# Twisted Arrow Construction for Segal Spaces

Chirantan Mukherjee and Nima Rasekh  University of Western Ontario, London,
Canada<EMAIL_ADDRESS>Max Planck Institute for Mathematics, Bonn, Germany
<EMAIL_ADDRESS>
(Date: June 2023)

###### Abstract.

We give an explicit description of the twisted arrow construction for
simplicial spaces and demonstrate individually that it preserves the defining
properties of a complete Segal space. Moreover, we show that for a Segal
space, the natural projection from the twisted arrow Segal space is a left
fibration.

###### Key words and phrases:

Higher category theory, complete Segal spaces, twisted arrow construction,
left fibrations

###### 2020 Mathematics Subject Classification:

18N60; 18N40; 18N50; 18N45.

##  1\. Introduction

###  1.1. Twisted arrow categories

Twisted arrow categories have proven extremely useful in the study of a
variety of categorical settings, such as computations of (lax) limits and
(co)ends [Mac71, Lor21], categorical logic [Law70], configuration spaces
[Seg73], algebraic $K$-theory [Wal85], and the study of exponentiable
fibrations [Joh99, BN00].

Category theory has since been generalized to what is now called higher
category theory, $(\infty,1)$-category theory or simply $\infty$-category
theory [Ber10]. Similar to the $1$-categorical situation, higher categorical
analogues of the twisted arrow construction have also found a variety of
applications, such as in derived geometry [Lur17, Lur11], algebraic $K$-theory
of higher categories [Bar17, BGN18, BOO+18], computing lax $\infty$-limits
[GHN17], and decomposition spaces [HK22].

As such the twisted arrow $\infty$-category has already been studied in a
variety of settings, such as via quasi-categories [Lur17], $2$-Segal spaces
[BOO+20] and internal $\infty$-categories [Mar21].

###  1.2. Construction of twisted arrow complete Segal spaces

In this work we focus on the twisted arrow construction in the context of
complete Segal spaces, which is a prominent model of $(\infty,1)$-categories
introduced by Rezk [Rez01] and further studied in a variety of settings
[Toë05, JT07]. First, in Theorem 3.8 we give a short and self-contained proof
that the twisted arrow construction preserves the three defining
characteristics of a complete Segal space (Reedy fibrancy, Segal condition and
completeness condition), only assuming basic aspects of the complete Segal
space model structure as given in the original paper [Rez01]. The proof for
each one of the conditions can already be found separately in the literature
(such as Reedy fibrancy in [BOO+20] or the Segal and completeness condition
[Mar21]), however using far more advanced and distinct machinery that are not
directly compatible with each other.

Second, we show that the natural projection map from the twisted arrow Segal
space is a left fibration (Theorem 4.3), which generalizes several existing
results. Indeed, it generalizes an analogous result for the twisted arrow
projection of quasi-categories [Lur17, Proposition 5.2.1.3], which implicitly
took the completeness condition for granted, which our result demonstrates to
be a superfluous assumption. Moreover, combining this result with [Mar21,
Proposition 4.2.5] also gives us the same result for internal Segal objects,
generalizing twisted internal $\infty$-categories defined there. Finally, the
twisted arrow construction in particular gives us a parameterized version of
slice-Segal spaces (Remark 4.5) and this result hence also generalizes similar
results regarding the existence of over-Segal spaces without assuming
completeness [Ras17, Theorem 3.44], [RS17, Proposition 8.13].

###  1.3. Acknowledgement

The first author would like to thank the University of Trento, Italy for
financial support in the form of Thesis Research Abroad Scholarship.

##  2\. Twisted Arrow Simplicial Spaces

Before we commence let us recall relevant facts regarding simplicial spaces,
following the definitions and notations in [Rez01]. A simplicial space $W$ is
a bisimplicial set, that is a functor
$W\colon{\mathbbe{\Delta}}^{op}\times{\mathbbe{\Delta}}^{op}\to Set$. We
denote the category of simplicial spaces by $s\mathscr{S}$ and the simplicial
space represented by $([n],[l])$ by $F(n)\times\Delta[l]$. To simplify
notation if $l=0$ ($n=0$), we use the notation $F(n)$ ($\Delta[l]$). More
generally, for a given simplicial set $S$ we also denote by $S$ the simplicial
space given as $S_{nl}=S_{l}$. This gives us a fully faithful inclusion from
simplicial sets to simplicial spaces, which preserves $\Delta[l]$ and allows
us to define a simplicial tensor: For a given simplicial space $X$ and
simplicial set $S$ we define the simplicial tensor as the Cartesian product
$X\times S$. The tensor also gives us a simplicial enrichment explicitly given
by
$\mathrm{Map}_{s\mathscr{S}}(X,Y)_{l}=\mathrm{Hom}_{s\mathscr{S}}(X\times\Delta[l],Y)$.
To simplify notation, we denote the simplicial set
$\mathrm{Map}_{s\mathscr{S}}(F(n),Y)$ by $Y_{n}$. Moreover, recall from [JT07,
Section 4] the inclusion functor $p_{1}^{*}:s\mathscr{S}\mathrm{et}\to
s\mathscr{S}$ defined as $p_{1}^{*}(S)_{nl}=S_{n}$, which in particular means
$p_{1}^{*}(\Delta[n])=F(n)$.

We now proceed to review relevant aspects of the twisted arrow construction
for simplicial sets, as covered in [Lur17, Section 5.2.1]. Let
$Q:{\mathbbe{\Delta}}\to{\mathbbe{\Delta}}$ be the functor defined as
$Q([n])=[n]^{op}\star[n]$ and define the functor
$\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}:s\mathscr{S}\mathrm{et}\to
s\mathscr{S}\mathrm{et}$ as $\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}(S)=S\circ
Q$. This in particular means that
$\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}(S)_{n}\cong S_{2n+1}$. Moreover, the
two natural inclusions $[n]^{op}\hookrightarrow[n]^{op}\star[n]$,
$[n]\hookrightarrow[n]^{op}\star[n]$ induce a projection map
$\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}(S)\to S^{op}\times S$, which we call
the twisted arrow projection. See [Lur17, Construction 5.2.1.1] for more
details regarding this definition.

We want to define $\mathrm{Tw}:s\mathscr{S}\to s\mathscr{S}$ such that it
satisfies the following two conditions:

  1. (1)

It should lift the original twisted arrow construction, meaning for a given
simplicial set $W$,
$\mathrm{Tw}(p_{1}^{*}W)=p_{1}^{*}\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}(W)$.

  2. (2)

The functor should be simplicially enriched and tensored.

We claim that these two conditions determine the functor uniquely. First of
all, following [Kel05, Theorem 4.51], a simplicially enriched functor
$\mathrm{Tw}:s\mathscr{S}\to s\mathscr{S}$ that preserves colimits and tensors
is uniquely determined by its restriction $\mathrm{Tw}\circ
F(\bullet):{\mathbbe{\Delta}}\to s\mathscr{S}$. Moreover, by the first
assumption we need to have
$\mathrm{Tw}(F(\bullet))=p_{1}^{*}\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}(\Delta[\bullet])$
as functors from ${\mathbbe{\Delta}}$ to $s\mathscr{S}$. Combining these two
facts we have the following definition.

######  Definition 2.1.

Let $\mathrm{Tw}:s\mathscr{S}\to s\mathscr{S}$ be the unique simplicially
enriched and tensored functor that lifts the following functor

${{\mathbbe{\Delta}}}$${s\mathscr{S}\mathrm{et}}$${s\mathscr{S}\mathrm{et}}$${s\mathscr{S}}$${s\mathscr{S}}$$\scriptstyle{F(\bullet)}$$\scriptstyle{\Delta[\bullet]}$$\scriptstyle{\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}}$$\scriptstyle{p_{1}^{*}}$$\scriptstyle{\mathrm{Tw}}$

.

Unwinding the definitions and the fact that
$\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}(\Delta[n])=\Delta[2n+1]$, it follows
that $\mathrm{Tw}(F(n)\times\Delta[l])=F(2n+1)\times\Delta[l]$. Moreover, we
have isomorphisms of simplicial sets $\mathrm{Tw}(W)_{n}\cong W_{2n+1}$. We
also have the following alternative characterizations of $\mathrm{Tw}$, the
proof of which is a straightforward computation.

######  Lemma 2.2.

$\mathrm{Tw}$ is equal to the functor
$\mathrm{Fun}({\mathbbe{\Delta}}^{op},\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}):s\mathscr{S}\to
s\mathscr{S}$.

Hence, $\mathrm{Tw}$ can also be obtained by applying
$\mathrm{Tw}^{s\mathscr{S}\mathrm{et}}$ level-wise. This in particular means
that similar to the case for simplicial sets, for every simplicial space $W$
we have the twisted arrow projection $\mathrm{Tw}W\to W^{op}\times W$.

######  Remark 2.3.

If we relax one of the conditions above, we get other possible candidates that
are not as suitable. For example, the functor $W_{n,l}\mapsto W_{2n+1,2l+1}$
does lift the twisted arrow construction for simplicial sets, however, it is
not simplicial.

##  3\. Preservation Properties of the Twisted Arrow Construction

We now show the twisted arrow construction preserves the defining properties
of a complete Segal space.

###  3.1. Reedy Fibrancy

We commence by showing that $\mathrm{Tw}$ preserves Reedy fibrancy. Recall
from [Ree74], but also [Rez01, 2.4] that a simplicial space $W$ is Reedy
fibrant if for all $n\geq 0$, the map

$\mathrm{Map}(F(n),W)\to\mathrm{Map}(\partial F(n),W)$

is a Kan fibration.

