# Subset Sum Instances in ZFC Limbo

S. Gill Williamson

###### Abstract.

Our main result, Theorem 3.5, proves the existence an infinity of subset sum
problems solvable in polynomial time. The only proof we have of this result
uses the ZFC independent Jump Free Theorem of Friedman [Fri97], thus putting
Theorem 3.5 in what we informally call ZFC limbo [Wil17c]. The mathematics we
use is elementary and at the level of a good undergraduate course in
combinatorics or design and analysis of algorithms. The statement of the Jump
Free Theorem is also easily understood at this level (but not the proof).

Department of Computer Science and Engineering, University of California San
Diego; http://cseweb.ucsd.edu/~gill/<EMAIL_ADDRESS>Keywords:
combinatorics; lattice graphs; ZFC independence; P equals NP; order type
equivalence; regressive regularity; subset sum problem

##  1\. Introduction

Let $N=\\{0,1,2,\ldots\\}$ and $k\geq 2$. In this note we derive some
connections between the space and time hierarchies of computer science and the
large cardinal hierarchies of set theory. Our main result, Theorem 3.5,
concerns families $S$ of functions $f$ with domains finite subsets of $N^{k}$
and ranges finite subsets of $N$ ("finite" functions for short). We define
these families $S$ below in terms of elementary set theoretic properties:
finite, reflexive, jump free, full, etc.

Let $k,p\geq 2$. For any pair $(f,E)$ and $E^{k}\subseteq{\it domain}(f)$,
$|E|=p$, the function $f\in S$ naturally partitions $E^{k}\subset N^{k}$ into
three blocks $E_{0}^{k}$, $E_{1}^{k}$, $E_{2}^{k}$ (Definition 3.2). We define
(Definition 3.3) sets of integers $\Delta[f]E_{0}^{k}$, $\Delta[f]E_{1}^{k}$,
$\Delta[f]E_{2}^{k}$ associated with these three blocks. This construction is
parameterized by functions $\rho$ and $\gamma$ (Definition 3.1) and associated
with integers $t\geq 1$ (Definition 3.4) so the sets
$H_{p}=\Delta[f]E_{0}^{k}\cup\Delta[f]E_{1}^{k}\cup\Delta[f]E_{2}^{k}$ are
"generic" in some sense. We consider sequences of sets of this type
$H_{2},H_{3},\ldots,H_{p},\ldots$. Note that $|H_{p}|=p^{k}$ (as a multiset).
We prove Theorem 3.5 that for fixed $t\geq 1,\,k\geq 2$ associated "jump free
family" $S$, there is at least one sequence of the type
$H_{2},H_{3},\ldots,H_{p},\ldots$ that, considered as instances to subset sum
target $0$, can be solved in polynomial time $O((p^{k})^{t})$ where $p^{k}$
measures the size of the instances. Our only known proof uses Friedman’s Jump
Free Theorem 2.8 which is independent of ZFC. We conclude with some
interesting consequences.

##  2\. Basic definitions and theorems

We denote by $N$ the set of all nonnegative integers. For
$z=(n_{1},\ldots,n_{k})\in N^{k}$, $\max\\{n_{i}\mid i=1,\ldots,k\\}$ is
denoted by $\max(z)$. Define $\min(z)$ similarly.

In this section and the next we discuss Friedman’s Jump Free Theorem. In
Section 3 we prove our main results.

######  Definition 2.1 (Cubes and Cartesian powers in $N^{k}$).

The set $E_{1}\times\cdots\times E_{k}$, where $E_{i}\subset N$, $|E_{i}|=p$,
$i=1,\ldots,k,$ is called a $k$-cube of length $p$. If $E_{i}=E,i=1,\ldots,k,$
then this cube is $E^{k}$, the $k$th Cartesian power of $E$.

######  Definition 2.2 (Equivalent ordered $k$-tuples).

Two k-tuples in $N^{k}$, $x=(n_{1},\ldots,n_{k})$ and
$y=(m_{1},\ldots,m_{k})$, are order equivalent tuples $(x\,ot\,y)$ if the sets
$\\{(i,j)\mid n_{i}<n_{j}\\}=\\{(i,j)\mid m_{i}<m_{j}\\}$ and $\\{(i,j)\mid
n_{i}=n_{j}\\}=\\{(i,j)\mid m_{i}=m_{j}\\}.$

Note that $ot$ is an equivalence relation on $N^{k}$. The number of
equivalence classes is $\sum_{j=1}^{k}\sigma(k,j)<k^{k},$ $k\geq 2$, where
$\sigma(k,j)$ is the number of surjections from a $k$ set to a $j$ set. We use
“$x\,ot\,y$” and “$x,\,y$ of order type $ot$” to mean $x$ and $y$ belong to
the same order type equivalence class.

We present some basic definitions due to Friedman [Fri97], [Fri98].

######  Definition 2.3 (Regressive value).

Let $Y\subseteq N$, $X\subseteq N^{k}$ and $f:X\rightarrow Y$. An integer $n$
is a regressive value of $f$ on $X$ if there exist $x$ such that
$f(x)=n<\min(x)$ .

######  Definition 2.4 (Field of a function and reflexive functions).

For $A\subseteq N^{k}$ define ${\rm field}(A)$ to be the set of all
coordinates of elements of $A$. A function $f$ is reflexive in $N^{k}$ if
${\rm domain}(f)\subseteq N^{k}$ and ${\rm range}(f)\subseteq{\rm field}({\rm
domain}(f))$.

######  Definition 2.5 (The set of functions $T(k)$ ).

$T(k)$ denotes all reflexive functions with finite domain. We sometimes denote
a function with domain $D$ by $f_{D}$

######  Definition 2.6 (Full and jump free families).

Let $Q\subseteq T(k).$

  1. (1)

full family: We say that $Q$ is a full family of functions on $N^{k}$ if for
every finite subset $D\subset N^{k}$ there is at least one function $f$ in $Q$
whose domain is $D$.

  2. (2)

jump free family: For $D\subset N^{k}$ and $x\in D$ define $D_{x}=\\{z\mid
z\in D,\,\max(z)<\max(x)\\}$. Suppose that for all $f_{A}$ and $f_{B}$ in $Q$,
where $f_{A}$ has domain $A$ and $f_{B}$ has domain $B$, the conditions $x\in
A\cap B$, $A_{x}\subseteq B_{x}$, and $f_{A}(y)=f_{B}(y)$ for all $y\in A_{x}$
imply that $f_{A}(x)\geq f_{B}(x)$. Then $Q$ will be called a jump free family
of functions on $N^{k}$.

The jump free condition is illustrated in Figure 1. Intuitively, referring to
Figure 1, suppose that the region $A_{x}$ is to be searched for the smallest
of some quantity and the result recorded at $x$. Next, the search region is
expanded to a superset $B_{x}$ with the search results for $A_{x}$ still valid
(i.e., $f_{A}(y)=f_{B}(y)$ for all $y\in A_{x}$). Then, clearly $f_{A}(x)\geq
f_{B}(x)$.

######  Definition 2.7 (Function regressively regular over $E$).

Let $k\geq 2$, $D\subset N^{k}$, $D$ finite, $f:D\rightarrow N$. We say that
$f$ is regressively regular over $E$ (alternatively $f$ is regressively
regular over $E^{k}$) $E^{k}\subset D$, if for each order type equivalence
class $ot\,$ of $k$-tuples of $E^{k}$ either (1) or (2) occurs:

  1. (1)

constant less than min $\mathbf{E}$: For all $x,y\,\in E^{k}$ of order type
$ot$, $f(x)=f(y)<\min(E).$

  2. (2)

greater or equal min: For all $x\in E^{k}$ of order type $ot$
$f(x)\geq\min(x).$

Figure 1. Basic jump free condition 2.6.

We use ZFC for the axioms of set theory: Zermelo-Frankel plus the axiom of
choice. The Jump Free Theorem, Theorem 2.8 stated below, can be proved in ZFC
+ ($\forall n$)($\exists$ $n$-subtle cardinal) but not in ZFC+ ($\exists$
$n$-subtle cardinal) for any fixed $n$ (assuming this theory is consistent). A
proof of the Jump Free Theorem is in Section 2 of [Fri97], “Applications of
Large Cardinals to Graph Theory.” This proof references certain results from
[Fri98]. Most students can just take the proof of the Jump Free Theorem as
given for this discussion. The proof is complicated (but well checked by
logicians). Wikpedia is a good source for learning some basics about ZFC and
large cardinals.

######  Theorem 2.8 (Jump Free Theorem ([Fri97], [Fri98])).

Let $p,k\geq 2$ and $S\subseteq T(k)$ be a full and jump free family of
functions on $N^{k}$. Then some $f_{D}\in S$ has at most $k^{k}$ regressive
values on some $E^{k}\subseteq D$, $|E|=p$. In fact, some $f\in S$ is
regressively regular over some $E$ of cardinality $p$.

We note that the statement “$f\in S$ is regressively regular over some $E$ of
cardinality $p$” implies that ”some $f\in S$ has at most $k^{k}$ regressive
values on some $E^{k}\subseteq{\rm domain}(f)$, $|E|=p$”.

##  3\. Subset sum instances and the Jump Free Theorem

Let $S\subseteq T(k)$ be a full and jump free family of functions on $N^{k}$.
The Jump Free Theorem asserts that for each $p\geq 2$ there exists an $f\in S$
such that $f$ is regressively regular over some
$E^{k}\subseteq\mathrm{domain}(f)$ and $|E|=p$. Definition 3.1 introduces two
families of functions that will be used to construct instances to the subset
sum problem. $Z$ denotes the integers.

