# Isometry groups of formal languages for generalized Levenshtein distances
111The work of the author was supported by the RSF, project no. 22-11-00075

Vladimir Yankovskiy  Faculty of Mechanics and Mathematics of Moscow State
University, Moscow 119991 Russia, Leninskie Gory, MSU.

###### Abstract

This article is a partial answer to the question of which groups can be
represented as isometry groups of formal languages for generalized Levenshtein
distances. Namely, it is proved that for any language the modulus of the
difference between the lengths of its words and the lengths of their images
under isometry for an arbitrary generalized Levenshtein distance that
satisfies the condition that the weight of the replacement operation is less
than twice the weight of the removal operation is bounded above by a constant
that depends only on the language itself. From this, in particular, it follows
that the isometry groups of formal languages with respect to such metrics
always embed into the group $\Pi_{n=1}^{\infty}S_{n}$. We also construct a
number of examples showing that this estimate is, in a certain sense,
unimprovable.

MSC 2020: 05E18, 20B25, 20H15

##  1 Introduction

This paper is a partial answer to the question of which groups can be
represented as isometry groups of formal languages for generalized Levenshtein
distances. Namely, the following theorem is proved:

######  Theorem 1.

Let $L$ be an arbitrary formal language, $d$ be an arbitrary generalized
Levenshtein distance satisfying the condition that the weight of the
replacement operation is less than twice the weight of the removal operation.
Then there exists $m\in\mathbb{N}$ such that $||\phi(w)|-|w||\leqslant m$

Note that for generalized Levenshtein distancess with a replacement weight
greater than or equal to twice the insertion weight, this statement is not
true.

It follows from Theorem 1, in particular, that the isometry group of any
formal language with respect to the Levenshtein metric embeds in
$\Pi_{n=1}^{\infty}S_{n}$.

A number of examples are also constructed, demonstrating that this estimate
is, in a certain sense, unimprovable, moreover, for languages with different
growths.

######  Definition 1.

Growth of $L$ is a function of $n\mapsto|\\{w\in L||w|\leqslant n\\}|$

######  Theorem 2.

Let $G$ be an arbitrary finite group, $d$ an arbitrary generalized Levenshtein
distance. Then there are $n\in\mathbb{N}$ and language
$L\subset\\{0;1\\}^{24n}$ such that $|L|=n$, $Isom_{d}(L)\cong G$ and
$d(u,v)\in\\{4,6\\}$ for any two distinct words $u,v\in L$.

Here and below, $A^{k}$ denotes the set of all words of length $k$ over the
alphabet $A$.

######  Theorem 3.

Let $G_{1},G_{2},...$ be a countable sequence of arbitrary finite groups, $d$
be an generalized Levenshtein distance. Then there exists a language
$L\subset\\{0;1\\}^{*}$ with growth $O(n)$ such that
$Isom_{d}(L)\cong\Pi_{n=1}^{\infty}G_{n}$.

######  Theorem 4.

Let $d$ be an arbitrary generalized Levenshtein distance. Then for any integer
$k\geqslant 2$, there exists a language $L$ over an alphabet of $k$ characters
with growth $\Theta(k^{\sqrt{n}})$ such that $Isom_{d}(L)\cong
S_{k}^{\infty}\times\Pi_{n=1}^{\infty}S_{k^{n}}$.

The last fact is interesting because the isometry group obtained by Theorem 1
is maximal for all languages.

Also, special attention is paid to regular languages:

######  Definition 2.

Regular language is a formal language that can be obtained from finite
languages by applying a finite number of union operations ($U\cup V$), product
($UV=\\{uv|u\in U,v\in V\\}$) and Kleene stars
($L^{*}=\bigcup_{n=0}^{\infty}L^{n}$).

It is proved in the article that their isometry groups are also sufficiently
diverse and the given bound for them is unimprovable:

######  Theorem 5.

Let $G$ and $H$ be arbitrary finite groups, $d$ be an arbitrary generalized
Levenshtein distance. Then there is a regular language $L\subset\\{0;1\\}^{*}$
such that $Isom_{d}(L)\cong G\times H^{\mathbb{N}}$.

######  Theorem 6.

Let $d$ be an arbitrary distance from the family of generalized Levenshtein
distances. Then there exists a regular language $L$ such that
$Isom_{d}(L)\cong\Pi_{n=1}^{\infty}S_{2n}$.

The last fact is interesting because the isometry group obtained by Theorem 1
is maximal for all languages, not necessarily regular ones.

Here and below, $S_{n}$ denotes a symmetric group on $n$ elements, $C_{n}$
denotes a cyclic group of order $n$, $\Pi$ denotes the Cartesian product of
groups.

The isometry groups of finite languages have been studied before.

For example, in [3] it is proved that for an arbitrary finite alphabet $A$,
the isometry group $A^{n}$ with respect to the Hamming distance is isomorphic
to $S_{|A|}^{n}\times S_{n}$.

Another paper on a similar topic is [6], where it is proved that for an
arbitrary finite alphabet $A$ and $2\leqslant k_{1}<k_{2}$ the isometry group
of the language $\bigcup_{k=k_{1}}^{k_{2}}A^{k}$ with respect to the
¡¡internal Levenshtein distance¿¿ (the minimum number of operations of
insertions, deletions and replacements of characters that transform one word
into another in such a way that all ¡¡intermediate words¿¿ lie in the source
language) is isomorphic to $S_{|A|}^{n}\times C_{2}$

In the same article, the isometry groups of languages, including infinite
ones, are studied with respect to the family of generalized Levenshtein
distances. Their definitions and main properties will be given in Section 1.

The work consists of 9 sections (including introduction):

  * •

Section 2 formulates the definition of a family of generalized Levenshtein
metrics, and also classifies isometry groups of one-character languages.

  * •

Section 3 proves Theorem 1 and constructs a counterexample for the case when
the weight of the replacement is greater than or equal to twice the weight of
the insertion.

  * •

In Section 4, we carry out the preparatory work necessary for the proof of
Theorem 2.

  * •

Section 5 proves Theorem 2.

  * •

Section 6 proves Theorem 3.

  * •

Section 7 provides a proof

  * •

Section 7 proves Theorem 4.

  * •

Section 8 proves Theorem 5.

  * •

Section 9 proves Theorem 6.

##  2 Generalized Levenshtein distances

######  Definition 3.

Let $A$ be a finite alphabet and the words $u,v\in A^{*}$. Then generalized
Levenshtein distance with insertion weight $\gamma$ and replacement weight
$\theta$ between $v$ and $u$ is

$lev_{\gamma,\theta}(u,v)=\min\\{\gamma n+\theta m|u\text{ can be translated
into }v\text{ by n insertions or deletions and m replacements}\\}$

Obviously, for $\gamma,\theta>0$ $lev_{\gamma,\theta}$ is a distance on
$A^{*}$.

The most studied special cases of the generalized Levenshtein distance are:

  * $lev_{1,1}$

classical Levenshtein metric – the minimum number of insertions, deletions, or
substitutions required to tranform one word into another (first discussed in
[4])

  * $lev_{1,2}$

is the minimum number of insertions or deletions required to transform one
word into another.

  * $lev_{n,1}$

for $n\to\infty$ converges pointwise to the Hamming metric, the minimum number
of substitution operations required to transform one word into another.

Also, there is an alternative way to specify the generalized Levenshtein
distance by the recursive formula:

######  Proposition 1 ([5]).

$lev_{\gamma,\theta}(a,b)=\begin{cases}\gamma|a|&\quad|b|=0\\\
\gamma|b|&\quad|a|=0\\\ lev_{\gamma,\theta}(a.tail,b.tail)&\quad
a.head=b.head\\\
\min(\theta+lev_{\gamma,\theta}(a.tail,b.tail),\gamma+lev_{\gamma,\theta}(a.tail,b),\gamma+lev_{\gamma,\theta}(b.tail,a))&\quad
a.head\neq b.head\end{cases}$

where $a.tail$ is the suffix of $a$ containing all of its characters except
the first one, and $a.head$ is the first element of $a$.

In particular, from this formula, as well as the invariance of the Levenshtein
distance under ¡¡reflection¿¿ of words, it is true that
$lev_{\gamma,\theta}(uxv,uyv)=lev_{\gamma,\theta}(x,y)$.

The following inequalities also always hold:

  * •

$lev_{\gamma,\theta}(u,v)\leqslant(\theta-\gamma)\min(|u|,|v|)+\gamma\max(|u|,|v|)$

  * •

$lev_{\gamma,\theta}(u,v)\geqslant\gamma||u|-|v||$, and equality is achieved
if and only if the shorter word is contained in the longer one as a
subsequence.

######  Definition 4.

Let $(M_{1},d_{1})$ and $(M_{2},d_{2})$ be metric spaces.
We will call a bijection $\phi:M_{1}\to M_{2}$ homothety if there exists
$t\in\mathbb{R}$ such that for any words $u,v\in\mathbb{L}$
$d_{2}(\phi(u),\phi(v))=td_{1}(u,v)$.
We will call a bijection $\phi:M_{1}\to M_{2}$ isometry if for any words
$u,v\in\mathbb{L}$ $d_{2}(\phi(u),\phi(v))=d_{1}(u,v)$.

The set $Isom_{d}(M)$ of all isometries of the metric space $(M,d)$ into
itself forms a composition group. Moreover, the isometry group of a metric
space is always isomorphic to the isometry group of its image under homothety.

######  Proposition 2.

Let $d$ be a generalized Levenshtein metric and $L$ an arbitrary language.
Then there exists $\theta\in(0;2]$ such that the trivial mapping $(L,d)$ to
$(L,lev_{1,\theta})$ is a homothety.

###### Proof.

As $\theta$ we can take $\min(\frac{\theta_{0}}{\gamma_{0}},2)$ ∎

We will denote the distance $lev_{1,\theta}$ as $lev_{\theta}$, and the
isometry group of the language $L$ with respect to it as $Isom_{\theta}(L)$.

The class of isometry groups of languages over a one-element alphabet is
rather small.

######  Proposition 3.

Let $L\subset\\{a\\}^{*}$, $\theta\in(0;2]$. Then $Isom_{\theta}(L)$ is either
trivial or isomorphic to a cyclic group of order 2.

###### Proof.

$l:a^{n}\mapsto n$ is an isometry of $(\\{a\\}^{*},lev_{\theta})$ onto the
metric space $(\mathbb{N},d(m,n)=|n-m|)$.
From this we can conclude that $Isom(L)\cong Isom(l(L))$.
Suppose $N\subset\mathbb{N}$, $|N|\geqslant 2$ (everything is obvious for
$|N|=1$). Let $n_{0},n_{1},...$ be elements of $N$ sorted in ascending order,
$\phi$ be an isometry from $N$.
$\phi(n_{0})\in\\{n_{0},n_{|N|}\\}$. Otherwise, if $\phi(n_{0})=n_{i}\neq
n_{|N|}$, then

$|n_{i+1}-n_{i-1}|=|\phi^{-1}(n_{i+1})-\phi^{-1}(n_{i-1})|<$
$<|\phi^{-1}(n_{i+1})-n_{0}|+|n_{0}-\phi^{-1}(n_{i-1})|=$
$=|(n_{i+1}-n_{i}|+|n_{i}-n_{i-1}|=|n_{i+1}-n_{i-1}|$

It’s impossible.
Let $\phi(n_{0})=n_{0}$. Let us prove by induction that $\phi$ is trivial:

  * Base:

For $N=0$ $\phi(n_{0})=n_{0}$ (no other items).

