observed radio peaks are due to $\nu_{m}$.
While there are scenarios in which $\nu_{m}$ is below $\nu_{a}$, resulting in
the peak of the SED being $\nu_{a}$ instead of $\nu_{m}$, for typical physical
parameters this occurs at late times (months after the GRB onset) and is
mostly observed at low radio frequencies (e.g. van der Horst et al., 2008;
Beniamini & van der Horst, 2017). We confirmed this by checking that the
slopes in both light curves and SEDs indeed match the theoretical expectations
for $\nu_{m}$ being the peak (Granot & Sari, 2002). Furthermore, if the radio
peak is caused by $\nu_{a}$ instead of $\nu_{m}$, the derived $\Psi$ parameter
(see Section 3.1) would deviate strongly from the typical range and could have
an unphysical value (see also Section 5.4). The latter is also true if the
observed radio peak is caused by the reverse shock instead of the forward
shock (see also Beniamini & van der Horst, 2017).

A fraction $\xi_{e}$ of the total number of electrons is accelerated into a
power-law distribution of electron energies or Lorentz factors, with a minimum
Lorentz factor $\gamma_{m}$, and emit synchrotron radiation. In many broadband
modeling studies, $\epsilon_{e}$ is assumed to describe the entire population
of electrons, with $\xi_{e}$ equal to 1. This assumption is commonly made
since $\xi_{e}$ is degenerate with the energy, density, and $\epsilon_{e}$ and
$\epsilon_{B}$ in standard broadband modeling (Eichler & Waxman, 2005); and
broadband modeling efforts including a variable $\xi_{e}$ result in poor
constraints on its value (e.g., Cunningham, 2021). However, it has been shown
that $\xi_{e}\ll 1$ is necessary for prompt emission models that are based on
synchrotron emission from a kinetic dominated outflow, such as internal shocks
(Daigne & Mochkovitch, 1998; Bošnjak et al., 2009; Beniamini & Piran, 2013).
In the following subsections we show that constraints on $\xi_{e}$ at the
forward shock can be obtained from the peaks in radio afterglows, with a less
stringent degeneracy that hampers such constraints from broadband modeling.

###  3.1 $\Psi$ as A Proxy for $\epsilon_{e}$ and $\xi_{e}$

Beniamini & van der Horst (2017) presented a framework in which they defined a
parameter $\Psi$ as a proxy for $\epsilon_{e}$, which can be derived from
observational properties of the peaks in radio light curves. Based on
equations for the peak flux and peak frequency as a function of time and the
aforementioned physical parameters (Granot & Sari, 2002), Beniamini & van der
Horst (2017) derived equations for the parameter $\Psi$ from the ratio of the
peak flux and peak frequency. We have modified the equations for $\Psi$ by
including the parameter $\xi_{e}$, which was set equal to 1 in Beniamini & van
der Horst (2017), to investigate the relationship between $\epsilon_{e}$ and
$\xi_{e}$. Following Beniamini & van der Horst (2017), we show the equations
for a homogeneous circumburst medium, such as expected for the interstellar
medium (ISM), as well as a stellar wind (Wind):

$\begin{split}\Psi_{\text{ISM}}&=\left(\frac{261.4~{}(1+z)^{1/2}~{}\nu_{p}~{}t_{p}^{3/2}~{}E_{\gamma,iso,53}^{1/2}}{10^{15}~{}d_{28}^{2}~{}F_{\nu_{p}}~{}\text{max}(1,t_{p}/t_{j})^{1/2}}\right)^{1/2}\\\
&=\frac{(p-2)}{0.177~{}(p-1)}\left(\frac{p-0.67}{p+0.14}\right)^{1/2}\left(\frac{1-\epsilon_{\gamma}}{\epsilon_{\gamma}}\right)^{-1/4}n_{0}^{-1/4}\epsilon_{e}~{}\xi_{e}^{-3/2}\end{split}$
(1)

$\begin{split}\Psi_{\text{Wind}}&=\left(\frac{249.4~{}(1+z)~{}\nu_{p}~{}t_{p}}{10^{15}~{}d_{28}^{2}~{}F_{\nu_{p}}}\right)^{1/2}\\\
&=\frac{(p-2)}{0.277~{}(p-1)}\left(\frac{p-0.69}{p+0.12}\right)^{1/2}A_{*}^{-1/2}\epsilon_{e}~{}\xi_{e}^{-3/2}\end{split}$
(2)

Equations 1 and 2 have been structured such that the top line of each equation
contains observable parameters and the bottom line contains the physical
parameters. The observed peak parameters in these equations are the peak
frequency $\nu_{p}$ in GHz, the peak flux $F_{\nu_{p}}$ in mJy, and the peak
time $t_{p}$ in days. These three are determined from the afterglow light
curves or SEDs. For the other observable parameters, the redshift $z$ is
obtained from optical spectroscopy of the afterglow (or in some cases the host
galaxy), as well as the luminosity distance $d_{28}$ in units of $10^{28}$ cm;
$E_{\gamma,\text{iso},53}$ is the isotropic equivalent energy of the burst in
units of $10^{53}$ erg; and the jet break time $t_{j}$, in days, is derived
from achromatic light curve breaks where possible. For the physical
parameters, the density $n_{0}$ in a homogeneous medium is given in cm-3, and
for a stellar wind the density is characterized by $A_{*}$ following Chevalier
& Li (1999); $\epsilon_{\gamma}$ is the energy efficiency of the prompt
emission, with $\epsilon_{\gamma}=E_{\gamma,iso}/(E_{\gamma,iso}+E_{K,iso})$;
and $p$ the aforementioned electron energy distribution power-law index. As
can be seen from Equations 1 and 2$,\Psi$ has a weak dependence on most
physical parameters except for $\epsilon_{e}$ and $\xi_{e}$, which allows us
to use $\Psi$ as a proxy for the combination of parameters
$\epsilon_{e}~{}\xi_{e}^{-3/2}$.