######  Definition 3.1 (Families of functions).

For each fixed $k\geq 2$ we define a two families of functions, $\rho$ and
$\gamma$ with domains $E^{k}$, $E$ a finite subset of $N$.

$\mathbb{R}_{k}=\\{\rho\mid\rho:E^{k}\rightarrow N,\,E^{k}\subset
N^{k},\,\rho(x)\geq\min(x),\,x\in E^{k}\\}.$
$\mathbb{I}_{k}=\\{\gamma\mid\gamma:E^{k}\rightarrow Z,E^{k}\subset N^{k}\\}.$

In the next definition we consider functions $f$ and sets of the form $E^{k}$
contained in the domain of $f$. We use $f$ to partition $E^{k}$ into three
blocks which we call $E^{k}_{0}$, $E^{k}_{1}$, and $E^{k}_{2}$. These
partitions will be used to make a connection between sets of instances defined
in Definition 3.3 and regressive regularity, Definition 2.7.

######  Definition 3.2 ($f\in S$ partitions $E^{k}$).

Let $k,p\geq 2$, $f\in S$ , $E^{k}\subseteq{\rm domain}(f)$, $|E|=p.$ We
define the blocks $E^{k}_{0}$, $E^{k}_{1}$, $E^{k}_{2}$ of a partition of
$E^{k}$ as follows:

$E^{k}_{0}=\\{x\in E^{k}:f(x)<\min(E)\\}$ $E^{k}_{1}=\\{x\in E^{k}:\min(E)\leq
f(x)<\min(x)\\}$ $E^{k}_{2}=\\{x\in E^{k}:\min(x)\leq f(x)\\}.$

We map the blocks of the partition 3.2 into sets of integers as follows:

######  Definition 3.3 (Sets of instances from $E^{k}$).

Let $k,p\geq 2$, $f\in S$ , $E^{k}\subseteq{\rm domain}(f)$, Let the $\gamma$
and the $\rho$ be as in Definition 3.1. Using Definition 3.2 we define
$\Delta[f]E_{0}^{k}=\\{f(x)-\min(E):x\in E_{0}^{k}\\}$,
$\Delta[f]E_{1}^{k}=\\{\gamma(x):x\in E_{1}^{k}\\}$ and
$\Delta[f]E_{2}^{k}=\\{\rho(x)-\min(x):x\in E^{k}_{2}\\}$. Let $H_{p}$ denote
sets of the form

$H_{p}=\Delta[f]E_{0}^{k}\cup\Delta[f]E_{1}^{k}\cup\Delta[f]E_{2}^{k}$

where $|E|=p$.

We define a special class of functions $\rho$ of Definition 3.1.

######  Definition 3.4 ($t$-log bounded $\rho$ over $E^{k}$, $|E|=p$).

For $t\geq 1$ the function $\rho$ is $t$-log bounded over $E^{k}$ where
$E=\\{e_{0},\ldots,e_{p-1}\\}\,$ if

$|\\{\rho(x)-\min(x):0<\rho(x)-\min(x)<e_{0}k^{k},\,x\in E^{k}\\}|\leq
t\log_{2}(p^{k}).$

Recalling that $\rho(x)\geq\min(x)$ and $\rho(x)$ can be arbitrarily large, we
can choose

$|\\{\rho(x)-\min(x):\rho(x)-\min(x)\geq e_{0}k^{k},\,x\in E^{k}\\}|$

large enough to make $\rho$ $t$-log bounded.

######  Theorem 3.5 (Instances to subset sum target zero).

Let $t\geq 1,k\geq 2$ be fixed. Let $p\geq 2$ and let $S\subseteq T(k)$ be a
full, reflexive and jump free family. Referring to Definition 3.3, consider
sets of the form
$H_{p}=\Delta[f]E_{0}^{k}\cup\Delta[f]E_{1}^{k}\cup\Delta[f]E_{2}^{k}$ where
$|E|=p$ and $f\in S$. We regard sequences of the form
$H_{2},H_{3},\ldots,H_{p},\ldots$, as sequences of instances (size measured by
$\,p^{k}$) to the subset sum problem target zero. For each fixed $t\geq 1$,
$k\geq 2$, $S\subseteq T(k)$ there is a sequence
$H_{2},H_{3},\ldots,H_{p},\ldots$ that can be solved in polynomial time where
$p^{k}$ is the size of the instances.

###### Proof.

For each $p\geq 2$ use the Jump Free Theorem 2.8 to choose $f\in S$
regressively regular over some $E^{k}\subseteq{\mathrm{d}omain(f)}$, $|E|=p$.

By regressive regularity, $\Delta[f]E_{1}^{k}$ is empty. There are at most
$k^{k}$ terms in $\Delta[f]E^{k}_{0}$ and all are negative with absolute value
less than $e_{0}=\min(E)$. The terms in $\Delta[f]E^{k}_{2}$ are all greater
than or equal to zero. Zero is the target, so we assume all terms in
$\Delta[f]E^{k}_{2}$ are positive. Since $E^{k}_{2}\subseteq E^{k}$ we have

$|\\{\rho(x)-\min(x):0<\rho(x)-\min(x)<e_{0}k^{k},\,x\in E_{2}^{k}\\}|\leq
t\log_{2}(p^{k}).$

Thus, to solve the subset sum problem we compare all possible sums of
$2^{k^{k}}$ subsets of negative terms with all $2^{t\log_{2}(p^{k})}$ sums of
subsets of positive terms. Positive terms greater than or equal $e_{0}k^{k}$
are larger than any sums of negative terms. Thus, we have
$2^{k^{k}}(p^{k})^{t}$ total comparisons of sums. Thus, for each fixed $t$ and
$k$ the sequence of instances to subset sum target zero,
$H_{2},H_{3},\ldots,H_{p},\ldots$ can be solved in polynomial time where
$p^{k}$ is the size of the instances. ∎

##  4\. Conclusions

Theorem 3.5 uses the ZFC independent Jump Free Theorem (for each fixed $t\geq
1$). We know of no other proof thus putting Theorem 3.5 in what we call ”ZFC
Limbo.” If a ZFC proof could be found that the subset sum problem is solvable
in polynomial time $O(n^{\tau})$ where $n$ is the length of the instance
($p^{k}$ for fixed $k$ here), then that result would prove Theorem 3.5 for
$t=\tau$ and thus show it is provable within ZFC (i.e. remove it from ZFC
limbo). We conjecture, however, that Theorem 3.5 cannot be proved in ZFC alone
(for each fixed $t$). The basis for this conjecture is that the subset sum
problem arises from Theorem 3.5 and thus from the Jump Free Theorem in a very
natural, generic way. In particular, the sets of instances 3.3 are constructed
to be an alternative description of regressive regularity. Of course, if our
conjecture is true, “subset sum is solvable in polynomial time” cannot be
proved in ZFC (perhaps because it is false).

Acknowledgments: The author thanks Professor Sam Buss (University of
California San Diego, Department of Mathematics), Professor Emeritus Rod
Canfield (University of Georgia, Department of Computer Science), and
Professor Emeritus Victor W. Marek (University of Kentucky, Department of
Computer Science) for numerous helpful suggestions.

## References

  * [Fri97] Harvey Friedman.  Applications of large cardinals to graph theory.  Technical report, Department of Mathematics, Ohio State University, 1997\.
  * [Fri98] Harvey Friedman.  Finite functions and the necessary use of large cardinals.  Ann. of Math., 148:803–893, 1998.
  * [RW99] Jeffrey B. Remmel and S. Gill Williamson.  Large-scale regularities of lattice embeddings of posets.  Order, 16:245–260, 1999.
  * [Wil17a] S. Gill Williamson.  Lattice exit models.  arXiv:1907.11707v2 [math.CO], 2017.
  * [Wil17b] S. Gill Williamson.  On the difficulty of proving P=NP in ZFC.  arXiv:1708.08186 [math.CO], 2019.
  * [Wil17c] S. Gill Williamson.  Combinatorics in ZFC limbo.  JOC Vol. 10, Num. 3, 579-593 2019\.


# How people talk about each other:
Modeling Generalized Intergroup Bias and Emotion

Venkata S. Govindarajan1 Katherine Atwell2 Barea Sinno3
Malihe Alikhani2 David I. Beaver1 Junyi Jessy Li1
1 Department of Linguistics, The University of Texas at Austin
2 Department of Computer Sciences, University of Pittsburgh
3 Department of Political Science, Rutgers University
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Current studies of bias in NLP rely mainly on identifying (unwanted or
negative) bias towards a specific demographic group. While this has led to
progress recognizing and mitigating negative bias, and having a clear notion
of the targeted group is necessary, it is not always practical. In this work
we extrapolate to a broader notion of bias, rooted in social science and
psychology literature. We move towards predicting interpersonal group
relationship (IGR) — modeling the _relationship between the speaker and the
target_ in an utterance—using fine-grained interpersonal emotions as an
anchor. We build and release a dataset of English tweets by US Congress
members annotated for interpersonal emotion – the first of its kind, and
‘found supervision’ for IGR labels; our analyses show that subtle emotional
signals are indicative of different biases. While humans can perform better
than chance at identifying IGR given an utterance, we show that neural models
perform much better; furthermore, a shared encoding between IGR and
interpersonal perceived emotion enabled performance gains in both tasks.