Before we proceed to the proof, we make the following observation regarding
Reedy fibrancy of $\mathrm{Tw}W$. Let us define
$\partial_{\mathrm{Tw}}F(2n+1)$ as the following coequalizer diagram

${{\underset{0\leq i<j\leq n}{\coprod}F(2n-3)}}$${{\underset{0\leq i\leq
n}{\coprod}F(2n-1)}}$${\partial_{\mathrm{Tw}}F(2n+1)}$

.

Using the fact that $\mathrm{Tw}$ preserves colimits and $\partial F(n)$ is
precisely given via an analogous coequalizer diagram [GJ09, Proposition 2.3],
it follows that $\mathrm{Map}(\partial
F(n),\mathrm{Tw}W)\cong\mathrm{Map}(\partial_{\mathrm{Tw}}F(2n+1),W)$, which
means in order to establish the Reedy fibrancy of $\mathrm{Tw}(W)$ we only
need to show the map

$\mathrm{Map}(F(2n+1),W)\to\mathrm{Map}(\partial_{\mathrm{Tw}}F(2n+1),W)$

is a Kan fibration.

######  Proposition 3.1.

Let $W$ be a Reedy fibrant simplicial space. Then $\mathrm{Tw}W$ is also Reedy
fibrant.

###### Proof.

Following the explanation above, we only need to establish that
$\mathrm{Map}(F(2n+1),W)\to\mathrm{Map}(\partial_{\mathrm{Tw}}F(2n+1),W)$ is a
Kan fibration. However, $W$ is Reedy fibrant and Reedy model structure is
simplicial and so we only need to show that $\partial_{\mathrm{Tw}}F(2n+1)\to
F(2n+1)$ is a cofibration, meaning we want to show that for all $k\geq 0$, the
map $i:\partial_{\mathrm{Tw}}F(2n+1)_{k}\to F(2n+1)_{k}$ is injective.

We can explicitly describe $\partial_{\mathrm{Tw}}F(2n+1)_{k}$ as a quotient
of $\coprod_{0\leq i\leq n}F(2n-1)_{k}$. Take two $k$-cells $\alpha,\beta$ in
two copies of $F(2n-1)$. If $\alpha,\beta$ are in the same copy then $i$ is
injective by construction and so there is nothing to check. If $\alpha$ and
$\beta$ live in two different copies, then by definition we have the following
pullback square

${F(2n-3)_{k}}$${F(2n-1)_{k}\ni\beta}$${\alpha\in
F(2n-1)_{k}}$${F(2n+1)_{k}}$$\scriptstyle{f}$${\lrcorner}$$\scriptstyle{g}$

.

Now, the fact that $\alpha$ and $\beta$ map to the same element in
$F(2n+1)_{k}$ and the universal property of pullbacks implies that there
exists a $\gamma\in F(2n-3)_{k}$ such that $g(\gamma)=\alpha$ and
$f(\gamma)=\beta$ and so $\alpha\sim\beta$, meaning $\alpha=\beta$ in
$\partial_{\mathrm{Tw}}(F(2n+1))$. This gives us the desired injection and so
we are done. ∎

###  3.2. Segal Condition

Next we want to show that the twisted arrow Segal space is a Segal space.
Recall that a Reedy fibrant simplicial space $W$ is a Segal space if the maps

$\displaystyle
W_{n}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\underbrace{W_{1}\underset{{W_{0}}}{\times}\cdots\underset{{W_{0}}}{\times}W_{1}}_{n-\;factors}$

are Kan equivalences for all $n\geq 2$.

######  Proposition 3.2.

If $W$ is a Segal space, then $\mathrm{Tw}W$ is also a Segal space.

###### Proof.

By Proposition 3.1 $\mathrm{Tw}W$ is a Reedy fibrant simplicial space. Hence,
we just need to show that
$\mathrm{Tw}(W)_{n}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathrm{Tw}(W)_{1}\underset{\mathrm{Tw}(W)_{0}}{\times}\cdots\underset{\mathrm{Tw}(W)_{0}}{\times}\mathrm{Tw}(W)_{1}$
is a Kan equivalence for $n\geq 2$. By induction it suffices to show that
$\mathrm{Tw}(W)_{n}\to\mathrm{Tw}(W)_{n-1}\underset{\mathrm{Tw}(W)_{0}}{\times}\mathrm{Tw}(W)_{1}$
is an equivalence, which unwinds to showing that $W_{2n+1}\to
W_{2n-1}\underset{W_{1}}{\times}W_{3}$ is an equivalence. Now, the fact that
the Kan model structure is right proper and the following commutative diagram

${W_{2n-1}}$${W_{1}}$${W_{3}}$${W_{2n-1}}$${W_{1}}$${W_{1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{Tw}(d_{0})}$$\scriptstyle{\pi_{1}}$

implies that we have the following Kan equivalence

$W_{2n-1}\underset{W_{1}}{\times}W_{3}\to
W_{2n-1}\underset{W_{1}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}\cong
W_{2n-1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}.$

This means in the following commutative triangle

${W_{2n+1}}$${W_{2n-1}\underset{W_{1}}{\times}W_{3}}$${W_{2n-1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

the right hand map is a weak equivalence. Moreover, the left hand diagonal map
is an equivalence by the Segal condition and so the desired result follows by
$2$-out-of-$3$. ∎

######  Remark 3.3.

The result also follows as a special case of [BOO+20, Theorem 2.9] as every
Segal space is in particular a $2$-Segal space.

###  3.3. The Homotopy Category of Twisted Arrow Segal Spaces

Before we proceed to the completeness condition, we will separately analyze
the homotopy category of a twisted arrow Segal space, in the sense of [Rez01,
Section 5]. Concretely, for a given Segal space $W$, we want to construct an
equivalence $F_{W}:\mathrm{Ho}\mathrm{Tw}(W)\to\mathrm{Tw}\mathrm{Ho}(W)$.

We will start with the construction of $F_{W}$. Unwinding definitions, the set
of objects of $\mathrm{Ho}\mathrm{Tw}(W)$ is given by $W_{10}$, whereas the
set of objects of $\mathrm{Tw}\mathrm{Ho}(W)$ is given by $W_{10}/\sim$, where
$\sim$ is the homotopy relation. Hence, $F_{W}$ on objects maps a morphism to
its homotopy class.

Similarly, a morphism in $\mathrm{Ho}\mathrm{Tw}(W)$ is given by a homotopy
class $[\sigma]$, where $\sigma\in W_{30}$, whereas a morphism in
$\mathrm{Tw}\mathrm{Ho}(W)$ is given by a pair of homotopy classes of
morphisms $([g],[h])$, where $g,h\in W_{10}$. So, we define $F_{W}$ on
morphisms as $F_{W}([\sigma])=([d_{2}d_{2}\sigma],[d_{0}d_{0}\sigma])$.

We now verify that $F_{W}$ indeed preserves identities and composition. For a
given object $f:x\to y$ in $\mathrm{Ho}\mathrm{Tw}(W)$ we have

$\displaystyle F([s_{1}s_{1}f])$
$\displaystyle=([d_{2}d_{2}s_{1}s_{1}f],[d_{0}d_{0}s_{1}s_{1}f]\coloneqq([id_{x}],[id_{y}]).$

and for two given composable morphisms $[\sigma],[\sigma^{\prime}]$ in
$\mathrm{Ho}\mathrm{Tw}(W)$ and choice of composition $\sigma^{\prime\prime}$
of $\sigma,\sigma^{\prime}$ we have

$\displaystyle F([\sigma^{\prime}])\circ F([\sigma])$
$\displaystyle=([d_{2}d_{2}\sigma^{\prime}],[d_{0}d_{0}\sigma^{\prime}])\circ([d_{2}d_{2}\sigma],[d_{0}d_{0}\sigma])$
$\displaystyle=([d_{2}d_{2}\sigma^{\prime\prime}],[d_{0}d_{0}\sigma^{\prime\prime}])$
$\displaystyle=F([\sigma^{\prime\prime}]).$

Having defined $F_{W}$, we now prove it is an equivalence. Notice, by
construction, $F_{W}$ is surjective on objects, hence we will now proceed to
proving that it is also fully faithful.

######  Lemma 3.4.

The functor $F_{W}:\mathrm{Ho}\mathrm{Tw}(W)\to\mathrm{Tw}\mathrm{Ho}(W)$ is
fully faithful.

###### Proof.

Throughout this proof fix two objects $f,g$ in $\mathrm{Ho}\mathrm{Tw}(W)$.
First we show that $F_{W}$ is full. Let $([k],[h])$ be a morphism from
$F_{W}(f)$ to $F_{W}(g)$ and fix representatives $k,h\in W_{10}$. By the Segal
condition, the triple $(k,f,h)$ in
$W_{1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}$ gives us an
element $\sigma\in W_{30}$, such that $d_{0}d_{0}\sigma=k$ and
$d_{2}d_{2}\sigma=h$, meaning that $F_{W}([\sigma])=([k],[h])$, proving that
$F_{W}$ is full.

We now show $F_{W}$ is faithful. Let $[\sigma],[\sigma^{\prime}]$ be two
morphisms from $f$ to $g$, such that
$F_{W}([\sigma])=F_{W}([\sigma^{\prime}])$. We need to show that
$[\sigma]=[\sigma^{\prime}]$, which by definition means constructing a path in
the space $*\underset{W_{1}\times W_{1}}{\times}W_{3}$ from $\sigma$ to
$\sigma^{\prime}$.

