11institutetext:  IRT SystemX, Paris-Saclay, France 11email:
<EMAIL_ADDRESS>22institutetext: Université Paris-Saclay,
France 33institutetext: LTCI, Telecom ParisTech 33email:
<EMAIL_ADDRESS>44institutetext: Atos, Les Clayes-
sous-BoisŸ 44email<EMAIL_ADDRESS>
#  Asymptotic Performance Analysis of Blockchain Protocols

Antoine Durand 1122 Elyes Ben-Hamida 11 David Leporini 44 Gérard Memmi 33

###### Abstract

In the light of the recent fame of Blockchain technologies, numerous proposals
and projects aiming at better practical viability have emerged. However,
formally assessing their particularities and benefits has proven to be a
difficult task. The aim of this work is to compare the fundamental differences
of such protocols to understand how they lead to different practical
performances. To reach this goal, we undertake a complexity analysis of a wide
range of prominent distributed algorithms proposed for blockchain systems,
under the lens of Total Order Broadcast protocols. We sampled protocols
designed for very different settings and that use a broad range of techniques,
thus giving a good overview of the achievements of state-of-the-art
techniques. By analyzing latency and network usage, we are able to discuss
each protocol’s characteristics and properties in a consistent manner. One
corollary result to our work is a more robust criteria to classify protocols
as permissioned or permissionless.

###### Keywords:

Blockchain Complexity Analysis Asymptotic Performance Distributed Computing
Permissionless Permissioned

##  1 Introduction

Nakamoto’s famous work on Bitcoin [1] showed the potential for applications of
Byzantine agreement protocols. One of Bitcoin’s main achievements is being
efficient at large scale, such that the protocol stays practical even in the
so-called permissionless setting, i.e., when participation is open to anyone
and not restricted. This contrasts with the existing line of research for
fault tolerant distributed systems that were mostly suited for small-scale
systems. Since Bitcoin, numerous propositions and publications from both
scholars and industrials emerged, ranging from proofs of concept, small
improvements to new full-fledged protocols supporting various environments
such as large data center or network of smart objects (aka Internet of Things)
[2]. Many of these propositions are targeted toward improving performance in
terms of confirmation time and transactions per seconds. Indeed, these metrics
are directly visible by the end users, and Bitcoin is notoriously weak in
those regards compared to others existing protocols such as PBFT [3].

###  1.1 Motivation

Although progress has been made to understand how Bitcoin-like protocols
relates to existing research [4, 5], the contribution and relevance of each
proposition is difficult to compare and assess. Additionally, considering the
highly competitive climate of cryptocurrencies, the good faith of each
authors’ description and evaluation may be questioned.

As a result, it is difficult for the users to draw a coherent view of the
blockchain ecosystem without making an exhaustive technological deep dive.
Worse, each project’s practical viability may greatly vary even across similar
protocols, due to different implementation, usage and context.

In order to shed some light on this issue, we start from the simple
observation that most blockchain protocols aim at implementing some variant of
Total Order Broadcast (TO Broadcast), also referred to as Atomic Broadcast.
Informally, a TO Broadcast is a protocol that allows processes to reliably
broadcast messages, such that messages order are agreed on by the whole
network. Hence, we can readily use TO Broadcast as an abstraction layer of the
functionality offered by blockchain protocols, which allows for a generic
discussion on their properties. In particular, it will serve as a basis to
define common performance metrics.

This work will evaluate the efficiency of prominent blockchain protocols, when
seen as a way to implement TO broadcast. The aim of this paper is to compare
each protocol efficiency related to its network usage. As such, we will
evaluate latency and communication complexity as functions of network
parameters. To better relate them with Bitcoin, we will particularly focus on
their scalability in terms of number of participants. This helps us to better
understand the different trade-offs adopted by existing algorithms to reach
their goals in terms of performance metrics that are relevant to the end user.

###  1.2 Paper results

We have analyzed a wide range of algorithms: Bitcoin [1, 6], Snow White [7],
Spectre [8], Ouroboros [9], Algorand [10, 11], Tendermint [12] and
HoneyBadgerBFT [13].

In this paper we were able to compare each algorithm through a generic
framework, by instantiating a few parameters describing each assumed model.
Thus, these protocols can be consistently analyzed and compared in terms of
communication complexity (overall number of bits sent through the network) and
latency (the time it takes for all correct nodes to deliver a broadcast
message). This allows us to make a simple but robust analysis that justifies
the informal observations that have been made on these algorithms. This can be
useful, for instance, to determine the appropriate algorithm for a given use
case. Also, this work opens research opportunities for protocols achieving yet
unexplored latency/communication trade-offs, as well as the potential for
interesting impossibility results. Additionally, we are able to make a more
robust argumentation on deciding whether an algorithm can be deemed as suited
for the permissionless setting, i.e. its performances should scale with the
number of participants.

###  1.3 Related Work

To the best of our knowledge, the closest work to our is from Fitziet al.
[14]. Like us, they define performance-related metrics (namely, delay and
throughput) that they evaluate for several protocols. However the main focus
of their work is a technique to improve the performance of some blockchain
protocols. As such, their tools are mostly used to asses their proposal in
regards to a few bitcoin-like protocols, and our analysis is much more
complete. Vukolic [15] gave a good informal overview on how Proof-of-Work
(PoW) and the classical BFT paradigm relates to performance/scalability trade-
offs, but only made high level observations without quantification. Although
Cachin & Vukolić [16] reviewed several algorithms, their work focus on
resiliency and fault tolerance and not performance. Gervais et al. [17]
analyzed how tuning the performance-related parameters of PoW blockchains
impacts their security. In contrast, we analyze performance as a function of
security parameters and network model. Dib et al. [18] reviewed and compared
the performance of several algorithms, but their work is focus on the
permissioned setting, and displays practical numbers for the end user rather
than asymptotic complexity. BLOCKBENCH [19] is a framework from Dinh et al. to
evaluate the practical performance of blockchain _implementations_ , that
moreover targets only the permissioned setting.

Otherwise, as it is standard practice, several works provide a complexity
analysis of their algorithms. However they cannot always be compared since
they may be calculated for different metrics, and under different models. In
fact, a non negligible part of our work was to cast (and check) the proper
values into our framework. Moreover, except for a few mentioned cases, the
respective authors of the selected protocols made only a partial complexity
analysis, e.g. only for one of the metrics.

###  1.4 Paper Organization

This Paper is organized as follows: In Section 2, we make a short description
of the concepts relevant to our analysis. In Section 3, we describe the
adopted methodology in terms of parameters and metrics used to make the
evaluation. In Section 4, we detail the network behavior of each protocol and
give the steps leading to the results. In Section 5, we give a short analysis
of the results and motivate our arguments. Finally, we conclude this paper in
Section 6.

##  2 Background

###  2.1 Notations

We denote by $negl(\cdot)$ a negligible function, i.e. such that for every
polynomial $p$,
$\exists\kappa_{0},\forall\kappa>\kappa_{0},negl(\kappa)\leq\frac{1}{p(\kappa)}$._With
overwhelming probability_ is abbreviated w.o.p.$|\cdot|$ is the size operator,
i.e. the number of elements for a set, or the length for a tuple. We use
standard definition of $\Omega(\cdot),\Theta(\cdot),O(\cdot)$ notations for
complexity, i.e. $O(f(n))$ is of order at most $f(n)$, $\Omega(f(n))$ is of
order at least $f(n)$, $\Theta(f(n))$ is of order exactly $f(n)$. We will
further clarify their usage in our context in Subsection 3.2. We use the terms
protocol and distributed algorithm interchangeably.

###  2.2 Definitions

A Total Order Broadcast protocol, also called Atomic Broadcast, is a
distributed algorithm that implements two primitives, $\texttt{TO-
broadcast}(m)$ and $\texttt{TO-deliver}(m)$ with $m$ some message. We say that
a process $p$ TO-broadcast $m$ when it executes $\texttt{TO-broadcast}(m)$,
and that $p$ TO-delivers $m$ when it executes $\texttt{TO-deliver}(m)$. To be
a Total Order Broadcast, the protocol has to satisfy the following properties
[20]:

  * •

_Validity_ : If a correct process TO-broadcast a message $m$, then some
correct process eventually TO-delivers $m$.

  * •

_Agreement_ : If a correct process TO-delivers a message $m$, then all correct
processes eventually TO-delivers $m$.

  * •

_Integrity_ : If $p$ is a correct process, then for every TO-delivery of $m$
by $p$, $m$ has been TO-broadcast.

  * •

_Total order_ : If two correct processes TO-deliver two messages $m_{1}$ and
$m_{2}$, then both processes TO-delivers the messages in the same order.

Such protocol can be used to implement State Machine Replication, although the
latter generally refer to a context were external clients sends commands to a
network of replicas, which are consistent states machines. However the
differences are mostly related to application semantics, and thus both
concepts have been used interchangeably.

The studied protocols does not all solve exactly the same problem. In
particular, they do not make the same assumptions about the underlying network
model, and do not achieve the same properties, although they are similar. To
make a consistent analysis, as mentioned earlier, we consider them all as
Total Order Broadcast protocols. We consider they implement the primitives
$\texttt{To-Broadcast}(m)$ and notify messages delivery by calling
$\texttt{TO-deliver}(m)$. We will further specify how these primitives are
mapped to each protocol in order to solve total order Broadcast, and will make
note on the model variations adopted by each protocol. A notable difference
specific to Bitcoin and PoW-based algorithms is the weakening of the safety
properties, which are only guaranteed w.o.p., whereas more traditional
algorithms may not even be probabilistic.

##  3 Methodology

This section first describe the common model adopted, through parameters that
capture the relevant assumptions made by all protocols. Then we explain how we
related them to the security parameter to make meaningful statements, and we
finally define our metrics.

$n$ number of participants $\Delta$ bound on network delays $\alpha$ byzantine
power $\kappa$ security parameter Figure 1: Summary of parameters

###  3.1 Network Model

####  3.1.1 Common basis

All protocols analyzed are based on the Interactive Turing machines [21]
model. Further assumptions on the network are appropriately stated through
additional parameters that are given to each protocol at its instantiation.
Other unrelated assumptions are not made explicit through this analysis. Each
algorithm may require additional independent parameters that modify its
behaviour.

More precisely, the protocols assume a set $\Pi$ with $|\Pi|=n$ of processes
(also called nodes) with asynchronous clocks that exchange messages through
asynchronous reliable point-to-point channels. Asynchronous means that there
is no bound on the number of time steps taken for a message to be sent and
received. Reliable means that no messages are dropped or modified. The
communication graph is bidirectional and complete i.e. any node can send a
message to any node.

