# CaMEL: Case Marker Extraction without Labels 🐫

Leonie Weissweiler  Center for Information and Language Processing, LMU Munich
Valentin Hofmann  Masoud Jalili Sabet  Center for Information and Language
Processing, LMU Munich  Hinrich Schütze  Center for Information and Language
Processing, LMU Munich

###### Abstract

We introduce CaMEL (Case Marker Extraction without Labels), a novel and
challenging task in computational morphology that is especially relevant for
low-resource languages. We propose a first model for CaMEL that uses a
massively multilingual corpus to extract case markers in 83 languages based
only on a noun phrase chunker and an alignment system. To evaluate CaMEL, we
automatically construct a silver standard from UniMorph. The case markers
extracted by our model can be used to detect and visualise similarities and
differences between the case systems of different languages as well as to
annotate fine-grained deep cases in languages in which they are not overtly
marked.

T2AcmrTempora-TLF 🐫

CaMEL: Case Marker Extraction without Labels 🐫

##  1 Introduction

What is a case? Linguistic scholarship has shown that there is an intimate
relationship between morphological case marking on the one hand and semantic
content on the other (see Blake (1994) and Grimm (2011) for overviews). For
example, the Latin case marker -ibus111In this paper, we use italic when
talking about case markers as morphemes in a linguistic context and monospace
(accompanied by $ to mark word boundaries) when talking about case markers in
the context of our model. Transliterations of Cyrillic examples are given
after slashes. (Ablative or Dative Plural) can express the semantic category
of location. It has been observed that there is a small number of such
semantic categories frequently found cross-linguistically (Fillmore, 1968;
Jakobson, 1984), which are variously called case roles or deep cases.
Semiotically, the described situation is complicated by the fact that the
relationship between case markers and expressed semantic categories is seldom
isomorphic, i.e., there is both case polysemy (one case, several meanings) and
case homonymy or case syncretism (several cases, one marker) (Baerman, 2009).
As illustrated in Figure 1, the Latin Ablative marker -ibus can express the
semantic category of instrument besides location (case polysemy), and it is
also the marker of the Dative Plural expressing a recipient (case syncretism).
In addition, there is case synonymy (one case, several markers), which further
complicates morphosemiotics; e.g., in Latin, -is is an alternative marker of
the Ablative Plural.

-ibus
---
Abl |  Dat
I |  L |  R
-ами
---
Inst
I
-ах
---
Loc
L
-ам
---
Dat
R
Figure 1: Morpho-semiotic foundation of this study. The Latin case marker
-ibus is used for both the Ablative (Abl) and the Dative (Dat), which in turn
express the three semantic categories of instrument (I), location (L), and
recipient (R). This is an example of both case polysemy (one case: Abl,
several meanings: I and L) and case syncretism (several cases: Abl and Dat,
one marker: -ibus). Russian, on the other hand, has an isomorphic relationship
between Instrumental (Inst), Locative (Loc), and Dative (Dat), the case
markers corresponding to them (-ами/-ami, -ах/-ax, -ам/-am), and the expressed
semantic categories (I, L, R).

The key idea of this paper is to detect such complex correspondences between
case markers and expressed semantic categories in an automated way.
Specifically, we build on prior work by Cysouw (2014), who lays the
theoretical foundation for our study by showing that deep cases can be induced
from cross-linguistic usage patterns of case markers. As opposed to Latin,
Russian has separate cases (with separate case markers) for the semantic
categories of instrument (-ами/-ami), location (-ах/-ax), and recipient
(-ам/-am). Thus, knowing the Russian case marker corresponding to Latin -ibus
reduces the uncertainty about the expressed case role (Figure 1). This
reduction of uncertainty can be particularly helpful in a low-resource setting
where other means of analysis are unavailable.

In this work, we rely on the Parallel Bible Corpus (PBC; Mayer and Cysouw,
2014), a massively multilingual corpus, to investigate the relationship
between surface cases and their deep meanings cross-linguistically. To put our
idea into practice, we require an exhaustive set of case markers as well as a
set of parallel noun phrases (NPs) that we can further analyze with respect to
deep cases using the set of case markers. Both requirements pose a serious
challenge for languages with limited available resources. We therefore
introduce CaMEL (Case Marker Extraction without Labels), a novel and
challenging task of finding case markers using only (i) a highly parallel
corpus covering many languages, (ii) a noun phrase chunker for English, and
(iii) word-level pre-computed alignments across languages.

Our work uses the parallel nature of the data in two ways.

First, we leverage the word-level alignments for the initial step of our
pipeline, i.e., the marking of NPs in all languages (even where no noun phrase
chunker is available). To do so, we mark NPs in 23 different English versions
of the Bible and project these annotations from each English to each non-
English version using the word-level alignments, resulting in parallel NPs
that express the same semantic content across 83 languages. Based on the
projected annotations, we leverage the frequencies of potential case markers
inside and outside of NPs as a filter to distinguish case markers from lexical
morphemes and other grammatical morphemes typically found outside of NPs.

Second, we leverage the alignments for a fine-grained analysis of the semantic
correspondences between case systems of different languages.

We make three main contributions.

  * •

We define CaMEL (Case Marker Extraction without Labels), a new and challenging
task with high potential for automated linguistic analysis of cases and their
meanings in a multilingual setting.

  * •

We propose a simple method for CaMEL that is efficient, requires no training,
and generalises well to low-resource languages.

  * •

We automatically construct a silver standard based on human-annotated data and
evaluate our method against it, achieving an F1 of 45%.

To foster future research on CaMEL, we make the silver standard, our code, and
the extracted case markers publicly
available222https://github.com/LeonieWeissweiler/CaMEL.

##  2 Related Work

Unsupervised morphology induction has long been a topic of central interest in
natural language processing (Yarowsky and Wicentowski, 2000; Goldsmith, 2001;
Schone and Jurafsky, 2001; Creutz and Lagus, 2002; Hammarström and Borin,
2011). Recently, unsupervised inflectional paradigm learning has attracted
particular interest in the research community (Erdmann et al., 2020; Jin et
al., 2020), reflected also by a shared task devoted to the issue (Kann et al.,
2020). Our work markedly differs from this line of work in that we are
operating on the level of case markers, not full paradigms, and in that we are
inducing morphological structure in a massively multilingual setting.

There also have been studies on extracting grammatical information from text
by using dependency parsers (Chaudhary et al., 2020; Pratapa et al., 2021) and
automatically glossing text (Zhao et al., 2020; Samardžić et al., 2015) as
well as compiling full morphological paradigms from it (Moeller et al., 2020).
By contrast, our method is independent of such annotation schemata, and it is
also simpler as it does not aim at generating full grammatical or
morphological descriptions of the languages examined. There has been cross-
lingual work in computational morphology before (Snyder and Barzilay, 2008;
Cotterell and Heigold, 2017; Malaviya et al., 2018), but not with the
objective of inducing inflectional case markers.

Methodologically, our work is most closely related to the SuperPivot model
presented by Asgari and Schütze (2017), who investigate the typology of tense
in 1,000 languages from the Parallel Bible Corpus (PBC; Mayer and Cysouw,
2014) by projecting tense information from languages that overtly mark it to
languages that do not. Based on this, Asgari and Schütze (2017) perform a
typological analysis of tense systems in which they use different combinations
of tense markers to further divide a single tense in any given language. Our
work differs in a number of important ways. First, we do not manually select a
feature to investigate but model all features in our chosen sphere of interest
(i.e., case) at once. Furthermore, we have access to word-level rather than
verse-level alignments and can thus make statements at a more detailed
resolution (i.e., about individual NPs). Finally, we extract features not only
for a small selection of pivot languages, but even for languages that do not
mark case “non-overtly”, i.e., in a way that deviates to a large degree from a
simple 1–1 mapping (see discussion in §1).

##  3 Linguistic Background

There is ongoing discussion in linguistic typology about the extent to which
syntactic categories are shared and can be compared between the world’s
languages (see Hartmann et al. (2014) for an overview). While this issue is
far from being settled, there is a general consensus that (while not being a
language universal) there is a core of semantic categories that are
systematically found cross-linguistically, and that are expressed as
morphosyntactic case in many languages. Here, we adopt this assumption without
any theoretical commitment, drawing upon a minimal set of deep cases detailed
in Table 1. The set is loosely based on the classical approach presented by
Fillmore (1968).