###  3.2 $\chi$ as A Proxy for $\gamma_{m}$ and $\xi_{e}$

Following similar steps to deriving $\Psi$ as a proxy for $\epsilon_{e}$ and
$\xi_{e}$, we derive another proxy, $\chi$, for a parameter related to
electron acceleration in GRB jets: $\gamma_{m}$, the minimum Lorentz factor of
the electron distribution; and also $\chi$ has a dependence on $\xi_{e}$. We
use the relation between $\gamma_{m}$ and $\epsilon_{e}$ as detailed in Sari
et al. (1998), with the addition of a linear dependence of $\gamma_{m}$ on
$\xi_{e}$. Adopting standard equations for the shock Lorentz factor and the
equations for $\Psi$ given above, we derive $\chi$, which can be estimated
from observables, is linear in $\gamma_{m}$, and weakly dependent on other
physical parameters except for $\xi_{e}$ (with $t_{d}$ the time in days):

$\begin{split}\chi_{\text{ism}}&=266~{}\Psi_{\text{ism}}~{}E_{\gamma,\text{iso},53}^{1/8}~{}(z+1)^{3/8}~{}t_{d}^{-3/8}\\\
&=\left(\frac{p-0.67}{0.66~{}(p+0.14)}\right)^{1/2}\left(\frac{\epsilon_{\gamma}}{1-\epsilon_{\gamma}}\right)^{3/8}~{}n_{0}^{-1/8}~{}\gamma_{m}~{}\xi_{e}^{-1/2}\end{split}$
(3)

$\begin{split}\chi_{\text{wind}}&=475~{}\Psi_{\text{wind}}~{}E_{\gamma,\text{iso},53}^{1/4}~{}(z+1)^{1/4}~{}t_{d}^{-1/4}\\\
&=\left(\frac{p-0.69}{0.65(p+0.12)}\right)^{1/2}\left(\frac{\epsilon_{\gamma}}{1-\epsilon_{\gamma}}\right)^{1/4}~{}A_{*}^{-1/4}~{}\gamma_{m}~{}\xi_{e}^{-1/2}\end{split}$
(4)

The top line in these equations contains observables and $\Psi$, and the
bottom line of each equation contains the physical parameters. One can see
that $\chi$ is a proxy for the combination $\gamma_{m}~{}\xi_{e}^{-1/2}$.

##  4 Results

We have calculated the values for $\Psi$ and $\chi$, and their uncertainties,
for all GRBs in Samples 1 and 2. They are listed in Table 2 for both the
homogeneous (ISM) and stellar wind (Wind) ambient medium structure. Below we
discuss the results for our GRB samples. We note that in Table 1 we show the
results for two different peaks in the case of four GRBs in Sample 2, for
instance light curves at two different frequencies or SEDs at two different
times. The $\Psi$ and $\chi$ values derived for those different peaks are
consistent with each other for all four GRBs. In all the sample statistics
below we use the peak that is best sampled or defined by the data, and the
other peak is indicated in italic font in Tables 1 and 2.

###  4.1 Results for $\Psi$

Figure 1: Histogram in log space of $\Psi$ for our GRB sample, for the
homogeneous (ISM; blue) and stellar wind (Wind; orange) cases.

Figure 2: $\Psi$ versus peak flux for a homogeneous (top panel) and stellar
wind (bottom panel) medium, for the two samples of GRBs presented in this
paper.

Figure 1 shows a histogram of $\Psi$ for the ISM (blue) and Wind (orange)
circumburst medium cases, for all the GRBs listed in Table 2. The weighted
averages for $\Psi$, as well as the widths of the distributions, are
consistent between the two samples of GRBs, i.e. those from Beniamini & van
der Horst (2017) and our additional sample. We find that the weighted averages
for our total sample (Samples 1 and 2 combined) are $\Psi_{\text{ISM}}$ = 0.11
and $\Psi_{\text{wind}}$ = 0.14, and the distribution widths in log space are
$\sigma_{\text{$\Psi$,ISM}}=0.32$ and $\sigma_{\text{$\Psi$,Wind}}=0.28$.

We test for a possible sample bias or correlation between $\Psi$ and the peak
flux; see Figure 2. We find a Spearman’s rank correlation coefficient of -0.2
or -0.07 between the two parameters, with a p value of 0.08 or 0.6, for the
ISM and Wind cases, respectively. This implies that there is no significant
relationship between $\Psi$ and the peak flux. We note that our additional
Sample 2 has on average higher peak flux values than Sample 1 from Beniamini &
van der Horst (2017). This is due to the fact that we selected those GRBs for
which a peak could be accurately determined from well-sampled light curves or
SEDs, while the light curves from Chandra & Frail (2012) were not necessarily
sampled as well.