Before we can proceed we need to better understand the data of such a path. By
definition a path is a map $\Delta[1]\to*\underset{W_{1}\times
W_{1}}{\times}W_{3}$, which unwinds to a map

$H=F(3)\times\Delta[1]\coprod_{F(1)\times\Delta[1]\coprod
F(1)\times\Delta[1]}(F(1)\coprod F(1))\to W.$

As the Reedy model structure is left proper, the projection map
$F(3)\times\Delta[1]\to H$ is a Reedy equivalence. Moreover, by definition the
inclusion

$H^{\prime}=(G(3)\times\Delta[1])\coprod_{G(3)\coprod G(3)}(F(3)\coprod
F(3))\to F(3)\times\Delta[1]$

is an equivalence in the Segal space model structure and so the composition
$H^{\prime}\to H$ is one as well.

We now use this equivalence to construct a path from $\sigma$ to
$\sigma^{\prime}$. To simplify notation we will denote
$d_{0}d_{0}\sigma=k,d_{2}d_{2}\sigma=h,d_{0}d_{0}\sigma^{\prime}=k^{\prime},d_{2}d_{2}\sigma^{\prime}=h^{\prime}$.
The assumption implies that $[k]=[k^{\prime}]$ and $[h]=[h^{\prime}]$, meaning
there are paths $\gamma_{k},\gamma_{h}:\Delta[1]\to W_{1}$ going from $k$ to
$k^{\prime}$ and $h$ to $h^{\prime}$.

Now, the data of $(\gamma_{k},f,\gamma_{h},\sigma,\sigma^{\prime})$, where $f$
denotes the constant path, precisely assemble into a morphism $H^{\prime}\to
W$, which lifts to a morphism $H\to W$, as $W$ is a Segal space. This is
precisely the desired path from $\sigma$ to $\sigma^{\prime}$ and we are done.
∎

######  Proposition 3.5.

If $W$ is a Segal space, then the functor
$F_{W}:\mathrm{Tw}\mathrm{Ho}(W)\to\mathrm{Ho}\mathrm{Tw}(W)$ is an
equivalence.

###### Proof.

The functor $F_{W}$ is surjective by construction and fully faithful by Lemma
3.4, hence we are done. ∎

###  3.4. Completeness Condition

In this subsection we show that $\mathrm{Tw}$ also preserves the completeness
condition. Recall, from [Rez01], that a Segal space $W$ is complete if the
morphism $W_{0}\to W_{hoequiv}$ is a Kan equivalence. Here $W_{hoequiv}$ is
the collection of path-components in $W_{1}$ consisting of equivalences.
Further details can be found in [Rez01].

In order to proceed we need a better understanding of
$\mathrm{Tw}(W)_{hoequiv}$ relying on our understanding of the homotopy
category from the previous subsection.

######  Lemma 3.6.

If $W$ is a Segal space, then a morphism $\sigma$

${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{f}$$\scriptstyle{k}$$\scriptstyle{g}$$\scriptstyle{h}$

in the twisted arrow Segal space $\mathrm{Tw}W$ is a homotopy equivalence if
$k$ and $h$ are homotopy equivalences in $W$.

###### Proof.

We need to show that the class $[\sigma]$ in $\mathrm{Ho}\mathrm{Tw}(W)$ is an
isomorphism if and only if $[k],[h]$ are isomorphisms in $\mathrm{Ho}(W)$. By
Proposition 3.5, $[\sigma]$ is an isomorphism if and only if its image in
$\mathrm{Tw}\mathrm{Ho}(W)$ is an isomorphism. By definition of the twisted
arrow category, this is equivalent to $[k]$ and $[h]$ being isomorphisms, and
hence we are done. ∎

######  Proposition 3.7.

Let $W$ be a Segal space. Then we have the following pullback square

${\mathrm{Tw}(W)_{hoequiv}}$${\mathrm{Tw}(W)_{1}}$${W_{hoequiv}^{op}\times
W_{hoequiv}}$${W_{1}^{op}\times W_{1}}$${\lrcorner}$

.

###### Proof.

We denote the pullback by $Pb$ and show that $Pb=\mathrm{Tw}(W)_{hoequiv}$.
Notice that both $Pb\hookrightarrow\mathrm{Tw}(W)_{1}$ and
$\mathrm{Tw}(W)_{hoequiv}\hookrightarrow\mathrm{Tw}(W)_{1}$ are inclusions of
path-components, as they are stable under pullback. Hence, it suffices to show
that $\mathrm{Tw}(W)_{hoequiv}$ and $Pb$ include the same path components of
$\mathrm{Tw}(W)_{1}$.

If $f\in(Pb)_{0}$, then by Lemma 3.6, $f\in(\mathrm{Tw}(W)_{hoequiv})_{0}$.
Conversely, if $f\in(\mathrm{Tw}(W)_{hoequiv})_{0}$, then by universal
property of pullback, $f\in(Pb)_{0}$,

${\mathrm{Tw}(W)_{hoequiv}}$${Pb}$${\mathrm{Tw}(W)_{1}}$${W_{hoequiv}^{op}\times
W_{hoequiv}}$${W_{1}^{op}\times W_{1}}$${\lrcorner}$

Thus, $(Pb)_{0}=(\mathrm{Tw}(W)_{hoequiv})_{0}$, which further implies,
$Pb=\mathrm{Tw}(W)_{hoequiv}$. ∎

We are finally ready to prove the main result of this section.

######  Theorem 3.8.

If $W$ is a complete Segal space, then $\mathrm{Tw}W$ is a complete Segal
space.

###### Proof.

By Proposition 3.2 we only need to check the completeness condition. By
Proposition 3.7 and the pasting property of pullbacks, we have the following
diagram, where the left hand square is a pullback

${\mathrm{Tw}(W)_{0}}$${\mathrm{Tw}(W)_{hoequiv}}$${\mathrm{Tw}(W)_{1}}$${W_{0}^{op}\times
W_{0}}$${W_{hoequiv}^{op}\times W_{hoequiv}}$${W_{1}^{op}\times
W_{1}}$${\lrcorner}$${\lrcorner}$$\scriptstyle{\simeq}$

.

Since $W$ is a complete Segal space, $W_{0}^{op}\times
W_{0}\xrightarrow{\simeq}W_{hoequiv}^{op}\times W_{hoequiv}$ is an equivalence
of spaces, which implies that
$\mathrm{Tw}(W)_{0}\xrightarrow{\simeq}\mathrm{Tw}(W)_{hoequiv}$ is an
equivalence of spaces. ∎

##  4\. The Twisted Arrow Construction as a Left Fibration

In this final section we demonstrate that if $W$ is a Segal space, then the
twisted arrow projection $\mathrm{Tw}W\to W^{op}\times W$ is a left fibration.
First we need some technical lemmas.

######  Lemma 4.1.

If $W$ is a Segal space, then $\mathrm{Tw}W\to W^{op}\times W$ is a Reedy
fibration.

###### Proof.

We need to show that the map

$\mathrm{Map}(F(k),\mathrm{Tw}W)\to(W_{k}^{op}\times
W_{k})\underset{\mathrm{Map}(\partial F(k),W^{op}\times
W)}{\times}\mathrm{Map}(\partial F(k),\mathrm{Tw}W)$

is a Kan fibration.

By construction the left hand side is given by $\mathrm{Map}(F(2k+1),W)$.
Moreover, by the observation in the beginning of Subsection 3.1, we have
$\mathrm{Map}(\partial
F(k),\mathrm{Tw}W)\cong\mathrm{Map}(\partial_{\mathrm{Tw}}F(2k+1),W)$ and so
the right hand side simplifies to

$\mathrm{Map}(\partial_{\mathrm{Tw}}F(2k+1)\coprod_{\partial
F(k)\coprod\partial F(k)}F(k)\coprod F(k),W).$

As $W$ is Reedy fibrant, it hence suffices to show that

$\partial_{\mathrm{Tw}}F(2k+1)\coprod_{\partial F(k)\coprod\partial
F(k)}F(k)\coprod F(k)\to F(2k+1)$

is a cofibration or, in other words, a level-wise injection of sets. However,
this directly follows from the same argument given in Proposition 3.1 and so
we are done. ∎

######  Lemma 4.2.

If $W$ is a Segal space, then the following diagram is a homotopy pullback
square,

${\mathrm{Tw}(W)_{1}}$${\mathrm{Tw}(W)_{0}}$${W_{1}^{op}\times
W_{1}}$${W_{0}^{op}\times W_{0}}$${\lrcorner}$

.

###### Proof.

By definition $\mathrm{Tw}(W)_{0}=W_{1}$, and so the strict pullback is given
by $\mathrm{Tw}(W)_{1}\simeq
W_{1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}$. The desired
equivalence $\mathrm{Tw}(W)_{1}=W_{3}\simeq
W_{1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}$ now follows
from the Segal condition. ∎

######  Theorem 4.3.

If $W$ is a Segal space, then $\mathrm{Tw}W\to W^{op}\times W$ is a left
fibration.

###### Proof.

By Lemma 4.1, the map is a Reedy fibration and so it suffices to check the
appropriate homotopy pullbacks. By [Ras17, Lemma $3.29$], as $W$ is a Segal
space, it is sufficient to check the homotopy pullback condition for $n=1$,
which we have done in Lemma 4.2. ∎

######  Remark 4.4.

Following this result, [Ras17, Lemma 3.38] gives us an alternative proof that
the twisted arrow construction preserves complete Segal spaces. In fact this
is the approach used in [Mar21, Proposition 4.2.5], via other methods, to
obtain a similar result.