####  3.1.2 Additional Assumptions

Next, we define a few more parameters that convey the additional assumptions
made on the network model by the different algorithms. However the reader
should keep in mind that this does not fully capture all subtleties of the
models considered by each protocol.

##### Synchrony

We denote by $\Delta$ the maximum number of time steps taken by a message to
reach its destination. Strongly synchronous protocols require $\Delta$ in
their parameters, or a bound thereof. Partially synchronous protocols,
depending on how they are formulated, either: require $\Delta$ in their
parameters, but will simply not terminate if the given value is lower that
$\Delta$, or not require it at all. For asynchronous protocols, $\Delta$ is a
unit for asynchronous rounds defined as follow: Messages are assigned with a
round number $r$ such that all $r-1$ messages must be delivered before sending
any $r+1$ message.

In any case, evaluating the performance is only relevant when the protocols
terminate, hence we will ignore messages schedules that prevent termination.

It should be noted that the nodes bandwidth is only indirectly taken into
account: the bounds on network delay are supposed to hold such that all
messages can be fully transmitted.

##### Byzantine faults [22]

Furthermore, all protocols tolerate some level of byzantine failure, that is,
nodes that behave arbitrarily. We say that a node is correct or honest if it
follows the protocol. Otherwise, we interchangeably use the terms malicious,
faulty or Byzantine. To do so, all algorithms need some sense of a sybil
resistant resource i.e., a quantity that the adversary cannot artificially
increase. This is translated by the assumption that no more than a fraction
$\alpha$ of the resource is owned by malicious nodes. Therefore, we define a
generic parameter $\alpha$, which denotes the fractional byzantine ”power”
owned by the adversary. More precisely, depending on each assumption:

  * •

If there is at most $t$ byzantine nodes in $\Pi$, then $\alpha=\frac{t}{n}$

  * •

If PoW is assumed, then $\alpha$ is the fraction of the total hashrate that
byzantine node can produce.

  * •

If nodes have stake, then $\alpha$ is the the fraction of the total stake that
is owned by byzantine nodes.

The reader should keep in mind that although the adversary power is modeled
only through a unique parameter, the practical means and difficulty to acquire
a given $\alpha$ is vastly different depending on the case considered above,
e.g. it means something completely different to control $t$ specific computers
rather than being able to produce $x$ hashrate. Moreover, further assumptions
can be made to make the adversary more or less powerful, especially regarding
its capacity to dynamically corrupt nodes.

###  3.2 Security & protocol parameters

Moreover, all protocols require a parameter to tune the desired security
level. This parameter will be treated separately in this paper, and hence will
be specifically named $\kappa$. Then, _with overwhelming probability_ means
with probability $1-negl(\kappa)$.

Then, note that the precise requirement for each protocol is of the form: ”for
all admissible model parameters $n,\Delta,\alpha$, all stated properties holds
with probability $1-negl(\kappa)$”.

However, this statement only deals with probability of failure when $\kappa$
is varying and the model parameters are fixed. As a result, all $O(\cdot)$
notations used to prove only refer to $\kappa$ and treat the model parameters
as constants.

This is problematic for our purposes because we also want to express the
performance variation when the model parameters are varying (and in
particular, $n$). Hence, we cannot make the same usage of $O(\cdot)$ notations
for our metrics111This would result mostly in $O(\kappa^{c})$ values, for some
$c\geq 0$.

One solution could be forbid ourselves $O(\cdot)$ notations where the
constants also depends on some model parameter. However, it is common to have
the performance metrics depend on some complex variable which is shown to be
$O(\kappa)$, but its precise dependencies to the protocol parameters are
difficult to track, or even unknown.

As a result, we choose a middle ground where $O(\cdot)$ notations may still
depend on the model parameters, but we will note in subscript on _which_ of
them it does depend. For instance, $\Omega_{\alpha,\Delta}(\kappa)$ is a
variable of order at most $\kappa$, and that does not depend on $n$.

It should be noticed that this still hides bounded expressions of the model
parameters, for instance, we have that $\alpha=O_{\alpha}(1)=O(1)$ because
$\alpha$ is bounded by a constant independent of $\alpha$, namely, $1$. In
practice we will avoid such simplifications and keep the subscript.

Interestingly, each protocol may not require to be instantiated with the same
parameters for all nodes. This is the case for instance for $\kappa$ in
Bitcoin, where each node decide how many confirmations are deemed secure. On
the contrary, in Algorand $\kappa$ has to be uniquely chosen before the
protocol starts.

##### Additional parameter

Beyond the previous model parameters, each protocol can be instantiated with
additional protocol parameters. Since there will be at most one such parameter
relevant for our analysis for each protocol, we we will name it $p$ in the
following. This parameter is may be used to make efficiency and/or security
trade-offs, or its optimal value may be unknown. Their precise meaning will be
detailed in their corresponding section.

###  3.3 Metrics

We analyze the standard metrics used to evaluate distributed algorithms
efficiency: latency and communication complexity. These are even more relevant
in our context because they are directly experienced by the end user.

##### Latency

is the number of time steps between a message TO-broadcast and its TO-delivery
by the last honest node. Note that, assuming synchrony, one round is $\Delta$
time steps. In the context of cryptocurrencies, it is directly the time the
end user has to wait before his transaction is effective.

##### Communication complexity

is the amortized number of bits received by honest players for a message to be
TO-delivered by all honest players. It should be noticed that communication
complexity is usually given in terms of overhead, but we take here the total
value, to give a sense of the network capacity required to run the protocol.
This metric will be given as a function of the total message size $b$. As a
side effect, the use of $b$ allows informally to distinguish a ”constant
overhead” cost, i.e. that occurs independently of the data to broadcast, from
an ”efficiency” cost which reflects the overhead that scales with the amount
of data transmitted. We also remark that, depending on the algorithm, the
bound on network delays $\Delta$ may also cover messages of size $b$. Thus, in
practice, increasing $b$ could in turn increase $\Delta$.

We will use the term confirmation time as a synonym to latency, and bit
complexity as a synonym to communication complexity.

##  4 Analysis

We now present our complexity results. This section is divided in three
subsections, corresponding to different kinds of algorithms and in fine
analysis methodology. The first subsection treats blockchain style algorithms
through a common framework from Garay et al. [6]. The second subsection
evaluates protocols that follows the existing line of research for Byzantine
Fault Tolerant (BFT) algorithms. The last subsection gathers protocol that
work differently from the others and required specific treatment.

###  4.1 Using the Bitcoin backbone protocol

For Bitcoin, Snow White, and Ouroboros we will rely on the backbone protocol
formalism from Garay et al. [6]. These protocols follows the classical
blockchain approach, where leaders periodically broadcast new blocks,
resulting in a tree structure where a common chain can (hopefully) be
extracted. In Bitcoin, protocol participants are miners.

Their goal is to satisfy three properties: $k_{p}$-Common Prefix,
($k_{g}$,$\tau$)-Chain Growth and ($k_{q}$,$\mu$)-Chain Quality, abbreviated
CP, CG and CQ respectively. They can in turn be used to prove to prove
$k$-Persistence and $u$-Liveness. As a side note, in the first version of the
backbone protocol [23], a weaker version of CP was considered which did not
allow for a black box reduction. Pass & Shi [24] further introduced an
additional property, future self-consistency, solving this issue. We refer
here to the strong version of CP, that includes future self-consistency, as in
the cited version of the backbone protocol. In this framework, $k$-Persistence
and $u$-Liveness are properties of a Robust Transaction Ledger, which implies
State Machine Replication [25]. The $k$ parameter is the number of
confirmations that a block must get to be committed, and $u$ is the number of
round that a transaction takes before being committed.

Informally, CP states that pruning $k_{p}$ block from a chain results in a
prefix that is shared by all nodes. CG states that every $k_{g}$ consequent
rounds each node adds at least $\tau$ blocks, i.e. $\tau$ is a round/block
conversion parameter. CQ states that in every chain segment of length $k_{q}$,
there is at least a proportion $\mu$ of honest blocks. The proof for
persistence is straightforward from CP. However the proof for liveness yields
a value from the confirmation time as a function of the properties parameters,
namely, $u=max(k_{g},\frac{k_{q}+k_{p}}{\tau})$.

We sketch the general idea of this proof here: After $u\geq k_{g}$ rounds,
there are at least $\tau u$ new blocks in the honest players chain (CG).
Therefore, if $\tau u>k_{q}+k_{p}$, after $u$ rounds, the honest players’
chain grew by at least $k_{q}+k_{p}$ blocks. In this new chain, the first
segment of length $k_{q}$ is a common prefix to all honest players, because it
is $k_{p}$ blocks deep (CP). Moreover, there is at least $k_{q}\mu\geq 1$
blocks in this segment that are from honest players, which contains the
transaction. Thus it is reported as stable by all honest players. Therefore,
if $u\geq k_{g}$ and $\tau u>k_{q}+k_{p}$, then the protocol satisfy
$u$-Liveness.

With foresight, we have that $k_{q}+k_{p}$ will be a security parameter. Thus,
we will additionally detail their value depending on the model parameters.

For these protocols, the analysis regarding communication complexity is
simpler: Each block of size $b$ is broadcast once, and subsequent required
confirmations does not change the amortized cost since each of them will
(eventually) commit $b$ additional bits. The minimal communication cost is
$bn$. Furthermore, this cost is only increased by orphaned blocks. However,
the properties of a robust ledger alone does not allow to bound the number of
orphaned blocks. We observe that the three Bitcoin-Backbone based protocols
satisfy an additional property: In $k_{b}$ rounds, there are at most
$k_{b}\tau_{b}$ blocks produced. Along with CG with $\tau>0$, this bounds the
ratio of orphaned blocks by $\Theta_{\alpha}(1)$.

####  4.1.1 Bitcoin

[1]

For a baseline comparison, we start the analysis of Bitcoin. We build our
analysis on the work from Garay et al. [6], where they showed that Bitcoin
satisfy the three properties Common Prefix, Chain Quality and Chain Growth.

In this work the authors articulate their proofs on the assumption of a
_typical execution_ , that roughly states that parties produce blocks at a
rate close enough to their expected value (i.e., hashpower). Then they show
that any execution of $k$ rounds is typical with probability
$1-e^{-\Theta_{\epsilon}(k)}$, where $\epsilon$ is a variable that quantify
how close to its expected value is the block production. In turn, the proofs
will rely on $\epsilon$ being ”not too large”, which is implicitly assumed in
the ”honest majority assumption”. This assumption gives a bound on $\epsilon$
that depends on $\alpha$ and $\Delta/p$, where $p$ is the expected block
creation time. Hence, an execution longer than
$\Omega_{\alpha,\Delta/p}(\kappa)$ rounds is typical w.o.p.