###  4.2 Results for $\chi$

Figure 3: Histogram in log space of $\chi$ at 1 day for our entire GRB sample,
for the homogeneous (ISM; blue) and stellar wind (Wind; orange) cases.

Figure 4: $\chi$ at 1 day versus $E_{\gamma,\text{iso},53}$ for a homogeneous
(top panel) and stellar wind (bottom panel) medium, for the two samples
presented here.

Figure 3 shows a histogram of $\chi$ at 1 day for the ISM (blue) and Wind
(orange) ambient medium cases, for all GRBs listed in Table 2. We find that
the weighted averages are $\chi_{\text{ISM}}$ = 28 and $\chi_{\text{wind}}$ =
63, and the distribution widths in log space are
$\sigma_{\text{$\chi$,ISM}}=0.36$ and $\sigma_{\text{$\chi$,Wind}}=0.29$,
similar to the distribution widths for $\Psi$. We note that the $\chi$ values
and weighted averages will change as a function of time following the time
dependence in Equations 3 and 4, but the widths of the distributions in log
space will remain constant.

We examined possible correlations between $\chi$ and any of the observables.
We only find a marginal correlation between $\chi$ and
$E_{\gamma,\text{iso},53}$, shown in Figure 4. The Spearman’s rank correlation
coefficient between these two parameters is 0.5 or 0.4, with a p value of
$1\cdot 10^{-5}$ or $4\cdot 10^{-4}$, for the ISM and Wind cases,
respectively. This marginal correlation seems to be driven by a couple of GRBs
with relatively low and high values for $\chi$ and $E_{\gamma,\text{iso},53}$.
Given the marginal significance of the correlation and the strong dependence
on a few data points, we do not discuss this possible correlation further.

##  5 Discussion

GRB afterglows uniquely enable us to study the evolution of the synchrotron
spectral peak frequency $\nu_{m}$, due to the relatively high $\gamma_{m}$
compared to other jet sources, such as active galactic nuclei and X-ray
binaries (e.g., Jorstad et al., 2001; Miller-Jones et al., 2006). In those
other cases of relativistic jet emission, the Lorentz factor of the shocks
that are accelerating electrons are too low for $\nu_{m}$ to be significantly
above the synchrotron self-absorption frequency $\nu_{a}$, which is typically
associated with the SED peak in those sources. In particular, $\nu_{m}$ is
proportional to the bulk Lorentz factor of the blast wave to the fourth power
(Sari et al., 1998), which has a large impact for GRBs that have Lorentz
factors of tens to hundreds at early times (Rees & Meszaros, 1992). Being able
to measure $\nu_{m}$ and the associated peak flux allows for constraints in
GRBs on $\gamma_{m}$, $\epsilon_{e}$ and $\xi_{e}$, as discussed below.

###  5.1 Constraints on $\gamma_{m}$, $\epsilon_{e}$ and $\xi_{e}$

Figure 5: Representation of the lower and upper limits for $\xi$ and
$\epsilon_{e}$, based on $\gamma_{m}\gtrsim 2$ taken at a time of 50 days
after the burst (vertical red lines) and $\epsilon_{\gamma}\gtrsim 0.01$
(horizontal grey lines). Solid red lines represent the lower limit of $\xi$
for $\gamma_{m}\gtrsim 2$, and dashed red lines are for $\xi$ plus one
standard deviation of $\xi$.

As laid out in Section 3, $\Psi$ is a proxy for
$\epsilon_{e}~{}\xi_{e}^{-3/2}$, and $\chi$ at a given time is a proxy for
$\gamma_{m}~{}\xi_{e}^{-1/2}$. For further calculations, we assume values of
$\epsilon_{\gamma}=0.15$ (Beniamini et al., 2015), $p=2.3$ (e.g., Ryan et al.,
2015), $n_{0}=1$ and $A_{*}=1$ (e.g., Granot & van der Horst, 2014), with the
note that while some of these physical parameters may have relatively broad
distributions, $\Psi$ and $\chi$ have weak dependencies on those physical
parameters (see also discussion below). From the weighted averages of $\Psi$
and $\chi$, we calculate the most likely values of the following combinations
of physical parameters:

  * •

$\epsilon_{e}~{}\xi_{e}^{-3/2}\sim 0.16$ for ISM and $0.21$ for Wind

  * •

$\gamma_{m}~{}\xi_{e}^{-1/2}\sim 53~{}t_{d}^{-3/8}$ for ISM and
$96~{}t_{d}^{-1/4}$ for Wind

One could translate these numbers to values for $\epsilon_{e}$ and
$\gamma_{m}$ directly by making assumptions on $\xi_{e}$. For instance, the
most likely value of $\epsilon_{e}$ is $0.16-0.21$ for $\xi_{e}=1$, decreasing
to $(5.1-6.6)\cdot 10^{-3}$ for $\xi_{e}=0.1$ and $(1.6-2.1)\cdot 10^{-4}$ for
$\xi_{e}=0.01$. However, there are lower limits one can set on $\xi_{e}$ based
on the constraints on $\gamma_{m}~{}\xi_{e}^{-1/2}$. The peak frequency
$\nu_{m}$ has been identified as such in some well-observed GRB afterglows,
such as GRB 970508 and GRB 030329, out to $\sim 50$ days (e.g. Frail et al.,
2000; Frail et al., 2005; van der Horst et al., 2005; Granot & van der Horst,
2014). At those times, the GRB blast wave was still relativistic, and we put a
conservative lower limit of $\gamma_{m}\gtrsim 2$ at $50$ days. This implies
lower limits on $\xi_{e}$ of $\gtrsim 2.5\cdot 10^{-2}$ or $\gtrsim 3.1\cdot
10^{-3}$ for ISM or Wind, respectively.