Going beyond deep cases, Cysouw (2014) envisages a more fine-grained analysis
of what is conventionally clustered in a deep case or semantic role. Briefly
summarised, the theoretical concept is this: if every language has a slightly
different case system, with enough languages it should be possible to divide
and cluster NPs at any desired level of granularity, from the conventional
case system down to a specific usage of a particular verb in conjunction with
only a small set of nouns. For example, the semantic category of location
could be further subdivided into specific types of spatial relationships such
as ‘within’, ‘over’ and ‘under’. Taken together, it would then be possible to
perform theory-agnostic typological analysis of case-like systems across truly
divergent and low-resource languages by simply describing any language’s case
system in terms of its clustering of very fine-grained semantic roles into
larger systems that are overtly marked.

The approach sketched in the last paragraph is not limited to case systems but
has been applied to person marking (Cysouw, 2008), the causative/inchoative
alternation (Cysouw, 2010), and motion verbs (Wälchli and Cysouw, 2012). The
variety of linguistic application areas highlights the potential of developing
methods that are much more automated than the work of Cysouw and
collaborators. While we stay at the level of traditional deep cases in this
paper, we hope to be able to extend our method into the direction of a more
general analysis tool in the future.

The remainder of the paper is structured as follows. Section 4 describes our
method in detail. Section 5 gives an overview of our results. Finally, Section
6 presents two exploratory analyses.

Deep Case | Description | Example
---|---|---
Nominative | The subject of the sentence |  He is the Messiah!
Genitive | An entity that possesses another entity | Are you the Judean People’s Front?
Recipient | A sentient destination | I gave the gourd to Brian.
Accusative | The direct object of the sentence | Consider the lilies.
Locative | The spatial or temporal position of an entity | They haggle in the market.
Instrumental | The means by which an activity is carried out | The graffiti was written by hand.
Table 1: Descriptions and examples for the deep cases distinguished in this
paper, which loosely follow the core deep cases proposed in the classical
approach of Fillmore (1968).

##  4 Methodology

###  4.1 Data

We work with the subset of the PBC (Mayer and Cysouw, 2014) for which the
SimAlign alignment algorithm (Jalili Sabet et al., 2020) is available,
resulting in 87 languages for our analysis. From the corpus, we only extract
those verses that are available in all languages, thus providing for a
relatively fair comparison, and remove Malagasy, Georgian, Breton, and Korean,
as they have much lower coverage than the other languages. This leaves us with
83 languages and 6,045 verses as our dataset. We also select 23 English
versions from the PBC that cover the same 6,045 verses. For each of the 6,045
verses, we then compute $83\times 23=1909$ verse alignments: 83 (for each
language) multiplied with 23 (for each English version). In the following, we
will describe the components of our pipeline (Figure 2).

NP AnnotationNP ProjectionCandidate Set CreationFrequency
FilteringInside/Outside FilteringRestriction to Word Endings Figure 2:
Overview of our pipeline.

###  4.2 NP Annotation

Because our intermediate goal is to induce complete lists of case markers in
all languages we cover, the first step is to restrict the scope of our search
to NPs. We hope that this will allow us to retrieve case markers for nouns and
adjectives while disregarding verb endings that might otherwise have similar
distributional properties. As we are working with 83 languages, most of which
are low-resource and lack high-quality noun phrase chunkers, we first identify
NPs in English using the spaCy noun phrase chunker (Honnibal et al., 2020) and
then project this annotation using the alignments to mark NPs in all other
languages. The exception to this are German and Norwegian Bokmål, for which
noun phrase chunkers are available directly in spaCy. Because both the spaCy
noun phrase chunker and the alignments are prone to error, we make use of 23
distinct English versions of the Bible and mark the NPs in each of them with
the goal of lessening the impact of noise.

###  4.3 NP Projection

We project the NP annotation of a given English version to a second language
using the alignments. Specifically, we find the NP in the target language by
following the alignments from all words in the English NP while maintaining
the word order of the target sentence. We treat each annotated version of the
corpus resulting from the different English versions as a separate data
source. As an example, Figure 3 shows two English versions and the NP
projections for Latin and German. While the alignments, particularly those
from English to Latin, are not perfect, they result in complementary errors.
The first wrongly aligns the first mention of pastor bonus, resulting in only
pastor being marked as an NP. The second misses the alignment of life and
animam. In these two cases, the other alignment corrects the error.

There are two major results from this process.

First, we obtain the set $N$ of all NPs marked in English, each with all of
its translations in the other languages. An example of an entry in this set,
taken from Figure 3, would be the fine shepherd, pastor bonus, der
vortreffliche Hirte, …, while the fine shepherd, pastor, der vortreffliche
Hirte, … would be another, slightly defective, example.

Second, we obtain a pair of multisets, $W^{l}_{\operatorname{in}}$ and
$W^{l}_{\operatorname{out}}$, one for each language $l$.
$W^{l}_{\operatorname{in}}$ (resp. $W^{l}_{\operatorname{out}}$) is the
multiset of all word tokens that appear inside (resp. outside) of NPs of
language $l$. In the following, we will use $M(w)$ to refer to the frequency
of word $w$ in the multiset $M$.

Egosumpastorbonus;bonuspastoranimamsuamponitproovibus.Iamthefinesheperd;thefinesheperdsurrendershissoulonbehalfofthesheep.IchbindervortrefflicheHirte;dervortrefflicheHirtegibtseineSeelezugunstenderSchafehin.Egosumpastorbonus;bonuspastoranimamsuamponitproovibus.Iamthegoodsheperd;thegoodsheperdsacrificeshislifeforthesheep.IchbindervortrefflicheHirte;dervortrefflicheHirtegibtseineSeelezugunstenderSchafehin.
Figure 3: Example of alignments and NP projections (English to Latin and
English to German) with two different English versions (top and bottom).

For each language, we want to remove false positives from the word types
contained within NPs (which are an artefact of wrong alignments) by using the
frequency of each word type inside and outside of NPs.

In principle, this could be done by means of a POS tagger and concentrating on
nouns, adjectives, articles, prepositions, and postpositions, but as we do not
have access to a reliable POS tagger for most languages covered here, we use
the relative frequency information gained from our NP annotations. More
specifically, we assign each word type $w\in W^{l}_{\operatorname{in}}\cup
W^{l}_{\operatorname{out}}$ to $I_{l}$ (the set of words for language $l$ that
are NP-relevant) if
$|W^{l}_{\operatorname{in}}(w)|>|W^{l}_{\operatorname{out}}(w)|$, and to
$O_{l}$ (the set of words for language $l$ that are not NP-relevant)
otherwise. This enhances the robustness of our method against occasional
misannotations: for Latin, ovibus ‘sheep’, from our previous example, occurred
once outside an NP but 45 times inside and is now an element of
$I_{\operatorname{Latin}}$, while intellegent ‘they understand’ occurred once
inside an NP but 22 times outside and is therefore an element of
$O_{\operatorname{Latin}}$.

###  4.4 Candidate Set Creation

From each language, we create a set of candidate case markers
$\operatorname{candidates}(w)$ for a word $w$ by collecting all character
$n$-grams of any length from $w$ that are also members of $I_{l}$. We
explicitly mark the word boundaries with $ so that $n$-grams in the middle of
words are distinct from those at the edges. For example, candidates extracted
from ovibus would be $ovi, ibus$, but also $ovibus$ and i. Our first candidate
set is computed as $C_{1}^{l}=\bigcup\\{\operatorname{candidates}(w)\mid w\in
I_{l}\\}$.

###  4.5 Frequency Filtering

We define $I_{l}(c)$ as the number of words in $I_{l}$ that contain the
candidate $c$, and $O_{l}(c)$ analogously for $O_{l}$. As a first step, we
filter out all $n$-grams with a frequency in $I_{l}$ lower than a threshold
$\theta$.333We set $\theta=97$ based on grid search. This results in
$C_{2}^{l}=\\{c\mid c\in C_{1}^{l},I_{l}(c)\geq\theta\\}$.