  * Step:

suppose that $\phi(n_{i})=n_{i}$ for all $i<k$. Then $n_{k}$ is the only
nearest element to $n_{k-1}$ not contained in $\\{n_{0},...,n_{k-1}\\}$.
Therefore, $\phi(n_{k})$ is the only nearest element to
$\phi(n_{k-1})=n_{k-1}$ not contained in
$\phi(\\{n_{0},...,n_{k-1}\\})=\\{n_{0},...,n_{k-1}\\}$, i.e.
$\phi(n_{k})=n_{k}$.

Let $\phi(n_{0})=n_{|N|}$. Let us prove by induction that $\phi:n_{k}\mapsto
n_{|N|-k}$.

  * Base:

For $N=0$ $\phi(n_{0})=n_{0}=n_{|N|}$ (no other elements).

  * Step:

suppose that $\phi(n_{i})=n_{|N|-i}$ for all $i<k$. Then $n_{k}$ is the only
nearest element to $n_{k-1}$ not contained in $\\{n_{0},...,n_{k-1}\\}$.
Therefore, $\phi(n_{k})$ is the only nearest element to
$\phi(n_{k-1})=n_{|N|-k+1}$ not contained in
$\phi(\\{n_{0},...,n_{k-1}\\})=\\{n_{|N|},...,n_{|N|-k+1}\\}$, i.e.
$\phi(n_{k})=n_{|N|-k}$.

Thus, there can be no other isometries, except for these two (and the second
one is far from always realized).
Hence $Isom(L)$ is either trivial or isomorphic to a cyclic group of order 2.

∎

However, already for the two-element alphabet, the isometry groups are more
complicated, which will be demonstrated in the following sections.

##  3 Proof of Theorem 1

######  Lemma 1 ([2]).

There is no infinite language over a finite alphabet in which no word occurs
as a subsequence of another.

We will denote by $M(L)$ the set of all minimal words in the language $L$ with
respect to the inclusion order as a subsequence.

######  Lemma 2.

Let $L$ be a formal language, $\theta\in(0;2)$. Then if $Isom_{\theta}(L)$
acts transitively on $L$, then $L$ is finite.

###### Proof.

Let $L$ be an infinite language. $M(L)$ is finite by Lemma 1. Therefore, by
the Dirichlet principle, $L$ has an infinite sublanguage with a unique minimal
word. Let’s denote it as $L_{0}$ and the only minimal word in it as $w_{0}$.
Language $\\{w\in L_{0}||w|>\frac{2}{2-\theta}|w_{0}|\\}$ is also infinite.
So, arguing similarly, it also has an infinite sublanguage with a single
minimal word. Let’s denote it as $L_{1}$ and the only minimal word in it as
$w_{1}$.
Language $\\{w\in L_{1}||w|>2|w_{1}|\\}$ is also infinite. So, arguing
similarly, it also has an infinite sublanguage with a single minimal word.
Let’s denote it as $L_{2}$, and the only minimal word in it as $w_{2}$.
Now suppose that $Isom_{\theta}(L)$ acts transitively on $L$. Then there
exists $\phi\in Isom_{\theta}(L)$ such that $\phi(w_{1})=w_{0}$. Then the
chain of inequalities is fulfilled:

$lev_{\theta}(w_{0},w_{1})+lev_{\theta}(w_{1},w_{2})=lev_{\theta}(w_{0},w_{2})=lev_{\theta}(\phi(w_{0}),\phi(w_{2}))\leqslant$
$\leqslant\max(|\phi(w_{0})|,|\phi(w_{2})|)+(\theta-1)\min(|\phi(w_{0})|,|\phi(w_{2})|)\leqslant$
$\leqslant\theta|w_{0}|+\max(lev_{\theta}(w_{0},\phi(w_{0})),lev_{\theta}(w_{0},\phi(w_{2})))+(\theta-1)\min(lev_{\theta}(w_{0},\phi(w_{0}))),lev_{\theta}(w_{0},\phi(w_{2})))=$
$=\theta|w_{0}|+\max(lev_{\theta}(w_{1},w_{0}),lev_{\theta}(w_{1},w_{2}))+(\theta-1)\min(lev_{\theta}(w_{1},w_{0}),lev_{\theta}(w_{1},w_{2}))=$
$=\theta|w_{0}|+|w_{2}|-|w_{1}|+(\theta-1)(|w_{1}|-|w_{0}|)<lev_{\theta}(w_{0},w_{1})+lev_{\theta}(w_{1},w_{2})$

Contradiction. ∎

Let now $m=\max\\{|\phi(v)|-|v||v\in M(L),\phi\in Isom_{\theta}(L)\\}$ (this
set is finite by Lemmas 1 and 2). Let us prove by contradiction that
$||w|-|\phi(w)||\leqslant m$.

Let $||w|-|\phi(w)||>m$. Without loss of generality, we assume that
$|w|<|\phi(w)|$. Now let $u\in M(L)$ be contained in $w$ as a subsequence.

Then

$lev_{\theta}(w,u)=lev_{\theta}(phi(w),phi(u))\geqslant|\phi(w)|-|\phi(u)|>|w|+m-|\phi(u)|\geqslant$
$\geqslant|w|+|\phi(u)|-|u|-|\phi(u)|=|w|-|u|=lev_{\theta}(w,u)$

Contradiction.$\square$

######  Corollary 1.

The isometry group of any formal language $L$ embeds in
$\Pi_{n=1}^{\infty}S_{n}$.

###### Proof.

Let $n_{1},n_{2},...$ be the orders of the orbits of the natural action of
$Isom(L)$ on $L$ (the orbits are finite by Lemma 2). Since $Isom(L)$ acts
effectively on $L$, it embeds in
$\Pi_{i=1}^{\infty}S_{n_{i}}\leqslant\Pi_{n=1}^{\infty}S_{n}$ . ∎

######  Proposition 4.

There is a regular language $L\subset\\{0,1\\}^{*}$ such that the isometry
group $Isom_{2}(L)\cong D_{\infty}$, and it acts transitively on $L$.

###### Proof.

Consider a regular language $L=1^{*}\cup 0^{*}$ and a bijection
$\phi:L\to\mathbb{Z}$ given by the formulas

$\phi(0^{n})=n$ $\phi(1^{n})=-n$

for all $n\in\mathbb{N}_{0}$.
Let us show that $\phi$ is an isometry of the spaces $(L,lev_{2})$ and
$(\mathbb{Z},d(x,y)=|x-y|)$. Really

$lev_{2}(0^{n},0^{m})=|m-n|$ $lev_{2}(1^{n},1^{m})=|m-n|$
$lev_{2}(0^{n},1^{m})=m+n$

for all $n,m\in\mathbb{N}_{0}$.
Moreover, the isometry group of the metric space $(\mathbb{Z},d(x,y)=|x-y|)$
acts transitively on it and is isomorphic to $D_{\infty}$. ∎

##  4 Word stretching

We will denote the Hamming distance as $h$.

######  Definition 5.

The word stretching is the operation $st:A^{*}\times A^{*}\to A^{*}$ given
recursively:

$st(\Lambda,w_{2})=\Lambda$ $st(w_{1}a,w_{2})=st(w_{1},w_{2})w_{2}a$

for all $w_{1},w_{2}\in A^{*},a\in A$

######  Lemma 3.

Let the words $w_{1},w_{2}\in A^{*}$, where $|w_{1}|=|w_{2}|$.
Let $k\in\mathbb{N}$, where $k>h(w_{1},w_{2})$.
Let the symbols $a,b\in A$, and $a\neq b$.
Let $\theta\in(0;2]$.
Then the equality holds:

$lev_{\theta}(st(w_{1},a^{k}ba^{k}),st(w_{2},a^{k}ba^{k}))=h(w_{1},w_{2})$

###### Proof.

Let us show that there is a minimal ¡¡path¿¿ from $st(w_{1},a^{k}ba^{k})$ to
$st(w_{2},a^{k}ba^{k})$ that does not contain insertions or deletions.
Indeed, suppose that the shortest ¡¡path¿¿ contains a deletion. $k$ deletions
cannot contain this ¡¡path¿¿ (otherwise it will not be the shortest one). This
means that the ¡¡segment¿¿ of the word between this deletion and one of the
insertions closest to it (on one side or the other) is shifted by a distance
less than $k$.
At the same time, at positions that are not multiples of $k+1$ (numbering
starts from 1), there are no symbols different from $a$, and at positions
comparable to $k+1$ modulo $2k+2$ there are no symbols different from b .
So, if there are $t$ significant symbols on this ¡¡segment¿¿, then there are
also $t-1$ ¡¡central¿¿ $b$ symbols on it that turned out to be shifted. It
would take $2(t-1)$ replacement operations to put them in order. This means
that the total cost of operations on the interval (including the initial
deletion and insertion) will be at least $2(t-1)\theta+2$. In this case,
element-by-element replacement of all significant characters on the segment
without any insertions and deletions will cost $t\theta\leqslant
2(t-1)\theta+2$.
So, gradually changing the pairs ¡¡insert / delete¿¿ to replace significant
characters, we get the desired ¡¡path¿¿. ∎

##  5 Proof of Theorem 2

######  Lemma 4.

Let $\Gamma(V,E)$ be a finite simple cubic graph. Then there exists a language
$\\{w_{v}\\}_{v\in V}\subset\\{0;1\\}^{|E|}$ such that

$h(w_{u},w_{v})=\begin{cases}4&\quad\\{u,v\\}\in E\\\ 6&\quad(u,v)\not\in
E\end{cases}$

###### Proof.

Let $E=\\{e_{0},...,e_{m}\\}$. Let’s define $i:V\times E\to\\{0,1\\}^{*}$ as
follows:

$i(v,e)=\begin{cases}1&\quad v\in e\\\ 0&\quad v\not\in e\end{cases}$

Let’s define the word $w_{v}=i(v,e_{0})i(v,e_{1})...i(v,e_{|E|})$.
It is easy to see that the Hamming distances between words turned out to be as
required. ∎

######  Theorem 7.

([1]) A group is finite if and only if it is isomorphic to the automorphism
group of some cubic graph.