######  Remark 4.5.

The previous result implies that for a given object $x$ in a Segal space $W$,
the fiber of $\mathrm{Tw}W\to W^{op}\times W$ via the inclusion $\\{x\\}\times
W\hookrightarrow W^{op}\times W$ is a left fibration $\mathrm{Tw}(W)_{x}\to
W$, such that
$(\mathrm{Tw}(W)_{x})_{0}=W_{1}\underset{W_{0}}{\times}\Delta[0]$.

This means $\mathrm{Tw}(W)_{x}\to W$ simply recovers the under-Segal space
left fibration $W_{x/}\to W$ as described in [Ras17, Definition 3.41] and so
Theorem 4.3 is a direct generalization of [Ras17, Theorem 3.44].

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# Periodic Activities of Fast Radio Burst Repeaters from Precessing Magnetars
with Evolving Obliquity

Xin-Ming Feng1, Yuan-Pei Yang1,2 and Qiao-Chu Li3,4
1South-Western Institute for Astronomy Research, Yunnan University, Kunming
650504, China
2Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023,
China
3School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China
4Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210023, China
E-mail<EMAIL_ADDRESS>(YPY)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Fast radio bursts (FRBs) are cosmological radio transients with millisecond
durations and extremely high brightness temperatures. One FRB repeater, FRB
180916.J0158+65 (FRB 180916B), was confirmed to appear 16.35-day periodic
activities with 5-day activity window. Another FRB repeater, FRB 121102, and
two soft gamma-ray repeaters (SGRs), SGR 1935+2154 and SGR 1806-20, also show
possible periodic activities. These periodicities might originate from the
precession process of young magnetars due to the anisotropic pressure from the
inner magnetic fields as proposed in the literature. In this work, we analyze
a self-consistent model for the rotation evolution of magnetars and obtain the
evolutions of magnetar precession and obliquity. We find that if the FRB
repeaters and the SGRs with (possible) periodic activities originate from the
magnetar precession, their ages would be constrained to be hundreds to tens of
thousands of years, which is consistent with the typical ages of magnetars.
Assuming that the FRB emission is beaming in the magnetosphere as proposed in
the literature, we calculate the evolution of the observable probability and
the duty cycle of the active window period. We find that for a given magnetar
the observable probability increases with the magnetar age in the early stage
and decreases with the magnetar age in the later stage, meanwhile, there are
one or two active windows in one precession period if the emission is not
perfectly axisymmetric with respect to the deformation axis of a magnetar,
which could be tested by the future observation for repeating FRB sources.

###### keywords:

(transients:) fast radio bursts – stars: evolution – stars: magnetic field –
stars: neutron – stars: rotation.

††pubyear: 2024††pagerange: Periodic Activities of Fast Radio Burst Repeaters
from Precessing Magnetars with Evolving Obliquity–References

##  1 Introduction

Fast radio bursts (FRBs) are radio transients with millisecond durations and
extremely high brightness temperatures. Since the “Lorimer Burst” (FRB 010724)
was discovered in 2007 (Lorimer et al., 2007), hundreds of FRBs have been
published so far, and dozens of them appear repeating behaviors (e.g.,
Thornton et al., 2013; Spitler et al., 2016; Chatterjee et al., 2017;
Bannister et al., 2019; Ravi et al., 2019; Prochaska et al., 2019; Marcote et
al., 2020). FRBs are found to be almost isotropic distributed in all sky, and
the dispersion measures (DMs) of most FRBs exceed the DM contributions from
the Milky Way, suggesting a cosmological origin (e.g., Thornton et al., 2013).
Together with the short durations, their brightness temperatures are required
to be extremely high, $T_{\rm B}\gtrsim 10^{35}{\rm K}$. Therefore, the
radiation mechanism of FRBs must be coherent (e.g., Zhang, 2020). Possible
coherent radiation mechanisms applying for FRBs include curvature radiation by
charged bunches (e.g., Kumar et al., 2017; Yang & Zhang, 2018, 2023; Lu et
al., 2020; Yang et al., 2020; Cooper & Wijers, 2021), synchrotron maser (e.g.,
Waxman, 2017; Metzger et al., 2019; Beloborodov, 2020; Lyubarsky, 2020), and
plasma radiation (e.g., Philippov et al., 2020; Yang & Zhang, 2021; Mahlmann
et al., 2022). On the other hand, the physical origin of FRBs is still not
well understood. Based on the distance of the emission region from the FRB
engine, e.g., a neutron star, there are two broad-classes models to explain
their physical origin: 1. FRBs are generated in the magnetosphere of a neutron
star (Kumar et al., 2017; Katz, 2018; Lu & Kumar, 2018; Yang & Zhang, 2018,
2021; Wadiasingh & Timokhin, 2019; Kumar & Bošnjak, 2020; Lu et al., 2020;
Wang, 2020; Wang et al., 2020; Lyutikov & Popov, 2020; Wadiasingh et al.,
2020; Zhang, 2017; Dai, 2020; Geng et al., 2020; Ioka, 2020). 2. FRBs are
produced by the synchrotron maser mechanism in the shock of an outflow far
away from the neutron star (Lyubarsky, 2014; Waxman, 2017; Metzger et al.,
2019; Beloborodov, 2020; Marcote et al., 2020; Yu et al., 2021; Wu et al.,
2020).

The recent CHIME/FRB catalog represented the first large FRB sample, including
more than ten FRB repeaters and a few hundreds of one-off FRBs (CHIME/FRB
Collaboration et al., 2021). It is interesting that the bursts from FRB
repeaters have relatively larger pulse duration, narrower bandwidth, and lower
brightness temperature compared with the one-off FRBs (Pleunis et al., 2021;
Luo et al., 2023; Zhu-Ge et al., 2023). Some FRB repeaters appear (possible)
periodic activities: 1) FRB 180916.J0158+65 (FRB 180916B) shows a 16.35-day
periodic activity with a 5-day activity window (Chime/Frb Collaboration et
al., 2020; Pastor-Marazuela et al., 2021; Bethapudi et al., 2023); 2) FRB
121102 shows a possibly longer period of 161 day with an active window of
about 100 days (Rajwade et al., 2020; Cruces et al., 2021). Besides, some
Galactic magnetars also appear possible periodic activities in SGRs. For
example, SGR 1806-20 and SGR 1935+2154 were found to have possible periodic
activities of 398.2 days (Zhang et al., 2021) and 127 day111In one earlier
work (Zou et al., 2021), a possible 238-day activity period of SGR 1935+2154
was reported using the data of Fermi/GBM from 2014 July to 2021 October. (Xie
et al., 2022), respectively.

Some astrophysical scenarios are involved in explaining the periodic
activities of FRB repeaters (see the review of Zhang (2020), and some detailed
comments on different scenarios are also discussed in Katz (2021) and Wei et
al. (2022)): 1) an FRB repeater is in a binary star systems, and the activity
period corresponds to the orbital period of the binary system (Dai & Zhong,
2020; Ioka & Zhang, 2020; Li et al., 2021). Such a scenario predicts that some
observation properties, e.g., dispersion measure and rotation measure, of the
FRB repeater might be periodic if the turbulence in the environment is not
significant (Wang et al., 2022; Yang et al., 2023; Zhao et al., 2023). 2) the
periodic activity of an FRB repeater is due to the precession of an isolated
deformed neutron star, meanwhile, the FRB emission is beaming in the
magnetosphere of the neutron star (Levin et al., 2020; Zanazzi & Lai, 2020;
Sridhar et al., 2021; Li & Zanazzi, 2021; Cordes et al., 2022). 3) the
periodic activity is produced by the forced precession contributed by orbital
motion (Yang & Zou, 2020) or fallback disk (Tong et al., 2020). 4) periodicity
corresponds to the slow rotation of an isolated neutron star, like a radio
pulsar (Beniamini et al., 2020; Xu et al., 2021). 5) the periodic activities
originate from the thermal-viscous instability of the accretion disk around
the neutron star in a binary system (Geng et al., 2021). 6) the periodicity
originates from the precession of the jet or accretion disc of a compact
object (neutron star or black hole) (Sridhar & Metzger, 2022; Katz, 2022). It
is worth noting that although FRB 180916B exhibits a 16.35-day periodicity in
its burst activity, its RM appears unrelated to the periodic activity and
evolves during a long term much larger than the activity period (Mckinven et
al., 2023). Thus, the activity period seems more likely to originate from the
intrinsic process of the FRB engine, e.g., the precession or rotation of a
compact object.

In this work, we consider that the observed periodic activities of FRB
repeaters and SGRs are due to the magnetar precession, and constrain the
magnetars’ ages under the picture of the self-consistent dynamic evolution of
the magnetar precession. We also discuss the observable probability and the
activity window due to the obliquity evolution. The paper is organized as
follows. We calculate the evolution of the magnetar rotation and precession in
Section 2, and constrain the ages of some special sources, including FRB
repeaters and SGRs in Section 3. We discuss the observable probability and the
activity window in Section 4. The results are discussed and summarized in
Section 5.

##  2 Rotation and precession evolution of magnetars

In this section, we analyze the self-consistent evolutions of the rotation and
precession of magnetars. We define the magnetar radius as $R$, the angular
velocity as $\Omega$, the moment of inertia as $I$, the surface magnetic field
as $B$, and the obliquity angle between the spin axis and magnetic axis as
$\theta$. Since a young magnetar is deformed by the internal magnetic field,
it appears free precession when its spin axis is not parallel with the
deformation symmetry axis, as shown in Fig. 1.