##  1 Introduction

Currently, most work studying bias in NLP situates bias as negative or
pejorative language use towards an individual or group based on traits like
race, gender, etc (Kaneko and Bollegala, 2019; Sheng et al., 2019; Sap et al.,
2020; Webson et al., 2020; Pryzant et al., 2020; Sheng et al., 2020). While
these approaches greatly advance our understanding of bias in language and its
impact and mitigation in NLP, focusing on specific demographic dimensions or
an individual’s intent is limiting and not always practical. Research in
psychology and social science suggests a different perspective. Bias can be
seen as a relationship between people and groups, situated in context (Van
Dijk, 2009); as such, bias refers to differences in behavior (in this case
language use) as a result of differences in the relationship between speaker
and target. The language we produce is biased in one way or another, whether
we intend to or not, and whether that bias is positive, negative, or not
clearly associated with any valuation (Beaver and Stanley, 2018).

In psychological work on Linguistic Intergroup Bias (Maass, 1999), bias
originates from the relationship between the speaker and target of an
utterance, i.e. their interpersonal dynamics, and manifests later in subtle
ways. Consider the utterances (tweets) in 1, drawn from our collected data in
which the identity of the speaker and target are masked:

. . In-group: We stand w @Doe, who has seen a lot worse than cheap insults
from an insecure bully. #MLKDAY weekend. .̱ Out-group: Parents and families
live in constant fear for their children with food allergies. A worthy
bipartisan cause - thank you @Doe for your leadership on this issue.

Both express support and admiration towards the target referent _Doe_ –
however, the second example uses words indicative that the speaker and target
do not share a relevant social identity (in this case, their political party),
expressed by words like _bipartisan_. The intensity of admiration expressed is
also greater in 1 than 1. Thus, these two seemingly similar statements differ
along interpersonal dimensions that are instructive as to how the bias of the
speaker seeps into the utterance.

We now introduce two new tasks that directly model language use in terms of
two interpersonal dimensions: (i) interpersonal group relationship (IGR)
prediction, where we seek to understand how people talk about others who they
consider to be in their same social group (in-group), versus those they
consider outside their social group (out-group), and (ii) _perceived_
interpersonal emotion detection, where we situate these differences in terms
of the emotion expressed in text _towards or in connection with_ a target
individual described in the utterance. Note that _interpersonal_ emotion is
different from a more standard, utterance level emotion detection task, as
illustrated in row 2 of Table 1 which has seemingly opposing emotions.

We present a first-of-its-kind, _annotated_ dataset for fine-grained
interpersonal emotion detection, consisting of 3,033 tweets from members of
the US Congress; all of these tweets mention another Congress member, hence
providing us with ‘found supervision’ for IGR prediction (whether the speaker
and the target belong to the same political party). Our analyses show that
while positive interpersonal emotions appear in both in- and out-group
situations, negative emotions like anger and disgust are overwhelmingly
present in the latter. Meanwhile, human judgments for in vs. out-group
membership on this dataset are overly reliant on the polarity of emotion;
specifically, human judges are much _less_ likely to attribute positive
emotions towards out-group targets.

Baseline performances for perceived interpersonal emotion detection shows that
this is a challenging task, as is consistent with existing work in emotion
detection in general Demszky et al. (2020). In particular, emotions in this
dataset are often expressed with considerable subtlety, likely a
characteristic of official political speech. To investigate whether IGR and
emotions are intertwined and useful towards each other, we further developed a
multi-task model for the prediction of both. We found compelling evidence that
multi-tasking IGR and interpersonal emotion improves performance on both tasks
with over 10% improvement in detection of disgust in out-group contexts, and
3% improvement in IGR prediction.

To summarize the contributions of this paper, we tackle _generalized
intergroup bias_ , a notion of bias rooted in social psychology that applies
to all the various differences in the ways that people talk about others in
their in-group or out-group. Standard bias tasks in NLP, and the broader goal
of debiasing models could thus be set in a more general context. We present
the first dataset to study both interpersonal group membership and emotion,
which allows us to analyze both human and model behavior in terms of how the
two interact with each other. We release our code and data online at
github.com/venkatasg/interpersonal-bias.

Tweet | Interpersonal Emotion | In/Out group?
---|---|---
As @Doe says, the times have found each and every one of us to Defend our Democracy For The People. Worth reading every line. | Admiration | In-group
Freedom has no greater nor tougher champion than @Doe. My prayers are with him and his family. | Admiration & Sadness | In-group
You don’t get to decide what’s “fine,” @Doe. The constitution does. #DefendOurDemocracy #WednesdayThoughts | Anger & Disgust | Out-group
Thank you again Senator @Doe for leading the SRF WIN Act[…] I’m proud to be a co-sponsor | Admiration & Joy | Out-group
Table 1: Example utterances from our dataset with in/out group and
interpersonal emotion labels

##  2 Interpersonal Contexts & Emotions

Our aim is to build a generalized, data-driven approach towards studying bias
situated in interpersonal utterances, which we define as any utterance where
there is a target individual being talked about or referred to. Our goal is to
model two novel tasks described below; examples are shown in Table 1.

#### Interpersonal Group relationship

IGR is defined by the relationship between the speaker and target of an
utterance. People belong to multiple social groups as part of their identity,
however usually only some identities are salient in an utterance in context.
We define _in-group_ utterances as ones where the speaker and target are in
the same social group, and _out-group_ utterances as one where they are in
different social groups. Given an utterance $u$ written by an individual $s$
with target $t$, the IGR prediction task classifies whether $s$ and $t$ belong
to the same social group within the context of $u$.

#### Interpersonal Emotion

We define _perceived_ interpersonal emotion as the emotion expressed by a
speaker $s$ _towards, or in connection with_ the target $t$ of the utterance
$u$, as perceived by a reader. We use the Plutchik wheel of emotions, which is
widely adopted in the community, as the basis of our emotion taxionomy
Plutchik (2001); we use the 8 fundamental emotions (_admiration, anger,
disgust, fear, interest, joy, sadness, surprise_) instead of the full 24
emotions in the wheel due to data sparsity. Interpersonal emotions may be
different, or a subset of, emotion for the whole of an utterance, as
illustrated in rows 2, 3 and 4 of Table 1. Given an utterance $u$ written by
an individual $s$ with target $t$, the interpersonal emotion detection task
identifies the perceived emotion of $s$ towards the target $t$.

##  3 Data Collection

In our area of focus, we require natural language data which satisfies the
following criteria: _(1)_ Each utterance must have at least one target about
whom the utterance mainly concerns. _(2)_ The relationship between the speaker
and the target must be inferred based on metadata or other information.
Specifically, we are interested in aspects of their social identity that they
share or differ on.

The dataset we collect comes from tweets by members of US Congress where other
members are mentioned in the same tweet. We use this as a convenient testbed:
each member’s group affiliation (i.e., their party identity) is public, thus
we can easily know whether the speaker is tweeting to a target in their own
party or not.111For simplicity, we do not consider other factors such as the
home state of a congress member. In other words, this dataset gives us “found
supervision” for our first task of IGR prediction. For our second task, we
annotate a subset of these tweets for perceived interpersonal emotion; this
is, to our knowledge, the first dataset dedicated to interpersonal emotion.

###  3.1 Data Sources and Preprocessing

Social media text like tweets offer a fertile ground for our study. A focus on
tweets with _mentions_ in them satisfies our first criterion – people
generally use mentions to say something about or towards another individual on
twitter. Tweets by members of US Congress are a matter of public record, and
we can infer the social relationship (in terms of party affiliation) between
speaker and target using publicly available information. We prioritize working
with a dataset of tweets by members of US Congress (downloaded using the
Twitter API) between 2010 and 2021, spanning two presidencies, during which
both parties held power in Congress. We filter these tweets to exclude
retweets, and include those tweets that mention _at most_ one other member of
Congress whose party affiliation is known. We believe these 2 assumptions are
sufficient to arrive at a dataset of tweets where the speaker is talking
towards/about _one_ target. Thus, we restrict ourselves to two social groups
in this sphere — Democrat and Republican parties in the US. We sample an equal
number of in-group and out-group tweets from a large sample consisting of all
tweets by members of Congress. Apart from years 2010–2012 and 2021 which
contained fewer tweets due to sparsity issues, we sampled at least 300 tweets
each year.

###  3.2 Interpersonal Emotion Annotation

While we can infer whether a tweet is in-group or out-group based on the
identity of speaker and target whose political affiliations are known, we
still require annotated data on perceived interpersonal emotions.
Interpersonal emotions vary in subtle ways from sentiment or overall sentiment
of utterances: an utterance can have negative sentiment overall, but still
convey positive emotions towards the target of the sentence (expressing
admiration at someone’s death for instance). For this reason, we devise an
annotation schema for annotating _the emotion expressed by speaker $s$ towards
target $t$_.

#### Instructions

Annotators are presented with a tweet, with the identity of the speaker
unknown and that of the target masked with a placeholder name @Doe to minimize
potential biases of the annotators’ prior knowledge of party affiliation
intruding into the annotation:

. If @Doe can get her hair done in person, Congress can vote in person…

Annotators are instructed to read the tweet and select only the most notable
emotion(s) they think are expressed by the tweet author _in connection with_
@Doe. To aid annotators, we provide examples of the 8 Plutchik emotions (_joy,
admiration, fear, suprise, sadness, digust, anger and interest_) expressed as
interpersonal emotions in tweets. Annotators are also shown a schematic of the
Plutchik wheel of emotions, which acquaints them with how the emotions are
related to one another in our framework. Annotators are allowed to select more
than one emotion to account for emotion co-occurrence. We also explicitly tell
annotators that more than one of the emotions can be present in the tweets, to
encourage them to select all interpersonal emotions expressed. They are also
allowed to not choose any emotion.