Let $G$ be a finite group, $\Gamma(E,V)$ be the cubic graph corresponding to
it by Theorem 7, $L_{0}$ be the language constructed for $\Gamma$ by Lemma 2,
$L_{1}=st(L_{0},1^{7}01^{7})$.
Since $diam(L_{1})=6$, by Lemma 1, $Isom_{\theta}(L_{1})\cong G$. The length
of words in $L_{1}$ is equal to $16|E|=24|V|=24|L_{1}|$. $\square$

##  6 Proof of Theorem 3

Let $G_{1},G_{2},...$ be a countable sequence of finite groups.
$L_{1},L_{2},...$ are the corresponding languages constructed by Corollary 1.
We construct the languages $L_{1}^{\prime},L_{2}^{\prime},...$ using
recursion:

$L_{0}^{\prime}=\\{\Lambda\\}$

$L_{n+1}^{\prime}=(01)^{l_{n}+7}L_{n+1}$

where $l_{n}$ is the length of words in $L_{n}^{\prime}$.

It is easy to see that the isometry group $L_{n}^{\prime}$ coincides with the
isometry group $L_{n}$. Moreover, for any words $w_{n}\in L_{n}^{\prime}$ and
$w_{m}\in L_{m}^{\prime}$, $m>n$
$lev_{\theta}(w_{n},w_{m})=l(L_{m}^{\prime})-l(L_{n}^{\prime})$. This equality
is true because, to get a shorter word from a longer one, it is enough to
remove everything except the $(01)^{l(L_{n})}$ prefix, and then select the
correct character from each $01$ block.
It follows from this that for any sequence $\phi_{1},\phi_{2},...$ of
isometries of languages $L_{1}^{\prime},L_{2}^{\prime},...$ it is true that
$\phi:w_{n}\mapsto\phi(w_{n})$ for all $n\in\mathbb{N}$, $w_{n}\in
L_{n}^{\prime}$ is an isometry of $\bigcup_{n=1}^{\infty}L_{n}^{\prime}$.
Moreover, $\bigcup_{n=1}^{\infty}L_{n}^{\prime}$ has no other isometries.
Indeed, $\Lambda$ goes into itself as the only word without neighbors at a
distance of $4$ or $6$. So the length of words in this language (that is, the
distance to $\Lambda$) is an invariant.
In other words,
$Isom_{\theta}(\bigcup_{n=1}^{\infty}L_{n}^{\prime})=\Pi_{n=1}^{\infty}Isom_{\theta}(L_{n}^{\prime})=\Pi_{n=1}^{\infty}Isom_{\theta}(L_{n})=\Pi_{n=1}^{\infty}G_{n}$.
$\bigcup_{n=1}^{\infty}L_{n}^{\prime}$ has growth $O(n)$ since for any
$n\in\mathbb{N}$ the length of all words in $L_{n}^{\prime}$ is greater
$24|L_{n}|$, which means $|\\{w\in L||w|\leqslant k\\}|\leqslant
1+\frac{k}{24}$. $\square$

##  7 Proof of Theorem 4

Let $A=\\{a_{1},...,a_{k}\\}$ be an arbitrary alphabet. Let’s build languages
$L_{1},L_{2},...$ recursively:

$L_{0}=\Lambda$

$L_{n}=(a_{1}...a_{k})^{l(k,n-1)}st(A^{k^{n}},a_{1}^{k^{n+1}}a_{2}a_{1}^{k^{n+1}})$

where $l(k,n)$ is the length of $L_{n}$.
Note that $l(k,n)=l(k,n-1)+2k^{n}(k^{n+1}+2)=O(k^{2n})$. In this case,
$|L_{n}|=k^{k^{n}}$. Hence, the language $\bigcup_{n=1}^{\infty}L_{n}$ has
growth $\sum_{l(k,t)\leqslant n}k^{k^{t}}=\Theta(k^{\sqrt{n}})$
Moreover, it is easy to see that the isometry group $L_{n}$ is isomorphic to
$S_{k}^{k^{n}}\times S_{2^{n}}$. Moreover, the distance between any words
$w_{n}\in L_{n}^{\prime}$, $w_{m}\in L_{m}$, where $m>n$, is equal to
$l(L_{m})-l(L_{n})$, because to get a shorter words from a longer one, it is
enough to remove everything except the prefix $(a_{1}...a_{k})^{l(L_{n})}$,
and then select the correct character from each $a_{1}...a_{k}$.
It follows from this that for any sequence $\phi_{1},\phi_{2},...$ of
isometries of languages $L_{1},L_{2},...$ it is true that
$\phi:w_{n}\mapsto\phi(w_{n})$ for all $n\in\mathbb{N}$, $w_{n}\in L_{n}$ is
an isometry of $\bigcup_{n=1}^{\infty}L_{n}$.
Moreover, $\bigcup_{n=1}^{\infty}L_{n}$ has no other isometries.
Indeed, $\Lambda$ goes into itself as the only word without neighbors at a
distance of $1$. So the length of words in this language (that is, the
distance to $\Lambda$) is an invariant.
In other words,
$Isom_{\theta}(\bigcup_{n=1}^{\infty}L_{n})=\Pi_{n=1}^{\infty}Isom_{\theta}(L_{n})=S_{k}^{\infty}\times\Pi_{n=1}^{\infty}S_{k^{n}}$.
$\square$.

##  8 Proof of Theorem 5

######  Lemma 5.

Let $L\subset\\{0,1\\}^{n}$, $\theta\in(0;2]$. Then
$Isom_{\theta}(L((01)^{n})^{*})\cong Isom(L)^{\mathbb{N}}$.

###### Proof.

Let $u,v\in L$ and $p,q\in\mathbb{N}$. Let us show that

$lev_{\theta}(u(01)^{pn},v(01)^{qn})=\begin{cases}2n|p-q|&\quad p\neq q\\\
d(u,v)&\quad p=q\end{cases}$

Indeed, in the first case, a longer word is obtained from a shorter one by
removing everything superfluous. In the second case, it is enough to perform
transformations only on different prefixes.
From this, in particular, it follows that any transformation of the form
$\phi:u(01)^{kn}\mapsto\psi(k)(u)(01)^{kn}$, where $\psi$ – - an arbitrary
function $\mathbb{N}\to Isom_{\theta}(L)$, is an isometry of
$(L((01)^{n})^{*},lev_{\theta})$.
To verify that there are no other isometries, note that $u(01)^{kn}$ has
exactly $2|L|$ ”neighbours” at a distance of $2tn$, for $t\leqslant k$ and
exactly $|L|$ for $t>k$.

∎

Using Theorem 2, we can construct uniform languages
$L_{1}\subset\\{0,1\\}^{n}$ and $L_{2}\subset\\{0,1\\}^{m}$ such that
$Isom_{\theta}(L_{1})\cong G$, while $Isom_{\theta}(L_{2})\cong H$.
By Lemma 3 $Isom_{\theta}(L_{2}((01)^{m})^{*})\cong H^{\mathbb{N}}$.
Now consider the language $L_{1}\cup(01)^{n+m}L_{2}((01)^{m})^{*}$. It is easy
to see that $lev_{\theta}(u,(01)^{n+m}v(01)^{mk})=2(n+(k+2)m)$.
Hence, for every $\phi\in Isom_{\theta}(L_{1})$ and $\psi\in
Isom_{\theta}(L_{2}((01)^{mk})^{*})$ the map $\chi$ taking all $u\in L_{1}$
into $\phi(u)$, and all $(01)^{n+m}v$, where $v\in L_{2}((01)^{mk})^{*}$, — in
$(01)^{n+m}\psi(v)$, is an isometry of
$L_{1}\cup(01)^{n+m}L_{2}((01)^{mk})^{*}$.
Moreover, there can be no other isometries, since the word $v$ from the new
language belongs to $L_{1}$ if and only if it has no ”neighbors” at a distance
greater than $n$ but less than $n+2m$.
So $Isom_{\theta}(L_{1}\cup(01)^{n+1}L_{2}((01)^{mk})^{*})\cong G\times
H^{\mathbb{N}}$. Moreover, $L_{1}\cup(01)^{n+1}L_{2}((01)^{mk})^{*}$ is
regular by construction. $\square$

##  9 Proof of Theorem 6

Consider the language $L=(010)^{*}(110)(010)^{*}\cap(\\{0,1\\}^{6})^{*}$. Let
us show that for arbitrary two words $u,w\in L$ such that $|u|-|v|=6$, $v$ can
be obtained from $u$ with exactly $6$ deletions. Note that at positions
dividing $3$ (we assume that the numbering of positions starts from $0$), each
word can contain only one unit. Let these special units in the words $u$ and
$v$ be at positions $3i$ and $3j$, respectively. Then, if $i=j$, it suffices
to remove the suffix of length $6$ (of the form $010010$) from $u$. Otherwise,
you must first remove the subword $u_{3i}u_{3i+1}u_{3i+2}=110$ from $u$, and
then from the subword $u_{3j}u_{3j+1}u_{3j+2}u_{3j+3}=0100$ (numbering in the
new order after deletion) remove $u_{3j}$ and $u_{3j+2}u_{3j+3}$. Thus we get
$v$.
From the above it follows by induction that for $u,v\in L$

$lev_{\theta}(u,v)=\max(||u|-|v||,2)$

. Thus we see that the distance between words does not depend on anything
other than their length. This means that for any sequence of permutations
$\sigma_{i}$ of elements $L\cap A^{6i}$ the map
$\phi:u\mapsto\sigma_{\frac{|u|}{6}}(u)$ is an isometry .
At the same time, the absence of other isometries follows from the fact that
each word $u\in L$ has exactly $\frac{2|u|}{3}$ neighbors at a distance of
$6$.
That is, since $|L\cap A^{6i}|=2i$,
$Isom_{\theta}(L)\cong\Pi_{n=1}^{\infty}S_{2n}$. $\square$

##  10 Thanks

I express my gratitude to Anton Klyachko and Alexander Olshansky for valuable
comments on my work, as well as to Alexei Talambutsa for information about
Higman’s paper [2], which contains the lemma used by me in the proof of
Theorem 1.