Given the limits on $\xi_{e}$, we calculate ranges for $\epsilon_{e}$ of
$6.6\cdot 10^{-4}\lesssim\epsilon_{e}\lesssim 0.16$ for a homogeneous medium
and $3.6\cdot 10^{-5}\lesssim\epsilon_{e}\lesssim 0.21$ for a stellar wind
medium.

In Figure 5, we show a representation of the limits (solid vertical red lines)
on, and dependence (solid diagonal grey lines) of, $\xi_{e}$ and
$\epsilon_{e}$, for our conservative limit on $\gamma_{m}$. This figure
includes uncertainty regions, bound by diagonal dashed lines, based on the
standard deviation in log-space of the $\Psi$ distributions. It also shows the
lower limits on $\xi_{e}$, and therefore on $\epsilon_{e}$, for taking into
account the standard deviation in log-space of the $\chi$ distributions
(dashed vertical red lines). In that case, the allowed ranges for both
$\xi_{e}$ and $\epsilon_{e}$ are significantly smaller. We also note that
while $\epsilon_{e}$ and $\xi_{e}$ are allowed to both be small, this would
have observable consequences: having a low fraction of electrons accelerated
into a power-law distribution means that there is a significant thermal
electron population, which could be observed at radio wavelengths (Ressler &
Laskar, 2017; Warren et al., 2018). However, since a clear signature of such a
thermal electron population has not been observed, $\xi_{e}$ is likely not
much lower than $\sim 0.1$, which means that $\epsilon_{e}$ should have a
lower limit of almost $0.01$. As we show in the following sub-section, energy
budget considerations constrain $\epsilon_{e}$ and $\xi_{e}$ even further, and
exclude $\xi_{e}$ values that would lead to a significant thermal electron
population.

####  5.1.1 Constraints from $\epsilon_{\gamma}$

Nava et al. (2014) and Beniamini et al. (2015) put constraints on
$\epsilon_{\gamma}$ and $\epsilon_{e}$ based on modeling the X-ray and high-
energy gamma-ray light curves. They used the prompt gamma-ray fluence and the
afterglow light curve as measured by the _Fermi Gamma-ray Space Telescope_
(_Fermi_) Large Area Telescope (LAT), and the fact that the latter is in a
spectral regime of fast cooling (i.e., $\nu_{m}>\nu_{c}$) and without any
synchrotron self-Compton cooling effects. They used scalings for the afterglow
flux as a function of physical parameters (Granot & Sari, 2002), with
$\xi_{e}=1$, to show that $\epsilon_{\gamma}$ and $\epsilon_{e}$ have narrow
distributions, for $\epsilon_{e}$ with similar distribution widths as those
that we have derived from radio peaks. They found a value of
$\epsilon_{\gamma}\simeq 0.15$ for $\epsilon_{e}\simeq 0.1$, and a scaling
between these two parameters. In this spectral regime, the scaling of the
afterglow flux with almost all the physical parameters is identical between a
homogeneous and stellar wind medium:

$\begin{split}F_{\nu}&=f(p,\nu,t,z,d_{L})~{}\epsilon_{e}^{p-1}~{}\xi_{e}^{-(p-2)}~{}E_{iso}^{(p+2)/4}~{}\epsilon_{B}^{(p-2)/4}\\\
&=f(p,\nu,t,z,d_{L},E_{\gamma,iso})~{}\epsilon_{e}^{p-1}~{}\xi_{e}^{-(p-2)}~{}\left(\frac{1-\epsilon_{\gamma}}{\epsilon_{\gamma}}\right)^{(p+2)/4}~{}\epsilon_{B}^{(p-2)/4}\end{split}$
(5)

in which $f$ is a function of observables and $p$.