Figure 1: Schematic configuration of the free precession of a magnetar in the
inertial frame. $\hat{p}$ indicates the direction of the magnetic axis.
$\hat{\Omega}$ corresponds to the direction of rotation angular velocity.
$\hat{Z}$ denotes the direction of the angular momentum. $\hat{m}_{f}$
indicates the direction of the radiation beaming center, and it moves around
$\hat{p}$ in the long term due to the precession process. $\hat{n}$ is the
line of sight. We further define $\alpha$ as the angle between $\hat{p}$ and
$\hat{m}_{f}$, $\chi$ as the half-opening angle of the radiation beam,
$\theta$ as the obliquity, $\varphi$ as the angle between $\hat{p}$ and
$\hat{Z}$, and $\zeta$ as the angle between $\hat{n}$ and $\hat{p}$.

The anisotropic pressure from the internal magnetic field could deform the
magnetar into an oblate or prolate shape, which is usually described by the
ellipticity parameter $\epsilon=(I_{zz}-I_{xx})/I_{xx}$, where $I_{xx}$ and
$I_{zz}$ are the moments of inertia of the magnetar about $x$\- and $z$-axes
in body frame, respectively, and $I_{yy}=I_{xx}\neq I_{zz}$ is assumed. For
the deformation caused by the anisotropic internal magnetic pressure, the
value of ellipticity parameter $\epsilon$ is given by (Haskell et al., 2008;
Mastrano et al., 2011)

$\epsilon=2.5\times 10^{-7}\beta\left(\frac{B}{10^{14}~{}{\rm G}}\right)^{2},$
(1)

where $\beta=(1-0.385/\Lambda)$, and $\Lambda$ is the ratio between poloidal
and total magnetic energy inside the neutron star. Usually one requires
$0\leqslant\Lambda\leqslant 1$, where $\Lambda=0$ and $\Lambda=1$ represent a
star with a purely toroidal and a purely poloidal field, respectively. In this
paper, we consider a magnetar with a mass of $M=1.4~{}{\rm M_{\odot}}$ and a
radius of $R=10^{6}~{}{\rm cm}$. If the toroidal magnetic field dominates the
internal magnetic field, the neutron star would be prolate; if the polar field
dominates, the neutron star would be oblate. Generally, toroidal field
dominates the internal field, leading to the magnetar being prolate, so
$\beta<0$ ($\epsilon<0$) (Haskell et al., 2008; Mastrano et al., 2011; Lander
& Jones, 2020). In reality, the internal magnetic field would be more complex,
it usually yields $|\beta|\ll 1$ in Eq.(1) (see Zanazzi & Lai, 2020). In the
following discussion, we will discuss the different scenarios with $|\beta|$
are taken as 0.5 and 0.1 respectively. One should notice that Eq. (1) might be
invalid once the neutron star has cooled enough for its interior to be
superconducting.

Generally, the magnetar rotation is significantly affected by the magnetic
dipole radiation with the torque of $J_{\rm E}=-T_{\rm
E}B^{2}\Omega^{3}(1+\sin^{2}\theta)$, where $T_{\rm E}=R^{6}/(4c^{3})$, and
the term of $(1+\sin^{2}\theta)$ is corrected by involving the force-free wind
plasma for the charge-filled magnetosphere, leading to the torque non-zero
even when the spin axis and the magnetic axis are aligned (Contopoulos et al.,
1999; Gruzinov, 2005; Timokhin, 2006; Spitkovsky, 2006). In addition to the
magnetic dipole radiation, due to the deformation of the magnetar, the
magnetar can generate gravitational wave (GW) when the deformation axis is not
parallel with the spin axis, and the torque of GW radiation is given by
$J_{\rm G}=-T_{\rm G}\epsilon^{2}\Omega^{5}\sin^{2}\theta(1+15\sin^{2}\theta)$
(Chau & Henriksen, 1970; Melatos, 2000), where $T_{\rm G}=2GI^{2}/(5c^{5})$.
Generally, the torque of GW radiation $J_{\rm G}$ is too small compared with
the torque of the magnetic dipole radiation $J_{\rm E}$, thus, $J_{\rm G}$ is
ignored in the following calculation. The evolution equations of the angular
velocity and obliquity is given by (Mondal, 2021; Chau & Henriksen, 1970;
Melatos, 2000; Kalita et al., 2020)

$\displaystyle I\dfrac{d\Omega}{dt}$ $\displaystyle=-T_{\rm
E}B^{2}\Omega^{3}(1+\sin^{2}\theta),$ $\displaystyle
I\Omega\dfrac{d\theta}{dt}$ $\displaystyle=-T_{\rm
E}B^{2}\Omega^{3}\sin\theta\cos\theta,$ (2)

where the moment of inertia is taken as $I=10^{45}~{}{\rm g~{}cm^{2}}$. Notice
that here we also ignore the contributions from the radiation and ejecta by
magnetar activity (e.g., bursts, flares, etc.), because they might not
significantly change the angular momentum of the magnetar during the long-term
evolution due to the possible random beaming or isotropic emission/ejection.

The magnetic field evolution of a magnetar has been generally considered to be
dominated by the Ohmic dissipation and Hall drift (Goldreich & Reisenegger,
1992), and the evolution equation could be approximately written as (e.g.,
Colpi et al., 2000; Aguilera et al., 2008; Glampedakis et al., 2011)

$\frac{dB}{dt}\simeq-\frac{B}{\tau_{\rm Ohm}}-\frac{1}{B_{\rm
i}}\frac{B^{2}}{\tau_{\rm Hall}},$ (3)

where $B_{\rm i}$ is the initial magnetic field strength, and $\tau_{\rm Ohm}$
and $\tau_{\rm Hall}$ are the typical timescales of Ohmic dissipation and Hall
drift decay, respectively. The typical timescales of Ohmic dissipation and
Hall drift decay are taken as $\tau_{\rm Ohm}=10^{6}~{}{\rm yr}$ and
$\tau_{\rm Hall}=2\times 10^{3}~{}{\rm yr}~{}(B_{\rm i}/10^{15}~{}{\rm
G})^{-1}$, respectively in the following calculations (e.g., Aguilera et al.,
2008; Mondal, 2021).

Next, we discuss the free precession of a magnetar. Since the deformation is
caused by the internal magnetic field, the deformation axis is approximately
the magnetic axis. If the spin axis is not parallel to the deformation axis,
the neutron star will precess around $\hat{p}$ axis, as shown in Fig. 1, and
the precession period is given by (e.g., Levin et al., 2020; Zanazzi & Lai,
2020)

$P_{\rm prec}=\frac{P}{|\epsilon\cos\theta|},$ (4)

where $P=2\pi/\Omega$ is the magnetar rotation period. Once the initial
conditions are given, the evolutions of obliquity $\theta$, surface magnetic
field $B$, rotation period $P$, and precession period $P_{\rm prec}$ could be
solved using the differential equations of Eq.(2), Eq.(3) and Eq.(4).

In order to obtain the evolutions of rotation and precession, one needs know
the initial obliquity. Lander & Jones (2020) found that the obliquity $\theta$
tends to $\pi/2$ shortly after birth due to the viscous damping, then
decreases towards zero over hundreds of years with the aligning effect of the
exterior torque (magnetic dipole radiation). The early evolution of $\theta$
is mainly affected by the internal viscous damping and the magnetic dipole
radiation,

$\frac{d\theta}{dt}=\frac{\dot{E}_{\rm
visc}}{I\epsilon\sin\theta\cos\theta\Omega^{2}}+\frac{\dot{E}_{\rm
EM}^{(\theta)}}{I\Omega^{2}},$ (5)

where $\dot{E}_{\rm
EM}^{(\theta)}\simeq-(R^{6}/4c^{3})\Omega^{4}B^{2}(1+\sin^{2}\theta)\sin\theta\cos\theta$
is the effective electromagnetic energy-loss rate considering that
$d\theta/dt$ should vanish for $\theta=0,\pi/2$ (Mestel, 1968; Lander & Jones,
2020), $\dot{E}_{\rm visc}$ is the viscous energy-loss rate dominated by shear
and bulk viscosities. Notice that $\dot{E}_{\rm visc}<0$ and $\dot{E}_{\rm
EM}^{(\theta)}<0$, so that internal damping gives $d\theta/dt<0$ for the
oblate deformation ($\epsilon>0$) and $d\theta/dt>0$ for the prolate
deformation ($\epsilon<0$). The typical timescale for the electromagnetic term
is

$\displaystyle\tau_{\rm EM}\sim\frac{4Ic^{3}}{R^{6}\Omega^{2}B^{2}}\simeq
2.7\times 10^{3}~{}{\rm s}\left(\frac{B}{10^{15}~{}{\rm
G}}\right)^{-2}\left(\frac{P}{1~{}{\rm ms}}\right)^{2}.$ (6)

The viscous energy-loss rate is mainly dominated by bulk viscosities, and the
typical timescale is estimated by (Lander & Jones, 2018)

$\displaystyle\tau_{\rm visc}=8~{}{\rm s}\left(\frac{T}{10^{10}~{}{\rm
K}}\right)^{6}\left(\frac{P}{1~{}{\rm
ms}}\right)^{4}\left(\frac{B}{10^{15}~{}{\rm G}}\right)^{-2}$
$\displaystyle\times\left[1+0.02\left(\frac{T}{10^{10}~{}{\rm
K}}\right)^{-12}\left(\frac{P}{1~{}{\rm
ms}}\right)^{-2}\left(\frac{B}{10^{15}~{}{\rm G}}\right)^{4}\right].$ (7)