## References

  * [1] R. Frucht  “Graphs of degree three with a given abstract group”  Canadian Journal of Mathematics, Vol. 1, 1949, pp. 365–378
  * [2] G. Higman  “Ordering by divisibility in abstract algebras”  Proceedings of the London Mathematical Society, Vol. 2, 1952, pp. 326–336
  * [3] A.A. Markov  “On transformations, not spreading distortions”  Selected works, Vol. II, 1956, pp. 70–94
  * [4] V.I. Levenshtein  “Binary codes capable of correcting deletions, insertions, and reversals.”  Soviet physics. Doklady, Vol. 163, 1965, pp. 845–848
  * [5] R. Wagner and M. Fisher  “The string to string correction problem”  Journal of the ACM, Vol. 21, 1974, pp. 168–178
  * [6] P.E. Ruth and M.E. Ladser  “Levenshtein graphs: resolvability, automorphisms and determining sets”  arXiv:2107.06951, 2021


# Squared torsion $f(T,\mathcal{T})$ gravity and its cosmological implications

Simran Arora 0000-0003-0326-8945<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Aaqid Bhat<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

We present the coupling of the torsion scalar $T$ and the trace of energy-
momentum tensor $\mathcal{T}$, which produces new modified $f(T,\mathcal{T})$
gravity. Moreover, we consider the functional form
$f(T,\mathcal{T})=\alpha\mathcal{T}+\beta T^{2}$ where $\alpha$ and $\beta$
are free parameters. As an alternative to a cosmological constant, the
$f(T,\mathcal{T})$ theory may offer a theoretical explanation of the late-time
acceleration. The recent observational data to the considered model especially
the bounds on model parameters is applied in detail. Furthermore, we analyze
the cosmological behavior of the deceleration, effective equation of state and
total equation of state parameters. However, it is seen that the deceleration
parameter depicts the transition from deceleration to acceleration and the
effective dark sector shows a quintessence-like evolution.

Keywords: $f(T,\mathcal{T})$ gravity; acceleration; observational constraints;
equation of state

##  I Introduction

The in-depth verification of late time acceleration has led to immense
research towards its explanation. It is commonly known by the observations of
type Ia Supernovae [1, 2], BAO [3, 4], CMB [5], and $H(z)$ measurements [6].
The dark energy, which tried to explain the late-time acceleration as the
outcome of a type of energy connected to the cosmological constant, is one of
the successful primary models. In order to navigate the path beyond the
typical dark energy models, one can go beyond the general theory of relativity
by modifying the geometry. Alternative theories such as $f(R)$ gravity [7, 8,
9], a coupling between matter and curvature through $f(R,\mathcal{T})$ gravity
[10, 11], where $\mathcal{T}$ is the trace of energy momentum tensor, $f(R,G)$
[12, 13] ($G$ is the Gauss-Bonnet) have all attempted to explain the dark
energy phenomenon in the context of curvature.
As a result, more general geometries than the Riemannian, which may be valid
at solar system level, may provide an explanation for the behavior of matter
at large scales in the universe. There has been a rising interest in
teleparallel gravity, a different type of modified gravity that uses torsion
instead of curvature. The basic idea behind the teleparallel approach is to
replace the metric of spacetime by a set of tetrad vectors which is the
physical variable describing the gravitational properties. Moreover, this
mathemcatical development employs a different connection known as the
Weitzen$\ddot{o}$ck connection.
When one extends the action of the modified gravity based on torsion, a
separate and intriguing class of modified gravity arises named as the
teleparallel equivalent of general relativity or $f(T)$ gravity. However, a
number of analyses in $f(T)$ gravity such as cosmological solutions [14], late
time acceleration [15, 16], thermodynamics [17], cosmological perturbations
[18], cosmography [19] have been applied in the literature. For a thorough
analysis of $f(T)$ gravity, one can check [20].
Another new suggestion in modified gravity is to employ the coupling between
the torsion and trace of energy-momentum tensor known as $f(T,\mathcal{T})$
theory, in a similar fashion as $f(R,T)$ gravity. The $f(T,\mathcal{T})$
gravity has been proposed in [21], and its consistency with cosmological data
and the necessary physical conditions for a coherent cosmological theory still
has to be validated. The coupling of torsion and matter expands the
possibilities for describing the characteristics of dark energy or, more
specifically, what is driving the observed acceleration. This theory has been
investigated in the context of reconstruction and stability [22, 23], late-
time acceleration and inflationary phases [21], growth factor of sub-horizon
modes [24], quark stars [25].
The goal of the current study is to construct the extended coupled-matter
modified gravity by starting with TEGR rather than GR. The construction of
$f(T,\mathcal{T})$ gravity, which allows for arbitrary functions of the
torsion scalar $T$ and the trace of the energy-momentum tensor $\mathcal{T}$,
is the focus of the current effort. In this paper, we investigate a squared-
torsion $f(T,\mathcal{T})$ model, that raises a question on the viability of
such a theory as a candidate to account for late-time acceleration. Further,
the parameters are constrained using the set of observational datasets and in
particular, we check the late-time accelerating behavior holds true for
$f(T,\mathcal{T})$ using the cosmological parameters.
The plan of the work is the following: Starting from the background of $f(T)$,
we introduce the framework of $f(T,\mathcal{T})$ gravity in section II.
Section III is devoted to the cosmological framework and the solutions to the
field equations. Specifically, in section IV, we deal with the observational
data and methodology used to constrain the parameters involved. The late-time
accelerated phase is examined in section V through cosmological evolution.
Finally, a conclusion is given in section VI.

##  II Field equations

The fundamental preliminaries for the reconstruction of the $f(T)$ and
$f(T,\mathcal{T})$ theories of gravity are presented in this section.
One needs a new connection, the $Weitzenb\ddot{o}ck$ connection [26] to obtain
the torsion-based theory defined as
$\tilde{\Gamma}^{\alpha}_{\mu\nu}=e_{a}^{\,\,\alpha}\partial_{\nu}e^{a}_{\,\,\mu}$,
where $e^{a}_{\,\,\mu}$ and $e_{a}^{\,\,\alpha}$ are tetrads (or vierbeins).
These vierbeins relate to the metric tensor $g_{\mu\nu}$ at each point x of
the spacetime manifold as

$g_{\mu\nu}(x)=e^{a}_{\,\,\mu}(x)e^{b}_{\,\,\nu}(x)\eta_{ab}.$ (1)

Here, $\eta_{ab}=diag(1,-1,-1,-1)$ is the Minkowski metric tensor. Hence, the
torsion tensor describing the gravitational field is

$T^{\alpha}_{\,\,\mu\nu}=\Gamma^{\alpha}_{\,\,\nu\mu}-\Gamma^{\,\,\alpha}_{\mu\nu}=e_{a}^{\,\,\alpha}\left(\partial_{\mu}e^{a}_{\,\,\nu}-\partial_{\nu}e^{a}_{\,\,\mu}\right).$
(2)

We define the contortion and the superpotential tensor through the components
of the torsion tensor

$\displaystyle K^{\mu\nu}_{\,\,\alpha}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(T^{\mu\nu}_{\,\,\alpha}-T^{\nu\mu}_{\alpha}-T_{\alpha}^{\,\,\mu\nu}\right),$
(3) $\displaystyle S_{\alpha}^{\,\,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(K^{\mu\nu}_{\,\,\alpha}+\delta^{\mu}_{\alpha}T^{\lambda\mu}_{\,\,\lambda}-\delta^{\nu}_{\alpha}T^{\lambda\mu}_{\,\,\lambda}\right).$
(4)

Using equations (2) and (4), one obtain the torsion scalar [21, 27, 20]

$T=S_{\alpha}^{\,\,\mu\nu}T^{\alpha}_{\,\,\mu\nu}=\frac{1}{2}T^{\alpha\mu\nu}T_{\alpha\mu\nu}+\frac{1}{2}T^{\alpha\mu\nu}T_{\nu\mu\alpha}-T_{\alpha\mu}^{\,\,\,\,\alpha}T^{\nu\mu}_{\,\,\,\,\nu}.$
(5)

One can define the gravitational action for teleparallel gravity by

$S=\int d^{4}x\,e\,[T+\mathcal{L}_{m}],$ (6)

where $e=det(e^{a}_{\,\,\mu})=\sqrt{-g}$ and $\mathcal{L}_{m}$ is the matter
Lagrangian. In fact, one can extend $T$ to $T+f(T)$, the so called $f(T)$
gravity. Moreover, it can be generalized to become a general function of both
the torsion scalar and the trace of the energy-momentum tensor $\mathcal{T}$,
which results in the $f(T,\mathcal{T})$ gravity.
The gravitational action for $f(T,\mathcal{T})$ gravity is given by

$S=\frac{1}{16\pi G}\int d^{4}x\,e[T+f(T,\mathcal{T})]+\int
d^{4}x\,e\,\mathcal{L}_{m}$ (7)

Varying the action with respect to the vierbeins yields the field equations

$(1+f_{T})\left[e^{-1}\partial_{\mu}(ee_{a}^{\,\,\alpha}S_{\alpha}^{\,\,\lambda\mu})-e_{a}^{\,\,\alpha}T^{\mu}_{\nu\alpha}S_{\mu}^{\,\,\nu\lambda}\right]+e_{a}^{\,\,\lambda}\left(\frac{f+T}{4}\right)+\\\
\left(f_{TT}\partial_{\mu}T+f_{T\mathcal{T}}\partial_{\mu}\mathcal{T}\right)e_{a}^{\,\,\alpha}S_{\alpha}^{\,\,\lambda\mu}-f_{\mathcal{T}}\left(\frac{e_{a}^{\,\,\alpha}\overset{em}{T}_{\alpha}^{\,\,\lambda}+p_{m}e_{a}^{\,\,\lambda}}{2}\right)=\\\
4\pi Ge_{a}^{\,\,\alpha}\overset{em}{T}_{\alpha}^{\,\,\lambda}.$ (8)

where $f_{\mathcal{T}}={\partial f}/{\partial\mathcal{T}}$,
$f_{T\mathcal{T}}={\partial^{2}f}/{\partial T\partial\mathcal{T}}$, and
$\overset{em}{T}_{\alpha}^{\,\,\lambda}$ is the energy-momentum tensor.

We incorporate the flat FRW metric as usual to apply the aforementioned theory
in a cosmological framework to obtain modified Friedman equations. The FRW
metric read

$ds^{2}=dt^{2}-a(t)^{2}\delta_{ij}dx^{i}dx^{j},$ (9)

where $a(t)$ is the scale factor. Further, (8) give rise to modified Friedmann
equations:

$H^{2}=\frac{8\pi
G}{3}\rho_{m}-\frac{1}{6}\left(f+12H^{2}f_{T}\right)+f_{\mathcal{T}}\left(\frac{\rho_{m}+p_{m}}{3}\right),$
(10) $\dot{H}=-4\pi
G(\rho_{m}+p_{m})-\dot{H}(f_{T}-12H^{2}f_{T\mathcal{T}})\\\
-H(\dot{\rho_{m}}-3\dot{p_{m}})f_{T\mathcal{T}}-f_{\mathcal{T}}\left(\frac{\rho_{m}+p_{m}}{2}\right).$
(11)

Here, $\mathcal{T}=\rho_{m}-3p_{m}$ in the above equation is true for the
perfect matter fluid.