CP, CQ and CG are proven assuming a typical execution, with $k_{p}=k_{q}\geq
2kf$ and $\tau=(1-\epsilon)f$ for some variable $f$, hence we have
$u$-Liveness with
$u\geq\frac{4k}{1-\epsilon}=\Theta_{\alpha,\Delta/p}(\kappa)$.

####  4.1.2 Ouroboros

[9]

Ouroboros is a strongly synchronous Proof-of-Stake (PoS) protocol. It is
currently implemented for cryptocurrency Cardano (https://www.cardano.org).

Ouroboros does not assume PoW, and consequently has to generate its own
randomness for the leader election. This is done inductively by leveraging
previous broadcast to implement a common coin tossing. This part is somewhat
complex but fortunately our analysis is blind to its details. Moreover, it
gives the useful property of having one unique designated leader per round.

As a result, there is at most one block per round, and CP, CG and CQ can be
proven using solely properties of the distribution of honest and malicious
leader in the block tree. This is done through the analysis of characteristics
strings, which are an encoding of the schedule of honest and malicious
leaders. Then they show that a string of length $k$ is forkable with
probability $negl(k)$. This is, of course, for an appropriate notion of fork
that is latter used to prove CP CQ and CG. In particular, since the
distribution of characteristics strings only depends on $\alpha$, we have that
$k=\Omega_{\alpha}(\kappa)$ Moreover, $\tau=1-\alpha$, because there is at
least one block for each round with an honest leader.

As with Bitcoin, strong synchrony is assumed, therefore the latency is
$\Delta\Omega_{\alpha}(\kappa)$

####  4.1.3 Snow White

[7]

Like Ouroboros, Snow White is a strongly synchronous protocol that does not
assume PoW. Its particularity however is its tolerance of honest nodes that do
not show up.

Snow White is able to cope with honest nodes being asleep, as long as there
are more awake honest nodes than malicious ones. In contrast, asleep honest
nodes in Bitcoin are counted as malicious, at least until they are implicitly
dropped during difficulty adjustment. In that sense Snow White tolerates
stronger adversaries, because for $\alpha$ malicious nodes, it requires only
$\alpha$ honest ones and $1-2\alpha$ asleep ones instead of just $1-\alpha$
honest nodes.

The notations used are slightly different, because all time-related values are
directly expressed in terms of number of time steps instead of number of
rounds. As a result, the Chain Growth $\tau$ parameter directly translate
blocks to time steps, and we can simply rely on $k_{g}+k_{p}$ being
$O(\kappa)$.

Snow White works somewhat similarly to Ouroboros, in the sense that it
inductively generates its own public randomness to elect leaders. However,
each node has a fixed probability $p$ to be elected at each time step, instead
of having unique leaders at each round. The authors then prove Chain Growth
with $\tau^{-1}=O_{\alpha,np}(1)+\Delta O(1)$. Roughly speaking, the first
term is the expected time it takes for an online node to find a block, and the
second one is the time for the block to be sent. Thus we have a total latency
of $\Delta\Omega(\kappa)+\Omega_{\alpha,np}(\kappa)$

###  4.2 Traditional BFT style algorithms

The following algorithms are more inspired from the existing literature on BFT
consensus algorithms, and share some common concepts that we outline here for
convenience. First, they all follow the framework of a per-block consensus
instance, which is final and essentially independent from the others. This
instance is implemented as a procedure that each node starts with its own
input, and then the output is the next block to be added.

These protocols often make use of simpler common primitives. One of them is
Reliable BroadCast (abbreviated RBC), that let a designated sender to
broadcast a value such that honest nodes either agree on the value sent, or
that the broadcast was unsuccessful. Formally, a Reliable Broadcast protocol
is exactly TO-Broadcast, without the Total Order property [22].

Another primitive used is the Binary Agreement (abbreviated BA) primitive. In
a BA, each processor propose its own input value $v_{i}\in\\{0,1\\}$, and
deliver another value, such that:

  * •

_Agreement_ : If a correct processes delivers a value $v$, then all correct
processes delivers $v$.

  * •

_Validity_ : If a correct process delivers $v$, then $v$ was proposed by a
correct process.

  * •

_Termination_ : All correct processes eventually decides.

####  4.2.1 Tendermint

[12]

Tendermint is a partially synchronous TO Broadcast algorithm, targeting high
performance in a small-scale permissioned setting.

Its normal case operation is reminiscent of PBFT [3], except that leaders are
changed at each command (i.e. block). For each block, there is a leader that
will execute RBC implemented through two all-to-all voting rounds. Of course,
to optimize bandwidth, only the hash of the block is included in the voting
messages. This require the $b$ bytes of the batch to be sent to all $n$ nodes,
and then the Reliable Broadcast cost of $n^{2}$ bytes, taking three of
communication steps. That is, in a fault-free execution, $(bn+n^{2})\Theta(1)$
bit complexity and $\Delta\Theta(1)$ latency.

Note that Tendermint, as PBFT, is optimized for the normal case operation.
However, since the leader is arbitrarily designated, it may be malicious. In
this case, the properties of the Reliable broadcast still ensure safety, but
termination can be prevented. This is why, after a timeout, it will be the
turn of the next designated leader to proceed for a new broadcast. The choice
for the new leader being in a fixed round-robin order, all $\alpha
n=\Theta_{\alpha}(n)$ malicious nodes may be leaders first and delay the block
commit by the same factor. Therefore, the worst case communication cost is
$(bn^{2}+n^{3})\Theta_{\alpha}(1)$ bits and $\Delta n\Theta_{\alpha}(1)$
latency.

####  4.2.2 HoneyBadgerBFT

[13] HoneyBadgerBFT (abbreviated HBBFT) is an asynchronous probabilistic
algorithm that solves TO-broadcast.

It is built on the Asynchronous Common Subset (ACS) primitive. Informally, ACS
is used to agreed on a subset of the union of the nodes’ inputs. It is defined
as follow [13]:

  * •

_Validity_ : If a correct node deliver a set $v$, then $|v|\geq(1-\alpha)n$
and $v$ contains the inputs of at least $(1-2\alpha)n$ correct nodes.

  * •

_Agreement_ : If a correct node deliver $v$, then every node deliver $v$.

  * •

_Termination_ : All correct nodes eventually delivers.

To prevent the adversary from selectively censoring transactions included in
the output set, HBBFT uses a threshold encryption scheme before broadcasting
transactions.

The ACS is a reduction to Reliable Broadcast and binary Byzantine Agreement
from Ben-Or et al. [26]. Roughly, each node runs in parallel an instance of
RBC, to send their value. Then, for each RBC, there is a BA instance to agree
whether it did terminate. Then, the nodes outputs the union of RBC values for
which its termination has been agreed. To ensure the Validity property, the
protocol has to ensure that at least $(1-\alpha)n$ RBC instances terminated.

The RBC terminates in three rounds and has $|m|nO(1)+n^{2}log(n)O(\kappa)$ bit
complexity, with $|m|$ the message size. The BA is an asynchronous
probabilistic protocol with $n^{2}O(\kappa)$ bit complexity. At each round it
has $1/2$ probability to terminate. As stated above, HBBFT has to wait for
$(1-\alpha)n=n\Theta_{\alpha}(1)$ parallel instances of them to terminate, the
time for this to happen is $log(n)O_{\alpha}(1)$ rounds.

However one of HBBFT achievements is that at the end of this procedure, it
commits data from all nodes input. That is, if all nodes have an input of size
$b$, then the batch committed will be of size $nb\Theta(1)$ bits. Hence, at
each batch of size $b$ committed, there is $O(1)bn+n^{3}log(n)O(\kappa)$ bits
received by honest players.

Note that the authors do provide an analysis, but they state their results in
terms of _overhead_ , i.e. the total cost divided by $b$. Furthermore, by
specifying a batching policy $b=n^{2}log(n)O(\kappa)$, they obtain the $nO(1)$
figure which is a constant per-node overhead. This emphasis on the low
overhead is less visible in our results, although it is translated by the fact
that HBBFT complexity on the $b$ factor is $bn$ instead of $bn^{2}$ for other
BFT-style protocols, i.e. a reduction factor of $n$ consistent with the
authors analysis.

###  4.3 Others Blockchain based algorithms

The two following protocols adopts significantly different paradigms, such
that they are treated separately.

####  4.3.1 Spectre

[8]

Spectre is a PoW-based partially synchronous distributed algorithm aimed
specifically at a cryptocurrency application.

In Spectre, the PoW merely serves as a network-level synchronization
primitive. As such, there is little restriction on the mining process, and the
mining hardness $p$ is only a protocol parameter for tuning network behavior.
More precisely, the only assumption regarding $p$ is that blocks are produced
less quickly that the nodes can receive them, hence it is _not_ related to
$\kappa$.

The Spectre protocol maintains a public ledger that grows over time, and a
sub-protocol $RobustTxO$ outputs the linearized log with the safety and
liveness properties. Once again, participants to the protocol are miners. A
message is a transaction, and we execute TO-Broadcast when including it in a
block. A transaction is TO-Delivered when it is output by $RobustTxO$. Then,
under some conditions that we detail below, Spectre is a (variant of a) TO
Broadcast, because [8]: Proposition 5 (Safety) implies Agreement and validity,
Proposition 4 (Consistency) implies Total Order, and Proposition 6 (Weak
Liveness) implies Termination.

Because it is targeted for a cryptocurrency application, Spectre achieves a
weaker version of TO-Broadcast in two regards. First, the total order is
weakened into a partial order, meaning that the messages are only ordered if
the application requires so. Secondly it has a weakened liveness property.
Translated to the TO-Broadcast abstraction, it means that Termination may not
be satisfied under some conditions on the nodes’ inputs. This weakening is
acceptable for Spectre because those conditions express nodes honesty
according to the application context, i.e. honest cryptocurrency users that
never try to double-spend will enjoy termination.

Roughly speaking, transactions are extracted from the blocks graph, and it is
possible for each of them to compute whether their probability to be undone is
greater than a given $\epsilon$. Then, the $\Omega_{\alpha}(\kappa)$
confirmations are still required, but since block creation is more flexible,
the $\delta$ is no longer a multiplicative factor but an additive overhead
that ensure a consistent view across all honest nodes.