For a large range of $p$ values, the afterglow flux has a very weak dependence
on $\epsilon_{B}$ and also a fairly weak dependence on $\xi_{e}$. It does have
a strong dependence on $\epsilon_{e}$ and $\epsilon_{\gamma}$, roughly linear
in both. Since the afterglow flux measured in the _Fermi_ LAT is narrowly
distributed (Nava et al., 2014), and the flux strongly depends on only
$\epsilon_{e}$ and $\epsilon_{\gamma}$, any change in one of the parameters
leads to a change in the other as well. For instance, $\epsilon_{e}=10^{-2}$
corresponds to $\epsilon_{\gamma}\simeq 10^{-2}$, and $\epsilon_{e}=10^{-3}$
to $\epsilon_{\gamma}\simeq 7\cdot 10^{-4}$. The latter value for
$\epsilon_{\gamma}$ is unphysical, because it requires an extremely large
kinetic energy in the GRB blast wave. We put a conservative lower limit of
$\epsilon_{\gamma}\gtrsim 0.01$: for an observed gamma-ray energy of
$E_{\gamma,iso}\simeq 10^{54}$ erg, a gamma-ray efficiency of 0.01 implies
$E_{iso}\simeq 10^{56}$ erg and a true, collimation-corrected energy of $\sim
10^{53-54}$ erg, which is well above the energetics that can be provided by
GRB progenitor and central engine models (Beniamini et al., 2017). Therefore,
we use this lower limit on $\epsilon_{\gamma}$ to set a lower limit on
$\epsilon_{e}$ of 0.01. This is indicated in Figure 5 as a horizontal grey
line, and corresponds to a lower limit of $\xi_{e}\gtrsim 0.1$.

Given the constraints on $\xi_{e}$, ranging from $\sim 0.1$ to 1, we put
constraints on the range of most likely values for $\gamma_{m}$ at a given
observer time: $(16-53)t_{d}^{-3/8}$ for ISM and $(30-96)t_{d}^{-1/4}$ for the
Wind case, using the fact that $\gamma_{m}$ is proportional to $\xi_{e}^{1/2}$
(see Equations 3 and 4).

###  5.2 Comparison to Broadband Modeling

The methodology presented in Beniamini & van der Horst (2017) and this paper
can be used as a diagnostic tool for broadband modeling studies. Since we
focus on only radio light curves and SEDs, the amount of information is
limited compared to full-blown modeling across the electromagnetic spectrum.
However, the constraints on physical parameters derived from radio afterglow
peaks should be consistent with those from broadband modeling. Granot & van
der Horst (2014) compiled broadband modeling studies by various authors using
different methodologies, finding a range of $\epsilon_{e}$ between $4\cdot
10^{-3}$ and $0.3$ (all for $\xi_{e}=1$). This is a significantly broader
range than the widths of our distributions for $\Psi$ or $\epsilon_{e}$, but
this is likely due to the wide variety of methodologies used.

Aksulu et al. (2022) studied 22 well-sampled long GRB afterglows to obtain
estimates for the physical parameters, using a state-of-the-art, simulations-
based modeling methodology. The standard deviation of their distributions for
$\epsilon_{e}$ are 0.24 for ISM and 0.28 for Wind, close to the widths of our
distributions for $\Psi$. They find $\epsilon_{e}$ values between 0.1 and 1,
with mean values of 0.34 for a homogeneous medium and 0.28 for a stellar wind
medium, which are higher than our findings. We note that a larger density by a
factor of a few (Wind) up to $\sim 10$ (ISM) would result in consistent
$\epsilon_{e}$ values; and Aksulu et al. (2022) indeed find a distribution of
densities with a peak that is significantly larger than 1 for both $n_{0}$ and
$A_{*}$.

The findings presented in this paper, and recent broadband modeling studies
(such as Aksulu et al., 2022) reporting similar distribution widths for, and
values of, $\epsilon_{e}$, shows the validity of using radio peaks as
diagnostic tools for broadband modeling. This also means that the distribution
widths and values for $\epsilon_{e}$, $\gamma_{m}$ and $\xi_{e}$ presented in
this paper can be used for direct comparison with expectations from electron
acceleration simulations for relativistic shocks (e.g., Sironi & Spitkovsky,
2011; Bykov et al., 2012; Marcowith et al., 2016). However, this requires
careful consideration of the effects of all physical parameters on the
narrowness of the derived distributions.

###  5.3 Universality of physical parameters

The aforementioned constraints on physical parameters are assuming that the
other physical parameters at play are universal among GRB afterglows. Changing
$p$ from $2.3$ to $2.1$ or $3$, following distributions presented in the
literature (e.g. Starling et al., 2008; Ryan et al., 2015), does not have a
significant effect on the value of $\chi$, but does change $\Psi$ by a factor
of $\sim 2.7$ at most, i.e. $\sim 0.4$ in log-space. On the other hand,
varying $\epsilon_{\gamma}$ does not have a significant effect on $\Psi$ but a
small effect on $\chi$: changing $\epsilon_{\gamma}$ by a factor 3, which is
quite extreme compared to current constraints (Nava et al., 2014; Beniamini et
al., 2015), results in a factor $\sim 1.6$ change in $\chi$, i.e. $\sim 0.2$
in log-space.

Varying the density has an effect on both $\Psi$ and $\chi$. The magnitude of
variations is quite uncertain, but broadband modeling studies suggest about
one order of magnitude larger or smaller (Granot & van der Horst, 2014; Aksulu
et al., 2022). This leads to changes in $\Psi$ of factors 1.9 (ISM) or 3.2
(Wind), i.e. 0.27 (ISM) or 0.5 (Wind) in log-space; and to changes in $\chi$
of factors 1.3 (ISM) or 1.8 (Wind), i.e. 0.12 (ISM) or 0.25 (Wind) in log-
space.