Comparing the modified Friedmann equations (10) and (11) to General Relativity
equations

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{eff}\right),$ (12) $\displaystyle\dot{H}$
$\displaystyle=$ $\displaystyle-4\pi
G\left(\rho_{m}+p_{m}+\rho_{eff}+p_{eff}\right).$ (13)

we obtain

$\rho_{eff}=\frac{1}{16\pi
G}[f+12f_{T}H^{2}-2f_{\mathcal{T}}(\rho_{m}+p_{m})]$ (14)

$p_{eff}=\frac{1}{16\pi
G}[f+12f_{T}H^{2}-2f_{\mathcal{T}}(\rho_{m}+p_{m})]+\\\
(\rho_{m}+p_{m})\left[\frac{(1+\frac{f_{T}}{8\pi
G})}{1+f_{T}12H^{2}f_{TT}+H(\frac{d\rho_{m}}{dH})(1-3{c_{s}}^{2})f_{T\mathcal{T}}}-1\right]$
(15)

The effective and total equation-of-state parameter is defined as follows;

$\displaystyle\omega_{eff}$ $\displaystyle=$
$\displaystyle\frac{p_{eff}}{\rho_{eff}},$ (16) $\displaystyle\omega$
$\displaystyle=$ $\displaystyle\frac{p_{eff}+p_{m}}{\rho_{eff}+\rho_{m}}.$
(17)

We consider $p_{m}=0$ for the dust universe which implies
$\omega=\frac{\omega_{eff}}{1+\frac{\rho_{m}}{\rho_{eff}}}$. The conservation
equation involving the effective energy and pressure reads

$\dot{\rho}_{eff}+\dot{\rho_{m}}+3H(\rho_{m}+\rho_{eff}+p_{m}+p_{eff})=0.$
(18)

##  III Cosmology

This section examines the cosmological impacts of $f(T,\mathcal{T})$ gravity
while emphasizing on a specific model. We consider the functional form
$f(T,\mathcal{T)}=\alpha\mathcal{T}+\beta T^{2}=\alpha\rho_{m}+\beta
T^{2}=\alpha\rho_{m}+\gamma H^{4}$, where $\alpha$ and $\gamma=36\beta$ are
constants [21]. For simplicity, we use $8\pi G=c=1$. The model defines a
straightforward deviation from GR inside the framework of $f(T,\mathcal{T)}$.
The case $\alpha=0$, the model behaves as a power-law cosmology in $f(T)$
theory [19]. In this case, we obtain $f_{T}=\frac{\gamma T}{18}$,
$f_{TT}=\frac{\gamma}{18}$ ,$f_{\mathcal{T}}=\alpha$, $f_{T\mathcal{T}}=0$.
Hence, using the above expressions and equations (10) & (11), we have the
following:

$\displaystyle\rho_{m}$ $\displaystyle=$
$\displaystyle\frac{3\left(1-\frac{\gamma
H^{2}}{2}\right)}{1+\frac{\alpha}{2}}H^{2},$ (19) $\displaystyle\dot{H}$
$\displaystyle=$ $\displaystyle-\frac{3(1+\alpha)\left(1-\frac{\gamma
H^{2}}{2}\right)}{(\alpha+2)(1-\gamma H^{2})}H^{2},$ (20) $\displaystyle q$
$\displaystyle=$ $\displaystyle\frac{3(1+\alpha)\left(1-\frac{\gamma
H^{2}}{2}\right)}{(\alpha+2)(1-\gamma H^{2})}-1.$ (21)

Moreover, the effective dark energy density and pressure from equations (14)
and (15) can be obtained as

$\displaystyle\rho_{eff}$ $\displaystyle=$
$\displaystyle\frac{3H^{2}(\alpha+\gamma H^{2})}{\alpha+2},$ (22)
$\displaystyle p_{eff}$ $\displaystyle=$
$\displaystyle-\frac{3H^{2}(\alpha+\gamma H^{2})}{(\alpha+2)(\gamma
H^{2}-1)}.$ (23)

which gives $\omega_{eff}=\frac{1}{1-\gamma H^{2}}$.
Now, we replace the term $d/dt$ by $d/dlna$ via the expression
$d/dt=H\frac{d}{dlna}$, ($a=\frac{1}{1+z}$) such that solution of equation
(20) is

$H(z)=H_{0}\sqrt{\frac{\sqrt{1-2^{-2A}(2z+2)^{2A}\left(2\gamma
H_{0}^{2}-\gamma^{2}H_{0}^{4}\right)}+1}{\gamma H_{0}^{2}}}$ (24)

where $A=\frac{3(\alpha+1)}{\alpha+2}$.

##  IV Observational constraints and Methodology

In this section, we will conduct a statistical study utilising the Monte Carlo
Markov Chain (MCMC) approach, where we compare the predictions with data sets
to the cosmic observations, to assess the viability of a model. In particular,
we use Type Ia Supernovae (SNeIa) data, Baryon acoustic oscillation (BAO) data
and the Observational Hubble data (H(z)).

###  IV.1 SNeIa data

Since Type Ia Supernovae act as “standard candles” and allow us to estimate
cosmic distance. They are widely applied to impose constraints on the dark
energy sector. In particular, we use the Pantheon compilation of 1048 points
spanning the redshift range $0.01<z<2.26$ [28]. The $\chi^{2}$ function is
given as

$\chi^{2}_{SN}=\Delta\mu C^{-1}_{SN}\Delta\mu^{T},$ (25)

where $\Delta\mu=\mu_{i}-\mu_{th}$ is the difference between the observational
and theoretical distance modulus, and $C^{-1}_{SN}$ corresponds to the inverse
covariance matrix of the data. Further, we define $\mu=m_{B}-M_{B}$, where
$m_{B}$ is the observed apparent magnitude at a given redshift, while $M_{B}$
is the absolute magnitude (Retrieving the nuisance parameters according to the
new approach called BEAMS with Bias Correction (BBC) [29]). The theoretical
value is computed as

$\displaystyle\mu_{th}$ $\displaystyle=$ $\displaystyle
5log_{10}\left[\frac{d_{L}}{1Mpc}\right]+25,$ (26) $\displaystyle d_{L}$
$\displaystyle=$ $\displaystyle c(1+z)\int_{0}^{z}\frac{dy}{H(y,\theta)}.$
(27)

where $\theta$ is the parameter space.

###  IV.2 Hubble data

We make use of Hubble parameter measurements derived from the differential age
method (often known as cosmic chronometer (CC) data). Here, we consider 31
points compiled in [30]. The $\chi^{2}$ function is given as

$\chi^{2}_{Hz}=\sum_{i=1}^{31}\frac{\left[H(z_{i})-H_{obs}(z_{i})\right]^{2}}{\sigma(z_{i})^{2}},$
(28)

where $H_{obs}$ is the observed value, $\sigma(z_{i})$ is the observational
error.

###  IV.3 BAO

Baryon acoustic oscillations (BAO) are pressure waves generated by
cosmological perturbation in the baryon-photon plasma at the recombination
epoch and appear as distinct peaks on large angular scales. We use BAO
measurements from the Six Degree Field Galaxy Survey (6dFGS), Sloan Digital
Sky Survey (SDSS), the LOWZ samples of Baryon Oscillation Spectroscopic Survey
(BOSS) [31, 32]. The expressions used for BAO data are

$\displaystyle d_{A}(z)$ $\displaystyle=$ $\displaystyle
c\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})},$ (29) $\displaystyle D_{v}(z)$
$\displaystyle=$
$\displaystyle\left[\frac{d_{A}(z)^{2}cz}{H(z)}\right]^{1/3},$ (30)
$\displaystyle\chi_{BAO}^{2}$ $\displaystyle=$ $\displaystyle X^{T}C^{-1}X.$
(31)

Here, $d_{A}(z)$ is the comoving angular diameter distance, and $D_{v}(z)$ is
the dilation scale and $C$ is the covariance matrix [33].

###  IV.4 Results

The statistical results for the model is shown as contour plots in Figs. 1 and
2. Also, table 1 corresponds to the values obtained for the parameter space to
the combination of data sets. We noticed weaker constraints in case of $BAO$
whereas stronger constraints for $SNeIa$ and $joint(Hz+BAO+SNeIa)$ analysis.
We assumed $\gamma=0.0006$ [21] so that $\frac{\gamma H^{2}}{2}<1$. We can
observe that the BAO data is anti-correlated with other data sets. We also get
$H_{0}$ constraint consistent with Planck results [34] favored by
$\Lambda$CDM.

Figure 1: One-dimensional and two-dimensional marginalized confidence regions (68% CL and 95% CL) for $\alpha$, $H_{0}$ obtained from the $Hubble$, $BAO$ and $Pantheon$ data for the f(T,$\mathcal{T}$) gravity model. Figure 2: One-dimensional and two-dimensional marginalized confidence regions (68% CL and 95% CL) for $\alpha$, $H_{0}$ obtained from the $Hubble+BAO+Pantheon$ data for the f(T,$\mathcal{T}$) gravity model. Figure 3: Plot shows the expansion rate of the universe with theoretical predictions (red curve) and $\Lambda$CDM (black curve with $\Omega_{\Lambda_{0}}=0.7$ and $\Omega_{m_{0}}=0.3$). The blue dots represent 31 Hubble points with the corresponding error bars. Figure 4: Plot shows the $\mu$-redshift relation of Pantheon SN sample with theoretical predictions (red curve) and $\Lambda$CDM (black curve). with $\Omega_{\Lambda_{0}}=0.7$ and $\Omega_{m_{0}}=0.3$). The blue dots represent 1048 Pantheon points with the corresponding error bars. Table 1: Best-fit values of model parameters obtained from observational datasets Datasets | $\alpha$ | $H_{0}$ | $z_{t}$ | $q_{0}$ | $\omega_{0}$
---|---|---|---|---|---
$BAO$ | $8.55^{+0.63}_{-0.53}$ | $69.20^{+1.0}_{-1.0}$ | $0.36^{+0.04}_{-0.04}$ | $-0.36^{+0.03}_{-0.04}$ | $-0.57^{+0.024}_{-0.024}$
$Hz$ | $4.21^{+0.82}_{-0.82}$ | $68.01^{+0.71}_{-0.71}$ | $0.60^{+0.141}_{-0.101}$ | $-0.45^{+0.04}_{-0.04}$ | $-0.63^{+0.024}_{-0.026}$
$SNeIa$ | $3.85^{+0.79}_{-0.91}$ | $67.66^{+0.48}_{-0.48}$ | $0.65^{+0.198}_{-0.092}$ | $-0.46^{+0.02}_{-0.04}$ | $-0.64^{+0.019}_{-0.024}$
$Hz+BAO+SNeIa$ | $3.83^{+0.0009}_{-0.0009}$ | $67.67^{+0.0010}_{-0.0010}$ | $0.65^{+0.0004}_{-0.0017}$ | $-0.46^{+0.0001}_{-0.937}$ | $-0.64^{+0.00003}_{-0.00003}$

##  V Cosmological Evolution

The plots in this section demonstrate how the universe can have very
intriguing dynamics depending on the values of the parameters. The Hubble
function, presented in figure 3 is a monotonically increasing function of
redshift throughout the entire evolution of the universe.
Figure 5 shows that the universe begins its history from deceleration ($q>0$)
and shows accelerating phase ($q<0$) after a transition redshift $z_{t}$. The
deceleration parameter is defined as $q=-\frac{\dot{H}}{H^{2}}-1$. This
evolution is consistent with the recent universe behaviour as it went through
three stages: a decelerating dominated phase, an accelerating expansion phase,
and a late-time accelerating phase. Keep in mind that the universe terminates
in a de Sitter expansion at asymptotically lower redshifts. We find that the
present value of deceleration parameter ($q_{0}$) [35, 36] and $z_{t}$ [37,
38] is in good agreement with $SNeIa$ and $Hz+BAO+SNeIa$ data sets.