More precisely, Spectre implements a procedure $RiskAcceptTx$ that gives a
bound on the probability that safety will not hold for this transaction. The
authors then show than the bound returned is smaller than a given $\epsilon$
after $log(\frac{1}{\epsilon})O(1)$ honest blocks are created. However, since
this procedure additionally require the node local values for $\alpha$ and
$\Delta$, we have that the number of honest blocks required is
$O_{\alpha,\Delta}(\kappa)$. The time taken for this to happen is obtained by
multiplying it by $\frac{1}{(1-\alpha)p}=(p\Theta_{\alpha}(1))^{-1}$. Then for
all nodes to be aware of these blocks an additional $\Delta$ overhead is
added, resulting in $\Delta O(1)+\frac{O_{\alpha,\Delta}(\kappa)}{p}$ latency.

Communication complexity on the other hand is simpler to analyze. In fact, the
analysis is similar to the one for others blockchain style algorithms. Since
Spectre selectively outputs transactions individually, the adversary only mean
to increase the communication complexity is to use all its hashpower to send
invalid transactions, effectively increasing the amortized cost by a factor
$\frac{1}{1-\alpha}=\Theta_{\alpha}(1)$.

####  4.3.2 Algorand

[10], [11]

Algorand is a strongly synchronous protocol with a per-block agreement
approach.

Its byzantine agreement primitive vastly differs from traditional BFT
algorithms. In fact, like Ouroboros and Snow White, Algorand rely on a public
randomness computed in previous blocks. It is used to elect a committee
(instead of a leader) _at each round_ that will have sufficiently many honest
nodes, w.o.p. These committees run a byzantine agreement which does not
require a private state from the nodes (except from their private key), since
committees would not be able to pass it on to the next committee properly.
This property is called _player replaceability_ by the authors. Except from
this property, common techniques from BFT algorithms can be used to reach
agreement in a constant number of rounds. In particular, since it has been
ensured that committees have a constant fraction of honest nodes, standard
quorum-based arguments are still valid.

More precisely, at each round, each user has a fixed probability $p$ to be
part of the committee. To bound the number of malicious nodes in a committee,
the authors leverage the fact that $\forall\beta<\alpha$, the probability of
having at most $\beta k$ malicious nodes in a uniformly sampled committee of
size $k$ is $1-negl(k)$. Thus, the committees expected size is
$\Omega_{\alpha}(\kappa)$.

As a result, each round that would be equivalent to a $n^{2}$ all-to-all
communication in a traditional BFT algorithm is now a ”committee-to-all”
$n\Omega_{\alpha}(\kappa)$ communication.

To optimize bandwidth, Algorand still does some sort of leader election. More
precisely, the expected committee size for the block proposers is the smallest
such that there is at least one proposer w.o.p., that is $\Theta(\kappa)$.

As a result, Algorand’s communication complexity is
$bn\Theta(\kappa)+n\Omega_{\alpha}(\kappa)$ and its latency $\Delta O(1)$.

##  5 Discussion

Table 1: Performance summary Algorithm | Latency | Communication complexity
---|---|---
Nakamoto [6] | $\Delta\Theta_{\alpha,\Delta/p}(\kappa)$ | $bn\Theta_{\alpha}(1)$
Ouroboros [9] | $\Delta\Omega_{\alpha}(\kappa)$ | $bn\Theta_{\alpha}(1)$
SnowWhite [7] | $\Delta\Omega(\kappa)+\Omega_{\alpha,np}(\kappa)$ | $bn\Theta_{\alpha}(1)$
Spectre [8] | $\Delta O(1)+\frac{O_{\alpha,\Delta}(\kappa)}{p}$ | $bn\Theta_{\alpha}(1)$
Algorand [10] | $\Delta O(1)$ | $bn\Theta(\kappa)+n\Omega_{\alpha}(\kappa)$
Tendermint [12] | $\Delta n\Theta_{\alpha}(1)$ | $(bn^{2}+n^{3})\Theta_{\alpha}(1)$
HoneyBadgerBFT [13] | $\Delta log(n)O_{\alpha}(1)$ | $bnO(1)+n^{3}log(n)O(\kappa)$

Table 1 summarizes the different results for the studied protocols. These
clearly outline the different styles of algorithms that required different
methodologies in Section 4. Pure Blockchain style algorithms are the most
efficient regarding communication complexity. Their cost is even optimal in a
sense, because $bn$ bits are required to make the messages known, and some
(worst case) overhead due to byzantine nodes seems unavoidable. On the other
hand, we can see that their latency suffers from their security parameter,
which itself may be an unbounded function of the network parameters.

We see that Algorand stands out in this table, because it is the only one
whose latency does not depend on $\kappa$, but their bit complexity does. This
is easily understandable in the fact that blocks in Algorand are final and
that no fork can occur. Intuitively, we can see Algorand as doing all the
random picking of participants in parallel, at the same time, instead of
sequentially.

##### Blockchain communication complexity

It may be surprising that blockchain protocols have such low communication
cost compared to traditional $O(n^{3})$ algorithms. This can be explained by
the fact that blockchain style algorithms share a common component: public,
unbiased randomness. Note that, for instance, the hash of a block is indeed a
public random, but not unbiasable, because the adversary is able to contribute
to the input of the hash, thus allowing her to make the random value suits her
needs. This component is needed to leverage an important mechanism of
blockchain protocols: Within a random sample of the protocol participants,
each node has a probability $\alpha<1$ of being malicious. Therefore, by
repeatedly taking nodes at random, the total number of malicious nodes can be
bounded w.o.p. This way, a finite number of unreliable broadcasts can be
sufficient to reach agreement with high probability, hence the probabilistic
safety.

##### Impact on the meaning of permissioned/permissionless

Permissioned and permissionless are terms that are well understood informally,
but their precise meaning can be problematic. Indeed, even if ”being able to
join the system at any moment” is somewhat clear, deciding whether a given
algorithm is permissionless is arguably arbitrary. This is due to a possible
argument that a simple transformation could relax any permissioned algorithm
into a permissionless one. Indeed, in a permissioned setting, nodes are
determined and can be given arbitrary weights (i.e. having nodes count as
multiple ones), thus translating the fault tolerance to a fraction of the
total weight. Then, defining the weights as the nodes stake recorded in the
blockchain itself, we should obtain a legitimate PoS protocol. This
transformation has already been used as an argument in practice [27].
Moreover, the inverse transformation (from permissionless to permissioned)
being also trivial, it would seem that the distinction
permissioned/permissionless would be more specific to the usage of the
algorithm rather than the algorithm itself. However, this seems to be false in
practice, as algorithms are de facto targeted towards a specific setting.

Our work allow for an answer to this issue. In our sense, we say that an
algorithm is suitable for the permissionless setting if it is scalable and
stay efficient even with a very high number of participants. In our opinion,
it should have a bit complexity linear in $n$, and a latency independent from
$n$. We notice that Pass & Shi [28] proposed a definition from permissionless
environments, in which they formally define a model were executions are
permissionless/permissioned. In that sense, our statement is not a
_definition_ of a permissionless environment in itself, but a criteria for
categorizing algorithms as suited to the permissionless setting or not.

##  6 Conclusion

In this work, a wide range of popular blockchain protocols was analyzed. Our
metrics and methodology has been carefully tuned to accurately describe the
behavior of each protocol.

We showed that, when considered as variants of Total Order Broadcast
protocols, permissionless blockchain protocols had a very efficient network
usage, but were suffering latency-wise from their security parameter. This
work also can be used as a ground methodology to assess new protocols
performances. Finally we discussed some impacts of these results on the
classification of permissioned and permissionless algorithms.

## ACKNOWLEDGMENTS

We are thankful to Petr Kuznetsov, Guillaume Hebert, Thomas Domingos, and
Vincent Coquet for their help in fruitful discussions. This work was carried
as part of the Blockchain Advanced Research & Technologies (BART) Initiative.
We are also thankful to the Technical University of Munich (TUM). This
research work has been carried out under the leadership of the Institute for
Technological Research SystemX, and therefore granted with public funds within
the scope of the French Program Investissements d’Avenir.

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# Time delay of fast radio burst population with respect to the star formation
history

Hai-Nan Lin Department of Physics, Chongqing University, Chongqing 401331,
China  Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing
University, Chongqing 401331, China  Xin-Yi Li Department of Physics,
Chongqing University, Chongqing 401331, China  Chongqing Key Laboratory for
Strongly Coupled Physics, Chongqing University, Chongqing 401331, China  Rui
Zou Department of Physics, Chongqing University, Chongqing 401331, China
Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University,
Chongqing 401331, China

###### Abstract

In spite of significant progress in the research of fast radio bursts (FRBs)
in recent decade, their origin is still under extensive debate. Investigation
on the population of FRBs can provide new insight into this interesting
problem. In this paper, based on the first CHIME/FRB catalog, we construct a
Bayesian framework to analyze the FRB population, with the selection effect of
the CHIME telescope being properly taken into account. The energy function is
modeled as the power-law with an exponential cutoff. Four redshift
distribution models are considered, i.e., the star formation history (SFH)
model, and three time-delayed models (Gaussian delay, log-normal delay, and
power-law delay). The free parameters are simultaneously constrained using
Bayesian inference method, and the Bayesian information criterion (BIC) is
used in model comparison. According to BIC, the log-normal delay model fits
the data best. The power-law delay model and Gaussian delay model can also
give reasonable fits, although they are not as good as the log-normal delay
model. However, the SFH model is strongly disfavored compared with the three
time-delayed models. The energy function is tightly constrained and is almost
independent of the redshift models, with the best-fitting power-law index
$\alpha\approx 1.8$, and cut-off energy $\log(E_{c}/{\rm erg})\approx 42$. The
FRB population shows on average $3\sim 5$ billion years time delay with
respect to the SFH. Therefore, the hypothesis that the FRB population traces
the SFH is conclusively ruled out.

fast radio bursts – galaxies: star formation – cosmological parameters

##  1 Introduction

Fast radio bursts (FRBs) are energetic (up to $\sim 10^{43}$ erg) and short-
duration ($\sim$ milliseconds) radio pulses coming from the Universe, see e.g.
(Platts et al., 2019; Petroff et al., 2019; Xiao et al., 2021; Zhang, 2023)
for recent review. The first FRB (which is now named FRB 20010724) was found
from the archival data of the 64-metre Parkes radio telescope in 2007 (Lorimer
et al., 2007), but it had not aroused people’s attention until four other
similar signals was discovered six years later (Thornton et al., 2013).
Thereafter, researches in FRB fields have achieved rapid progress, especially
in recent decade. The localization of host galaxy and the direct measurement
of redshift confirm the cosmological origin (Keane et al., 2016; Chatterjee et
al., 2017). Phenomenologically, FRBs can be divided into two categories, i.e.,
repeater and non-repeater. In 2016, the Arecibo 305-metre radio telescope
detected several similar radio signals from the same sky position of FRB
20121102 (Spitler et al., 2016), making it to be the first discovered
repeating FRB. Later on, it was found that the source of FRB 20121102 is
extremely active, which emits more than one thousand bursts in about 60 hours
spanning less than two months (Li et al., 2021). With the operation of the
Canadian Hydrogen Intensity Mapping Experiment (CHIME, Amiri et al. 2018), we
are able to detect FRBs in real time, thus extensively enlarging the FRB
sample (Amiri et al., 2021). Up to now, about eight hundred FRB sources have
been published (Xu et al., 2023), a majority of which are discovered by the
CHIME telescope. With the join of high-sensitive radio telescopes such as the
Square Kilometre Array (SKA) and the Five-hundred-meter Aperture Spherical
radio Telescope (FAST), we expect that the number of FRB sources, as well as
the number of bursts of individual repeating sources will significantly
increase in the near future.