These variations in $\Psi$ and $\chi$ should be compared to their distribution
widths presented in Section 4: 0.32 (ISM) and 0.28 (Wind) for $\Psi$, and
$0.36$ (ISM) and 0.29 (Wind) for $\chi$. One could conclude from the
considerations above that these distribution widths can be explained largely
or even completely by variations in the ambient medium density, and allow for
(almost) universal values for the physical parameters related to the electron
acceleration process in GRB blast waves. However, it is important to keep in
mind possible biases in the sample selection that may lead to such
conclusions. While there is no significant correlation between $\Psi$ and the
radio brightness (see Figure 2), the GRBs in our sample have been selected
because a radio peak could be detected. Chandra & Frail (2012) showed that
their sample of radio afterglows was sensitivity limited, and the added sample
presented in this paper is biased towards well-sampled radio afterglows. It is
possible that a larger range of radio peak fluxes would lead to a broader
range of microphysical parameters as well. The peak flux does not depend on
$\epsilon_{e}$, but is proportional to $\xi_{e}$, so it is possible that GRBs
with lower peak fluxes have lower $\xi_{e}$, and also lower $\epsilon_{e}$,
values. Addressing this possible bias will require more dedicated observations
with current telescopes and deeper observations with new-generation radio
observatories such as the Square Kilometer Array (SKA; e.g., Burlon et al.,
2015) and the next-generation Very Large Array (ngVLA; e.g., McKinnon et al.,
2019).

###  5.4 Evolution of Microphysical Parameters

Figure 6: $\Psi$ for GRB 030329 from light curves at different frequencies
(indicated in figure, in GHz), for homogeneous (ISM; blue) and stellar wind
(Wind; orange) cases, and with vertical lines indicating $\nu_{m}$ crossing
$\nu_{a}$ and the blast wave becoming sub-relativistic.

Up to this point we have assumed that the microphysical parameters are
constant as a function of time. Broadband modeling studies (e.g., Filgas et
al., 2011; van der Horst et al., 2014) and particle acceleration simulations
(e.g., Lemoine, 2015) indicate that $\epsilon_{e}$ may be time variable,
although the variability could be mostly in the magnetic field rather than the
electron acceleration. Given that the $\epsilon_{e}$ distribution is narrow,
it is unlikely that there is a strong temporal evolution of $\epsilon_{e}$,
unless it is always similar around the typical radio peak observations.

One way to investigate the possible temporal evolution of $\epsilon_{e}$ is to
calculate this parameter from SEDs at different times or light curves at
different radio frequencies; and covering a large time span, so variability
can be measured or strongly constrained. A good case study for this is GRB
030329, which was observed across several radio frequencies, with well-
determined peaks in at least 7 of them. Figure 6 shows the values of $\Psi$ as
a function of time, based on the light curves presented in van der Horst et
al. (2008), and assuming that the light curve peak is the passage of $\nu_{m}$
across the observing band. The first four light curve peaks result in similar
values for $\Psi$, but at later times the value of $\Psi$ deviate
significantly. This is not surprising, given that broadband modeling has shown
that $\nu_{m}$ crosses $\nu_{a}$ at $\sim 20$ days, making $\nu_{a}$ the peak
of the spectrum after that time; and the blast wave becomes sub-relativistic
and eventually non-relativistic at some point, violating the assumptions for
our calculations of $\Psi$. Since the derived $\Psi$ values before $20$ days
do not evolve, there does not seem to be any temporal evolution of
$\epsilon_{e}$ in this particular GRB afterglow. This will need to be further
tested with other GRBs that have multiple radio peaks spanning a large time
range.