###  4.6 Inside/Outside Frequency Filtering

For this step, we make use of the observation that case is a property of
nouns. Hence, a case marker is expected to occur much more frequently within
NPs. This will serve to distinguish the case markers from verb inflection
markers, which should otherwise have similar distributional properties. To
implement this basic idea, for each candidate $c$ in language $l$, we first
construct the contingency table shown in Table 2.

Table 2: Contingency table for candidate case marker $c$ in language $l$ for inside/outside filtering. A morphological marker that occurs significantly more often inside NPs than outside of NPs is likely to be a nominal case marker.  |  $c$ | $\neg c$
---|---|---
NP |  $I_{l}(c)$ | $\sum_{c^{\prime}\neq c\in C_{2}^{l}}I_{l}(c^{\prime})$
$\neg$ NP |  $O_{l}(c)$ | $\sum_{c^{\prime}\neq c\in C_{2}^{l}}O_{l}(c^{\prime})$

We use the table to test whether a candidate is more or less likely to appear
inside NPs by comparing the frequencies of the candidate inside and outside
NPs to those of all other candidates. Shown in the cells are the frequencies
used for the test for each candidate. The columns correspond to the frequency
of the candidate in question versus all other candidates while the rows
distinguish the frequencies inside versus outside NPs. We carry out a Fisher’s
Exact Test (Fisher, 1922) on this table, which gives us a $p$-value and an
odds ratio $r$. $r<1$ if the candidate is more likely to occur outside an NP,
and $r>1$ if it is more likely to occur inside. The $p$-value gives us a
confidence score to support this ratio (lower is better). We keep for
$C_{\operatorname{final}}^{l}$ only those candidates for which $p<\phi$ and
$r>\chi$.444We set $\phi=0.08$ and $\chi=0.34$ based on grid search. For
example, ibus$ makes it past this filter with $p($ibus$$)=2.869\cdot 10^{-6}$
and $r($ibus$$)=1.915$ – it is significant and it occurs inside NPs more often
than outside NPs. In contrast, t$ is discarded as it has $p($t$$)=3.18\cdot
10^{-149}$ and $r($t$$)=0.249$ – it is significant, but it has been found to
occur much more likely outside than inside NPs.

###  4.7 Restriction to Word Endings

Suffixoidal inflection is cross-linguistically more common than prefixoidal
and infixoidal inflection (Bauer, 2019). This is also reflected in our
dataset, where not a single language has prefixoidal or infixoidal inflection.
We hence restrict the set of considered $n$-grams to ones at the end of words.

##  5 Evaluation of Retrieved Case Markers

We evaluate our method for case marker extraction without labels using a
silver standard.

###  5.1 Silver Standard

As we are, to the best of our knowledge, the first to introduce this task, we
cannot rely on an existing set of gold case markers for each language we
cover. As most of the languages included are low-resource, reliable
grammatical resources do not always exist, which makes the handcrafting of a
gold standard difficult. Therefore, and also to ensure relative comparability,
we evaluate against a silver standard automatically created from the UniMorph
(Sylak-Glassman 2016, Kirov et al. 2018, McCarthy et al.
2020)555https://unimorph.github.io dataset. The UniMorph data consists of a
list of paradigms, which we first filter by their POS tag, keeping only nouns
and adjectives and filtering out verbs and adverbs. An example of a paradigm
is given in Table 3. While the Nominative Singular (left column) is included
in addition to the inflected forms (middle column), the straightforward
approach of extracting the suffixes of the inflected forms is not optimal for
every language, as the Nominative Singular form can differ from the root. We
therefore proceed as follows.

First, we form a multiset of all inflected forms. In our example, this would
result in {Abflug, Abfluges, Abflug, Abflug, Abflüge, Abflüge, Abflügen,
Abflüge}666Notice that we do not handle diacritics in any special way. This
limitation is owed to the exploratory nature of our work. We see diacritics as
an exciting problem for the future.. Next, we iterate over this multiset,
removing one word each time if it occurs only once. This is meant to make the
algorithm more robust against outlier words which do not share a common base
with the rest of the paradigm. We then extract the longest common prefix for
the remaining elements. We build a frequency list of these prefixes, which in
our example has only one element, Abfl, with a frequency of 3. We take the
most frequent element from the frequency list and compare it to the Nominative
Singular, Abflug. Of these two candidates, we take the longer one. We thereby
prioritise precision over recall as roots that are too short quickly result in
many different suffixes that are too long, due to the high overall number of
paradigms. Finally, we iterate over the inflected forms again, extracting the
suffix if the chosen root is a prefix, which in our example yields one new
suffix: es$, as Abflüge and Abflügen are not prefixed by Abflug. We examine
the results for each language and exclude the languages where either basic
knowledge of the language or common sense makes it apparent that sets are much
too large or too small, resulting in a diverse set of 19 languages to evaluate
our methods against. We note that this process automatically excludes
adpositions and clitics, which is in line with our focus on suffixoidal
inflection (Section 4.6). We make our silver standard publicly available.

Nominative | inflected | unused
---|---|---
Singular | forms | information
| base |  | suffix |
Abflug | Abfl | ug |  | N | NOM | SG
Abfl | ug | es | N | GEN | SG
Abfl | ug |  | N | DAT | SG
Abfl | ug |  | N | ACC | SG
Abfl | üge |  | N | NOM | PL
Abfl | üge |  | N | GEN | PL
Abfl | ügen |  | N | DAT | PL
Abfl | üge |  | N | ACC | PL
Table 3: Example of silver standard creation. Marked in orange is the
Nominative Singular form, in red the base (“base”) as determined by the
algorithm, and in green the only suffix (“suffix”) that is extracted from this
paradigm. Additional, unused information in the UniMorph data is marked in
grey.

###  5.2 Results

Our results are provided in Table 4. We observe that precision is higher, at
times even substantially, than recall for most languages contained in the
silver standard. Looking at Table 5 as an example, we can see that low
precision is mostly due to retrieved case markers being longer (ение$/enie) or
shorter (й$/j) than the correct ones. It is one of the main challenges in this
task to select the correct length of a case marker from a series of substring
candidates. The shorter substrings will automatically be more frequent and
often correct, but this is not easily solved by a frequency threshold, which
excludes other correct candidates that are naturally less frequent.
Additionally, we observe that some recall errors are due to an incorrect
length of $n$-grams in the silver standard (ьям/’jam), highlighting that this
issue also exists in its creation process, and suggesting that our performance
might even improve when measured against handcrafted data.

Language | P | R | F1
---|---|---|---
Albanian | .74 | .47 | .58
Belarusian | .43 | .41 | .42
Bengali | .50 | .40 | .44
Czech | .50 | .58 | .54
German | .54 | .47 | .50
Greek | .67 | .19 | .30
Icelandic | .83 | .31 | .45
Indonesian | .31 | .42 | .36
Irish | .42 | .30 | .35
Latin | .65 | .56 | .60
Lithuanian | .18 | .38 | .24
Nynorsk | .79 | .48 | .59
Bokmål | .67 | .45 | .54
Polish | .52 | .33 | .40
Russian | .54 | .54 | .54
Slovenian | .41 | .28 | .33
Swedish | .68 | .25 | .36
Ukrainian | .45 | .48 | .47
Average | .54 | .41 | .45
Table 4: Precision (P), Recall (R) and F1 on the task of case marker extraction without labels for languages contained in our silver standard. Nynorsk and Bokmål are two varieties of Norwegian. Intersection |  Algorithm Only |  Silver Standard Only
---|---|---
у, я, ом, ого, о, в, ой, и, ми, ам, ей, ю, ы, ов, ых, а, м, х, ами |  ий, ные, ое, ение, ии, го, ый, ка, ые, к, ки, ия, ние, й, ния, ие |  ыми, ах, ев, ьям, ому, ья, н, ьях, ями, ям, е, ях, ьев, ем, ым, ьями
u, ja, om, ogo, o, v, oj, i, mi, am, ej, ju, y, ov, yx, a, m, x, ami |  ij, nye, oe, enie, ii, go, yj, ka, ye, k, ki, ija, nie, j, nija, ie |  ymi, ax, ev, ’jam, omu, ’ja, n, ’jax, jami, jam, e, jax, ’ev, em, ym, ’jami
Table 5: The output of our algorithm for Russian compared to the silver
standard. We show suffixes that occur in the intersection of algorithm output
and silver standard (“Intersection”), those that occur only in the algorithm
output (“Algorithm Only”) and those that occur only in the silver standard
(“Silver Standard Only”). To allow for a clear and concise presentation, the
table does not observe the convention of using $ for boundaries.