We set the corresponding time-scales equal, $\tau_{\rm EM}\sim\tau_{\rm
visc}$, and obtained the critical rotation period is about $P_{\rm cr}\sim
18~{}{\rm ms}$ for the about typical parameters. Thus, the critical timescale
is $\tau_{\rm cr}\sim 10$ days. For a neutron star with age $\lesssim\tau_{\rm
cr}$, the evolution of the obliquity $\theta$ is dominated by bulk
viscosities. In particular, for the prolate deformations with $\epsilon<0$,
one has $d\theta/dt>0$, leading to the obliquity $\theta$ tending to $\pi/2$
shortly after birth. Since the critical timescale $\tau_{\rm cr}$ is small
compared with the whole age of a neutron star, in the following calculation,
the initial value of the obliquity could be set close to $\pi/2$. In the
following numerical calculations, we take it as $\pi/2-\delta$ with
$\delta\sim 0.1$. We take the initial period $P_{\rm i}$ from $5~{}{\rm ms}$
to $80~{}{\rm ms}$ and take the initial magnetic field $B_{\rm i}$ from
$10^{14}{\rm G}$ to $10^{16}{\rm G}$. For the parameter $\beta$ in the
ellipticity equation Eq.(1), we take $|\beta|=0.5$ and $|\beta|=0.1$.

Fig. 2 shows the evolutions of the obliquity $\theta$, the surface magnetic
field $B$, the spin period $P$ and the spin periodic derivative $\dot{P}$ with
different initial surface magnetic fields of $B_{\rm
i}=10^{14},10^{15},10^{16}{\rm G}$. The initial period is taken as $P_{\rm
i}=10~{}{\rm ms}$, and the deformation parameter is taken as $|\beta|=0.1$.
The obliquity decreases over a long term, and the stronger the initial surface
magnetic field, the faster the obliquity evolution. The spin period increases
during the long term and trends to a constant value at the later stage. One
should notice that the evolution of the magnetic field significantly affects
the spin evolution for magnetars. As shown in Eq.(3), the magnetic field of a
magnetar would significantly decay after $\sim 10^{3}~{}{\rm yr}~{}(B_{\rm
i}/10^{15}~{}{\rm G})^{-1}$ due to the Hall drift. This is the reason why the
spin period tends to a constant value at late times. For the above typical
parameters, the surface magnetic field decreases significantly after
$\gtrsim(100-10^{4})~{}{\rm yr}$. Similarly, in Fig. 3, we show the evolutions
of $\theta$, $B$, $P$ and $\dot{P}$ with different initial spin periods of
$P_{\rm i}=5,10,20~{}{\rm ms}$. The initial surface magnetic field is $B_{\rm
i}=10^{15}{\rm G}$, and the deformation parameter is $|\beta|=0.1$. One can
see that the faster the initial spin, the faster the obliquity evolution,
meanwhile, the final obliquity depends on the initial spin period $P_{\rm i}$.
The evolution of the surface magnetic field and the late evolution of the spin
period are almost independent of $P_{\rm i}$.

Figure 2: The evolutions of obliquity $\theta$, surface magnetic field $B$,
rotation period $P$ and period derivative $\dot{P}$. The initial parameters
are set to $B_{\rm i}=10^{14},10^{15},10^{16}~{}{\rm G}$, $P_{\rm i}=10~{}{\rm
ms}$ and $|\beta|=0.1$.

Figure 3: Same as Fig. 2. The initial parameters are set to $P_{\rm
i}=5,10,20~{}{\rm ms}$, $B_{\rm i}=10^{15}~{}{\rm G}$ and $|\beta|=0.1$.

Once the evolutions of $P$, $\theta$, and $B$ are obtained, one can calculate
the evolution of the free precession period $P_{\rm prec}$ based on Eq.(4). In
Fig. 4, the left panel and right panel correspond to $P_{\rm prec}$ evolution
for different initial surface magnetic fields and different initial periods,
respectively. In the left panel, $P_{\rm i}=5~{}{\rm ms}$ is taken, and in the
right panel, $B_{\rm i}=10^{15}~{}{\rm G}$ is taken. The solid and dashed
lines corresponds to $|\beta|=0.1$ and $|\beta|=0.5$, respectively. We find
that the evolution of the precession period satisfies a broken power law in
the long term. The evolution of the precession period increases slowly in the
early stage and increases fast in the later stage. We define the broken
timescale as the critical value between these two stages, and the broken
timescale mainly depends on $B_{\rm i}$ and is almost independent of $P_{\rm
i}$ and $|\beta|$ as shown in Fig. 4. For the above typical parameters, the
evolution of $P_{\rm prec}$ increase faster after $\sim 1000~{}{\rm yr}$ for
$B_{\rm i}=10^{15}~{}{\rm G}$. The larger the initial surface magnetic field
or the faster the initial spin, the smaller the precession period. Besides, a
larger deformation parameter $|\beta|$ would also cause a smaller precession
period.

Figure 4: The evolution of the precession period $P_{\rm prec}$ for different
initial magnetic fields (left panel) and different initial periods (right
panel) with $|\beta|=0.1$ (solid line) and $|\beta|=0.5$ (dashed line). In the
left panel, $P_{\rm i}=5~{}{\rm ms}$, and in the right panel, $B_{\rm
i}=10^{15}~{}{\rm G}$.

##  3 Constraint on the ages of magnetars with periodic activities

Based on the evolution of the precession period, we can constrain the ages of
magnetars that show periodic activities. Recently, some FRB repeaters and SGRs
were found to show (possible) periodic activities, such as FRB 180916B, FRB
121102, SGR 1806-20, and SGR 1935+2154 (Chime/Frb Collaboration et al., 2020;
Rajwade et al., 2020; Zhang et al., 2021; Zou et al., 2021; Xie et al., 2022).
Assuming that their periodicities are due to the free precession of a magnetar
(Levin et al., 2020; Zanazzi & Lai, 2020; Li & Zanazzi, 2021), one can
constrain the ages of these sources, as shown in Fig. 5 ($|\beta|=0.1$) and
Fig. 6 ($|\beta|=0.5$) in the parameter space of $(B_{\rm i},P_{\rm i})$.

Figure 5: Contour diagram of the magnetar ages for different sources with the
precession period of 16.35 days for FRB 180916B, 161 days for FRB 121102. We
take the initial period $P_{\rm i}$ with range $(20~{}{\rm ms},80~{}{\rm ms})$
and the initial magnetic field $B_{\rm i}$ with range $(10^{14}~{}{\rm
G},10^{16}~{}{\rm G})$, and $|\beta|=0.1$. In this figure, the white regions
represent no solution for the given precession periods.

Figure 6: Same as Fig. 5 but with $|\beta|=0.5$.

FRB 180916B was found to be localized in a spiral galaxy at redshift
$z=0.0338$ (Tendulkar et al., 2021). This source appears a 16.35-day periodic
activity, and its activity window is about 5 day (Chime/Frb Collaboration et
al., 2020), and the activity window was found to be frequency-dependent
(Pastor-Marazuela et al., 2021; Bethapudi et al., 2023). Taking $P_{\rm
prec}=16.35~{}{\rm day}$, the magnetar age could be constrained in the
parameter space $(B_{\rm i},P_{\rm i})$, as shown in the subpanels of Fig. 5
and Fig. 6, respectively. The observed $P_{\rm prec}=16.35~{}{\rm day}$
requites that the magnetar age ranges from $\sim 100~{}{\rm yr}$ to $\sim
2000~{}{\rm yr}$ for the initial period with $P_{\rm i}\sim(20~{}{\rm
ms},80~{}{\rm ms})$, initial surface magnetic field with $B_{\rm
i}\sim(10^{14}~{}{\rm G},10^{16}~{}{\rm G})$, and deformation parameter
$|\beta|\sim(0.1,0.5)$. Compared with the subpanels of Fig. 5 and Fig. 6, one
can find that the more significant the deformation (the larger $|\beta|$ or
$|\epsilon|$), the longer the constrained magnetar age. The reason is that for
a given observed precession period, the more significant the deformation (the
larger $|\beta|$ or $|\epsilon|$), the longer the magnetar age, as shown in
Fig. 4.

FRB 121102, the first known repeater, was localized in a star-forming dwarf
galaxy at $z=0.19273$ (Chatterjee et al., 2017; Tendulkar et al., 2017). This
source has a possible $\sim 161$-day periodic activity with 60% duty cycle
(Rajwade et al., 2020; Cruces et al., 2021). Taking $P_{\rm prec}=161~{}{\rm
day}$, the magnetar age could be constrained in the parameter space $(B_{\rm
i},P_{\rm i})$, as shown in the subpanels of Fig. 5 and Fig. 6, respectively.
The observed $P_{\rm prec}=161~{}{\rm day}$ requites that the magnetar age
ranges from $\sim 400~{}{\rm yr}$ to $\sim 7000~{}{\rm yr}$ for the initial
period with $P_{\rm i}\sim(20~{}{\rm ms},80~{}{\rm ms})$, initial surface
magnetic field with $B_{\rm i}\sim(10^{14}~{}{\rm G},10^{16}~{}{\rm G})$, and
deformation parameter $|\beta|\sim(0.1,0.5)$.