Table 2: Values of $\Psi$ and $\chi$ calculated for the entire sample of GRB afterglows. Italicized lines indicate GRBs with a second peak that are not included in sample calculations (and only included for comparison purposes). Name | $\Psi_{\text{ISM}}$ | $\Psi_{\text{Wind}}$ | $\chi_{\text{ISM}}$ | $\chi_{\text{Wind}}$
---|---|---|---|---
190829A | 0.25 $\pm$ 0.08 | 0.53 $\pm$ 0.16 | 35 $\pm$ 10 | 64 $\pm$ 19
171010A | 0.78 $\pm$ 0.04 | 0.38 $\pm$ 0.02 | 256 $\pm$ 14 | 238 $\pm$ 13
161219B | 0.059 $\pm$ 0.003 | 0.26 $\pm$ 0.01 | 8 $\pm$ 1 | 26 $\pm$ 1
_160625B_ | _0.20 $\pm$ 0.09_ | _0.06 $\pm$ 0.02_ | _116 $\pm$ 48_ | 7 _8 $\pm$ 32_
160625B | 0.34 $\pm$ 0.07 | 0.08 $\pm$ 0.02 | 193 $\pm$ 39 | 113 $\pm$ 23
160509A | 0.093 $\pm$ 0.007 | 0.050 $\pm$ 0.004 | 42 $\pm$ 3 | 45 $\pm$ 3
_160509A_ | _0.076 $\pm$ 0.005_ | _0.047 $\pm$ 0.003_ | _34 $\pm$ 2_ | _42 $\pm$ 3_
150413A | 0.45 $\pm$ 0.04 | 0.22 $\pm$ 0.02 | 195 $\pm$ 18 | 198 $\pm$ 18
141121A | 0.29 $\pm$ 0.06 | 0.29 $\pm$ 0.06 | 106 $\pm$ 22 | 161 $\pm$ 34
140713A | 0.076 $\pm$ 0.006 | 0.13 $\pm$ 0.01 | 16 $\pm$ 1 | 27 $\pm$ 2
140311A | 0.037 $\pm$ 0.007 | 0.050 $\pm$ 0.009 | 22 $\pm$ 4 | 47 $\pm$ 8
120326A | 0.11 $\pm$ 0.03 | 0.16 $\pm$ 0.04 | 36 $\pm$ 9 | 74 $\pm$ 19
_120326A_ | _0.09 $\pm$ 0.02_ | _0.14 $\pm$ 0.02_ | _31 $\pm$ 5_ | _65 $\pm$ 10_
111215A | 0.10 $\pm$ 0.03 | 0.07 $\pm$ 0.02 | 37 $\pm$ 10 | 37 $\pm$ 10
100901A | 0.08 $\pm$ 0.01 | 0.10 $\pm$ 0.02 | 29 $\pm$ 5 | 58 $\pm$ 11
100814A | 0.127 $\pm$ 0.005 | 0.099 $\pm$ 0.004 | 44 $\pm$ 2 | 51 $\pm$ 2
_100418A_ | _0.20 $\pm$ 0.12_ | _0.35 $\pm$ 0.20_ | _36 $\pm$ 21_ | _59 $\pm$ 33_
100418A | 0.20 $\pm$ 0.02 | 0.35 $\pm$ 0.04 | 36 $\pm$ 4 | 59 $\pm$ 7
100414A | 0.234 $\pm$ 0.009 | 0.101 $\pm$ 0.004 | 112 $\pm$ 4 | 100 $\pm$ 4
091020 | 0.131 $\pm$ 0.007 | 0.110 $\pm$ 0.006 | 46 $\pm$ 3 | 55 $\pm$ 3
090902B | 0.9 $\pm$ 0.1 | 0.26 $\pm$ 0.05 | 561 $\pm$ 104 | 375 $\pm$ 69
090715B | 0.05 $\pm$ 0.01 | 0.09 $\pm$ 0.02 | 27 $\pm$ 7 | 75 $\pm$ 19
090424 | 0.26 $\pm$ 0.04 | 0.31 $\pm$ 0.04 | 74 $\pm$ 10 | 134 $\pm$ 18
090423 | 0.22 $\pm$ 0.04 | 0.15 $\pm$ 0.03 | 138 $\pm$ 26 | 131 $\pm$ 25
090328 | 0.36 $\pm$ 0.02 | 0.23 $\pm$ 0.01 | 117 $\pm$ 5 | 127 $\pm$ 6
090313 | 0.052 $\pm$ 0.003 | 0.054 $\pm$ 0.003 | 30 $\pm$ 2 | 54 $\pm$ 3
071010B | 0.12 $\pm$ 0.01 | 0.13 $\pm$ 0.02 | 33 $\pm$ 4 | 52 $\pm$ 6
071003 | 0.13 $\pm$ 0.01 | 0.072 $\pm$ 0.006 | 55 $\pm$ 4 | 59 $\pm$ 5
070125 | 0.42 $\pm$ 0.01 | 0.209 $\pm$ 0.005 | 209 $\pm$ 5 | 221 $\pm$ 5
051022 | 0.41 $\pm$ 0.06 | 0.19 $\pm$ 0.03 | 170 $\pm$ 23 | 168 $\pm$ 23
050904 | 0.45 $\pm$ 0.02 | 0.155 $\pm$ 0.007 | 346 $\pm$ 15 | 231 $\pm$ 10
050820A | 0.17 $\pm$ 0.03 | 0.11 $\pm$ 0.02 | 78 $\pm$ 16 | 89 $\pm$ 18
050603 | 0.18 $\pm$ 0.02 | 0.083 $\pm$ 0.011 | 95 $\pm$ 12 | 82 $\pm$ 11
031203 | 0.291 $\pm$ 0.007 | 2.80 $\pm$ 0.08 | 152 $\pm$ 4 | 243 $\pm$ 7
030329 | 0.113 $\pm$ 0.001 | 0.206 $\pm$ 0.001 | 26 $\pm$ 0 | 66 $\pm$ 1
030226 | 0.089 $\pm$ 0.001 | 0.115 $\pm$ 0.002 | 37 $\pm$ 1 | 75 $\pm$ 1
021004 | 0.077 $\pm$ 0.002 | 0.078 $\pm$ 0.002 | 29 $\pm$ 1 | 40 $\pm$ 1
020819B | 0.53 $\pm$ 0.05 | 0.57 $\pm$ 0.05 | 117 $\pm$ 11 | 157 $\pm$ 14
020405 | 0.7 $\pm$ 0.1 | 0.7 $\pm$ 0.1 | 222 $\pm$ 45 | 360 $\pm$ 73
011211 | 0.12 $\pm$ 0.02 | 0.16 $\pm$ 0.02 | 47 $\pm$ 6 | 88 $\pm$ 11
011121 | 0.30 $\pm$ 0.03 | 0.36 $\pm$ 0.03 | 81 $\pm$ 7 | 151 $\pm$ 13
010222 | 0.5 $\pm$ 0.1 | 0.32 $\pm$ 0.07 | 254 $\pm$ 53 | 367 $\pm$ 76
000926 | 0.090 $\pm$ 0.004 | 0.079 $\pm$ 0.003 | 41 $\pm$ 2 | 63 $\pm$ 3
000911 | 0.18 $\pm$ 0.02 | 0.11 $\pm$ 0.01 | 84 $\pm$ 8 | 112 $\pm$ 11
000418 | 0.209 $\pm$ 0.003 | 0.128 $\pm$ 0.002 | 71 $\pm$ 1 | 69 $\pm$ 1
000301C | 0.091 $\pm$ 0.003 | 0.094 $\pm$ 0.003 | 33 $\pm$ 1 | 48 $\pm$ 2
991208 | 0.114 $\pm$ 0.001 | 0.105 $\pm$ 0.001 | 38 $\pm$ 0 | 58 $\pm$ 1
990510 | 0.09 $\pm$ 0.01 | 0.09 $\pm$ 0.01 | 36 $\pm$ 4 | 63 $\pm$ 8
981226 | 0.16 $\pm$ 0.03 | 0.25 $\pm$ 0.05 | 39 $\pm$ 8 | 70 $\pm$ 14
980703 | 0.129 $\pm$ 0.003 | 0.098 $\pm$ 0.002 | 42 $\pm$ 1 | 51 $\pm$ 1
970826 | 0.37 $\pm$ 0.09 | 0.27 $\pm$ 0.06 | 147 $\pm$ 34 | 199 $\pm$ 46
970508 | 0.263 $\pm$ 0.003 | 0.260 $\pm$ 0.003 | 63 $\pm$ 1 | 74 $\pm$ 1