###  5.3 Ablation Study

We conduct an ablation study to assess the effects of the different pipeline
components.

####  5.3.1 Evaluating NP Projection

In order to evaluate how well our method of projecting NP annotation using
alignments to languages without an available NP chunker (see Section 4.3)
works, we evaluate it against the monolingual spaCy chunkers for Norwegian
Bokmål and German, which are the only available languages besides English. We
do not directly compare annotated spans but instead their influence on our
method as we have intentionally designed our pipeline to be robust to some
noise. As $I_{l}$, the set of words considered to be NP-relevant, is the
essential output of the annotation projection, we compare two versions, the
set as a result of direct NP chunking and the set as a result of our
annotation procedure. Taking the former as the ground truth for evaluating the
latter (assuming that the directly chunked set has superior quality), we
observe an F1 of 88.5 % for German and 67.8 % for Norwegian Bokmål. While
these numbers seem low at first, the fact that our overall F1 on Norwegian
Bokmål (.54, see Table 4) is better than on German (.50) indicates that the
later elements of the pipeline are to a certain extent robust against
misclassification of NPs.

####  5.3.2 Ablating Pipeline Components

Figure 4: t-SNE plot of the contextual distribution of the Latin case marker
-ibus and the Polish case markers -ach and -om. Outliers omitted. The plot
shows NPs who in Latin are marked with the case marker ibus$ and in Polish
either with ach$ (orange) or om$ (green). Centroids are marked with an X. The
plot shows that the Polish case markers exhibit a more fine-grained
representation of the underlying semantic categories, which makes it possible
to disambiguate the homonymous Latin case marker.

We report the average Precison, Recall, and F1 across all languages in our
silver standard without individual filtering components in Table 6. Simple
frequency filtering (see “$\neg\theta$”), excluding $n$-grams within words
(see “middle”) and at the beginning of words (see “beginning”) are all
necessary for good performance. Inside/outside filtering based on $p$-value is
the most important component of the pipeline (see “$\neg\phi$”). Surprisingly,
inside/outside filtering based on odds ratio has almost no effect.

| ablation | P | R | F1
---|---|---|---|---
our method |  |  |  |
(Table 4) |  | .54 | .41 | .45
| $\neg\theta$ | .11 | .59 | .16
| $\neg\phi$ | .00 | .00 | .00
| $\neg\chi$ | .53 | .41 | .44
| middle | .11 | .41 | .17
| beginning | .33 | .41 | .35
Table 6: Precision (P), Recall (R) and F1 averaged over all languages on the
task of case marker extraction without labels when each step of our pipeline
is ablated. $\neg\theta$: no Frequency Filtering; $\neg\phi$: no
Inside/Outside Filterung based on $p$-value; $\neg\chi$: no Inside/Outside
Filtering based on odds ratio; middle: include middle of the word; beginning:
include beginning of the word.

##  6 Exploratory Analyses

We can use our automatically extracted case markers, in combination with the
parallel NPs that are extracted as part of the pipeline, for innovative
linguistic analyses. We present two examples in this section.

###  6.1 Marking of Deep Cases

First, we demonstrate how, given a parallel NP, the case markers can be used
to determine its deep case. We return to $N$ (see Section 4.3), our set of
parallel NPs extracted from the PBC, and for a selected subset of languages,
group them by their combination of case markers. The basic idea is to infer an
NP’s (potentially very fine-grained) deep case by representing it as its
combination of case markers across languages.

For example, we can disambiguate the Latin case marker -ibus by looking at the
different groups the NPs containing it form with Russian case markers. Recall
that -ibus can express location, instrument, and recipient and that Russian
expresses these categories by separate case markers: -ах/-ax for location,
-ами/-ami for instrument, and -ам/-am for recipient (see Figure 1) – all three
of which have been retrieved by our method. Given a Latin NP marked by the
ending -ibus, the parallel NP in Russian can help us determine its deep case.
Thus, for domibus, дворцах/dvorcax shows that the semantic category is
location, i.e., ‘in the houses’. For operibus bonis, добрыми делами/dobrymi
delami shows that the semantic category is instrument, i.e., ‘through the good
deeds’. Finally, for patribus, предкам/predkam shows that the semantic
category is a recipient, i.e., ‘for/to the parents’.

###  6.2 Similarities between Case Markers

We also demonstrate how we can use their distributional similarities over NPs
to show how case markers that are similar in this respect correspond to
similar combinations of deep cases. We first generate an NP-word cooccurrence
matrix over the NP vocabulary of all languages in which each row,
corresponding to an inflected word firm $w$ in language $l$, indicates which
NPs (corresponding to columns) cooccur with $w$. in the parallel data. We then
reduce the dimensionality of the matrix by means of t-SNE (Van der Maaten and
Hinton, 2008), allowing us to inspect systematic patterns with respect to the
“contexts” in which certain case markers occur (where “context” refers to
words the case marker is aligned to in other languages, not words the case
marker coccurs with in its own language). In a semiotic situation like the one
shown in Figure 1, this setup allows us to examine how the semantic region
expressed by a certain homonymous case marker in one language is split into
more fine-grained regions in another language that distinguishes the semantic
categories that are lumped together by the case marker (and which, if they are
at the right level of abstraction, can correspond to deep cases).

Figure 4 shows this scenario for the Latin Ablative marker -ibus. It
corresponds to two distinct case markers in Polish, -ach (Loc) and -om (Dat).
The figure shows that the region occupied by Latin -ibus splits into two
distinct clusters in Polish, allowing us to visually determine which
underlying case is expressed by the homonymous suffix -ibus. This underscores
the exploratory potential of our approach.

##  7 Conclusion and Future Work

We have introduced the new and challenging task of Case Marker Extraction
without Labels (CaMEL) and presented a simple and efficient method that
leverages cross-lingual alignments and achieves an F1 of 45% on 19 languages.
We introduce an automatically created silver standard to conduct our
evaluation. We have further demonstrated two ways in which our retrieved case
markers can be used for linguistic analysis.

We see two potential avenues for future work. The first is the further
improvement of case marker extraction. The main problem to tackle here is that
of small sets of overlapping substrings of which only one is the correct
marker, and developing some further measures by which they can be
distinguished. Furthermore, it would be useful to find data from more low-
resource languages and languages that have typological properties different
from the extensively studied large language families (Indo-European, Turkic,
Sino-Tibetan etc.). We could then verify that our method performs well across
languages and attempt to expand our silver standard to more languages while
still ensuring quality. The second area is that of further automating the
analysis of deep case and case syncretism. Ideally, we would develop a method
that can distinguish the different possible reasons for divergent case marking
in languages, with the eventual goal of creating a comprehensive overview of
case and declension systems for a large number of languages.

## Acknowledgements

This work was funded by the European Research Council (#740516). The second
author was also supported by the German Academic Scholarship Foundation and
the Arts and Humanities Research Council. The third author was also supported
by the German Federal Ministry of Education and Research (BMBF, Grant No.
01IS18036A). We thank the reviewers for their extremely helpful comments.