Although hundreds of FRBs have been found, most of them are not well
localized, leaving their origins largely unknown. But there is no dispute that
most FRBs are extragalactic origin, because the observed dispersion measure
(DM) greatly excesses the maximum contribution from the Milky Way. Only one
event (FRB 20200428)111This burst is now renamed as FRB 20200428D, since some
other FRBs are observed on the same day. is confirmed to originate from the
Milky Way. This specific burst is found to be associated with a short X-ray
burst from the Galactic magnetar SGR 1935+2154 (Andersen et al., 2020;
Bochenek et al., 2020). This implies that at least some, if not all FRBs
originate from magnetars. Although young magnetars are proposed as the leading
source candidates, recent observations suggest that there may be multiple FRB
progenitors (Moroianu et al., 2023). Similar to the classification of FRBs
themselves, FRB progenitors can also be classified into non-repeater and
repeater categories, thus leading to two classes of FRB models: catastrophic
models for the non-repeaters and non-catastrophic models for the repeaters.
Initially, all FRBs were considered as non-repeaters and are most likely
catastrophic events, until the discovery of the first repeating event FRB
20121102 (Spitler et al., 2016). Given the significantly higher event rate
density of FRBs than the catastrophic events in the Universe (Ravi et al.,
2019; Luo et al., 2020), it is highly probable that a majority of FRBs
originate from repeating sources. So far, at least dozens of FRB models have
been proposed, see e.g. Platts et al. (2019) for excellent review. Considering
the short-duration and extremely energetic properties of FRBs, the most
popular FRB models involve at least one compact object, such as neutron star,
magnetar or white dwarf in the center of FRB source. For instance, the merger
of binary white dwarfs (Kashiyama et al., 2013), the merger of binary neutron
stars (Wang et al., 2016; Totani, 2013), the coalescence of neutron star –
black hole systems (Mingarelli et al., 2015), the collapse of neutron star to
black hole or quark star (Falcke & Rezzolla, 2014; Shand et al., 2016), the
starquake of pulsar (Wang et al., 2018), just name a few.

The physical origin of FRBs is still under extensive debate, and there is no
conclusive evidence for preferring one model against the others. Statistical
analysis shows that repeaters and non-repeaters have very different
properties, hence may have different progenitors (Pleunis et al., 2021).
Different progenitors will lead to different FRB population distributions.
Therefore, investigating the population of FRBs is of great importance in
order to distinguish different FRB models. Figure 1 of Zhang et al. (2021a)
gives a comprehensive summary of the most popular FRB models and the
corresponding population distributions. For instance, the core-collapse models
predict that the FRB population should trace the star formation history (SFH),
the models involving the merger of binary compact objects predict that FRB
population should have a significant time delay with respect to the SFH, while
for some other models (such as the cosmic comb model and cosmic string model)
the FRB population is unpredictable.

The population of FRBs have been extensively studied in literature. Based on
two FRB samples detected by the Parkes and the Australian Square Kilometre
Array Pathfinder (ASKAP), Zhang et al. (2021a) found that the SFH model and
three time-delayed models can all match the data well, as long as the free
parameters are chosen properly. Using similar samples, James et al. (2022)
showed that FRB population evolves with redshift in a way consistent with, or
faster than the SFH. The large FRB sample observed by the CHIME telescope
(Amiri et al., 2021) allows us to study the FRB population in details. Based
on the first CHIME/FRB catalog, Zhang & Zhang (2022) showed that the data can
well match the time-delayed models, but the SFH model is conclusively
excluded. It is also possible that the CHIME/FRB sample is a hybrid sample
consists of both the time-delayed population and SFH population. Based on the
same CHIME/FRB sample, Qiang et al. (2022) investigated the SFH model as well
as some empirical models, and also arrived at the conclusion that the SFH
model is ruled out. Very recently, Lin & Zou (2024) revised the constraints on
the FRB population using the first CHIME/FRB catalog, and found that more than
one models can match the data if the parameters are optimized, but the SFH
model is strongly disfavored. All these studies rule out the hypothesis that
FRBs trace the SFH, but the concrete FRB population remains to be
unconstrained.

The observed FRB population is mainly influenced by two intrinsic
distributions of FRB sources, i.e. the energy distribution and redshift
distribution. Several independent works show that the intrinsic energy
distribution follows the power-law, and maybe with a high-energy exponential
cutoff, which is independent of the redshift distribution (Luo et al., 2018,
2020; Lu et al., 2020; Lin & Zou, 2024). Compared with the energy
distribution, the redshift distribution is less known. Different progenitor
theories predict different redshift distributions. The most motivated redshift
distribution model is the SFH model, which predicts that the occurrence rate
of FRBs should mirror the cosmic SFH. However, the SFH model has already been
ruled out by observations (Zhang & Zhang, 2022; Qiang et al., 2022; Lin & Zou,
2024). Due to the additional time delay caused by gravitational wave radiation
in the mergers of binary compact objects, this particular channel predicts a
redshift distribution that deviates significantly from that of the SFH model.
Therefore, the time-delayed models have been proposed (Zhang et al., 2021a;
Zhang & Zhang, 2022). Using the Kolmogorov-Smirnov (KS) test method, it has
been shown that the time-delayed models can to some extent match a large
sample of FRBs, as long as the model parameters are chosen properly (Zhang &
Zhang, 2022). However, the KS-test method used in Zhang & Zhang (2022) relies
on burdensome simulations, and the optimal parameters couldn’t be acquired.

The recently published CHIME/FRB catalog (Amiri et al., 2021) offers a large
FRB sample, facilitating an in-depth study of the FRB population. In this
paper, we reinvestigate the FRB population use the first CHIME/FRB catalog.
Three time-delayed redshift distribution models are considered: Gaussian
delay, log-normal delay and power-law delay. These models have already been
used to study the population of FRBs (Zhang et al., 2021a; Zhang & Zhang,
2022) and gamma-ray bursts (Wanderman & Piran, 2015; Sun et al., 2015). The
SFH model is also considered for comparison, although is has already been
ruled out. With the selection effect of telescope taking into account, we
construct a joint likelihood of energy, fluence and redshift. The model
parameters are optimized using the Bayesian inference method, and the Bayesian
information criterion (BIC) are used in model comparison.

The structure of the rest parts of this paper are organized as follows: In
Section 2, we introduce the FRB population models and construct a Bayesian
framework in order to constrained the models parameters. In Section 3, based
on the Bayesian framework, the model parameters are constrained using the data
of the first CHIME/FRBs catalog. Finally, relevant discussion and conclusions
are given in Section 4.

##  2 Methodology

The redshift of FRB is necessary to investigate the population. Unfortunately,
a majority of FRBs have no direct measurement of redshift. One way to solve
this problem is using the dispersion measure (DM) as a proxy of redshift. DM
is the integral of the number density of free electrons along line-of-sight,
which can be extracted directly from the dynamical spectra. FRBs at further
distance are expected to confront more electrons thus have larger DMs. The
observed DM of an extragalactic FRB, can in general be decomposed into four
terms (Deng & Zhang, 2014; Gao et al., 2014; Macquart et al., 2020),

${\rm DM_{obs}}={\rm DM_{MW,ISM}}+{\rm DM_{WM,halo}}+{\rm DM_{IGM}}+\frac{{\rm
DM_{host}}}{1+z}.$ (1)

The first term, $\mathrm{DM_{MW,ISM}}$, is the contribution from the Milky Way
interstellar medium, which can be estimated using Milky Way electron density
models, such as the NE2001 model (Cordes & Lazio, 2002) and YMW16 model (Yao
et al., 2017). The second term, $\mathrm{DM_{WM,halo}}$, represents the
contribution from the Milky Way halo. This term has not been well constrained
yet, but it is approximately estimated to be $\sim 50~{}{\rm pc~{}cm}^{-3}$
(Prochaska & Zheng, 2019; Macquart et al., 2020). The third term, ${\rm
DM_{IGM}}$, is the contribution from the intergalactic medium (IGM), which can
be calculated given a specific cosmological model. Finally, the last term $\rm
DM_{host}$ represents the dispersion caused by free electrons in the FRB host
galaxy.

In the standard ${\rm\Lambda}$CDM model, the average value of the ${\rm
DM_{IGM}}$ term can be calculated as (Deng & Zhang, 2014)

$\langle{\rm DM_{IGM}}(z)\rangle=\frac{3cH_{0}\Omega_{b}f_{\rm IGM}f_{e}}{8\pi
Gm_{p}}\int_{0}^{z}\frac{1+z}{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}dz,$
(2)

where $c$ is the speed of light, $m_{p}$ is the proton mass, $G$ is the
Newtonian gravitational constant, $f_{\rm IGM}=0.84$ is the baryon mass
fraction in IGM, and $f_{e}=7/8$ is the electron fraction. The Hubble constant
$H_{0}$, the vacuum energy density $\Omega_{\lambda}$ and the baryon density
$\Omega_{b}$ are fixed to the Planck 2018 results (Aghanim et al., 2020). Note
that the actual value of ${\rm DM_{IGM}}$ may significantly deviate from
equation (2) due to the fluctuation of matter density. Numerical simulations
demonstrate that the distribution of $\rm DM_{IGM}$ can be well modeled by the
quasi-Gaussian function (Macquart et al., 2020; Zhang et al., 2021b),

$p_{\rm
IGM}(\Delta)=A\Delta^{-\beta}\exp\left[-\frac{(\Delta^{-\alpha}-C_{0})^{2}}{2\alpha^{2}\sigma_{\rm
IGM}^{2}}\right],~{}~{}~{}\Delta>0,$ (3)

where $\rm\Delta\equiv DM_{IGM}/\left\langle DM_{IGM}\right\rangle$,
$\alpha=\beta=3$, the effective standard deviation $\sigma_{\rm
IGM}=Fz^{-1/2}$ with $F$ being a free parameter, $A$ is the normalization
constant, and $C_{0}$ is chosen to ensure that the mean value of $\Delta$ is
unity.