#### Annotation

To obtain reliable annotations, we prequalify annotators using a qualifying
task. Annotators were recruited on Mechanical Turk using a qualifying task
where they were asked to annotate 6 tweets using the schema shown above. We
restricted the qualification task to annotators living in the USA who had
attempted at least 500 HITS and had a HIT approval rate $\geq$ 98%. After
manual inspection, 6 annotators were qualified for bulk annotation. Each tweet
was annotated by three different annotators. To ensure annotators were paid a
fair wage of at least 10$ an hour, we paid annotators $0.50 per HIT. Each HIT
involved annotating 3 tweets, which we estimate to take on average 3 minutes
to complete. In total, 3,033 tweets between 2010 and 2021 were annotated with
perceived interpersonal emotion.

Figure 1: Emotions ordered by the number of examples where at least one rater
uses a particular label. The color indicates the average interrater
correlation.

#### Agreement

To measure agreement between annotators on the Plutchik-8 emotion wheel, we
use the Plutchik Emotion Agreement (PEA) score from Desai et al. (2020). The
PEA score addresses the issue of penalizing all disagreements equally, by
penalizing dissimilar emotion annotations higher than more similar ones
(according to the Plutchik wheel). Our PEA score is 0.73. The original PEA
formulation used the best(max) pair of emotion annotations between two
workers. Taking the _worst_ combination of emotions between two workers
(averaged over all tweets and workers), the PEA (min) score is 0.60. Overall,
we find moderate to high agreement on fine-grained interpersonal emotions. In
Figure 1 we also present interrater correlation, a metric used in Demszky et
al. (2020); we see that distributions are similar.

Emotion | Train | Dev | Test
---|---|---|---
Admiration | 467 | 64 | 58
Anger | 225 | 40 | 46
Disgust | 206 | 32 | 43
Fear | 1 | 0 | 0
Interest | 701 | 83 | 84
Joy | 801 | 107 | 106
Sadness | 72 | 11 | 11
Surprise | 1 | 0 | 0
_No Emotion_ | 519 | 56 | 63
Table 2: Distribution of emotions in train-dev-test split

#### Aggregation

We consider a tweet to have a certain emotion label if at least 2 out of 3
annotators agree that the particular emotion was present in the tweet. A total
of 638 tweets have no interpersonal emotion associated with them. We employ a
80-10-10 train-dev-test split on our data.

The number of annotated examples (tweets) per emotion is shown in Table 2. We
omit _fear_ and _surprise_ from future tables due to the absence of annotated
examples.

##  4 Preliminary Analysis

#### How are emotions distributed?

When observing the distribution of aggregated emotion labels themselves, a
clear pattern emerges as seen in Table 3. Negative emotions such as anger and
disgust are almost always expressed in out-group settings, while positive
emotions are present in both in-group and out-group settings. A similar
distribution of emotions was observed for Democrats and Republicans — members
of both parties reserved their public anger and disgust for members of the
other party. This reflects an innate bias in terms of the distribution of
interpersonal emotions per situation, and warrants future work to explore
negative interpersonal emotions in an in-group setting.

Emotion | All | In-Group | Out-Group
---|---|---|---
Admiration | 15.5 | 22.2 | 9.1
Anger | 8.2 | 1.0 | 15.1
Disgust | 7.4 | 0.3 | 14.2
Interest | 22.9 | 27.2 | 18.6
Joy | 26.7 | 32.2 | 21.4
Sadness | 2.5 | 2.6 | 2.4
_No Emotion_ | 16.8 | 14.5 | 19.1
Table 3: Proportion of emotions in different interpersonal contexts

Figure 2 shows the co-occurrence of interpersonal emotions in our dataset. We
can see that emotions that are farther apart and more dissimilar, such as
admiration and disgust, joy and sadness, co-occur infrequently. Emotions that
are closer such as anger and disgust, admiration and joy, co-occur much more
often. The only outlier is the higher than normal co-occurrence of admiration
with sadness — after a closer examination, this can be attributed to tweets
expressing admiration and sadness at the passing, or end of the career, of a
fellow congressperson.

#### Who were the targets of negative emotions?

On further analysis, it appears that most of the out-group disgust and anger
is directed at 3 handles – @speakerryan, @speakerpelosi, and @speakerboehner
who were all Speakers of the House of Representatives over most of the time
period of our dataset. 63.7% of disgust and 64.3% of anger is directed towards
these three twitter handles. 11.9% of all tweets in our dataset are directed
at these handles, indicating the preponderance of negative interpersonal
emotion directed at the Speaker of the house. However, we note that negative
emotions like anger and disgust were still expressed towards 51 and 45
different individuals in our dataset, respectively.

Figure 2: Co-occurence of emotions in our dataset. Figure 3: Distribution of
interpersonal emotions in unsupervised representations of tweets in our
dataset. Orange indicates the emotion was present for that tweet. Each point
represents one tweet from our dataset.

#### Can humans predict in/out-group?

While our data naturally comes with “gold” IGR labels, what is unexplored is
whether the distinction between in-group and out-group speech is prominent and
noticeable by humans. Additionally, it is also unclear if humans might have
their own expectation of how in/out-group speech should be characterized.

Concretely, we investigate if human annotators were capable of accurately
performing the IGR prediction task when the speaker and target are masked. Two
authors of this paper, one a social science graduate student, and the other a
computational linguistics graduate student, annotated 50 random tweets from
our validation data which they had not been exposed to earlier for in/out
group labels. Their Fleiss $\kappa$ agreement score was 0.64, indicating
moderate agreement.

To check how accurate their judgements were, we calculate for each annotator
their F1 score against our “gold” in/out group labels. Their F1 scores on
these 50 tweets were 0.67 and 0.63, which as we will discuss in Section 6,
only match simple baselines of supervised systems. Annotators comments
indicate that they overly relied on the sentiment of tweets to make the
classification — positive sentiment means in-group and negative sentiment
means out-group. While negative emotions are over-represented in out-group
situations as Table 3 shows, our dataset contains a substantial presence of
out-group tweets with positive interpersonal emotions as well. Annotators also
noticed some lexical cues like ‘bipartisan’ that are indicative of out-group
tweets.

#### Do pre-trained representations capture interpersonal emotions?

Pre-trained language models have been found to learn sentence representations
that cluster by domain without supervision (Aharoni and Goldberg, 2020). We
wished to investigate if any of our annotated properties cluster inherently in
reduced representations of the tweets in our data. To obtain unsupervised
representations, we use BERTweet (Nguyen et al., 2020), a language model pre-
trained on 850M English tweets. We take the 768 dimensional embeddings from
the final layer of the <s> token in BERTweet, and dimensionally reduce them to
2 dimensions using UMAP (Sainburg et al., 2021). Figure 3 shows the
distribution of tweets, color coded for interpersonal emotions. While there is
a lot of overlap between representations when stratified by emotion, we can
see that some emotions that are intuitively opposite, like admiration &
disgust, joy & sadness are moderately separable. This indicates that
interpersonal emotions do define some topic or domain level properties of a
tweet.

##  5 Experiments

We detail our experiments for the two novel tasks discussed in Section 2:
predicting the IGR (in-group or out-group) given a tweet, and predicting the
interpersonal emotion given a tweet. We present baselines for the two tasks
separately, and also present a multi-task model to gauge the extent to which
knowledge of IGR may help in predicting interpersonal emotion, and vice versa.

###  5.1 Interpersonal Group Relationship

#### Sentiment-Rule

Our first baseline is a rule-based one leveraging coarse sentiment: if a
tweet’s sentiment is predicted to be negative, classify it as out-group; if
positive, classify it as in-group; and if neutral, classify it as either in-
group or out-group randomly. We use a RoBERTa-Base model finetuned for
sentiment on tweets (Barbieri et al., 2020) to extract the sentiment of each
tweet in our dataset.

#### NB-SVM

As a second baseline, we build an SVM model that uses Naive-Bayes log-counts
ratios of unigrams and bigrams (Wang and Manning, 2012).

#### BERTweet

We use BERTweet (Nguyen et al., 2020), a language model pre-trained on 850M
English tweets as our dataset consists purely of English language tweets. A
classification head is placed on top of the language model. We also experiment
with a version where the language model parameters are frozen, and only the
classification head parameters are finetuned (BERTweet-ft).

The input to all models is only the tweet with no other context, and the
target masked with a placeholder @USER.

###  5.2 Interpersonal Emotion

#### EmoLex

As a baseline model for interpersonal emotion identification, we rely on
EmoLex (Mohammad and Turney, 2013). EmoLex consists of 14,182 crowdsourced
words associated with the 8 basic Plutchik emotions. Critically, these words
appear in emotional contexts, but are not necessarily emotion words
themselves. EmoLex counts occurrences of words from its lexicon in an
utterance, and assigns a normalized score for each emotion based on occurrence
frequency. We consider an emotion to be on, if it’s normalized score is $\geq$
0.001. While EmoLex has issues with regards to its context insensitivity and
the social biases built into its lexicon (Zad et al., 2021), we include it as
a baseline to understand to what extent interpersonal emotions can be deduced
using a lexicon.

#### BERTweet

We use the same BERTweet model as earlier. We add a dense output layer on top
of the pretrained model for the purposes of finetuning, with a sigmoid cross
entropy loss function to support multi-label classification. The loss is
weighted for each of the 8 emotion labels with the ratio of positive and
negative examples to increase precision. If none of the 8 emotion labels are
flipped on, we consider that to be the ‘No Emotion’ label, i.e. there is no
interpersonal emotion between speaker and target in the tweet. We experiment
with a version of the model where the language model parameters are frozen and
only the labelling head parameters are finetuned (BERTweet-ft).