Figure 5: Variation of the deceleration parameter $q$ as a function of the
redshift $z$ for different data sets. Figure 6: Variation of $\omega_{eff}$ as
a function of the redshift $z$ for different data sets

Determining the equation of state’s value and its evolution is another attempt
to comprehend the existence of dark energy. The equation of state
($\omega_{eff}$) in figure 6 show a similar evolution, moving towards negative
at lower redshifts. Moreover, we show the total equation of state parameter
($\omega$) in figure 7. Hence, the both the equation of state parameter lie in
the quintessence regime ($-1<\omega<0$), approaching the cosmological constant
($\omega=-1$) at smaller redshifts. We find that the present value of
$\omega_{0}$ is in good agreement with $SNeIa$ and $Hz+BAO+SNeIa$ data sets.

Figure 7: Variation of $\omega$ as a function of the redshift $z$ for
different data sets

##  VI Conclusion

Inspired by the teleparallel-formulation of general relativity, we attempted
to investigate the extension of $f(T)$ gravity based on the coupling between
the torsion scalar $T$ and the trace of energy-momentum tensor $\mathcal{T}$.
The essential point is that both $f(T)$ components, as well as the matter
energy density and pressure, contribute to the effective dark energy sector.
The additional freedom of the imposed Lagrangian in the $f(T,\mathcal{T})$
cosmology allows for a very wide range of conditions and behaviors.

In the current study, we investigated the cosmological implications of
$f(T,\mathcal{T})$ theory. We considered the squared-torsion model
$f(T,\mathcal{T})=\alpha\mathcal{T}+\beta T^{2}$, where $\alpha$ and $\beta$
are free parameters. We obtained the solution of modified Friedmann equations
in the form of Hubble parameter as a function of redshift $z$. Further, in
section III, we employed the recent observational data: $Hubble$, $BAO$,
$SNeIa$ and the $joint$ analysis to constrain the unknown model parameters. In
figures 1 and 2, we obtained the best-fit values of model parameters. In
comparison with the $\Lambda$CDM model, the obtained $H(z)$ and the $\mu(z)$
of the considered model are confronted to the cosmic data in figures 3 and 4
respectively.

Depending on the model parameters constrained, we discovered a wide range of
intriguing cosmological behaviors. For instance, we found evolution of
deceleration parameter, explicitly experiencing a change from a deceleration
to acceleration, capable of explaining the late-time universe. Additionally,
the effective EoS ($\omega_{eff}$) and total EoS ($\omega$) behaves in a
similar fashion demonstrating that the cosmic fluid has the characteristics of
quintessence dark energy. Moreover, we find that the present values of
$q_{0}$, $\omega_{0}$ and $z_{t}$ are in good agreement with $SNeIa$ and
$Hz+BAO+SNeIa$ data sets.
Finally, it is essential to point that $f(T,\mathcal{T})$ subject to
observational data can explain late-time accelerating universe and can be
applied to different regimes to establish a viable gravitational formalism.
Furthermore, the perturbation analysis could be extended to the vector and
tensor analysis which is useful in predicting the inflationary scenario. We
hope that this analysis will encourage readers to consider torsional modified
gravity as a candidate to describe the universe.

###### Acknowledgements.

SA acknowledges BITS-Pilani, Hyderabad Campus for the financial support. AB
acknowledges University Grants Commission (UGC) Maulana Azad National
Fellowship (MANF), New Delhi, India for awarding Junior Research Fellowship
(UGC-Ref.No.: 211610222082). SA & PKS acknowledge IUCAA, Pune, India for
providing support through the visiting Associateship program.

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Neutrino-antineutrino Asymmetry of C$\bm{\nu}$B on the Surface of the Round
Earth

Guo-yuan Huang1*** E-mail<EMAIL_ADDRESS>
1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany

###### Abstract

It has been claimed that the coherent scattering of relic neutrinos with the
Earth will result in a neutrino-antineutrino asymmetry of
$\mathcal{O}(10^{-4})$ on the Earth surface, which is five orders of magnitude
larger than the naive model expectation. In this work we show that this
overdensity was overestimated for the perfectly round Earth by solving the
exact solution with partial waves. The actual asymmetry is only around
$6\times 10^{-6}$ at $1~{}{\rm cm}$ above the ground. This value will quickly
drop to $\mathcal{O}(10^{-9})$ at the height of $10~{}{\rm m}$. To achieve the
proposed asymmetry of $\mathcal{O}(10^{-4})$, a special geography may be
needed as the experimental site.

##  1 Introduction

The detection of relic neutrinos remains one of the most fundamental
challenges of experimental particle physics. The most promising way so far is
by observing the inverse beta decays, proposed by S. Weinberg in 1962 [1],
even though there are still several technical obstacles towards a feasible
observation [2, 3, 4, 5, 6]. Besides this method, an alternative is to observe
the collective effects induced by the wind of relic neutrinos [7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Compared
to the radioactive process, the major advantage of coherent scatterings is
that the event rate can be enhanced by a factor as large as the Avogadro
constant $N_{\rm A}\approx 6.02\times 10^{23}$, which may help to compensate
the suppression of Fermi coupling constant $G_{\rm F}$. There are mainly two
effects that have been investigated in this direction: one is the mechanical
recoil induced by the net collision; the other is the Stodolsky effect [8],
i.e., a net torque due to the spin energy splitting in the presence of a
neutrino-antineutrino asymmetry. The challenge of the first effect is that the
mechanical recoil will be distributed among all the atoms in the ensemble, and
hence the ultimate induced acceleration is highly suppressed. Whereas, the
Stodolsky effect is proportional to the neutrino-antineutrino asymmetry
$\eta_{\nu\overline{\nu}}$, which is expected to be of the same order of
magnitude as the Baryon asymmetry $\mathcal{O}(10^{-9})$ from many models.
This number is too small to have any observable effects.

Surprisingly, it has been recently noticed that this asymmetry can be enhanced
to $\eta_{\nu\overline{\nu}}=\mathcal{O}(10^{-4})$ due to the reflection
effect on the Earth surface [25]. This is not difficult to understand as
neutrinos and antineutrinos feel opposite matter potentials of the Earth. In
particular, the total external reflection occurs for neutrinos as in optics
when the incidence angle of relic neutrino flux approaches the critical angle.
With an isotropic neutrino flux, the reflected waves were claimed to result in
a large neutrino-antineutrino asymmetry near the Earth surface, which is five
orders of magnitude larger than $\mathcal{O}(10^{-9})$. If confirmed, this
effect would be very encouraging for the observation of relic neutrinos.

The above conclusion was primarily drawn by approximating the Earth to the toy
model of a slab with an infinite thickness, and neutrinos were assumed to be
injected from vacuum above the slab surface. For the round Earth with a finite
size, this approximation appears to be oversimplified, because trajectories of
any incident neutrinos will eventually emerge from the Earth surface. A better
approximation of the round Earth might be a slab with a finite thickness. The
thickness of the slab can be roughly represented by the depth that neutrinos
can reach inside the Earth with a certain incidence angle. The neutrino
overdensity on the Earth surface is in principle given by the neutrino
wavefunction integrated over all incoming angles. In this case, there will be
neutrinos injected from both above and below one slab surface. Unfortunately,
we find that accounting for those neutrinos coming from the other side of the
slab will reduce the overdensity significantly. To reinforce our result, we
have calculated this overdensity without the slab approximation, by directly
considering the potential of the round Earth and solving the scattering
problem with partial waves.

The structure of the rest of this work is as follows. In Sec. 2, we
investigate the ideal reflection of relic neutrinos by a finite-thickness
slab, and discuss how the overdensity will be altered for an isotropic
neutrino injection. In Sec. 3, the neutrino wavefunction by collectively
scattering with the round Earth will be solved by summing up contributions of
different partial waves. We finally make our conclusion and remark on a
possible solution in Sec. 4.

##  2 Reflection of Neutrinos by a Finite-thickness Slab

To solve the neutrino wavefunction, one has to investigate the scattering of
incident neutrino waves with a spherical potential. However, if the effect of
curvature of the Earth surface is small enough, one may approximate the
scenario of concern to the coherent scattering of plane waves with a finite-
thickness slab. The depth of the slab will be around $d=R(1-\cos{\theta})\sim
R\,\theta^{2}/2$ with $\theta$ being the incidence angle with respect to the
horizon and $R=6371~{}{\rm km}$ being the Earth radius. The corresponding
zenith angle is just $\alpha=\pi/2-\theta$. A schematic diagram of the
scenario of concern is shown in Fig. 1.

Figure 1: A schematic plot for the coherent scattering of neutrinos with a
slab with the thickness $d$. The detector is placed at height $h$ to the slab
surface. The overdensity receives three contributions: (i) the reflected wave
solely $|\Psi_{\rm R}|^{2}$; (ii) the interference between the original and
reflected waves $2\mathrm{Re}(\Psi_{\rm 0}\Psi^{*}_{\rm R})$; (iii) the
transmitted wave from the other side of the slab $|\Psi_{\rm T}|^{2}$.

Considering the above approximation, we first investigate the case that a
collimated monoenergetic neutrino beam is incident on an infinitely wide slab
with the thickness $d$. By collectively interacting with the matter in the
slab, neutrino waves will undergo both reflection and transmission near the
slab surface. In particular, a positive potential in matter can result in the
total external reflection of neutrinos when the incidence angle is smaller
than the critical angle $\theta_{\rm c}$. The detailed outcome of the
scattering depends on the neutrino mass, the neutrino momentum and the matter
potential. For the convenience of later discussions, we set the neutrino mass
to be $m_{\nu}=0.1~{}{\rm eV}$ with a momentum $k_{\nu}\approx 2.7\times
10^{-4}~{}{\rm eV}$ following Ref. [25]. The matter potential describing the
coherent forward scattering of electron neutrinos is also chosen to be
$V_{\nu_{e}}=-V_{\overline{\nu}_{e}}\approx 1.8\times 10^{-14}~{}{\rm eV},$
contributed by both the charged-current and neutral-current interactions. Note
that we can safely ignore the recoil of the slab induced by coherent
scatterings considering the heaviness of the Earth.