Currently, the host term $\rm DM_{host}$ is still poorly known, primarily due
to the lack of comprehensive observations of the local surroundings of most
FRB sources. The value of $\rm DM_{host}$ can vary significantly from burst to
burst, ranging from several tens to several hundreds pc cm-3 (Niu et al.,
2022; Kirsten et al., 2022). To account for such a large variation, it is
usually modeled using the log-normal distribution (Macquart et al., 2020;
Zhang et al., 2020),

$p_{\rm host}({\rm DM_{host}}|\mu,\sigma_{\rm host})=\frac{1}{\sqrt{2\pi}{\rm
DM_{host}}\sigma_{\rm host}}\exp\left[-\frac{(\ln{\rm
DM_{host}}-\mu)^{2}}{2\sigma_{\rm host}^{2}}\right],$ (4)

where the two free parameters $\mu$ and $\sigma_{\rm host}$ are the mean and
standard deviation of ln $\rm DM_{host}$, respectively.

The first two terms of the right-hand-side of equation (1) can be subtracted
from the total observed $\mathrm{DM_{obs}}$, leaving behind the extragalactic
DM,

${\rm DM_{E}}\equiv{\rm DM_{obs}}-{\rm DM_{MW,ISM}}-{\rm DM_{WM,halo}}={\rm
DM_{IGM}}+\frac{{\rm DM_{host}}}{1+z}.$ (5)

According to the probability theory, the distribution of ${\rm DM_{E}}$ can be
written as the product of the distributions of $\rm DM_{IGM}$ and $\rm
DM_{host}$, and then marginalize over either term (Macquart et al., 2020),

$p_{E}({\rm DM_{E}}|z)=\int_{0}^{(1+z)\rm DM_{E}}p_{\rm host}({\rm
DM_{host}}|\mu,\sigma_{\rm host})p_{\rm IGM}({\rm DM_{E}}-\frac{\rm
DM_{host}}{1+z}|F,z)d{\rm DM_{host}},$ (6)

where the free parameters $F$, $\mu$ and $\sigma_{\rm host}$ can be tightly
constrained using 17 well-localized FRBs (Tang et al., 2023).

Based on equation (6), we can reconstruct the ${\rm DM_{E}}-z$ relation, as is
shown in Figure 2 of Tang et al. (2023). Then the ${\rm DM_{E}}-z$ relation
can be used to infer the redshift and calculate the energy of FRBs. Given the
observed fluence $(F_{\nu})$, the isotropic energy can be calculated as
(Macquart & Ekers, 2018; James et al., 2021)

$E=\frac{4\pi d_{L}^{2}}{(1+z)^{2+\beta}}F_{\nu}\Delta\nu,$ (7)

where $\beta$ is the spectrum index ($F_{\nu}\propto\nu^{\beta}$), $d_{L}$ is
the luminosity distance, and $\Delta\nu$ is the bandwidth in which the FRB is
detected. The energies of the first CHIME/FRB catalog are listed in the
appendix of Tang et al. (2023).

Now we have the distributions of fluence, energy and redshift. These three
observables are used simultaneously to constrain the FRB population. The
distribution of energy can be well modeled by power-law with a high-energy
exponential cutoff (Luo et al., 2018; Lu et al., 2020),

$p(E)\propto\left({\frac{E}{E_{c}}}\right)^{-\alpha}\exp\left(-\frac{E}{E_{c}}\right).$
(8)

Although the cutoff energy $E_{c}$ is not well constrained yet, the power-low
index $\alpha$ is typically confined to be in the range
$1.8\lesssim\alpha\lesssim 2.0$ (Luo et al., 2018; Lu & Piro, 2019; Lu et al.,
2020). Here we treat both $E_{c}$ and $\alpha$ as free parameters.

The redshift distribution of FRBs in the observer frame is related to the
intrinsic event rate density $dN/dtdV$ by (Zhang et al., 2021a; Zhang & Zhang,
2022)

$p(z)\propto\frac{dt}{dt_{\rm obs}}\frac{dN}{dtdV}\frac{dV}{dz},$ (9)

where $dt/dt_{\rm obs}=(1+z)^{-1}$ accounts for the time dilation, and in the
flat $\rm\Lambda$CDM model the redshift-dependent specific comoving volume
$dV/dz$ is given by

$\frac{dV}{dz}=\frac{4\pi
cd^{2}_{C}}{H_{0}\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}},$ (10)

where $d_{C}=d_{L}/(1+z)$ is the comoving distance.

The intrinsic event rate density $dN/dtdV$ of FRB is poorly constrained. The
most well-motivated model is the SFH of the Universe. The SFH has been
extensively studied, and many parametric forms have been proposed (Yuksel et
al., 2008; Madau & Dickinson, 2014; Madau & Fragos, 2017). Here we use the
updated version of two-segment empirical SFH model (Madau & Fragos, 2017),

${\rm SFH}=\frac{(1+z)^{2.6}}{1+((1+z)/3.2)^{6.2}}.$ (11)

If FRBs originate from core-collapse of massive stars, then the FRB population
is expected to follow the SFH. On the other hand, if FRBs originate from the
merger of compact binaries, the time delay with respect to SFH is expected,
because a binary system must go through an extended inspiral phase prior to
eventual coalescence. Assuming that the time delay follows the probability
distribution $f(\tau)$, then the intrinsic event rate density of FRBs is the
convolution of the SFH with $f(\tau)$ (Wanderman & Piran, 2015),

$\frac{dN}{dtdV}\propto\int_{z}^{\infty}{\rm SFH}(z^{\prime})\cdot
f(t(z)-t(z^{\prime}))\frac{dt}{dz^{\prime}}dz^{\prime},$ (12)

where the cosmic time is related to redshift through

$t(z)=\int_{z}^{\infty}\frac{1}{H_{0}(1+z)\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}dz.$
(13)

The distribution of time delay $f(\tau)$ is still unclear. Here, we follow
Zhang et al. (2021a) and consider three different distributions:

$\bullet$ Gaussian delay model:

$f(\tau)=\frac{1}{{\sigma\sqrt{2\pi}}}\exp\left(-\frac{\left(\tau-\tau_{0}\right)^{2}}{2\sigma^{2}}\right).$
(14)

$\bullet$ Log-normal delay model:

$f(\tau)=\frac{1}{{\tau\sigma\sqrt{2\pi}}}\exp\left(-\frac{\left(\ln\tau-\ln\tau_{0}\right)^{2}}{2\sigma^{2}}\right).$
(15)

$\bullet$ Power-law delay model:

$f(\tau)=\left(\frac{1-\alpha_{\tau}}{{\tau_{\rm
max}^{1-\alpha_{\tau}}-\tau_{\rm
min}^{1-\alpha_{\tau}}}}\right){\tau}^{-\alpha_{\tau}}.$ (16)

In the power-law delay model, $\tau_{\rm max}$ is the maximum merger delay
timescale and $\tau_{\rm min}$ is the minimum. We consider $\tau_{\rm
max}=1/H_{0}$ to be the Hubble time and set $\tau_{\rm min}=20$ Myr. The above
three time-delay models have been successfully applied in modeling the
redshift distribution of the non-collapsar short gamma-ray bursts (Wanderman &
Piran, 2015).

Equations (8) and (9) are valid if all FRBs are detectable. In the actual
case, however, the distributions of energy and redshift observed by a specific
telescope strongly depend on the selection effect of the telescope. The
selection effect of the CHIME telescope is hard to determine. Here we follow
Zhang & Zhang (2022) and model the detection efficiency as a function of the
specific fluence,

$\eta_{\rm det}(F_{\nu})=\left(\frac{\mathrm{log}F_{\nu}-\mathrm{log}F^{\rm
min}_{\nu,\rm th}}{\mathrm{log}F_{\nu,\rm th}^{\rm max}-\mathrm{log}F^{\rm
min}_{\nu,\rm th}}\right)^{n},\quad\mathrm{log}F^{\rm min}_{\nu,\rm
th}<\mathrm{log}F_{\nu}<\mathrm{log}F_{\nu,\rm th}^{\rm max}.$ (17)

This model assumes a ‘grey zone’ between the minimum specific threshold
fluence $F_{\nu,\rm th}^{\rm min}$, and the maximum specific threshold fluence
$F_{\nu,\rm th}^{\rm max}$. Within this range FRBs may not be fully detected.
FRBs with $F_{\nu}<F_{\nu,\rm th}^{\rm min}$ are undetectable hence $\eta_{\rm
det}=0$, while above $F_{\nu,\rm th}^{\rm max}$ all FRBs are detectable hence
$\eta_{\rm det}=1$. Following Zhang & Zhang (2022), we fix $\mathrm{log}F^{\rm
min}_{\nu,\rm th}=-0.5$ (as shown by the data), while $\mathrm{log}{F_{\nu,\rm
th}^{\rm max}}$ is treated as a free parameter.

By inverting equation (7), the specific fluence $F_{\nu}$ can be expressed as
a function of energy and redshift,

$F_{\nu}(E,z)=\frac{(1+z)^{2+\beta}}{4\pi d_{L}^{2}\Delta\nu}E.$ (18)

Note that the luminosity distance $d_{L}(z)$ is a function of redshift. Taking
the selection effect into account, the distributions of fluence, energy and
redshift observed by the CHIME telescope can be written as (Lin & Zou, 2024)

$p_{\rm det}(F_{\nu})\propto\frac{d}{dF_{\nu}}\iint\limits_{F_{\nu}(E,z)\leq
F_{\nu}}p(z)p(E)\eta_{\rm det}(F_{\nu}(E,z))dE\,dz,$ (19) $p_{\rm
det}(E)\propto\int_{z_{\rm min}}^{z_{\rm max}}p(z)p(E)\eta_{\rm
det}(F_{\nu}(E,z))dz,$ (20) $p_{\rm det}(z)\propto\int_{E_{\rm min}}^{E_{\rm
max}}p(z)p(E)\eta_{\rm det}(F_{\nu}(E,z))dE.$ (21)

Equations (19,20,21) are the probability density functions (PDFs) of fluence,
energy and redshift, respectively. They can be used to fit to the data of the
CHIME/FRBs catalog. But in order to avoid the arbitrariness of binning, we use
the cumulative distribution functions (CDFs) instead of the PDFs in the
fitting procedure. The CDF of a quantity $Q$ is defined by