###  5.3 Multi-Task Model

In § 4, we observed that the emotions anger and disgust are overwhelmingly
present in out-group situations. Thus, we hypothesize that IGR information
would be useful towards interpersonal emotion identification, and vice versa.
To test this hypothesis we train a multi-task model. The model is trained to
predict _both_ the IGR label and emotion using shared parameter finetuning.

We use the same BERTweet model as earlier. We add two dense output layers on
top of the pretrained model, one for classifying IGR and another for labelling
interpersonal emotion. Both heads share the same parameters below. These are
trained with same loss as earlier individual models. The model alternates
between finetuning for group relationship and emotion over every training
item.

###  5.4 Implementation

We use bertweet-base pretrained embeddings from Huggingface’s models hub (Wolf
et al., 2020). All models are finetuned for a maximum of 20 epochs with early
stopping. Early-stopping patience for models trained on each task separately
is 3. The patience for the multi-task model is set at 5 as the multi-tasking
setup led to slower convergence. The learning rate for the classification
heads was set at 5e-3 while the learning rate for the internal language model
parameters was set at 2e-5. Dropout probabilities in classification heads was
set at 0.1. The best performing model before early stopping on validation data
was chosen in all cases. We report F1 scores averaged over 3 random restarts
for all models, with the standard deviation in parentheses next to the mean.

##  6 Results and Analysis

Model | F1 | Model | F1
---|---|---|---
Majority class | 51.1 | BERTweet | 74.1 (3.3)
Sentiment-Rule | 56.3 | BERTweet-ft | 66.5 (1.6)
NB-SVM | 62.5 | Multitask | 77.3 (0.8)
Human | 66.7 |  |
Table 4: Results on test set, with SD in parentheses, for interpersonal group relationship prediction task. In-group | Out-group
---|---
thanks, love, count me | thanks, bipartisan, restore
birthday, my colleague | kind, resignation
Table 5: Top unigram and bigram features from NB-SVM model for each class. | Emo | BERTweet | BERTweet | Multi-
---|---|---|---|---
| Lex |  | -ft | task
Admir. | 37.5 | 70.3 (3.7) | 40.7 (1.1) | 68.9 (1.6)
Anger | 26.6 | 71.3 (11.2) | 23.0 (3.4) | 69.3 (3.3)
Disgust | 25.5 | 47.1 (21.6) | 13.0 (4.1) | 74.5 (7.1)
Interest | 0 | 53.1 (3.3) | 5.8 (2.4) | 51.5 (8.5)
Joy | 48.4 | 85.9 (1.9) | 71.3 (1.4) | 83.6 (1.3)
Sadness | 4.3 | 11.1 (9.6) | 0 | 33.6 (18.5)
_No Emotion_ | 22.2 | 49.1 (1.2) | 43.4 (3.8) | 71.6(1.2)
Table 6: F1 scores on test set, with SD in parentheses, for interpersonal
emotion labelling task.

#### Interpersonal Group relationship

In modeling IGR, we find that Sentiment-Rule performs not much better than
chance (Table 4). This underscores one strength of our data, which contains a
sizable number of out-group tweets with positive interpersonal emotion
attached to them. The NB-SVM model based on unigrams and bigrams performs
slightly better, and picks up on some obvious out-group lexical cues like the
lemma ‘bipartisan’, as shown in Table 5. The BERTweet model performs
substantially better, performing over 10 points better than humans. The model,
with only the classification head finetuned, leaving the language model
parameters intact(BERTweet-ft) performs about 10 points above chance —
indicating that there may be features advantageous towards this task in the
vanilla LM itself.

#### Interpersonal Emotion

We find that the EmoLex baseline, which relies purely on lexical cues,
performs dismally on our data, with poor performance in both in-group and out-
group settings(Table 6). This is a strong indication that emotions are
expressed more implicitly in this dataset. The BERTweet model performs
substantially better, indicating that interpersonal emotions, even if
implicit, can be learned.

Emotion | BERTweet | MultiTask
---|---|---
Admiration | 77.9 (2.6) | 72.8 (3.9)
Anger | 71.7 (9.9) | 69.4 (3.4)
Disgust | 48.2 (22.4) | 75.9 (6.5)*
Table 7: F1 scores on test set, SD in parentheses on out-group tweets. *
indicates statistical significance (p<0.05)

#### Multitask Model

As Table 4 shows, Multi-tasking the two tasks leads to a noticeable
improvement in F1 for IGR prediction, with the differences being statistically
significant using a bootstrap test (p$<$0.05; Berg-Kirkpatrick et al., 2012);
the multi-task model is also more stable with much lower variance across runs.
These results suggest that interpersonal emotion is useful towards IGR
prediction.

Table 6 shows that the performance of the multitask model on predicting
interpersonal emotions is significantly better that the BERTweet model
(p$<$0.05) on emotions like _disgust_ , which suggests that IGR is useful
towards the task of emotion identification. Furthermore, multitasking boosted
performance at predicting the _no emotion_ label by 20%. Table 7 compares the
multitask model’s performance against the BERTweet model in _out-group_
settings (where most of the gains were found) for 3 emotions — illustrating
the boost in performance afforded by joint modeling of IGR and emotion for
_digust_. The 3 emotions listed also showed significant differences in their
distribution in in-group and out-group settings.

#### Humans vs. Models

Comparatively, we find that model performance exceeds human performance on the
task of in-group versus out-group prediction, albeit not on the same dataset.
The model’s main driver of performance is its high accuracy on positive
intergroup emotion out-group tweets, such as those expressing admiration or
joy. Human annotators consistently fall back on the heuristic that sentences
with positive affect probably imply that the speaker is talking about someone
in their in-group. But it is not the case in the political domain, where
overtures to bipartisanship serve as useful signals. For instance, both 6 and
6 express admiration towards the target Doe, where the first is in-group while
the second is out-group. The call to civility is the only subtle linguistic
cue that this tweet may constitute out-group speech.

. . Admire @OfficialCBC Chairman @Doe’s moral voice on issues of racism and
restorative justice. He is a real leader for our nation and Congress. .̱ A
decade has passed, but our friendship is the same. Proud to work with @Doe to
#ReviveCivility. #tbt Read more about our efforts here:

Future work needs to look into what information the embeddings are using to
make their classification decision.

#### Model Errors

While the multitasking setup improves model performance on the task of
predicting IGR, and outperforms human labelers in our small pilot, it still
gets some easy examples wrong, such as labelling 6 as in-group even though it
expresses some disgust at the target. The model also falls into the same trap
as human labelers — for instance assuming that a tweet expressing admiration
must be in-group 6.

. Trump selected @USER for HHS Secretary. Price has undeniable history of
cutting access to healthcare to millions, especially women.

. Inspiring speech from @USER - we have a duty to represent our country with
respect & dignity. #NationalDayofCivility.

To ensure that model performance on IGR prediction is not limited by the size
of our training data, we experimented with training BERTweet models on larger
datasets. Since we have ‘found supervision’ for IGR labels, we only need to
increase training data size by sampling more tweets from relevant accounts
using the same procedure detailed in § 3.1. We found that F1 score does not
increase with more training data.

Future work needs to look into linguistically motivated ways to improve model
performance on the IGR task. Since we have observed that the multi-task setup
improves model performance, perhaps other multi-task setups, such as modeling
the overall affect towards the target expressed by the speaker might help in
modeling IGR better.

##  7 Related Work

#### Emotion and Stance Detection

A wealth of work has looked at corpora and models for the detection of
perceived emotion in social media text (Mohammad, 2012; Wang et al., 2012;
Mohammad and Kiritchenko, 2015; Abdul-Mageed and Ungar, 2017; Desai et al.,
2020; Demszky et al., 2020). However existing work doesn’t distinguish between
emotion of a sentence as a whole, versus interpersonal emotion towards a
target. The task closest to our study of interpersonal emotions is stance
detection: whether the author has a favourable, neutral, or negative position
towards a proposition or target. Mohammad et al. (2016) looked at stance in
five target domains are given: abortion, atheism, climate change, feminism and
Hillary Clinton. While stance detection focuses on a collection of utterances
with the same topic, our interest is in modeling interpersonal emotion towards
a target individual which is more fine-grained and can vary in each utterance.

#### Intergroup bias in Psychology

The Linguistic Intergroup Bias (LIB) theory Maass et al. (1989); Maass (1999)
states that there is a systematic asymmetry in language production qualities
of a speaker as a function of the social category to which the referent of an
utterance belongs. Through psycholinguistic experiments, LIB seeks to explain
why stereotypes are transmitted and persist in daily life: in an interpersonal
situation, socially desirable in-group behaviors and undesirable out-group
behaviors are encoded at a higher level of abstraction, whereas socially
undesirable in-group behaviors and desirable in-group behaviors are encoded at
a lower level of abstraction. Work in psychology and psycholinguistics
reproduced LIB in various domains such as political news reporting Anolli et
al. (2006) and crime reporting Gorham (2006); as well as work exploring how
LIB can be used as an indicator for a speaker’s prejudicial attitudes Hippel
et al. (1997), or as a predictor for racism Schnake and Ruscher (1998).

Contemporaneous studies on LIB, however, are hand-coded and have so far tended
to focus on narrow concepts such as abstractness of the verb and coarse
notions of sentiment. Nonetheless, the LIB hypothesis connects the two
dimensions of interpersonal dynamics studied here with a third dimension
directly related to semantic properties of the utterance.