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# The evolution effects of radius and moment of inertia for rapidly rotating
neutron stars

Lin Lan  Department of Astronomy, Beijing Normal University, Beijing 100875,
China<EMAIL_ADDRESS>He Gao  Department of Astronomy, Beijing Normal
University, Beijing 100875, China<EMAIL_ADDRESS>Shunke Ai  Department of
Physics and Astronomy, University of Nevada Las Vegas, Las Vegas, NV 89154,
USA  Shao-Ze Li  Department of Astronomy, Beijing Normal University, Beijing
100875, China<EMAIL_ADDRESS>
###### Abstract

A newly born millisecond magnetar is thought to be the central engine of some
gamma-ray bursts (GRBs), especially those that present long-lasting X-ray
plateau emissions. By solving the field equations, we find that when the
rotational speed of the magnetar is approaching the breakup limit, its radius
$R$ and moment of inertia $I$ would undergo an obvious evolution as the
magnetar spins down. Meanwhile, the values of $R$ and $I$ would sensitively
depend on the adoption of neutron star (NS) equation of state (EoS) and the NS
baryonic mass. With different EoSs and baryonic masses considered, the
magnetic dipole radiation luminosity ($L_{\rm dip}$) could be variant within
one to two orders of magnitude. We thus suggest that when using the X-ray
plateau data of GRBs to diagnose the properties of the nascent NSs, EoS and NS
mass information should be invoked as simultaneously constrained parameters.
On the other hand, due to the evolution of $R$ and $I$, the temporal behavior
of $L_{\rm dip}$ would become more complicated. For instance, if the spin-down
process is dominated by gravitational wave emission due to the NS asymmetry
caused by magnetic field distortion ($\epsilon\propto B_{p}^{2}$), the segment
$L_{\rm dip}\propto t^{0}$ could be followed by $L_{\rm dip}\propto
t^{-\gamma}$ with $\gamma$ larger than 3. This case could naturally interpret
the so-called internal X-ray plateau feature shown in some GRB afterglows,
which means the sharp decay following the plateau is unnecessarily
corresponding to the NS collapsing. This may explain why some internal X-ray
plateaus are followed by late time central engine activity, manifested through
flares and second shallow plateaus.

gamma-ray burst: general - gravitational waves

##  1 INTRODUCTION

With the successful launch of the _Neil Gehrels Swift Observatory_ (Gehrels et
al., 2004), early-time afterglows for gamma-ray bursts (GRBs) have been
revealed. They are important for unveiling the open questions for GRB physics,
such as their progenitors and central engines (Zhang, 2018, for a review).
Observationally, a good fraction of the X-ray light curves of (long and short)
GRBs were discovered to show a long-lasting plateau feature. Such a feature
seems to be consistent with the prediction of millisecond magnetar (namely
rapidly spinning, strongly magnetized neutron star) engine, which is thought
to be formed from a violent event such as the massive star collapse (for long
GRBs) or double neutron star merger (for short GRBs) (Paczynski, 1986; Eichler
et al., 1989; Usov, 1992; Woosley, 1993; Thompson, 1994; Dai & Lu, 1998a, b;
MacFadyen & Woosley, 1999; Zhang & Mészáros, 2001; Metzger et al., 2008;
Zhang, 2011).

The energy reservoir of a newly born millisecond magnetar is the total
rotational energy, which could be estimated as

$\displaystyle E_{\rm rot}=\frac{1}{2}I\Omega^{2}$ (1)

where $I$ is the moment of inertia and $\Omega=2\pi/P$ is the angular
frequency of the nascent neutron star (NS). In principle, during the spin-down
process, the nascent magnetar loses its rotational energy via both magnetic
dipole radiation and gravitational wave (GW) emission, so that the spin-down
law can be written as (Shapiro & Teukolsky, 1983; Zhang & Mészáros, 2001)

$\displaystyle\dot{E}$ $\displaystyle=-L_{\rm dip}-L_{\rm GW}$
$\displaystyle=-\frac{B_{p}^{2}R^{6}\Omega^{4}}{6c^{3}}-\frac{32GI^{2}\epsilon^{2}\Omega^{6}}{5c^{5}}.$
(2)

$B_{p}$ is the dipolar field strength at the magnetic poles on the NS surface.
$R$ and $\epsilon$ are the radius and ellipticity of the NS, respectively.
Depending on the NS properties, such as the values of $B_{p}$ and $\epsilon$,
the spin-down process will be dominated by one or the other loss term.

The Poynting flux generated by the EM dipolar emission could undergo the
magnetic energy dissipation processes with high efficiency (Zhang & Yan, 2011)
and power the X-ray plateau feature in GRB afterglows (Rowlinson et al., 2010,
2013; Gompertz et al., 2013, 2014; Lü & Zhang, 2014; Lü et al., 2015). In
principle, the physical parameters of the newly born magnetar could be
estimated by fitting the observed GRB X-ray plateau data. On the other hand,
one can also infer the dominant energy loss term (EM or GW radiation) by
analyzing the braking index of the magnetar, which can be obtained by fitting
the decay slopes of the X-ray plateau and its follow-up segment (Zhang &
Mészáros, 2001; Lasky & Glampedakis, 2016; Lasky et al., 2017; Lü et al.,
2018, 2019). Studying the population of braking indices is another feasible
way to look at this (Sarin et al., 2020a). Some GRBs were discovered to show
an extended plateau, followed by a sharp decay with a decay index larger than
3 (called internal plateau; Rowlinson et al., 2010, 2013; Lü et al., 2015;
Sarin et al., 2020a), which is usually interpreted as the magnetar collapses
into a BH 111Alternatively, it has been proposed that the internal plateau
might be an imprint of the r-process heating on fall-back accretion (Desai et
al., 2019), or be the high-latitude emission signature from a structured jet
(Ascenzi et al., 2020).. For these cases, one can make estimation for the
dipolar magnetic field strength $B_{p}$ and the initial spin period $P_{0}$ of
the magnetar from the observed X-ray plateau luminosity and its ending time
(Rowlinson et al., 2010, 2013; Gompertz et al., 2013, 2014; Lü et al., 2015;
Gao et al., 2016; Sarin et al., 2020a).

In the previous studies, $R$ and $I$ of magnetars are usually assumed to be
constant, and are usually assigned to some so-called fiducial values, such as
$R=10^{6}~{}\rm cm$ and $I=1.5\times 10^{45}~{}\rm g~{}cm^{2}$. However, as
the GRB central engine, the magnetars are supposed to be rapidly rotating
(especially for short GRBs where the magnetars are formed from merger), in
which case its radius and moment of inertia should be related to the
rotational speed (Lattimer, 2012). Nevertheless, the specific values of $R$
and $I$ should also depend on the NS equation of state (EoS) and the NS mass.

In this paper, we intend to study in detail how $R$ and $I$ evolve as the
magnetar spins down, how these evolution effects alter the magnetic dipole
radiation behavior, and to what extent the NS EoS and NS mass affect the value
of dipole radiation luminosity.

##  2 $R$ and $I$ evolution

For the purpose of this work, we first need to know how the radius $R$ and
moment of inertia $I$ for a given fast rotating neutron star change with its
rotational speed. Obviously, it depends on the EoS of neutron star. Here we
use the numerical methods to treat the equilibrium equations for a stationary,
axially symmetric, rigid rotating NS, within a fully general relativistic
framework. In this case, the spacetime metric can be written as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{2\nu}\,dt^{2}+r^{2}\sin^{2}\theta
B^{2}e^{-2\nu}(d\phi-\omega\,dt)^{2}$ (3)
$\displaystyle\mbox{}+e^{2\alpha}(dr^{2}+r^{2}\,d\theta^{2}),$

where the potentials $\nu$, $B$, $\omega$, and $\alpha$ depend only on $r$ and
$\theta$. We have the following asymptotic decay as (Butterworth & Ipser,
1976)

$\displaystyle\nu$ $\displaystyle=$
$\displaystyle-\frac{M}{r}+\frac{B_{0}M}{3r^{3}}+\nu_{2}P_{2}(\cos\theta)+\mathcal{O}\left(\frac{1}{r^{4}}\right),$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
1+\frac{B_{0}}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{4}}\right),$
$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\frac{2I\Omega}{r^{3}}+\mathcal{O}\left(\frac{1}{r^{4}}\right),$
(4)

where $M$ is the NS mass and $\Omega$ is the angular frequency. $B_{0}$ and
$\nu_{2}$ are real constant. When describing the interior of the NS as a
perfect fluid, its energy-momentum tensor becomes

$\displaystyle T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu}+pg^{\mu\nu},$ (5)

where $\rho$ presents the energy density, $p$ denotes the pressure, and
$u^{\mu}$ is the $4$-velocity.