$N_{\mathrm{det}}(>Q)=A\int_{Q}^{Q_{\mathrm{max}}}p_{\mathrm{det}}(Q)dQ,~{}~{}~{}Q=\left\\{F_{\nu},\,E,\,z\right\\},$
(22)

where the normalization factor is chosen to match the total number of data
points. The $\chi^{2}$ of quantity $Q$ can be written as

$\chi^{2}_{Q}=\sum_{i}^{n}\frac{[N_{\mathrm{det}}(>Q_{i})-N(>Q_{i})]^{2}}{\sigma_{i}^{2}},~{}~{}~{}Q=\left\\{F_{\nu},\,E,\,z\right\\},$
(23)

where $N(>Q_{i})$ is the number of FRBs whose $F_{\nu}$, $E$ or $z$ is larger
than that of the $i$-th FRB, and the uncertainty is given by
$\sigma_{i}=\sqrt{N(>Q_{i})}$ (Aschwanden, 2015). The total $\chi^{2}$ is
constructed as

$\chi^{2}_{\mathrm{total}}=\chi^{2}_{F_{\nu}}+\chi^{2}_{z}+\chi^{2}_{E}.$ (24)

So the likelihood of observing a sample of FRBs with $F_{\nu}$, $E$ and $z$
can be written as

$\mathcal{L}(\mathrm{FRBs|{\bm{\theta}}})\propto\mathrm{exp}\left(-\frac{1}{2}\chi^{2}_{\mathrm{total}}\right),$
(25)

where ${\bm{\theta}}$ is the set of free parameters, including $\alpha$ and
$E_{c}$ in the energy distribution, $\log F_{\nu,\rm th}^{\rm max}$ and $n$ in
the selection function, and the parameters in the time-delayed models as
specified in equations (14,15,16). Then the posterior PDFs of free parameters
can be obtained according to the Bayesian theorem,

$P(\mathrm{{\bm{\theta}}|FRBs})\propto\mathcal{L}(\mathrm{FRBs|{\bm{\theta}}})P_{0}({\bm{\theta}}),$
(26)

where $P_{0}$ is the prior of the parameters.

##  3 Data and results

In this section, we use the first CHIME/FRB catalog (Amiri et al., 2021) to
constrain the free parameters. The first CHIME/FRB catalog contains 536 bursts
in total, including 474 apparently non-repeating bursts and 62 repeating
bursts from 18 different FRB sources. All of the bursts have well measured
$\rm DM_{obs}$, but most of them have no redshift measurement. We employ the
${\rm DM_{E}}-z$ relation to infer the redshift and calculate the energy. In
contrast to some previous works (such as Zhang & Zhang 2022, Qiang et al. 2022
and Shin et al. 2023) which combined the non-repeating bursts and the first
burst from the repeating FRB sources, here we only select the non-repeating
bursts. There are many reasons why we do this so. First, repeating and non-
repeating FRBs may have different origins, so their population may be
different. Including the repeating FRBs into our sample may mingle the FRB
population. Second, for the repeating FRBs, burst energies from the same FRB
source vary significantly, so there is no unique way to define the FRB energy.
Third, in the first CHIME/FRB catalog, there are only 18 repeating FRB
sources, in contrast to more than four hundred apparently non-repeating FRB
sources. We expect that including the repeating FRBs into our data sample will
not significantly affect the results. But be caution that (even all)
apparently non-repeaters could become repeaters in the future. In addition, we
only consider the FRBs with $\rm DM_{E}>100$ pc cm-3 (in total 444 FRBs). This
is because the ${\rm DM_{host}}$ term is expected to be in the order of
magnitude $100~{}{\rm pc~{}cm^{-3}}$. For FRBs with $\rm DM_{E}<100$ pc cm-3,
the contribution of the $\rm DM_{host}$ term dominates over the $\rm DM_{IGM}$
term, so the inferred redshift may be strongly biased. Three FRBs with
inferred redshift $z_{\rm inf}>3$ are excluded, because the ${\rm DM_{E}}-z$
relation is obtained based on the assumption that hydrogen and helium are
fully ionized, which may be broken at $z>3$. Excluding five FRBs with no
measurement of fluence, the remaining 436 bursts are call the “Full sample”
for convenience. The inferred redshifts and energies of the Full sample are
listed in Tang et al. (2023).

In addition, we employ the criteria proposed in Amiri et al. (2021) to filter
the data that may be unsuitable for population analyses: (1) signal to noise
ratio $S/N>12$; (2) $\rm DM_{obs}>1.5max(DM_{NE2001},DM_{YMW16})$; (3) not
detected in far sidelobes; (4) not detected during non-nominal telescope
operations; (5) scattering time $\tau_{\rm scat}<$10 ms. The 236 FRBs
satisfying these criteria are called the “Gold sample”. For comparison, the
Full sample and the Gold sample are used to constrain the free parameters
separately. We uniformly divide redshift $z$ into 30 bins in the range
$(0,3)$, divide fluence $\log F_{\nu}$ (in unit of Jy ms) into 25 bins in the
range ($-0.5,2$), and divide energy $\log E$ (in unit of erg) into 30 bins in
the range $(37,43)$. The histograms of energy, redshift and fluence for the
Gold sample and the Full sample are shown in Figure 1. As is seen, there is no
significant difference between this two samples.

Figure 1: The histograms of isotropic energy ($\log E$), inferred redshift
($z$) and fluence ($\log F_{\nu}$) of the Gold sample (upper three panels) and
the Full sample (lower three panels). The colored lines illustrate the best-
fitting results of the four models considered in this paper, with the
corresponding model parameters listed in Table 1 and Table 2.

Based on the likelihood given in equation (25), we calculate the posterior
PDFs of the free parameters using the Python package emcee (Foreman-Mackey et
al., 2013). We take non-informative priors for all the free parameters in a
wide range, specifically, $\alpha\in U(0.1,5)$, $\mathrm{log}E_{c}\in
U(37,44)$, $n\in U(0.1,8)$, $\mathrm{log}F_{\nu,\rm th}^{\rm max}\in
U(0.01,2.0)$, $\tau_{0}\in U(0.1,5)$, $\sigma\in U(0.1,5)$, $\alpha_{\tau}\in
U(0.1,5)$, where $\tau_{0}$ and $\sigma$ are in unit of billion years (Gyr).
The optimal parameters (the median values of the posterior PDFs) and their
1$\sigma$ uncertainties are summarized in Table 1 for the Gold sample, and in
Table 2 for the Full sample. The best-fitting curves are over-plotted in the
histograms in Figure 1. The posterior PDFs of the model parameters and the
two-dimensional confidence contours are shown in Figure 2. Note that the best-
fitting parameters and best-fitting curves are derived using the CDF rather
than the PDF, so they are independent of the binning of data points. We plot
the best-fitting curves in the histogram just for visualization purpose,
because curves of different models have significant overlap in the CDF plot.

Table 1: The optimal parameters and their 1$\sigma$ uncertainties constrained from the Gold sample. The parameters $\tau_{0}$ and $\sigma$ are in unit of billion years (Gyr). Model | $\alpha$ | log$E_{c}$/erg | log$F_{\nu,\rm th}^{\rm max}$ | $n$ | Model parameter |  |  $\Delta$BIC
---|---|---|---|---|---|---|---
SFH | $2.14^{+0.01}_{-0.01}$ | $42.02^{+0.01}_{-0.01}$ | $0.65^{+0.01}_{-0.01}$ | $4.49^{+0.13}_{-0.12}$ | N/A | N/A | 588.64
Gaussian | $1.89^{+0.02}_{-0.01}$ | $42.13^{+0.02}_{-0.02}$ | $0.48^{+0.02}_{-0.01}$ | $5.22^{+0.28}_{-0.27}$ | $\tau_{0}=0.16^{+0.09}_{-0.04}$ | $\sigma=3.84^{+0.22}_{-0.21}$ | 26.61
log-normal | $1.88^{+0.01}_{-0.01}$ | $42.14^{+0.02}_{-0.02}$ | $0.47^{+0.01}_{-0.01}$ | $5.23^{+0.30}_{-0.27}$ | $\tau_{0}=4.09^{+0.58}_{-0.55}$ | $\sigma=1.49^{+0.09}_{-0.10}$ | 0(fiducial)
power-law | $1.85^{+0.01}_{-0.01}$ | $42.17^{+0.02}_{-0.02}$ | $0.45^{+0.01}_{-0.01}$ | $5.28^{+0.34}_{-0.28}$ | $\alpha_{\tau}=2.72^{+1.85}_{-1.68}$ |  | 12.43
Table 2: The optimal parameters and their 1$\sigma$ uncertainties constrained from the Full sample. The parameters $\tau_{0}$ and $\sigma$ are in unit of billion years (Gyr). Model | $\alpha$ | log$E_{c}$/erg | log$F_{\nu,\rm th}^{\rm max}$ | $n$ | Model parameter |  |  $\Delta$BIC
---|---|---|---|---|---|---|---
SFH | $2.14^{+0.01}_{-0.01}$ | $41.94^{+0.01}_{-0.01}$ | $0.70^{+0.01}_{-0.01}$ | $3.43^{+0.03}_{-0.03}$ | N/A | N/A | 2015.58
Gaussian | $1.82^{+0.01}_{-0.01}$ | $42.05^{+0.01}_{-0.01}$ | $0.46^{+0.01}_{-0.01}$ | $3.52^{+0.08}_{-0.08}$ | $\tau_{0}=0.14^{+0.06}_{-0.03}$ | $\sigma=4.96^{+0.03}_{-0.06}$ | 56.28
log-normal | $1.85^{+0.01}_{-0.01}$ | $42.04^{+0.01}_{-0.01}$ | $0.49^{+0.01}_{-0.01}$ | $3.43^{+0.07}_{-0.07}$ | $\tau_{0}=4.93^{+0.05}_{-0.11}$ | $\sigma=1.47^{+0.03}_{-0.03}$ | 0(fiducial)
power-law | $1.84^{+0.01}_{-0.01}$ | $42.06^{+0.01}_{-0.01}$ | $0.48^{+0.01}_{-0.01}$ | $3.46^{+0.08}_{-0.07}$ | $\alpha_{\tau}=2.66^{+1.96}_{-1.81}$ |  | 5.97

Figure 2: The contour plots and marginalized posterior PDFs of the parameter
space. Upper-left: SFH model; upper-right: Gaussian delay model; lower-left:
log-normal delay model; lower-right: power-law delay model.