##  6 Conclusions

We have presented a methodology to constrain physical parameters of electron
acceleration in GRB blast waves based on radio afterglow peaks, building on
the methodology presented in Beniamini & van der Horst (2017). Our methodology
provides independent constraints on $\gamma_{m}$, $\epsilon_{e}$ and
$\xi_{e}$, and we have shown that it can be used to complement and corroborate
modeling efforts across the electromagnetic spectrum. We have expanded the
sample to 49 GRB afterglows and confirmed the initial findings in Beniamini &
van der Horst (2017). We have also shown that one can use peaks of SEDs, as
well as light curves, to constrain the aforementioned physical parameters.

We derived distributions for $\Psi$, which is a proxy for
$\epsilon_{e}~{}\xi_{e}^{-3/2}$, and $\chi$, which is a proxy for
$\gamma_{m}~{}\xi_{e}^{-1/2}$ at a given time. For conservative lower limits
on $\gamma_{m}$, we derived allowed parameter ranges of $2.5\cdot
10^{-2}\lesssim\xi_{e}\lesssim 1$ and $6.6\cdot
10^{-4}\lesssim\epsilon_{e}\lesssim 0.16$ for a homogeneous medium, and
$3.1\cdot 10^{-3}\lesssim\xi_{e}\lesssim 1$ and $3.6\cdot
10^{-5}\lesssim\epsilon_{e}\lesssim 0.21$ for a stellar wind medium. We
increase the lower limits even further by adopting that the energy efficiency
of the prompt gamma-ray emission, $\epsilon_{\gamma}$, should be at least
$0.01$ to avoid unphysical blast wave energetics. This leads to lower limits
of $\xi_{e}\gtrsim 0.1$ and $\epsilon_{e}\gtrsim 0.01$, which is consistent
with the lack of a clear signature of a thermal electron population in radio
observations. This means that we have constrained the values of $\epsilon_{e}$
and $\xi_{e}$ to within about one order of magnitude. We note that the
$\epsilon_{e}$ values for $\xi_{e}=1$ are consistent with those found by
state-of-the-art broadband modeling studies.

The distributions for $\Psi$ and $\chi$ are fairly narrow, with a standard
deviation of $\sim 0.3$ in log-space. These distribution widths can
potentially be attributed to variety in ambient medium densities and the
power-law index of the accelerated electron energy distribution, leading to
very narrow distributions in $\epsilon_{e}$, $\xi_{e}$ and $\gamma_{m}$.
However, we cannot conclude that there is universality among GRBs when it
comes to the microphysical parameters describing electron acceleration, since
our sample may be biased towards GRBs with bright radio afterglows. Follow-up
studies with radio-dimmer GRBs, either with current or new instrumentation,
will probe this universality question further. We also showed how future
observations of radio peaks can address possible temporal evolution of
$\epsilon_{e}$ or $\xi_{e}$.

## Acknowledgements

The authors would like to thank the anonymous referee for their useful
comments that strengthened the manuscript. RAD would like to thank Michael
Moss for assistance with Python programming and feedback on this paper; and
Sarah Chastain for assistance with with data collection. RAD would also like
to thank the George Washington University Undergraduate Research Award and the
Physics Department Frances E. Walker Fellowship for financial support to carry
out the research presented in this paper. PB was supported by a grant (no.
2020747) from the United States-Israel Binational Science Foundation (BSF),
Jerusalem, Israel.

## Data Availability

The data underlying this article are available in the article and in its
online supplementary material. The modeled data sets were derived from sources
in the public domain.

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