##  8 Conclusion

Taking a cue from studies of bias in social science and psychology, we situate
bias in language use through the lens of interpersonal relationships between
the speaker and target of an utterance, and the speaker’s interpersonal
emotional state with respect to the target. Over a corpus of tweets by members
of US Congress, we introduce two novel tasks – interpersonal group
relationship prediction (IGR) and interpersonal emotion labelling, to better
understand variation in language as a function of social relationship between
speaker and target in interpersonal utterances. We find certain interpersonal
emotions like anger and disgust are over-represented in out-group situations,
with the majority of the negative emotions directed at leaders of the two
political parties. Through modeling studies, we find that transformer based
models perform better than humans at predicting IGR given an utterance,
raising the question as to what latent features of language the model uses to
make this decision. Finally, we also find that joint modelling of the two
dimensions is beneficial to prediction of certain interpersonal emotions in
out-group situations. Future work needs to look into what information is
useful for predicting IGR and emotions – with the Linguistic Intergroup Bias
literature offering a clue as to which higher level semantic features vary
systematically.

## Ethics Statement

For our corpus of tweets on which we performed annotations, we downloaded the
tweets using the official Twitter API. In accordance with the Twitter Terms of
Service, we release tweet IDs and usernames, but not the tweet text itself.
Our dataset was built through crowdsourced annotations on Amazon Mechanical
Turk. To ensure annotators were paid a fair wage of at least $10 an hour, we
paid annotators $0.50 per HIT. Each HIT involved annotating 3 tweets, which we
estimate to take on average 3 minutes to complete.

## Limitations

Our results show the importance of having reliable and accurate emotion
prediction models, which is an open problem in psychology and computer
science. Future work might look into identifying different fine-grained
emotional constructs and study their correlations with the underlying
linguistic biases. Future work may also look into the generalizability of the
results presented here in other domains of language use.

While we present the utterances as constituting natural speech by one speaker
(the congressperson who sent the tweet), it is likely most congresspeople
employ social media teams that help in crafting the language of some of their
tweets. However, we believe for the sake of interpersonal group membership,
the relationship between the speakers and their targets would not be affected.

Finally, while we show that transformer based models perform better at IGR
prediction than humans, we note that the human performance was on a small
subset of test data. While it is possible that these models discovered latent
features that could explain their better performance, the model could also be
using spurious features idiosyncratic to our dataset, rather than true
differences in in-group versus out-group speech.

## Acknowledgements

This research is partially supported by NSF grants IIS-2107524, IIS-2145479
and Good Systems,222 http://goodsystems.utexas.edu a UT Austin Grand Challenge
to develop responsible AI technologies. We also acknowledge the Texas Advanced
Computing Center (TACC)333https://www.tacc.utexas.edu at UT Austin for
providing the computational resources for many of the results within this
paper.

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# The Cosmic Timeline Implied by the JWST High-redshift Galaxies

Fulvio Melia
Department of Physics, The Applied Math Program, and Department of Astronomy,
The University of Arizona, AZ 85721, USA John Woodruff Simpson Fellow. E-mail:
<EMAIL_ADDRESS>
###### Abstract

The so-called ‘impossibly early galaxy’ problem, first identified via the
Hubble Space Telescope’s observation of galaxies at redshifts $z>10$, appears
to have been exacerbated by the more recent James Webb Space Telescope (JWST)
discovery of galaxy candidates at even higher redshifts ($z\sim 17$) which,
however, are yet to be confirmed spectroscopically. These candidates would
have emerged only $\sim 230$ million years after the big bang in the context
of $\Lambda$CDM, requiring a more rapid star formation in the earliest
galaxies than appears to be permitted by simulations adopting the concordance
model parameters. This time-compression problem would therefore be
inconsistent with the age-redshift relation predicted by $\Lambda$CDM.
Instead, the sequence of star formation and galaxy assembly would confirm the
timeline predicted by the $R_{\rm h}=ct$ universe, a theoretically advanced
version of $\Lambda$CDM that incorporates the ‘zero active mass’ condition
from general relativity. This model has accounted for many cosmological data
better than $\Lambda$CDM, and eliminates all of its inconsistencies, including
the horizon and initial entropy problems. The latest JWST discoveries at
$z\gtrsim 14$, if confirmed, would add further support to the idea that the
$R_{\rm h}=ct$ universe is favored by the observation over the current
standard model.

###### keywords:

cosmology: observations – cosmology: theory – large-scale structure of the
Universe – stars: Population III – galaxies high-redshift

††pagerange: The Cosmic Timeline Implied by the JWST High-redshift
Galaxies–References††pubyear: 2023

##  1 Introduction

A surprising number of high-redshift galaxy candidates ($z>12$) have already
been discovered by the James Webb Space Telescope (JWST) in just the first few
weeks of operation (see Table 1). Identified through the Early Release
Observations (ERO) (Pontoppidan et al., 2022), the Cosmic Evolution Early
Release Science (CEERS) (Finkelstein et al., 2022) and Through the Looking
GLASS (GLASS-JWST) (Treu et al., 2022) science programs, many of them surpass
the distance record previously set at $z=11.1$ by the Hubble Space Telescope
(HST) (Oesch et al., 2016). This is certainly true of candidates up to $z\sim
13$, whose redshift has been confirmed spectroscopically (Robertson et al.,
2022).

But the fact that some of these well-formed $\sim 10^{9}\,M_{\odot}$
structures (at $z\sim 16-17$) appear to have emerged only $\sim 230$ Myr after
the big bang contrasts with their predicted formation in the standard model of
cosmology, which we here take to be $\Lambda$CDM with the Planck optimized
parameters: a Hubble constant, $H_{0}=67.4\pm 0.5$ km s-1 Mpc-1, a matter
density $\Omega_{\rm m}=0.315\pm 0.007$, scaled to today’s critical density
($\equiv 3c^{2}H_{0}^{2}/8\pi G$), and a spatial curvature constant, $k\approx
0$ (Planck Collaboration et al., 2020). In discussing this ‘impossibly early
galaxy’ problem (Melia, 2014, 2020), two principal issues typically emerge.
The first is whether the gas budget in the early Universe, notably the
fraction of baryons condensed within an assumed dark-matter halo distribution,
was sufficient to account for this high-$z$ galaxy demographic (Behroozi &
Silk, 2018). The answer could be yes (Donnan et al., 2022), as long as all of
the available baryonic gas in halos was converted into stars. The $z\sim
16-17$ galaxy candidates fall close to the $\Lambda$CDM limit, but do not
exceed it.

Table 1: JWST highest-redshift galaxies and their derived properties

Name | $\quad\;\;z^{a}$ | $\log(M/M_{\odot})$ | SFR |  Stellar Ageb | Reference
---|---|---|---|---|---
|  |  |  ($M_{\odot}$ yr-1) | (Myr) |
1\. S5-z17-1 | $16.66^{+1.86}_{-0.34}$ | $\;8.8^{+0.8}_{-0.5}$ | $9.7^{+30.7}_{-6.2}$ | - | Harikane et al. (2022)
2\. CEERS-93316 | $16.4^{+0.1}_{-0.1}$ | $\;9.0^{+0.4}_{-0.5}$ | $10.0^{+10}_{-6.8}\,$ | $20^{+40}_{-10}$ | Donnan et al. (2022); Naidu et al. (2022); Harikane et al. (2022)
3\. S5-z12-1 | $13.72^{+0.86}_{-1.92}$ | $\;8.1^{+1.3}_{-0.3}$ | $2.2^{+15.5}_{-1.0}\,$ | - | Harikane et al. (2022)
4\. WHL0137-5021 | $12.8^{+1.1}_{-12.5}$ | $\;8.53^{+0.18}_{-0.32}$ | $5.1^{+1.9}_{-1.1}\;\;$ | $58^{+35}_{-35}$ | Bradley et al. (2022)
5\. WHL0137-5124 | $12.8^{+1.9}_{-12.4}$ | $\;8.65^{+0.20}_{-0.30}$ | $6.9^{+3.2}_{-1.9}\;\;$ | $59^{+35}_{-37}$ | Bradley et al. (2022)
6\. GLASS-z13 | $12.4\pm{0.2}$ | $\;9.0^{+0.3}_{-0.4}$ | $7^{+4}_{-3}\;\,$ | $71^{+32}_{-33}$ | Naidu et al. (2022); Harikane et al. (2022)
7\. GLASS-z12-1 | $12.22^{+0.04}_{-0.11}$ | $\;8.6^{+0.8}_{-0.4}$ | $3.0^{+11.3}_{-0.6}\;$ | - | Harikane et al. (2022); Donnan et al. (2022)
8\. Maisie’s Galaxy | $11.8^{+0.2}_{-0.3}$ | $\;8.5^{+0.29}_{-0.44}$ | $2.1^{+4.8}_{-2.0}\;\;$ | $18^{+18}_{-9}$ | Finkelstein et al. (2022); Harikane et al. (2022)
9\. GN-z11c | $11.09^{+0.08}_{-0.12}$ | $\;9.0\pm{0.4}$ | $24\pm{10}\;\,$ | $40^{+60}_{-34}$ | Oesch et al. (2016)
10\. GLASS-z11 | $10.6\pm{0.3}$ | $\;9.4\pm{0.3}$ | $12^{+9}_{-4}\;\;$ | $111^{+43}_{-54}\;\,$ | Naidu et al. (2022); Harikane et al. (2022); Donnan et al. (2022)
11\. WHL0137-3407 | $10.5^{+1.0}_{-10.5}$ | $\;8.78^{+0.17}_{-0.33}$ | $7.3^{+2.5}_{-1.2}\;\,$ | $70^{+50}_{-44}$ | Bradley et al. (2022)
12\. WHL0137-5347 | $10.2^{+0.9}_{-9.7}$ | $\;9.01^{+0.21}_{-0.37}$ | $14.6^{+5.8}_{-3.5}\;\,$ | $62^{+54}_{-43}$ | Bradley et al. (2022)
13\. WHL0137-5330 | $10.0^{+1.1}_{-7.9}$ | $\;8.77^{+0.15}_{-0.26}$ | $6.4^{+2.6}_{-1.8}\;\,$ | $83^{+52}_{-48}$ | Bradley et al. (2022)

11footnotetext: Photometric redshift (with $2\sigma$ uncertainties) for the
WHL sources calculated using the Calzetti et al. (2000) dust law. GLASS,
CEERS and Maisie’s Galaxy redshifts (with $1\sigma$ uncertainties) were fit
with the EAZY code (Brammer et al., 2008).22footnotetext: Mass-weighted age of
the current luminous stars, when available.