For a selected sample of EoSs [SLy (Douchin & Haensel, 2001), ENG (Engvik et
al., 1996), AP3 (Akmal & Pandharipande, 1997), WFF2 (Wiringa et al., 1988)]
within a range of maximum mass ($2.05M_{\odot}<M_{\rm TOV}<2.39M_{\odot}$), we
use the public code RNS (Stergioulas & Friedman, 1995) to solve the above
field equations for $R-P$ and $I-P$ relationships, where $P$ is the NS
spinning period. For completeness, for a given EoS, we also consider the
situations when the NSs have different baryonic mass $M_{b}$. The results are
presented in Figure 1.

For a given EoS, when the NS rotational speed is approaching the breakup
limit, $R$ and $I$ approach to constant. As the NS spins down, both $R$ and
$I$ would go through a rapid descent phase. When the spin is slow enough, $R$
and $I$ again tend to be constant. For a given rotational speed, greater
baryonic mass of the NS would lead to smaller $R$ and greater $I$. For our
selected EoSs, when the baryonic mass or rotational speed of a NS undergoes
changes, the radius of the NS can vary from 10 km to 15 km, and the moment of
inertia of the NS can vary from $1.5\times 10^{45}~{}\rm g~{}cm^{2}$ to
$4.5\times 10^{45}~{}\rm g~{}cm^{2}$.

Figure 1: Evolutionary behavior of the neutron star radius ($R$) and moment of
inertia ($I$) for different EoSs and baryonic mass.

##  3 Numerical Results

With the solutions of $R$ and $I$ for various rotational speed, for a NS with
given parameters (e.g., given the initial values of $B_{p}$, $\epsilon$,
$M_{b}$ and $P_{0}$), we can numerically solve its spin-down evolution
history, and calculate the evolution history of magnetic dipole radiation
luminosity. The result is obviously dependent on the NS EoS and different
baryonic mass. For each of our selected EoSs and baryonic mass, we calculate
the spin-down solutions for a large sample of NSs with various parameter
combinations, where the NS spin-down process is dominated either by EM
radiation or GW radiation. Here we assume that the magnetic dipole moment
$\mu\equiv B_{p}R^{3}$ is conserved.

Firstly, for EM radiation dominated case, we set initial value
$B_{p,0}=10^{15}$ G, $P_{0}=1{\rm ms}$ and $\epsilon=10^{-7}$. We test two
values for $M_{b}$ (e.g. $2.0M_{\odot}$ and $2.5M_{\odot}$), with which the NS
can be supported by most of the adopted EoSs. In Figure 2(a), we plot the
results for selected models that are relevant for reflecting the main
conclusions, which can be summarized as follows222Here we did not show any
cases in which the NS collapses during spin-down process.:

  * •

For the situations when realistic values are considered or the so called
fiducial values are assigned to $R$ and $I$ (e.g. $R=10^{6}~{}\rm cm$ and
$I=1.5\times 10^{45}~{}\rm g~{}cm^{2}$), the magnetic dipole radiation
luminosity could be different up to one order of magnitude. In this case, if
one use the so called fiducial values to estimate the dipolar magnetic field
strength and the initial spin period for the nascent NS, the results could be
systemically biased.

  * •

For a given EoS, different NS baryonic mass could evidently alter the
magnitude and evolution history of the dipole radiation luminosity. According
to the results in Figure 2(a), with higher baryonic mass, the luminosity
becomes smaller, and the spin-down timescale becomes larger. When the baryonic
mass is large enough, the magnetar would collapse into a black hole after
slightly spinning down, so that the dipole radiation would suddenly enter into
a rapid descent phase.

  * •

For a given baryonic mass, different EoSs could overall change the magnitude
of the dipole radiation luminosity. As shown in Figure 2(a), with a more stiff
EoS, the luminosity generally becomes larger, and the spin-down timescale
becomes smaller. Adopting different EoSs could change the fact that whether
and when the magnetar would collapse.

Figure 2: Numerical results for dipole radiation luminosity by considering the
$R$ and $I$ evolution effects in choice of different NS EoSs and baryonic mass
of NS.(a): EM dominated scenario. We set $B_{p,0}=10^{15}G$ and $P_{0}=1{\rm
ms}$ for initial condition, and $\epsilon=10^{-7}$; (b): GW dominated scenario
with a constant ellipticity. We set $B_{p,0}=10^{14}G$ and $P_{0}=1{\rm ms}$
for initial condition, and $\epsilon=10^{-3}$; (c): GW dominated scenario with
$\epsilon\propto B_{p}^{2}$. We set the initial condition as
$B_{p,0}=10^{12}G$, $\epsilon_{0}=10^{-4}$. Here we use the Kepler period for
each adopted $M_{b}$ and EoS as the initial spinning period $P_{0}$.

Then we consider the GW radiation dominated case, where intensity of GW
radiation depends on the NS asymmetries (manifested by the value of
$\epsilon$). Here we discuss two mechanisms to induce NS asymmetries: first,
the crust of a NS is solid and elastic, where the solid shape is related to
the history of neutron star formation and EoS (Haskell et al., 2006; Lasky,
2015), in which scheme the $\epsilon$ value is usually assumed to be a
constant during the spin-down process (Corsi & Mészáros, 2009; Lasky &
Glampedakis, 2016; Lü et al., 2017; Sarin et al., 2018; Lü et al., 2019; Lan
et al., 2020; Sur & Haskell, 2021). Second, the NS asymmetry is caused by the
distortion of NS magnetic field, in which scheme the ellipticity satisfies
$\epsilon\propto B_{p}^{2}$ (Bonazzola & Gourgoulhon, 1996; Haskell et al.,
2008; Dall’Osso et al., 2009; Gao et al., 2017a; Dall’Osso et al., 2018;
Lander & Jones, 2020). Similarly, two values of $M_{b}$ (e.g. $2.0M_{\odot}$
and $2.5M_{\odot}$) are tested. In figure 2(b) and Figure 2(c), we plot the
results for both scenarios, which can be summarized as follows:

  * •

For both scenario, with realistic values of $R$ and $I$ for different EoSs and
NS baryonic mass being considered, the magnetic dipole radiation luminosity
could be variant up to one order of magnitude. Similar to the EM radiation
dominated case, if one use the so called fiducial values of $R$ and $I$ to
estimate properties of the nascent NS, the results would be significantly
biased.

  * •

The history of magnetic dipole radiation luminosity for both scenario would
consist four segments. The first two segments are $L_{\rm dip}\propto t^{0}$
followed by $L_{\rm dip}\propto t^{-\gamma}$. The last two segments would be
$L_{\rm dip}\propto t^{-1}$ followed by $L_{\rm dip}\propto t^{-2}$. The
emergence of a new segment $L_{\rm dip}\propto t^{-\gamma}$ is due to the
evolution of $R$ and $I$. When $\epsilon$ is approximate to be a constant,
$\gamma$ is slightly smaller than 1; when $\epsilon\propto B_{p}^{2}$,
$\gamma$ is around or larger than 3.