Let us denote the normal vector of the slab surface to be the $z$-axis. Since
this system features a translational symmetry in the $x$-$y$ plane along the
slab surface, one can simplify the problem to a one-dimensional scattering
with a potential barrier of the thickness $d$. The scattering behavior will
hence rely on the relation between the $z$ component of the neutrino momentum
$k_{\nu\perp}$ and the potential $V$. If $k_{\nu\perp}<V$, the transmission
will be exponentially suppressed but not entirely vanishing because of the
quantum tunneling effect through the barrier. The reflected and transmitted
waves in vacuum can be parameterized by $\Psi_{\rm R}=C_{\rm
R}\exp{\mathrm{i}(k_{\nu{\mkern 3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern
3.0mu\vrule depth=0.0pt\mkern 3.0mu}}x-k_{\nu\perp}z)}$ and $\Psi_{\rm
T}=C_{\rm T}\exp{\mathrm{i}(k_{\nu{\mkern 3.0mu\vphantom{\perp}\vrule
depth=0.0pt\mkern 3.0mu\vrule depth=0.0pt\mkern 3.0mu}}x+k_{\nu\perp}z)}$,
respectively. Inside the matter, the wavefunction is
$\Psi^{\prime}=C^{\prime}_{1}\exp{\mathrm{i}(k_{\nu{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 3.0mu\vrule depth=0.0pt\mkern
3.0mu}}x-k^{\prime}_{\nu\perp}z)}+C^{\prime}_{2}\exp{\mathrm{i}(k_{\nu{\mkern
3.0mu\vphantom{\perp}\vrule depth=0.0pt\mkern 3.0mu\vrule depth=0.0pt\mkern
3.0mu}}x+k^{\prime}_{\nu\perp}z)}$ with
$k^{\prime}_{\nu\perp}=\sqrt{k^{2}_{\nu\perp}-2m_{\nu}V}$ being the effective
momentum in matter. Depending on the value of $V$, $k^{\prime}_{\nu\perp}$ can
be imaginary which indicates the exponential suppression of wavefunctions in
the barrier. The critical angle for total reflection is just $\sin\theta_{\rm
c}=\sqrt{2m_{\nu}V}/k_{\nu}$ set by $k^{\prime}_{\nu\perp}=0$. For the above
chosen parameters, the critical angle is $\sin\theta_{\rm c}\approx 2\times
10^{-4}$. One can solve those coefficients by matching the wavefunctions and
their derivatives at the boundaries of the slab. We find the following
solutions to the reflection and transmission coefficients,

$\displaystyle C_{\rm R}$ $\displaystyle=\frac{(k^{2}_{\nu\perp}-k^{\prime
2}_{\nu\perp})\sin(dk^{\prime}_{\nu\perp})}{(k^{2}_{\nu\perp}+k^{\prime
2}_{\nu\perp})\sin(dk^{\prime}_{\nu\perp})+2\,\mathrm{i}\,k_{\nu\perp}k^{\prime}_{\nu\perp}\cos(dk^{\prime}_{\nu\perp})}\;,$
(1) $\displaystyle C_{\rm T}$
$\displaystyle=\frac{2\,\mathrm{i}\,\mathrm{e}^{-\mathrm{i}dk_{\nu\perp}}k_{\nu\perp}k^{\prime}_{\nu\perp}}{(k^{2}_{\nu\perp}+k^{\prime
2}_{\nu\perp})\sin(dk^{\prime}_{\nu\perp})+2\,\mathrm{i}\,k_{\nu\perp}k^{\prime}_{\nu\perp}\cos(dk^{\prime}_{\nu\perp})}\;.$
(2)

The above expressions apply to all values of $V$. For $k_{\nu\perp}<V$, one
may find it convenient to replace $k^{\prime}_{\nu\perp}$ by
$\mathrm{i}\gamma$ with $\gamma=\sqrt{2m_{\nu}V-k^{2}_{\nu\perp}}$. In the
limit of infinity thickness, we have

$\displaystyle C_{\rm R}$
$\displaystyle=\frac{k_{\nu\perp}-k^{\prime}_{\nu\perp}}{k_{\nu\perp}+k^{\prime}_{\nu\perp}}\;.$
(3)

For the infinity thickness, the total reflection occurs in the case of
imaginary $k^{\prime}_{\nu\perp}$, i.e., $|C_{\rm R}|^{2}=1$. However, we
should emphasize that this is not true anymore for the finite thickness. In
particular, for small $d$, the reflection probability takes a perturbative
value in the Born approximation [14]

$\displaystyle|C_{\rm R}|^{2}$
$\displaystyle\approx\frac{(m_{\nu}V)^{2}d^{2}}{k^{2}_{\nu\perp}}\;.$ (4)

This probability is also suppressed by the smallness of the matter potential.
Around the critical angle, the condition for $|C_{\rm R}|^{2}$ to be order one
is given by $d>(m_{\nu}V)^{-1/2}\sim 5~{}{\rm m}$. In comparison, the depth
that neutrinos can travel inside the Earth at the critical angle $\theta_{\rm
c}\sim 2.2\times 10^{-4}$ is only $0.16~{}{\rm m}$, and correspondingly
$|C_{\rm R}|^{2}\sim 6\times 10^{-4}$, which is still within the valid regime
of the Born approximation. As a result, the total reflection over a flat slab
may not be fully applicable to our round Earth. Nevertheless, it is worthwhile
to first analyze the common features by exploring the overdensity in the
simplified scenario with a sufficiently large thickness. We will solve the
overdensity without such approximations in the next section.

Above the slab surface, the neutrino overdensity is contributed by the
reflected wave as well as the interference between the original and reflected
waves. In some cases, the interference term can even be dominant. The total
overdensity should be obtained by integrating over the zenith angle $\alpha$
of incoming neutrinos. Following Ref. [25], if we account for those neutrinos
only from above the slab, the overdensity reads

$\displaystyle\delta
n_{\nu(\overline{\nu})}=\int^{1}_{0}\left[\frac{|\Psi_{\rm
R}|^{2}}{2}+\mathrm{Re}(\Psi_{\rm 0}\Psi^{*}_{\rm
R})\right]\mathrm{d}\cos{\alpha}\;,$ (5)

The asymmetry is then given by $\eta_{\nu\overline{\nu}}=(\delta
n_{\nu}-\delta n_{\overline{\nu}})/n_{\nu}$. In Fig. 2, we plot the
overdensities $\delta n_{\nu}$ and $\delta n_{\overline{\nu}}$ in terms of the
height to the slab surface $h$. To analyze the importance of different terms,
we have also depicted separate contributions from the reflected wave
$|\Psi_{\rm R}|^{2}$ and the interference $2\mathrm{Re}(\Psi_{\rm
0}\Psi^{*}_{\rm R})$. The total overdensities for neutrinos and antineutrinos
are shown as the solid red and blue curves, respectively, whose difference can
reproduce the asymmetry in Ref. [25]. For neutrinos, the overdensity of
$\mathcal{O}(10^{-4})$ is dominated by the reflected wave solely (dashed red
curve), while the interference contribution (dotted red curve) is negligible.
For antineutrinos, both the reflection and the interference are important in
general.

Figure 2: The neutrino (in red) and antineutrino (in blue) overdensities in
terms of the height $h$ to the slab surface. Separate contributions are shown
from the reflected wave alone (dashed curves) and from the interference
between the original and reflected waves (dotted curves).

As has been mentioned, a loophole of the above analysis is that neutrinos
coming from the other side of the slab are not taken into account. We find
that accounting for those flux will in general reduce the overdensity of
neutrinos. In particular, we can observe the following fact: the overdensity
induced by the total reflection over one side of the slab is right equal to
the deficit of transmitted waves from the other side due to the probability
conservation, i.e., $|\Psi_{\rm R}|^{2}=1-|\Psi_{\rm T}|^{2}$. Hence, only the
negligible interference contribution remains if we add them up. Nevertheless,
the above argument cannot be simply applied to antineutrinos, for which the
interference is important. For antineutrinos with small heights in Fig. 2, the
deficit from the other side will make the total overdensity even more sizable.
However, this is not true in practice. A careful investigation shows that the
round Earth with a negative potential serves as a convex lens, which will also
compensate the original negative overdensity of antineutrinos. In general, the
overdensities for both neutrinos and antineutrinos (and hence the asymmetry)
should be severely suppressed compared to the case with flux injection only
from one side. However, the exact value can only be obtained by solving the
scattering with the spherical potential of the Earth.

##  3 Suppressed Asymmetry for Perfectly Round Earth

In this section, we prove that the neutrino-antineutrino asymmetry is highly
suppressed for the perfectly round Earth with an isotropic injection of the
cosmic neutrino flux. This result is valid because both the flux input and the
matter potential under assumption feature the spherical symmetry. We shall
calculate the neutrino number density $n_{\nu}$ at a given point $P$ above the
Earth. For illustration, a schematic diagram for the scattering of neutrinos
with a positive potential of the Earth is given in Fig. 3.

Figure 3: The scattering of a plane-wave neutrino with the matter potential of
the Earth. The detector is placed at the height of $P$. The neutrino density
at $P$ with an isotropic flux injection is equivalent to integrating the
wavefunction over the sphere $S$ with a collimated flux.

As the first step, we notice that the spherical symmetry enables us to
translate the neutrino number density at point $P$ with the isotropic flux
injection into an integration of the density over a sphere $S$ containing $P$
but with a collimated injection. The density value at each point on the sphere
is identical to the density at the fixed point $P$ contributed by the flux
from a different incoming angle (rotate that point to match with $P$). This
equivalence is critical for our later understanding from the classical
perspective.

Suppose that a monoenergetic plane wave is incident on the Earth with the
free-particle wavefunction
$\phi(\bm{x})=\mathrm{exp}(\mathrm{i}\bm{k}\cdot\bm{x})$ far away from the
potential. The resultant wavefunction by scattering with the Earth is denoted
as $\Psi(\bm{x})$. In the absence of the Earth, we should have
$\Psi(\bm{x})=\phi(\bm{x})$. The number density with an isotropic flux
injection hence reads

$\displaystyle
n_{\nu(\overline{\nu})}(r)=\frac{n_{0}}{4\pi}\int\left|\Psi(\bm{x})\right|^{2}\mathrm{d}\Omega=\frac{n_{0}}{4\pi
r^{2}}\varoiint\left|\Psi(\bm{x})\right|^{2}\mathrm{d}S\;,$ (6)

where the normalization factor $n_{0}$ is just the number density of neutrinos
in the absence of the Earth, $r$ is the distance from the Earth center to the
detector, and the second integration is performed over the sphere $S$.