33footnotetext: Discovered by HST prior to JWST.

The second concerns whether the dynamics of structure formation could account
for the highly compressed timeline implied by these discoveries (for the most
recent work on this topic, see Yajima et al. 2022; Keller et al. 2022; Kannan
et al. 2022; Inayoshi et al. 2022; Haslbauer et al. 2022; Mirocha & Furlanetto
2023; Whitler et al. 2023). It is the dynamics, of course, coupled to the
physical processes responsible for cooling the gas, that would have governed
how quickly stars could condense and assemble into billion solar-mass
structures. One must also fold into this discussion how the ‘stellar age’ of
the high-$z$ sources (column 5 in Table 1) should be interpreted. An
examination of the galaxy age versus star formation activity at $z>8$ (Furtak
et al., 2023; Whitler et al., 2023) suggests that the young stellar
populations producing much of the current luminosity are built upon older
components that formed at $z>15$, and are being observed during bursts of star
formation.

##  2 High-$z$ Galaxies in $\Lambda$CDM

A more indicative evolutionary history for these galaxies is therefore
provided by the broad range of simulations tracing the growth of initial
perturbations consistent with the measured anisotropies in the cosmic
microwave background. This study has evolved considerably over the past
decade, as each new set of observations has pushed the formation of galaxies
to progressively higher redshifts. The first generation of simulations
(Barkana & Loeb, 2001; Miralda-Escudé, 2003; Bromm & Larson, 2004; Ciardi &
Ferrara, 2005; Glover, 2005; Greif et al., 2007; Wise & Abel, 2008; Salvaterra
et al., 2011; Greif et al., 2012; Jaacks et al., 2012) began to elucidate how
the first (Pop III) stars probably formed by redshift $z\sim 20$, nestled in
the core of dark-matter halos with mass $M_{\rm halo}\sim 10^{6}\,M_{\odot}$
(Haiman et al., 1996; Tegmark et al., 1997; Abel et al., 2002; Bromm et al.,
2002). This delay after the big bang resulted from the combined influence of
several processes, including the initial gravitational collapse of the dark-
matter perturbations and the subsequent inefficient, radiative cooling of the
primordial gas. The baryonic matter cooled and condensed into stars only after
enough molecular hydrogen accumulated to enhance the energy loss rate (Galli &
Palla, 1998; Omukai & Nishi, 1998). In the standard model, the Universe would
have been $\sim 180$ Myr old at redshift 20. But not all of the halos and
their baryonic content would necessarily have taken this long to condense. The
more recent simulations (Yajima et al., 2022; Keller et al., 2022; Kannan et
al., 2022; Inayoshi et al., 2022; Haslbauer et al., 2022; Mirocha &
Furlanetto, 2023; Whitler et al., 2023), in particular, show that the halos
could have been distributed across this age, some appearing perhaps as early
as $\sim 120$ Myr after the big bang (more on this below).

A typical baryonic gas cloud could subsequently have formed a protostar at the
center of its halo host, eventually growing to become a $>100\,M_{\odot}$ main
sequence (Pop III) star (Kroupa, 2002; Chabrier, 2003). This is where a second
major difficulty emerges, however. The UV radiation from such massive stars
would have destroyed all of the $H_{2}$ in the original condensation,
suggesting that most of the minihalos likely contained—at most—only a handful
of Pop III stars (Yoshida et al., 2008). Thus, many of these early structures
were probably not the galaxies we see today. Nevertheless, a sufficiently
broad distribution of Pop III masses could have included many below $\sim
100\,M_{\odot}$, whose impact on the star formation would have been less
severe (Yajima et al., 2022; Keller et al., 2022; Kannan et al., 2022;
Inayoshi et al., 2022; Haslbauer et al., 2022; Mirocha & Furlanetto, 2023;
Whitler et al., 2023).

The outpouring of radiative and mechanical energy from this first phase of
stellar formation would have reheated and expelled the surrounding gas,
further delaying the formation of additional stars until the plasma had time
to cool again and condense to high densities. The time for this gas re-
incorporation would have been another $\sim 100$ Myr, i.e., roughly the
dynamical time for a first-galaxy halo to assemble (Yoshida et al., 2004;
Johnson et al., 2007). And now the Universe was almost 300 Myr old.

To its credit, this scenario does provide a plausible explanation for the size
of the observed galaxies. It suggests that a criticial mass of
$>10^{8}\,M_{\odot}$, along with an implied virial temperature $>10^{4}$ K,
would have allowed atomic line emission to cool the condensing gas (Wise &
Abel, 2007), and recollect and mix all components of the previously shocked
plasma (Greif et al., 2007). In other words, at least the mass inferred for
the earliest JWST galaxies (Table 1) is consistent with this theoretical
understanding.

But in the context of these simulations, there’s no getting around the fact
that the appearance of $\sim 10^{9}\,M_{\odot}$ galaxies at $z\sim 16-17$, if
confirmed spectroscopically, would create a significant problem. One needs a
billion new stars to have formed in only $\sim 70-90$ Myr by the time the
Universe was only $t\sim 230$ Myr old. None of the calculations to date have
been able to account for the formation of galaxies at this redshift, when the
expelled, hot gas hadn’t re-cooled and re-condensed yet.

This tension between the theoretical and observational timelines has motivated
the introducion of additional features and physical processes designed to
mitigate the disagreement as much as possible (Yajima et al., 2022; Keller et
al., 2022; Kannan et al., 2022; Inayoshi et al., 2022; Haslbauer et al., 2022;
Mirocha & Furlanetto, 2023; Whitler et al., 2023). For example, in their most
detailed simulations to date, Yajima et al. (2022) and Keller et al. (2022)
have demonstrated that the large scatter in cooling times and the presence of
systems with weaker Pop III supernovae that expel far less of the condensed
baryonic gas (Kitayama & Yoshida, 2005; Frebel & Norris, 2015) would have
allowed galaxies observed by JWST at $z\lesssim 14$ to still have formed in
the context of $\Lambda$CDM.

Figure 1: Comparison of galaxy growth curves with four different simulations
for GLASS-z13 (line 6 in Table 1), with a final stellar mass $M_{*}\sim
10^{9}\;M_{\odot}$: Salvaterra [6] corresponding to trajectory [6] in Figure
2; FOREVER22 (Yajima et al., 2022); SIMBA (Keller et al., 2022); and OBELISK
(Keller et al., 2022). The vertical dashed line indicates the age of this
galaxy ($\sim 345$ Myr) in the context of Planck-$\Lambda$CDM.

Kannan et al. (2022) have shown that a variable stellar initial mass function
may also have produced some galaxies earlier than previously thought. A top-
heavy stellar mass distribution appears to have a similar effect (Inayoshi et
al., 2022), while different star formation histories could reduce the actual
stellar masses of the galaxy candidates, thereby partially alleviating the
tension (Haslbauer et al., 2022). Mirocha & Furlanetto (2023) suggest that at
least three modifications to the HST-calibrated models would help lessen the
tension: (i) the adoption of halo mass-independent star formation (SFR)
efficiencies; (ii) a substantial scatter in galaxy SFRs at fixed halo masses;
and (iii) the non-trivial effects of dust, both on the inferred JWST colours
and on the produced stellar masses and ages. Finally, Whitler et al. (2023)
conclude that the tension may be eased if young stellar populations formed in
these early galaxies on top of older stellar populations.

Nevertheless, even with all of these modifications, the predicted galaxy
masses at $z\lesssim 14$ appear to fall short of those observed by factors of
a few. And they are significantly smaller than those of the galaxy candidates
at $z\sim 16-17$. Thus, if the JWST highest redshift sources are eventually
confirmed, even the more optimistic recent simulations would be unable to
explain their origin.

The four galaxy growth curves in Figure 1 compare the time required to reach
the inferred stellar mass of GLASS-z13 (the sixth entry in Table 1) by $t\sim
345$ Myr (corresponding to its redshift $z\sim 12.4$), based on four of the
main simulations discussed above. For the sake of giving $\Lambda$CDM the most
optimal evolutionary outcome, we shall adopt the Salvaterra et al. (2011) and
Jaacks et al. (2012) frameworks which, though not as detailed and nuanced as
the more recent work, predict a shorter growth time once star formation is
initiated, thus making it easier to fit the observed galaxies within the
$\Lambda$CDM timeline.