In previous works, it has been proposed to obtain the braking index of the
nascent NS by fitting the magnetic dipole radiation luminosity with

$\displaystyle L_{\rm dip}=L_{\rm sd}(1+\frac{t}{T_{\rm sd}})^{\frac{4}{1-n}}$
(6)

where $n$ is the braking index with $\dot{\Omega}\propto\Omega^{n}$, $L_{\rm
sd}=B^{2}_{\rm p}R^{6}\Omega_{0}^{4}/6c^{3}$ is defined as the NS spin-down
luminosity and $T_{\rm sd}\approx\frac{1}{2}I\Omega_{0}^{2}/L_{\rm sd}$ is
defined as the spin-down timescale of the NS. The obtained braking index is
then used to diagnose whether the magnetar spin-down is dominated by EM
radiation or GW radiation (Lasky et al., 2017; Lü et al., 2019), since when EM
radiation dominates, the torque equation would become
$\dot{\Omega}=-B_{p}^{2}R^{6}\Omega^{3}/6Ic^{3}$, so that the braking index
should be 3. On the other hand, when GW emission dominates, the torque
equation would become $\dot{\Omega}=-32GI\epsilon^{2}\Omega^{5}/5c^{5}$, so
that the braking index would be 5. According to our results, when the newly
born NS has a millisecond spin period, $R$ and $I$ are no longer constant, so
that if one still apply Equation 6 to fit the magnetic dipole radiation
luminosity, the obtained braking index could be slightly smaller than 3 for EM
emission dominated case or greatly smaller than 3 for GW emission dominated
case with constant $\epsilon\propto B_{p}^{2}$, or larger than 5 for GW
emission dominated case with constant $\epsilon$.

On the other hand, our results show that the sharp decay $t^{-(>3)}$ following
the X-ray plateaus (Rowlinson et al., 2010, 2013; Lü et al., 2015; Sarin et
al., 2020a) are unnecessarily related to the collapsing of the central
magnetar. Instead, it could be caused by the $R$ and $I$ evolution effect, if
the spin-down process of the nascent NS is dominated by GW radiation and the
ellipticity satisfies $\epsilon\propto B_{p}^{2}$. If our interpretation is
correct, this may explain why some GRBs that present the internal plateau
feature, still show late time central engine activity, manifested through
flares and second shallow plateaus (Troja et al., 2007; Perley et al., 2009;
Margutti et al., 2011; Gao et al., 2015, 2017b; Zhao et al., 2020). In the
magnetar collapsing scenario, only flares or plateaus close to the internal
plateau could be interpreted with fall-back accretion model (Chen et al.,
2017; Zhao et al., 2020).

##  4 Analytical analysis for the numerical results

Under certain approximations, we can try to explain the results from numerical
calculation by analytical method. By fitting the evolutionary behavior of the
$R$ and $I$ for different EoSs and baryonic mass, one can approximately get
that

$\displaystyle
R\simeq\left\\{\begin{array}[]{ll}R_{0}(\frac{\Omega}{\Omega_{\rm
k}})^{m},&\Omega_{1}<\Omega<\Omega_{\rm k};\\\
R_{0}(\frac{\Omega_{1}}{\Omega_{\rm k}})^{m},&\Omega\leq\Omega_{1}.\\\
\end{array}\right.$ (9)

Table 1: The evolution indexes of R and I with different EoSs and NS baryonic
mass $m$
---
| AP3 | ENG | SLy | WFF2
$M_{b}=2.0M_{\odot}$ | 0.34 | 0.35 | 0.21 | 0.33
$M_{b}=2.5M_{\odot}$ | 0.27 | 0.28 | 0.37 | 0.27
$M_{b}=3.0M_{\odot}$ | 0.38 | NO | NO | NO
$k$
| AP3 | ENG | SLy | WFF2
$M_{b}=2.0M_{\odot}$ | 0.14 | 0.13 | 0.13 | 0.12
$M_{b}=2.5M_{\odot}$ | 0.15 | 0.24 | 0.25 | 0.22
$M_{b}=3.0M_{\odot}$ | 0.31 | NO | NO | NO

$\displaystyle
I\simeq\left\\{\begin{array}[]{ll}I_{0}(\frac{\Omega}{\Omega_{\rm
k}})^{k},&\Omega_{1}<\Omega<\Omega_{\rm k};\\\
I_{0}(\frac{\Omega_{1}}{\Omega_{\rm k}})^{k},&\Omega\leq\Omega_{1},\\\
\end{array}\right.$ (12)

where $R_{0}$, $I_{0}$, and $\Omega_{\rm k}$ are the initial radius, initial
moment of inertia, and Keplerian angular velocity of the newly born NS,
respectively. It is worth noting that $\Omega_{\rm k}$ depends on the NS EoS
and baryonic mass. $m$ and $k$ are the power law index for $R$ and $I$
evolution with respect to the rotational speed of the NS. The best fitting
values of $m$ and $k$ for different EoSs and $M_{b}$ are collected in Table 1.
For our selected EoSs, $m$ ranges from 0.21 to 0.38 and $k$ ranges from 0.12
to 0.31. When the baryonic mass is large enough, the magnetar would collapse
into a black hole after slightly spinning down, in which case the values of
$m$ and $k$ are denoted as “NO”. With the assumption that the magnetic dipole
moment $\mu\equiv B_{p}R^{3}$ is conserved, the evolution of $L_{\rm dip}$
could be derived based on the Equation 2:

(I) For $L_{\rm dip}$ dominated case, one has

$\displaystyle L_{\rm dip}\simeq-\frac{(2+k)I_{0}\Omega^{k+1}}{2\Omega_{\rm
k}^{k}}\dot{\Omega}=\frac{B_{p,0}^{2}R_{0}^{6}\Omega^{4}}{6c^{3}},$ (13)

so that the complete solution of $\Omega(t)$ in Equation 13 could be written
as

$\displaystyle\Omega=\Omega_{0}\left[1+\frac{t}{T_{\rm
sd,em}}\right]^{\frac{1}{k-2}},$ (14)

where $\Omega_{0}$ is the initial angular frequency at $t=0$, and $T_{\rm
sd,em}$ is a corresponding characteristic spin-down timescale, which could be
derived as

$\displaystyle T_{\rm
sd,em}=\frac{3(2+k)I_{0}c^{3}}{(2-k)B_{p,0}^{2}R_{0}^{6}\Omega_{\rm
0}^{2}}(\frac{\Omega_{\rm 0}}{\Omega_{\rm k}})^{k}.$ (15)

Hence, the evolution history of $L_{\rm dip}$ could be expressed as

$\displaystyle L_{\rm dip}=L_{\rm sd,em}\left[1+\frac{t}{T_{\rm
sd,em}}\right]^{\frac{4}{k-2}},$ (16)

where $L_{\rm sd,em}$ is the initial luminosity of electromagnetic dipole
emission at $t=0$, which could be calculated as

$\displaystyle L_{\rm
sd,em}=\frac{B_{p,0}^{2}R_{0}^{6}\Omega_{0}^{4}}{6c^{3}}.$ (17)

It is clear that both the magnitude and the evolution history of $L_{\rm dip}$
depend on the realistic EoS and NS mass, as well as the evolution history of
the moment of inertia (with index $k$). Comparing Equation 6 with Equation 16,
one can obtain the braking index in this scenario as

$\displaystyle n=3-k.$ (18)

When $k>0$, we have $n<3$.

(II) For $L_{\rm GW}$ dominated case with $\epsilon$ keeping constant, one has

$\displaystyle L_{\rm GW}\simeq-\frac{(2+k)I_{0}\Omega^{k+1}}{2\Omega_{\rm
k}^{k}}\dot{\Omega}=\frac{32GI_{0}^{2}\epsilon^{2}\Omega^{6+2k}}{5c^{5}\Omega_{\rm
k}^{2k}},$ (19)

so that the complete solution of $\Omega(t)$ in Equation 19 could be written
as

$\displaystyle\Omega=\Omega_{0}\left[1+\frac{t}{T_{\rm
sd,gw}}\right]^{-\frac{1}{k+4}},$ (20)

where $T_{\rm sd,gw}$ could be derived as

$\displaystyle T_{\rm
sd,gw}=\frac{5(k+2)c^{5}}{64(k+4)GI_{0}\epsilon^{2}\Omega_{0}^{4}}(\frac{\Omega_{\rm
0}}{\Omega_{\rm k}})^{-k}.$ (21)

The evolution history of $L_{\rm dip}$ could be expressed as

$\displaystyle L_{\rm dip}=L_{\rm sd,em}\left[1+\frac{t}{T_{\rm
sd,gw}}\right]^{-\frac{4}{k+4}}.$ (22)

Similar to the EM dominated case, here both the magnitude and the evolution
history of $L_{\rm dip}$ depend on the realistic EoSs and NS mass, as well as
the evolution history of the moment of inertia (with index $k$). Comparing
Equation 6 with Equation 22, one can derive the braking index in this scenario
as

$\displaystyle n=5+k.$ (23)

When $k>0$, we have $n>5$.