In addition to FRB repeaters, some SGRs also show possible periodic X-ray
activities. A Galactic magnetar, SGR 1935+2154, has been active for a few
years. It is very interesting that FRB 200428 was detected from SGR 1935+2154
associated with one X-ray burst, which implies that at least a part of FRBs
originate from magnetars. Recently, SGR 1935+2154 was found to exhibit a
possible period of $127$ day in X-ray activity (Xie et al., 2022). We consider
that the periodic activities are due to the free precession of magnetar.
Combining the observation of $P=3.245~{}{\rm s}$ and $\dot{P}=1.43\times
10^{-11}{\rm s~{}s^{-1}}$ (Olausen & Kaspi, 2014), we can obtain the age,
initial magnetic field, and $|\beta|$ of this magnetar for a given initial
period. For $P_{\rm i}=5~{}{\rm ms}$, one has $T_{\rm age}=2136.6~{}{\rm yr}$,
$B_{\rm i}=5.66\times 10^{14}~{}{\rm G}$, and $|\beta|=0.0964$. For $P_{\rm
i}=10~{}{\rm ms}$, one has $T_{\rm age}=2103~{}{\rm yr}$, $B_{\rm
i}=5.37\times 10^{14}~{}{\rm G}$, and $|\beta|=0.105$.

Another active Galactic magnetar, SGR 1806-20, also shows a possible
$398.2$-day periodic activity (Zhang et al., 2021). Combining the observation
of $P=7.55~{}{\rm s}$ and $\dot{P}=49.5\times 10^{-11}{\rm s~{}s^{-1}}$
(Olausen & Kaspi, 2014), we can obtain the age, initial magnetic field, and
$|\beta|$ of this magnetar. For $P_{\rm i}=5~{}{\rm ms}$, one has $T_{\rm
age}=172.6~{}{\rm yr}$, $B_{\rm i}=4.40\times 10^{15}$, and
$|\beta|=0.000865$. For $P_{\rm i}=10~{}{\rm ms}$, one has $T_{\rm
age}=169.9~{}{\rm yr}$, $B_{\rm i}=4.33\times 10^{15}~{}{\rm G}$, and
$|\beta|=0.000859$.

##  4 Observable probability and active window of FRBs

Due to the magnetar rotation, precession, and beaming radiation, one can only
observe the radiation at some certain directions of the line of sight, as
shown in Fig. 1. In particular, for FRB repeaters, we assume that radio bursts
are generated in the magnetosphere as proposed in some magnetosphere models
(e.g., Yang & Zhang, 2018, 2021; Kumar & Bošnjak, 2020; Lu et al., 2020). When
the magnetosphere is triggered by the crust fracturing of a magnetar, Alfvén
waves are produced and propagate along the field lines, and a charge
starvation region is formed in the magnetosphere due to the parallel electric
fields induced by the Alfvén waves. Coherent radio bursts are produced by
coherent curvature radiation by charged bunches (Yang & Zhang, 2018; Kumar &
Bošnjak, 2020; Lu et al., 2020) or coherent plasma radiation due to nonuniform
pair production across magnetic field lines (Philippov et al., 2020; Yang &
Zhang, 2021; Mahlmann et al., 2022). We notice that for the free precession of
a biaxial neutron star if the radiation region is perfectly axisymmetric with
respect to the deformation axis of the moment of inertia tensor, the slow
precession of the neutron star about this axis would not produce any
observable changes. Therefore, there has to be some non-axisymmetry with
respect to the deformation axis to produce observable changes. In the
following discussion, we assume that the beaming direction of the FRB emission
region deviates from the deformation axis, as shown in Fig 1. We assume that
the FRBs are emitted from a beaming region with an angle $\alpha$ from the
magnetic axis $\hat{p}$. The half-opening angle of the radiation beam is
$\chi$. During the precession process, the beaming center direction
$\hat{m}_{f}$ moves around $\hat{p}$ in the long term. Due to the rotation and
the precession, the radiation beam will sweep across a solid angle that
corresponds to a region where the radiation is observable. For
$\pi/2-(\alpha+\chi)\leq\varphi$, the solid angle is
$\Psi=2[2\pi(1-\cos(\pi/2))-2\pi(1-\cos(\varphi-(\alpha+\chi)))]$, where the
factor of $2$ is involved due to the beaming radiation from the two opposite
poles considered, $\varphi$ is the angle between $\hat{p}$ and $\hat{Z}$. For
$(\alpha+\chi)\leq\varphi\leq\pi/2-(\alpha+\chi)$, the solid angle is
$2[2\pi(1-\cos(\varphi+\alpha+\chi))-2\pi(1-\cos(\varphi-(\alpha+\chi)))]$.
For $(\alpha-\chi)\leq\varphi\leq(\alpha+\chi)$, the solid angle is
$2[2\pi(1-\cos(\varphi+\alpha+\chi))]$. For $\varphi\leq(\alpha-\chi)$, the
solid angle is
$2[2\pi(1-\cos(\varphi+\alpha+\chi))-2\pi(1-\cos(\alpha-\chi-\varphi))]$. In
summary, the solid angle that corresponds to the radiation is observable is
given by

$\Psi=\begin{cases}4\pi\cos[\varphi-(\alpha+\chi)],&\frac{\pi}{2}-(\alpha+\chi)\leq\varphi\\\
8\pi\sin\varphi\sin(\alpha+\chi),&(\alpha+\chi)\leq\varphi\leq\frac{\pi}{2}-(\alpha+\chi)\\\
4\pi[1-\cos(\varphi+\alpha+\chi)],&(\alpha-\chi)\leq\varphi\leq(\alpha+\chi)\\\
8\pi\sin\alpha\sin(\varphi+\chi),&\varphi\leq(\alpha-\chi)\end{cases}.$ (8)

For a certain source, the observation direction is uniformly randomly
distributed in space. Thus, the corresponding observable probability is given
by

$\eta=\frac{\Psi}{4\pi}=\begin{cases}\cos[\varphi-(\alpha+\chi)],&\frac{\pi}{2}-(\alpha+\chi)\leq\varphi\\\
2\sin\varphi\sin(\alpha+\chi),&(\alpha+\chi)\leq\varphi\leq\frac{\pi}{2}-(\alpha+\chi)\\\
1-\cos(\varphi+\alpha+\chi),&(\alpha-\chi)\leq\varphi\leq(\alpha+\chi)\\\
2\sin\alpha\sin(\varphi+\chi),&\varphi\leq(\alpha-\chi)\end{cases}.$ (9)

Notice that the observable probability only involves the geometrical effect,
and other effects (e.g., burst frequency, flux threshold, etc.) are not
considered here. We define the angular momentum of the magnetar as $L$ for a
free body, and the direction of $L$ is $\hat{Z}$ shown in Fig. 1. In general,
the direction of the rotation angular velocity $\Omega$ is not collinear to
the direction of $L$. The components of $\Omega$ are given by

$\Omega_{xx}~{}=~{}\frac{L}{I_{xx}}\sin\varphi,$ (10)
$\Omega_{zz}~{}=~{}\frac{L}{I_{zz}}\cos\varphi,$ (11)

where $\Omega_{xx}$ and $\Omega_{zz}$ are the projections of $\Omega$ onto
x-axis and z-axis in the body frame. On the other hand, the obliquity angle is
given by

$\tan\theta~{}=~{}\frac{\Omega_{xx}}{\Omega_{zz}}.$ (12)

leading to,

$\tan\theta~{}=~{}\frac{I_{zz}}{I_{xx}}\tan\varphi~{}=~{}(1+\epsilon)\tan\varphi.$
(13)

The angle $\varphi$ evolves due to the evolution of the obliquity $\theta$ and
the ellipticity parameter $\epsilon$, leading to the evolution of the
observable probability $\eta$. Fig. 7 shows the relation between the
observable probability $\eta$ and the magnetar age $T_{\rm age}$. The red,
blue, and black solid lines denote the relations with $P_{\rm i}=10~{}{\rm
ms}$ and $B_{\rm i}=10^{14},10^{15},10^{16}{\rm G}$, respectively. The dotted,
solid, and dashed blue lines denote the relations with $B_{\rm i}=10^{15}{\rm
G}$ and $P_{\rm i}=5,10,20~{}{\rm ms}$. We take $|\beta|=0.1$, and
$\alpha=\pi/9$, $\chi=\pi/18$. We can see that for a given magnetar with an
initial magnetic field of $B_{\rm i}$ and a rotation period of $P_{\rm i}$,
the observable probability $\eta$ increases with the magnetar age $T_{\rm
age}$ in the early stage. The reason is that the solid angles swept by the two
radiation beams near the magnetic poles are overlapped at
$\theta\sim\varphi\sim\pi/2$ and the total solid angle increases as the
$\varphi$ becomes smaller. In the later stage, the two solid angles swept by
the two radiation beams are separated and the observable probability $\eta$
decreases with the magnetar age $T_{\rm age}$, because the solid angle swept
by one radiation beam becomes smaller as the $\varphi$ becoming smaller. We
notice that only the geometrical effect is considered in the above discussion.
Furthermore, since young magnetars with stronger magnetic fields and faster
spins tend to produce more frequent repeated bursts triggered by the crust
quake of neutron stars (Yang & Zhang, 2021; Li et al., 2022), the probability
of FRBs observed in the early stage might be much larger than that in the
later stage.

Figure 7: The evolution carve of observable probability $\eta$. We take
$|\beta|=0.1$, and $\alpha=\pi/9$, $\chi=\pi/18$. Here we consider that the
bursts are observable as long as they are within the radiation beam and only
the geometrical effects are considered here.