From Table 1, Table 2 and Figure 2, we can see that all parameters, except for
the power-law index $\alpha_{\tau}$ in the power-law delay model, can be
tightly constrained. The selection function strongly depends on the FRB
samples. Specifically, the parameter $n$ constrained from the Gold sample
($n\sim 5.2$) is much larger than that constrained from the Full sample
($n\sim 3.5$). The parameters related to the energy distribution and redshift
distribution are almost independent of the samples, implying that our method
is robust. The energy function parameters are consistent in three time-delayed
models, with $\alpha\sim 1.8$ and $\log E_{c}\sim 42$. The SFH model gives a
little larger value of $\alpha\sim 2.1$, but the cutoff energy $\log E_{c}$ is
consistent with the three time-delayed models. In the log-normal delay model,
the median time delay is $\tau_{0}=4.09_{-0.55}^{+0.58}$ Gyr for the Gold
sample, and $\tau_{0}=4.93_{-0.11}^{+0.05}$ Gyr. The Gaussian delay model
gives a much smaller value of $\tau_{0}$, i.e. $\tau_{0}=0.16_{-0.04}^{+0.09}$
Gyr for the Gold sample, and $\tau_{0}=0.14_{-0.03}^{+0.06}$ Gyr for the Full
sample. But be caution that $\tau_{0}$ is not the average time delay, because
the time delay is positive definite. The average time delay for the Gaussian
delay model can be calculated by $\bar{\tau}=\int_{0}^{\infty}\tau
f(\tau)d\tau/\int_{0}^{\infty}f(\tau)d\tau$. Thus we obtain $\bar{\tau}\sim
3.12$ Gyr for the Gold sample and $\bar{\tau}\sim 3.95$ Gyr for the Full
sample, which are comparable with the median time delays of the log-normal
delay model. As for the power-law delay model, the power-law index can’t be
well constrained, so we can’t give a reasonable estimation on the average time
delay.

To quantify the goodness of fit and pick up the optimal model, we calculate
the Bayesian information criterion (BIC), which is defined by (Schwarz, 1978)

${\rm BIC}=-2{\rm ln}\mathcal{L}_{\rm max}+k{\rm ln}(N),$ (27)

where $\mathcal{L}_{\rm max}$ represents the maximum likelihood estimated at
the best-fitting parameters, $k$ denotes the number of free parameters, and
$N$ is the total number of data points. A model with a smaller BIC value
indicates a superior fit. Note that it is the relative value of BIC, rather
than its absolute value, that holds significance. Choosing the model which has
the smallest BIC value as the fiducial model, the relative value of BIC is
defined as

$\Delta\rm BIC_{model}=BIC_{model}-BIC_{fiducial}.$ (28)

According to the Jeffreys’ scale (Jeffreys, 1998; Liddle, 2007), a model with
$\rm\Delta BIC\geq 5$ or $\rm\Delta BIC\geq 10$ indicates that there is
‘strong’ or ‘decisive’ evidence for disfavoring this model against the
fiducial model.

In the last column of Table 1 and Table 2, we list the $\rm\Delta BIC$ value
of each model. We can see that for both data samples, the model superiorities
are in the same order, namely Log-normal $>$ Power-law $>$ Gaussian $>$ SFH.
Both samples most prefer the log-normal model, so it is chosen as the fiducial
model. According to the $\Delta{\rm BIC}$ value, the power-law model and the
Gaussian model fit the data worse than the log-normal model, but they are much
better than the SFH model. Compared with the three time-delayed models, the
SFH model is decisively ruled out. This can also be clearly seen from Figure
1, in which the SHF curves significantly deviate from the data points. In
summary, the SFH model is decisively excluded by both samples, but all the
three time-delayed models can to some extent match both samples.

##  4 Discussion and Conclusions

In this paper, we investigated the FRB population using the first CHIME/FRB
catalog. The ${\rm DM_{E}}-z$ relation was used to infer redshift and energy
of each FRB. We constructed a Bayesian framework to fit the models
simultaneously to the distributions of energy, redshift and fluence. This
approach allows us to derive the best-fitting parameters when taking the
selection effect into account. The observables of FRBs are mainly influenced
by three factors: the intrinsic redshift distribution, the intrinsic energy
distribution, and the selection effect of detector. The intrinsic energy
distribution is modeled by the cutoff power-law. The selection effect is
modeled by a two-parametric function of fluence. The intrinsic redshift
distribution is what we are most interested in, and four models are
considered, namely the SFH model and three time-delayed models. For
comparison, two FRB samples are used to fit the models separately.

We constructed the joint likelihood of energy, redshift and fluence, and
calculate the posterior PDFs of the parameters using the Bayesian inference
method. The best-fitting parameters are summarized in Table 1 for the Gold
sample and Table 2 for the Full sample. Different samples mainly affect the
selection function, but the model parameters are insensitive to the samples,
implying the robustness of our method. The energy function is almost
independent of the redshift model we chosen. The best-fitting power-law index
$\alpha\approx 1.8$ is in good agreement with the previous results (Lu & Piro,
2019; Luo et al., 2018, 2020; Lu et al., 2020; Lin & Zou, 2024). In addition,
the cutoff energy is also tightly constrained to be $\log E\approx 42$, which
is consistent with the results of Lin & Zou (2024), but it is slightly larger
than the results of Zhang & Zhang (2022) and Qiang et al. (2022). Both FRB
samples support the log-normal model as the best model, but the other two
time-delayed models (Gaussian model and power-law model) can also match the
data to some extent. Compared with the three time-delayed models, the SFH
model is conclusively ruled out. In summary, there is no doubt that the FRB
population has significant time delay with respective to the SFH, but the
distribution of time delay remains to be further investigated.

The time-delayed models have already been exhaustively studied using the first
CHIME/FRB catalog by Zhang & Zhang (2022). There are two main differences
between our work and Zhang & Zhang (2022). The first important difference is
the method used to infer redshift. In our work, we reconstruct the ${\rm
DM_{E}}-z$ relation using Bayesian inference method, and the probability
distributions of ${\rm DM_{IGM}}$ and ${\rm DM_{host}}$ are considered. On the
contrary, Zhang & Zhang (2022) assumed that ${\rm DM_{host}}$ is a constant,
and directly calculated redshift by solving equation (2). As is shown in Tang
et al. (2023), the ${\rm DM_{E}}-z$ relation curve obtained using Bayesian
method is relatively lower than that directly calculated using equation (2).
Hence the inferred redshift (so the energy) we obtained here is larger than
that in Zhang & Zhang (2022). This is one of the reasons why the cutoff energy
we obtained is larger than that of Zhang & Zhang (2022) (another reason is
that Zhang & Zhang (2022) does not consider the k-correction in energy
calculation). The second major difference is the method of parameter
inference. In our work, we construct a joint likelihood of redshift, energy
and fluence, and optimize the parameters using the Bayesian inference method.
On the contrary, Zhang & Zhang (2022) divided the parameter space into grids,
and use the KS-test to check if each set of parameters can match the
observational data. The Bayesian inference method has advantage over the KS-
test method, because the latter can’t derive the optimal parameters. In spite
of these differences, we reach the same conclusion that the FRB population
does not trace the SFH. With the same CHIME/FRB sample, Hashimoto et al.
(2022) also showed that the SFH model is ruled out with high confidence level.

Similar work has been done by Shin et al. (2023), but they found that the SFH
model couldn’t be ruled out, which is in confliction with ours. The sample
used by Shin et al. (2023) has significant overlap with the Gold sample used
in our paper. So the different conclusions are unlikely to arise from the data
samples, but may be caused by the different methods we used. In our paper, we
first use the well-localized FRBs to reconstruct the ${\rm DM_{E}}-z$
relation, and then use it to infer the redshift and energy of the CHIME/FRBs.
In contrast, the well-localized FRBs were not used by Shin et al. (2023). The
parameters related to the ${\rm DM_{E}}-z$ relation ($\mu_{\rm host}$ and
$\sigma_{\rm host}$) were constrained simultaneously with the parameters
related to the energy and redshift distributions. They also included the
spectral index $\alpha$ as a free parameter. So the parameter space in Shin et
al. (2023) is larger than that in our paper, leading to relatively larger
uncertainties of parameters. In addition, the redshift distribution models we
considered here are different from that of Shin et al. (2023). In Shin et al.
(2023), the authors parameterized the redshift distribution as
$p(z)\propto[{\rm SFH}(z)]^{n}$. The best-fitting value
$n=0.96_{-0.67}^{+0.81}$ is well consistent with unity, so they concluded that
the CHIME/FRB population is consistent with SFH. But the uncertainty of $n$ is
very large, so the time delay is still possible. Note that the power-law form
$p(z)\propto[{\rm SFH}(z)]^{n}$ is too simple, and the peak redshift (i.e. the
redshift at which $p(z)$ reaches its maximum) is independent of $n$. So the
power-law parametrization of $p(z)$ is hard to reveal the time delay of FRB
population with respect to SFH, even if the time delay really exists.

The uncertainty of our method mainly arises from the inference of redshift. We
infer redshift of each FRB using the ${\rm DM_{E}}-z$ relation, which is
reconstructed from 17 well-localized FRBs at low redshift ($z<1$). We assume
that the parameters of the ${\rm DM_{E}}-z$ relation is redshift-independent
and directly extrapolate them to high redshift. Due to the lack of high-
redshift FRBs, the ${\rm DM_{E}}-z$ relation may has large uncertainty at
$z>1$. James et al. (2021) pointed out that there is a critical point beyond
which the one-to-one correspondence between ${\rm DM_{E}}$ and redshift may be
broken. If this is indeed the case, it is impossible to use the ${\rm
DM_{E}}-z$ relation to infer redshift. When calculate the burst energy, we
include a k-correction term to transform it to the rest frame, see equation
(7). However, the k-correction depends on the power-law index of energy
spectra, which has a large uncertainty for the CHIME/FRB catalog. To check if
the k-correction will significantly affect our results or not, we recalculate
the burst energy without the k-correction, i.e. $E=4\pi
d_{L}^{2}F_{\nu}\Delta\nu/(1+z)$, and redo our Bayesian analysis. We find that
including the k-correction or not can slightly affect the best-fitting
parameters, but does not change our main conclusions.

In summary, it is still premature to make a conclusive statement about the FRB
population based on the currently available data. We can confidently veto the
hypothesis that FRBs trace the SFH. There is strong evidence that the FRB
population has on average $3\sim 5$ billion years time delay with respect to
the SFH, but the accurate probability distribution of time delay is still
unconstrained. The log-normal delay model seems to match the data best, but
the power-law delay model and Gaussian delay model are also acceptable. We
anticipate that a large sample of well localized FRBs in the near future can
provide definitive insights into the FRB population.

This work has been supported by the National Natural Science Fund of China
under grant nos. 12275034 and 12347101, and the Fundamental Research Funds for
the Central Universities of China under grant no. 2023CDJXY-048.

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