(III) For $L_{\rm GW}$ dominated case with $\epsilon=\beta B_{p}^{2}$, one has

$\displaystyle L_{\rm GW}\simeq-\frac{(2+k)I_{0}\Omega^{k+1}}{2\Omega_{\rm
k}^{k}}\dot{\Omega}=\frac{32GI_{0}^{2}\beta^{2}B^{4}_{p}\Omega^{6+2k}}{5c^{5}\Omega_{\rm
k}^{2k}},$ (24)

so that the complete solution of $\Omega(t)$ in Equation 24 could be written
as

$\displaystyle\Omega=\Omega_{0}\left[1+\frac{t}{T_{\rm
sd,gw}}\right]^{\frac{1}{12m-k-4}},$ (25)

where $T_{\rm sd,gw}$ could be derived as

$\displaystyle T_{\rm
sd,gw}=\frac{5(k+2)R^{12}_{0}c^{5}}{64(k+4-12m)GI_{0}\beta^{2}\mu^{4}\Omega_{0}^{4}}(\frac{\Omega_{\rm
0}}{\Omega_{\rm k}})^{12m-k}.$ (26)

The evolution history of $L_{\rm dip}$ could be expressed as

$\displaystyle L_{\rm dip}=L_{\rm sd,em}\left[1+\frac{t}{T_{\rm
sd,gw}}\right]^{\frac{4}{12m-k-4}}.$ (27)

In this case, the evolution history of $L_{\rm dip}$ not only depends on the
evolution history of the moment of inertia $I$, but also the evolution history
of $R$. The decaying power law index $4/(12m-k-4)$ could be around or larger
than 3. Comparing Equation 6 with Equation 27, one can derive the braking
index in this scenario as

$\displaystyle n=5+k-12m,$ (28)

which could be much smaller than 5 (even smaller than 3).

##  5 Conclusion and Discussion

By solving the field equations, we find that when a NS’s rotational speed is
approaching the breakup limit, its radius and moment of inertia would undergo
an obvious evolution as the NS spins down. In this case, the deceleration
history of the NS would become more complicated for given initial dipole
magnetic field and ellipticity. Our main results could be summarized as
follows:

  * •

When realistic values of $R$ and $I$ for different EoSs and NS baryonic mass
are considered, the magnetic dipole radiation luminosity could be variant
within one to two orders of magnitude.

  * •

With the consideration of $R$ and $I$ evolution effects for a rapidly spinning
NS, its magnetic dipole radiation light curve would present new segments. For
instance, when GW radiation dominates the spin down power and $\epsilon\propto
B_{p}^{2}$, the history of magnetic dipole radiation luminosity would consist
four segments, i.e., $L_{\rm dip}\propto t^{0}$ followed by $L_{\rm
dip}\propto t^{-\gamma}$, and then followed by $L_{\rm dip}\propto t^{-1}$ and
$L_{\rm dip}\propto t^{-2}$. The new segment $L_{\rm dip}\propto t^{-\gamma}$
with $\gamma$ larger than 3 is due to the evolution of $R$ and $I$. In this
case, if one apply $L_{\rm dip}=L_{\rm sd}(1+t/T_{\rm sd})^{4/(1-n)}$ to fit
the dipole radiation luminosity, the obtained braking index could be much
smaller than 3 333Other reasons why the braking index could be smaller than 3
has been discussed in Lasky et al. (2017)..

  * •

In the case when EM radiation power is comparable to the GW radiation power,
the dipole radiation light curve would become even more complicated.
Especially when the initial EM radiation power is slightly larger, for
$\epsilon\propto B_{p}^{2}$ scenario, there would be a transition from EM loss
dominating to GW loss dominating, and then back again as the NS spins down. In
this case, the history of dipole radiation could consist five segments, i.e.,
$L_{\rm dip}\propto t^{0}$ followed by $L_{\rm dip}\propto t^{-2}$ and then
followed by $L_{\rm dip}\propto t^{-\gamma}$ ($\gamma$ is larger than 3), and
then followed by $L_{\rm dip}\propto t^{-1}$, finally followed by $L_{\rm
dip}\propto t^{-2}$. For those complicated situations, Equation 6 may not be a
good formula to fit the dipole light curve. But if one still applies Equation
6 to find the braking index, the result could vary from $n<3$ to $n>5$,
depending on how many segments being covered by the observations.

According to our results, 1) we suggest that when using the EM observations
such as the X-ray plateau data of GRBs to diagnose the properties of the
nascent neutron stars, NS EoS and mass information should be invoked as
simultaneously constrained parameters; 2) if the spin-down process of the
nascent NS is dominated by GW radiation and the ellipticity satisfies
$\epsilon\propto B_{p}^{2}$, the sharp decay following the X-ray plateau could
be caused by the $R$ and $I$ evolution effect rather than due to the NS
collapsing. If this is true, it may explain why some GRBs that present the
internal plateau feature still show late time central engine activity,
manifested through flares and second shallow plateaus. Future EM and GW joint
detection could help to distinguish these two scenarios.

Finally, we would like to point out several caveats of our results. First, the
physical conditions for a nascent NS would be very complicated, and some
conditions may significantly alter the radiation lightcurve, so that the R/I
evolution effect we discussed here would be reduced or even completely
suppressed. For instance, it has been proposed that the evolution of the
inclination angle between the rotation and magnetic axes of the NS could
markedly revise the X-ray emission (Çıkıntoğlu et al., 2020). Moreover, the
nascent NS is likely to undergo free precession in the early stages of its
lifetime when the rotation and magnetic axes of the system are not orthogonal
to each other, which would lead to the fluctuations in the X-ray light curve
(Suvorov & Kokkotas, 2020, 2021). On the other hand, the X-ray plateau
emission luminosity could emerge from a plerion-like model of the nascent NS,
in which model the magnetized, relativistic wind from a millisecond magnetar
injects shock-accelerated electrons into a cavity confined by the GRB
blastwave. The plerion model introduces an anticorrelation between the
luminosity and duration of the plateau, and also shows a sudden drop in the
X-ray emission when the central magnetar collapses into a black hole (Strang &
Melatos, 2019).

Secondly, even if the NS EoS and mass information were invoked as we suggest,
one may still face many difficulties when using the X-ray plateau data of GRBs
to diagnose the properties of the nascent NS. For instance, precise estimation
for the luminosity of X-ray plateaus, in many cases, is very difficult to
obtain, because there are no redshift measurement for most GRBs and the
uncertainties introduced by the cosmological k-correction could also be
significant (Bloom et al., 2001). On the other hand, a constant efficiency is
usually assumed to convert the spin-down luminosity to the observed X-ray
luminosity, which might be incorrect. It has been proposed that the efficiency
might be strongly dependent on the energy injecting luminosity, which leads to
a larger braking index from afterglow fitting compared to the case with
constant efficiency (Xiao & Dai, 2019). Furthermore, if one considers both
energy injection and radiative energy loss of the blastwave, the X-ray light
curve would be altered without changing the braking index, and this model
could also fit the data well (Dall’Osso et al., 2011; Sarin et al., 2020b).

Finally, it is worth noticing that besides the magnetar scenario, many
alternative models have been proposed to interpret the X-ray plateau emission,
such as the structure jet model (Beniamini et al., 2020), the high-latitude
emission model (Oganesyan et al., 2020), and the late time energy injection
model (Matsumoto et al., 2020), etc. As suggestted by Sarin et al. (2019),
before attempting to derive NS parameters, one should first ensure the
magnetar scenario being the preferred hypothesis.

We thank the anonymous referee for the helpful comments that have helped us to
improve the presentation of the paper. LL, HG and SZL acknowledge the National
Natural Science Foundation of China under Grant No. 11722324, 11690024,
11633001, the Strategic Priority Research Program of the Chinese Academy of
Sciences, Grant No. XDB23040100 and the Fundamental Research Funds for the
Central Universities.

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