Based on the above model, we further analyze the active window distribution of
the periodic activities of an FRB repeater as follows. We consider the motion
process of a radiation beam in the rotation frame where the line of sight
rotates around $\hat{Z}$ axis and the radiation beam processes around
$\hat{p}$ axis, as shown in Fig. 8. We emphasize that the rotation frame
defined here is not the same as the body frame described in literature (e.g.,
Li & Zanazzi, 2021). In the rotation frame, $\hat{Z}$ and $\hat{p}$ are
considered to be fixed. The trajectory of the line of sight in the rotation
frame is shown in the dashed line. The red circle denotes the radiation beam,
and the pink ring band denotes the region swept by the radiation beam due to
the precession in the rotation frame. Thus, the active window corresponds to
the period when the radiation beam crosses over the line of sight.

Figure 8: Schematic configuration of the motion process of a radiation beam in
the rotation frame where $\hat{Z}$ and $\hat{p}$ are fixed. The dashed line
denotes the line of sight. The small circle denotes the radiation beam, and
the ring band denotes the region swept by the radiation beam due to the
precession in this frame. $\varphi$ is the angle between $\hat{p}$ and
$\hat{Z}$. $\zeta$ is the angle between the line of sight and $\hat{p}$.
$\alpha$ is the angle between $\hat{p}$ and $\hat{m}_{f}$. $\chi$ is the half-
opening angle of the radiation beam.
$\theta_{1}=\arccos[(\cos(\varphi-\zeta-\chi)-\cos\varphi\cos\alpha)/(\sin\varphi\sin\alpha)]$,
and
$\theta_{2}=\arccos[(\cos(\varphi-\zeta+\chi)-\cos\varphi\cos\alpha)/(\sin\varphi\sin\alpha)]$.

Figure 9: The phase distribution of the activity window in one precession
period. $\varpi=1$ corresponds to the active window and $\varpi=0$ corresponds
to the nonactive window. Different panels correspond to different directions
of the line of sight with $\zeta=\pi/6,\pi/18,\pi/20,0$, respectively.
$\varphi=\pi/3$, $\alpha=\pi/9$ and $\chi=\pi/18$ are taken here for example.

In Fig. 8, the radiation beam moves around $\hat{p}$ due to the precession.
$\theta_{2}$ and $\theta_{1}$ correspond to the angles for entering and
exiting the active window, respectively, which are given by

$\displaystyle\theta_{1}$
$\displaystyle=\arccos\frac{\cos(\varphi-\zeta-\chi)-\cos\varphi\cos\alpha}{\sin\varphi\sin\alpha},$
$\displaystyle\theta_{2}$
$\displaystyle=\arccos\frac{\cos(\varphi-\zeta+\chi)-\cos\varphi\cos\alpha}{\sin\varphi\sin\alpha}.$
(14)

There are three cases for the distribution of the active window: 1) If the
line of sight (dashed line) passes through the pink ring band only once, i.e.,
$(\alpha-\chi)\leq\zeta\leq(\alpha+\chi)$, there is only one active window in
one precession period and the duty cycle is $(2\theta_{2})/2\pi$. 2) If the
line of sight passes through the pink ring band two times, i.e.,
$\zeta\leq(\alpha-\chi)$, there are two active windows in one precession
period and the duty cycle is $(2/2\pi)\left(\theta_{2}-\theta_{1}\right)$. 3)
If the line of sight does not pass through the pink ring region, there is no
active window. In summary, the duty cycle is given by

$\Pi=\begin{cases}\frac{1}{\pi}\theta_{2},&(\alpha-\chi)\leq\zeta\leq(\alpha+\chi)\\\
\frac{1}{\pi}\left(\theta_{2}-\theta_{1}\right),&\zeta\leq(\alpha-\chi)\\\
0,&(\alpha+\chi)\leq\zeta\end{cases}.$ (15)

In particular, for $\alpha\leq\chi$, there is always one active window in one
precession period once $\zeta\leq\alpha+\chi$. We also notice that the above
equation is valid for $P\ll P_{\rm prec}$, which could be naturally satisfied
in this work. In Fig. 9, we plot the distribution of the active window in one
precession period for different directions of the line of sight directions,
and $\varphi=\pi/3$, $\alpha=\pi/9$ and $\chi=\pi/18$ are taken for example.
We define the parameter $\varpi$ to describe whether the FRB source is in the
active window: $\varpi=1$ corresponds to the active window and $\varpi=0$
corresponds to the nonactive window. The initial phase corresponds to the one
when the radiation beam direction $\hat{m}_{f}$ is farthest from the line of
sight. For given $\varphi$, $\alpha$, and $\chi$, the distribution of the
active window mainly depends on the direction of the line of sight $\zeta$, as
pointed out in the above discussion. This result suggests that there are one
or two active windows in one precession period.

##  5 Conclusion and discussion

Some FRB repeaters and SGRs appear (possible) periodic activities with periods
of tens to hundreds days, which might originate from the free precession of a
young magnetar. The young magnetar is deformed by the anisotropic internal
magnetic pressure, and the free precession is generated when the magnetic axis
is not parallel with the spin axis. In this work, we analyze the self-
consistent rotation evolution of deformed magnetars and calculate the
corresponding precession evolution. We find that the obliquity evolves from
$\pi/2$ to a small value over hundreds of years, and the spin period tends to
be a constant finally. Due to the evolution of the spin, the obliquity, and
the surface magnetic field, the precession period increases faster after a
broken timescale that mainly depends on the initial surface magnetic field
$B_{\rm i}$. On the other hand, the larger the initial surface magnetic field
or the faster the initial spin, the smaller the precession period. And a
larger deformation parameter $|\beta|$ can also cause a smaller precession
period.

Based on this model, we constrain the ages of magnetars that appear (possible)
periodic activities with the reasonable parameter space $(B_{\rm i},P_{\rm
i})$. For FRB 180916B with 16.35-day activity period, its source age is
constrained from 100 to 2000 yr. The source age of FRB 121102 with 161-day
activity period is constrained from 400 to 7000 yr. Some Galactic magnetars
also appear possible periodic activities in the activities of SGRs. Combining
the observation of $P=3.245~{}{\rm s}$ and $\dot{P}=1.43\times 10^{-11}{\rm
s~{}s^{-1}}$ of SGR 1935+2154, we can obtain the age, initial magnetic field,
and $|\beta|$ of this magnetar for a given initial period. For $P_{\rm
i}=5~{}{\rm ms}$, one has $T_{\rm age}=2136.6~{}{\rm yr}$, $B_{\rm
i}=5.66\times 10^{14}~{}{\rm G}$, and $|\beta|=0.0964$. For $P_{\rm
i}=10~{}{\rm ms}$, one has $T_{\rm age}=2103~{}{\rm yr}$, $B_{\rm
i}=5.37\times 10^{14}~{}{\rm G}$, and $|\beta|=0.105$. For another magnetar
SGR 1806-20, although no FRBs were detected from this source, it still had a
possible period of 398.2 days in X-ray band. Combining the observation of
$P=7.55~{}{\rm s}$ and $\dot{P}=49.5\times 10^{-11}{\rm s~{}s^{-1}}$, we can
obtain the age, initial magnetic field, and $|\beta|$ of this magnetar. For
$P_{\rm i}=5~{}{\rm ms}$, one has $T_{\rm age}=172.6~{}{\rm yr}$, $B_{\rm
i}=4.40\times 10^{15}~{}{\rm G}$, and $|\beta|=0.000865$. For $P_{\rm
i}=10~{}{\rm ms}$, one has $T_{\rm age}=169.9~{}{\rm yr}$, $B_{\rm
i}=4.33\times 10^{15}~{}{\rm G}$, and $|\beta|=0.000859$. The above
constraints on the source ages suggest that FRB repeaters and SGRs might be
produced by young magnetars with significant deformation.

At last, we discussed the observable probability and the active window of FRB
repeaters. We notice that the radiation beam is required to be derived from
the deformation axis in order to produce the observable precession signal. In
this picture, we calculated the evolution of the observable probability
$\eta$. We found that for a given magnetar the observable probability $\eta$
increases with the magnetar age $T_{\rm age}$ in the early stage and decreases
with the magnetar age $T_{\rm age}$ in the later stage. Meanwhile, the
stronger the initial magnetic field (or the faster the initial spin) the
faster the evolution of the observable probability. Thus, the observed burst
rate of an old source are predicted to be much lower than that of a young
source, implying that FRB repeaters with high burst rates originate from young
neutron stars. On the other hand, since the radiation beam is from the
deformation axis, there are one or two active windows in one precession period
and the distribution of the active window depends on the light of sight. Such
a prediction could be tested by the future observation.

## Acknowledgements

We thank the anonymous referee for providing helpful comments and suggestions
that have allowed us to improve this manuscript significantly. We also thank
Zi-Wei Li, Dongzi Li, Fa-Yin Wang, Jin-Jun Geng, and Xiang-Hua Li for their
helpful discussions. This work is supported by the National Natural Science
Foundation of China grant No. 12003028 and the National SKA Program of China
(2022SKA0130100). The author also acknowledges supports from the “Science &
Technology Champion Project” (202005AB160002) and from two “Team Projects”–the
“Innovation Team” (202105AE160021) and the “Top Team” (202305AT350002), all
funded by the “Yunnan Revitalization Talent Support Program”.

## Data Availability

This theoretical study did not generate any new data. The code performed for
the calculations is available upon request.

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