Figure 2: Growth of stellar mass in the high-$z$ galaxy candidates discovered
by JWST at $10<z<16$, as a function of cosmic time $t$, in $\Lambda$CDM. The
principal epochs are (i) the initial halo condensation and cooling due to
molecular hydrogen. This epoch typically extended over the period $0.4\lesssim
t\lesssim 180$ Myr, but could have been as short as $\sim 120$ Myr for some of
the objects; (ii) the formation of the first Pop III stars at $t\sim 120-180$
Myr, i.e., $z\gtrsim 20$ in this model, (iii) the transition to Pop II star
formation at $t\lesssim 280$ Myr, and the observed Epoch of Reionization (EoR)
from $z\sim 15$ down to $6$ (i.e., $280<t<927$ Myr). These galaxy growth
trajectories are primarily based on the observed SFRs and the hydrodynamical
simulations in Jaacks et al. (2012), cross-checked with independent and
alternative calculations in Salvaterra et al. (2011). The labels on the curves
correspond to the catalog listings in Table 1.

The various factors discussed above may now be seen more quantitatively with
the simulated galaxy growth trajectories shown in Figure 2, which also
includes several critical epochs in $\Lambda$CDM. We assume that the JWST
galaxy candidates in Table 1 followed a history of growth like those of
Salvaterra et al. (2011) and Jaacks et al. (2012) at $z\sim 8$, except that
their cosmic time $t$ is suitably translated to match the redshift at which
they are observed. This is justified by the fact that $t$ is actually the
proper time in the comoving frame, and the Birkhoff theorem ensures that the
local growth rate was not overly affected by the cosmic expansion exterior to
the bound system (Weinberg, 1972; Melia, 2020). In other words, once a galaxy
halo becomes gravitationally bound and star formation is initiated, its
evolution thereafter should be roughly translationally invariant in $t$.

In the Salvaterra et al. (2011) simulations, the doubling time (i.e., the
inverse of the specific star formation rate sSFR, defined as the stellar mass
created per unit time per billion solar-masses) and the evolutionary time at
which the galaxy is observed, appears to be universally equal to $\sim
0.1-0.3$. By comparison, the Jaacks et al. (2012) calculations show that the
SFR for high-$z$ galaxies is ‘bursty,’ with an average value between $z\sim
15$ and $z\sim 6$ following an exponentially increasing function with
characteristic timescale $t_{\rm c}\sim 70-200$ Myr, scaling with stellar mass
in the range $10^{6}<M_{*}<10^{10}\,M_{\odot}$. The trajectories plotted in
Figure 2 follow this exponential growth, using the observed galaxy mass and
redshift to fix the end points. Given that the sSFRs probably fluctuated
during their evolution (Furtak et al., 2023; Whitler et al., 2023), we take an
average of the star formation rates quoted in Table 1, i.e., $\langle{\rm
sSFR}\rangle\sim 12.4$ Gyr-1, as a fiducial value for each galaxy.

The overall impression one gets from this illustration is that the new JWST
high-$z$ galaxies, if confirmed, would not be consistent with the standard
picture in $\Lambda$CDM. The previously growing tension developing at $z\sim
12$ has now become a more serious discordance at $z\sim 16-17$. There does not
appear to be any possibility with our current physical theories of explaining
how a billion-solar mass aggregate of stars could have condensed even before
the primoridal gas was allowed to cool and form most of the very first Pop
III, and any of the Pop II, populations. In this regard, our conclusion
concerning the implausibility of forming the JWST high-$z$ galaxies in the
context of $\Lambda$CDM is fully consistent with the findings of an
alternative approach to this problem (Boylan-Kolchin, 2022), based on the use
of the Sheth-Tormen (Sheth & Tormen, 1999) mass function to determine the
abundance of massive halos at high redshifts (Wang et al., 2022).

##  3 The Timeline in $R_{\rm h}=ct$

But previous comparative tests between $\Lambda$CDM and a theoretically more
advanced version, known as the $R_{\rm h}=ct$ universe (Melia & Shevchuk,
2012; Melia, 2020), have already hinted at the possibility that the age-
redshift relation in the latter may be a better match to the Universe’s
evolutionary history than that in the former. For example, $\Lambda$CDM has
considerable difficulty accounting for the seeding and growth of billion-solar
mass black holes by redshift $z\sim 7-8$, while the timeline in $R_{\rm h}=ct$
matches them very well (Melia, 2013, 2018b). A more complete description of
this model, including its theoretical foundation and a comparison of its
predictions with the data, may be seen in Melia (2018a, 2020). A review of the
current problems with the standard model, pointing to a need for further
development, e.g., as suggested by $R_{\rm h}=ct$, is provided in Melia
(2022).

Figure 3: Same as Figure 2, except now for the $R_{\rm h}=ct$ Universe. The
EoR here corresponds to $906\;{\rm Myr}<t<2.07$ Gyr, and the Dark Ages extend
up to $\sim 906$ Myr. The first Pop III stars emerged at $z\gtrsim 79$ and the
transition to Pop II stars occurred at $z\sim 51$. The extended period between
the onset of Pop II star formation and the appearance of the first JWST
galaxies (shown here in red) is absent in Figure 2. In this cosmology, the
JWST galaxy candidates are seen towards the end of the dark ages, where one
would expect them to be if they were responsible for re-ionizing the
intergalactic medium. Most importantly, all of these primordial galaxies would
have started their growth well after the transition from Pop III to Pop II
star formation had been completed at $\sim 280$ Myr.

One of the essential features of $R_{\rm h}=ct$ that distinguishes it from
$\Lambda$CDM is its expansion factor, $a(t)\propto t$, which results in the
simple age-redshift relation $1+z={t_{0}/t}$ in terms of the current age,
$t_{0}$ of the Universe. In this cosmology, the gravitational radius $R_{\rm
h}$ is equivalent to the Hubble radius $c/H(t)$ (Melia, 2018c), so
$t_{0}={1/H_{0}}$. Thus, if for simplicity we use the same Hubble constant as
$\Lambda$CDM, we find that $t_{0}\approx 14.5$ Gyr. The JWST galaxy
trajectories recalculated with these relations are displayed in Figure 3,
along with the correspondingly adjusted temporal phases. The EoR redshift
range $6<z<15$ here corresponds to $906\;{\rm Myr}<t<2.07$ Gyr. That is, the
Dark Ages ended at $t\sim 906$ Myr, providing ample time for the Universe to
assemble billion-solar mass galaxies once Pop III and Pop II stars started
forming in numbers. Very tellingly, all of the high-$z$ galaxy candidates
discovered so far appear towards the end of the dark ages, where one would
expect them to be if they contributed—perhaps even dominated—the reionization
process. In this model, one would need to find a billion-solar mass galaxy at
$z\sim 50$ to run into a similar age-redshift inconsistency as that in
$\Lambda$CDM.

##  4 Conclusion

The time compression problem in the standard model has been worsening for
several years. Attempts at remedying the situation with the premature
formation of supermassive black holes have focused on two principal
modifications: (i) the creation of massive (i.e., $\sim 10^{5}\,M_{\odot}$)
seeds; and (ii) super-Eddington accretion rates. The first of these is still
speculative because it requires the collapse of an essentially zero angular
momentum, optically-thick, radiation-dominated plasma, which would have
experienced substantial support from its internal pressure (Melia, 2009); the
second appears to have been ruled out by measurements suggesting that the most
distant quasars are accreting at or below their Eddington rate (Mortlock et
al., 2011; De Rosa et al., 2011; Willott et al., 2010).

The JWST discovery of high-$z$ galaxy candidates may have worsened this timing
problem considerably if their redshifts are confirmed spectroscopically,
because billion-solar mass structures must have formed in only $\sim 70-90$
Myr in some cases, and even prior to formation of the very first stars in
others. The simulations completed to date in the context of $\Lambda$CDM have
difficulty accounting for this outcome at $z\sim 17$, reinforcing the view
that the standard model may not be able to account for the formation of
structure at cosmic dawn, if the redshifts of these candidate galaxies are
confirmed spectroscopically.

Instead, the timeline predicted by the $R_{\rm h}=ct$ cosmology would fit the
birth and growth of both high-$z$ quasars and galaxies very well, adding to
the growing body of evidence supporting the introduction of the zero active
mass condition from general relativity as an indispensible modification to
$\Lambda$CDM.

Of course, there are still at least two ways out of this dilemma. First, the
JWST candidate galaxies at $z\gtrsim 14$ may simply be mis-identified sources
at lower redshifts. According to Furtak et al. (2023), only about half of the
high-$z$ galaxy photometric redshifts may be safely ruled out as low-$z$
interlopers. The other half, including those of the most distant candidates,
still await spectroscopic confirmation. Second, it is possible that we may be
missing something in the basic theory, and this caveat cannot be ignored. The
initial cooling of the primordial gas may have been due to something other
than molecular hydrogen. An unknown process may have permitted the plasma to
cool more efficiently, allowing Pop III stars to form even earlier than $t\sim
120-180$ Myr.

Certainly, the most recent simulations of Yajima et al. (2022) and Keller et
al. (2022) indicate that such a process permitting the formation of Pop III
stars as early as $\lesssim 100$ Myr would lessen the tension with the $z\sim
16-17$ galaxies in $\Lambda$CDM. This may not completely resolve the problem,
but would go a long way to mitigating the overall disagreement between the
JWST observations and the standard model. Future simulations will probe such
possibilities in even greater detail, perhaps uncovering a solution based on
new physics in $\Lambda$CDM. As of today, however, the JWST discoveries—if
confirmed—would support the timeline in $R_{\rm h}=ct$, but not in
$\Lambda$CDM.

## Acknowledgments

I am very grateful to the anonymous referee for a detailed and very
constructive review, which has led to significant improvements in the
manuscript.

## DATA AVAILABILITY STATEMENT

No new data were generated or analysed in support of this research.

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