###  3.1 The Semiclassical Approximation

We attempt to prove that $n_{\nu}(r)=n_{0}$ for both neutrinos and
antineutrinos (no overdensity for the round Earth) in the semiclassical limit
by considering neutrinos as localized particles following certain
trajectories. We will find that the result holds because of several
conservation laws. In the semiclassical limit, the interference effect between
different waves has been ignored. It is then straightforward to validate the
result by explicitly counting the neutrino flux going into (in blue) and out
of (in red) the sphere $S$ as shown in Fig. 3. From the particle number
conservation, we may first establish the following relation

$\displaystyle n_{\rm in}\bm{v}_{\rm in}\cdot\mathrm{d}\bm{S}=n_{\rm
out}\bm{v}_{\rm out}\cdot\mathrm{d}\bm{S}^{\prime}\;,$ (7)

where $n_{\rm in(out)}$ is the number density of the neutrino flux going into
(out of) the sphere $S$, the flux strength is just given by $n\bm{v}$ with
$\bm{v}$ being the corresponding velocity, and $\mathrm{d}\bm{S}^{\prime}$ is
a small area on the sphere which the neutrino flux from $\mathrm{d}\bm{S}$ is
deflected into (e.g., by total reflection).

Let us denote the angles between the neutrino velocity and the normal vector
of the area as $\theta_{\rm in}$ and $\theta_{\rm out}$, respectively.
According to the energy conservation, ${v}_{\rm in}={v}_{\rm out}$ must hold.
Furthermore, the angular momentum conservation should be satisfied for the
spherical potential due to rotational symmetry, leading to $\theta_{\rm
in}=\theta_{\rm out}$. As a consequence, we obtain $n_{\rm
in}\cdot\mathrm{d}{S}=n_{\rm out}\cdot\mathrm{d}{S}^{\prime}$. The number
density with the isotropic injection is then

$\displaystyle n_{\nu(\overline{\nu})}(r)=\frac{1}{4\pi r^{2}}\varoiint(n_{\rm
in}+n_{\rm out})\mathrm{d}S=\frac{2}{4\pi r^{2}}\varoiint n_{\rm
in}\mathrm{d}S\;.$ (8)

Here, $n_{\rm in}=n_{0}/2$, which is independent of how neutrinos are
deflected by the potential in the Earth. Thus, we have $n_{\nu}(r)=n_{0}$ in
the semiclassical limit.

However, the above conclusion is meaningful only when the semiclassical
viewpoint of particles with trajectories is involved. In practice, there will
also be diffraction and interference effects between the incoming and outgoing
waves. To account for those contributions, we attempt to solve the neutrino
wavefunction exactly.

###  3.2 A Rigorous Treatment

The scattering problem of concern is very typical and can be solved following
the textbook [28]. The task is to find the solution of the Lippmann-Schwinger
equation

$\displaystyle\left|\Psi\right\rangle=\left|\phi\right\rangle+\frac{1}{E-\widehat{H}_{0}+\mathrm{i}\epsilon}\widehat{V}\left|\Psi\right\rangle\;,$
(9)

where $\phi$ is the free wavefunction, $\widehat{H}_{0}$ is the free
Hamiltonian, and $\widehat{V}$ is the potential. Because the Earth potential
is taken to be spherically symmetric, a general form of the neutrino
wavefunction outside the Earth can be found with the help of partial-wave
expansion,

$\displaystyle\Psi(r,\theta)=\sum_{l}\mathrm{i}^{l}(2l+1)A_{l}(r)P_{l}(\cos{\theta})\;,$
(10)

where $\theta$ is the zenith angle with the $z$-axis fixed by the direction of
asymptotic incoming wave, $P_{l}$ is the Legendre polynomial, and $l$
represents the angular momentum. Adopting Eq. (9), the radial wavefunction
$A_{l}(r)$ in the non-relativistic limit follows

$\displaystyle\frac{\mathrm{d}^{2}(rA_{l})}{\mathrm{d}r^{2}}+\left[k^{2}_{\nu}-2mV(r)-\frac{l(l+1)}{r^{2}}\right]rA_{l}=0\;,$
(11)

where $V(r)=V\Theta(R-r)$ with $R$ being the Earth radius. The solution can be
conveniently expressed in terms of the Bessel functions

$\displaystyle
A_{l}(r)=\mathrm{e}^{\mathrm{i}\delta_{l}}\left[\cos{\delta_{l}}j_{l}(k_{\nu}r)-\sin{\delta_{l}}n_{l}(k_{\nu}r)\right],$
(12)

where $j_{l}$ and $n_{l}$ are the spherical Bessel functions of first and
second kinds, respectively, and $\delta_{l}$ is the phase shift determined by
the spherical potential. Similar to the case of the slab, by matching the
wavefunction at boundaries, we obtain the phase shift

$\displaystyle\tan{\delta_{l}}=\frac{k_{\nu}\,j_{l}(k^{\prime}_{\nu}R)\,j_{l+1}(k_{\nu}R)-k^{\prime}_{\nu}\,j_{l}(kR)\,j_{l+1}(k^{\prime}_{\nu}R)}{k_{\nu}\,j_{l}(k^{\prime}_{\nu}R)\,n_{l+1}(k_{\nu}R)-k^{\prime}_{\nu}\,j_{l+1}(k^{\prime}_{\nu}R)\,n_{l}(k_{\nu}R)}\;,$
(13)

with $k^{\prime}_{\nu}=(k^{2}_{\nu}-2mV)^{1/2}$ being the effective momentum
in matter. The above phase shift will become vanishingly small around
$l\gtrsim k_{\nu}R\approx 8.7\times 10^{9}$, which makes sense as angular
modes with impact parameters larger than the Earth radius should not
contribute to the scattering.

Using the above representation, the neutrino number density obtained with Eq.
(6) for the isotropic injection turns out to be

$\displaystyle
n_{\nu(\overline{\nu})}(r)=\frac{n_{0}}{2}\int^{1}_{-1}\left|\Psi(r,\theta)\right|^{2}\mathrm{d}\cos\theta=n_{0}\sum_{l}(2l+1)|A_{l}(r)|^{2}\;,$
(14)

In deriving the above relation, the orthogonal relation of different angular
modes has been used. We can check the validity of the above relation in the
absence of the Earth potential, for which the phase shift is vanishing, i.e.,
$\delta_{l}=0$. This is easy to verify by noticing the sum rule
$\sum^{\infty}_{l=0}(2l+1)j^{2}_{l}(kr)=1$. For non-vanishing phase shifts, we
find the overdensity $\delta n_{\nu(\overline{\nu})}(r)\equiv
n_{\nu(\overline{\nu})}(r)-n_{0}$,

$\displaystyle\delta
n_{\nu(\overline{\nu})}(r)=n_{0}\sum^{\infty}_{l=0}(2l+1)\left[\sin^{2}\delta_{l}\,(n^{2}_{l}(kr)-j^{2}_{l}(kr))-\sin
2\delta_{l}\,j_{l}(kr)n_{l}(kr)\right].$ (15)

The total reflection is supposed to occur at very small incidence angles,
corresponding to $l\sim kR$, and hence the above summation should be dominated
by those terms around $l\lesssim kR$. However, we can first check the
asymptotic behavior of Bessel functions. In the limit of $kr\to\infty$ with
$l$ being small, the spherical Bessel functions can be approximated as

$\displaystyle
j_{l}(kr)\approx\frac{1}{kr}\cos\left[kr-\frac{(l-1)\pi}{2}\right],\;\;\;n_{l}(kr)\approx\frac{1}{kr}\sin\left[kr-\frac{(l-1)\pi}{2}\right].$
(16)

Those two terms in Eq. (15) are then proportional to $\cos(2kr-l\pi)$ and
$\sin(2kr-l\pi)$, respectively. Because of the largeness of $r$, those rapidly
oscillating terms will be easily averaged to zero in the presence of a tiny
dispersion in $k$. Furthermore, even if $k$ is strictly monoenergetic,
contributions of adjacent $l$ in Eq. (15) are found to cancel each other,
e.g., $\sin(2kr-l\pi)+\sin(2kr-(l+1)\pi)=0$.

Figure 4: The overdensities of neutrinos (red curve) and antineutrinos (blue
curve) in terms of the height $h$ for the round Earth.

In application, we have to directly sum over all the partial waves without the
above approximation. To increase the evaluation speed, we have adopted the
asymptotic expansions of Bessel functions with large orders of $l$ [29].
However, even with this numerical boost, the summation cannot be finished in a
reasonable time. Nevertheless, we find that a converged result will be
obtained when the summation is performed starting from $l\sim kR\approx
8.73\times 10^{9}$ to smaller values around $l\sim kR-0.005kR$. This makes
sense as smaller $l$’s correspond to larger incidence angles, which are less
important for the overdensity.

In Fig. 4, we show the resultant overdensities as functions of the height $h$
for both neutrinos (red curve) and antineutrinos (blue curve). For
$h<0.1~{}{\rm m}$, the overall overdensity is of $\mathcal{O}(10^{-6})$, which
is two orders of magnitude smaller than the previous estimation of
$\mathcal{O}(10^{-4})$. For $h>1~{}{\rm m}$, the overdensity quickly decreases
with respect to the height. At $h=10~{}{\rm m}$, the neutrino-antineutrino
asymmetry is found to be $\eta_{\nu\overline{\nu}}\approx-4\times 10^{-9}$.
The fact that the overdensities for neutrinos and antineutrinos are roughly
(not exactly) opposite to each other is due to that the effective scattering
is largely within the perturbative regime. As we mentioned, this overdensity
arises from the interference between different waves. Away from the ground,
the interference between incoming and outgoing waves is expected to be
vanishingly small for the round Earth due to the decoherence effect, which
explains the behavior with respect to the height.

##  4 Concluding Remarks

The feasibility of observing relic neutrinos via collective effects will be
greatly enhanced, if there is a large neutrino-antineutrino asymmetry on the
surface of the Earth. However, we have shown that the overdensities of both
neutrinos and antineutrinos should be suppressed due to several conservation
laws including those of the particle number, the energy and the angular
momentum. The residual factor that can still affect the overdensity is the
interference effect between different waves. By solving the scattering of
plane-wave neutrinos with the spherical potential of the Earth, we have
obtained the overdensity on the Earth surface without the flat slab
approximation. The overall magnitude at small heights is only of
$\mathcal{O}(10^{-6})$, and this value will become vanishingly small away from
the ground.

Nevertheless, we should emphasize that the conclusion holds for a perfectly
round Earth. Thus, one possible way out is to choose a patch of ground
disobeying the rotational symmetry. On the one hand, to appreciate the total
reflection one should have a rather smooth ground in sight (of area $2hR$
within the horizon). For $h=10~{}{\rm m}$, the dimension of this ground is
around $11~{}{\rm km}$. One the other hand, to deviate from the rotational
symmetry one can require the ground beyond the horizon to be sufficiently
bumpy. In Fig. 1, this requirement is equivalent to making side B of the slab
bumpy such that the total reflection does not occur on that side. In this
manner, one may have a sizable overdensity on the surface of side A without
cancellations.

## Acknowledgments

The author would like to thank Andreas Trautner and Manibrata Sen for
insightful discussions during the refinement of this manuscript.

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