# Conditional and Unique Coloring of Graphs

P. Venkata Subba Reddy and K. Viswanathan Iyer
Dept. of Computer Science and Engg.
National Institute of Technology
Tiruchirapalli - 620015, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
author for correspondence

###### Abstract

For integers $k,r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a
proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of
degree $d(v)$ in $G$ is adjacent to at least $\min\\{r,d(v)\\}$ differently
colored vertices. Given $r$, the smallest integer $k$ for which $G$ has a
conditional $(k,r)$-coloring is called the $r$th order conditional chromatic
number $\chi_{r}(G)$ of $G$. We give results (exact values or bounds for
$\chi_{r}(G)$, depending on $r$) related to the conditional coloring of some
graphs. We introduce _unique conditional colorability_ and give some related
results.

Keywords. cartesian product of graphs; conditional chromatic number; gear
graph; join of graphs; uniquely colorable graphs

Subject Classification 68R10, 05C15

##  1 Introduction

Let $G=(V(G),E(G))$ be a simple, connected, undirected graph. For a vertex
$v\in V(G)$, the neighborhood of $v$ in $G$ is defined as $N_{G}(v)$= {$u\in
V(G):(u,v)\in E(G)$}, and the degree of $v$ is denoted by $d(v)$=$|N_{G}(v)|$.
The closed neighborhood of $v$ is defined as $N_{G}[v]=N_{G}(v)\cup\\{v\\}$.
For an integer $k>0$, a proper $k$-coloring of a graph $G$ is a surjective
mapping $c\colon V(G)\to\\{1,2,\ldots,k\\}$ such that if $(u,v)\in E(G)$ then
$c(u)\neq c(v)$. The smallest $k$ such that $G$ has a proper $k$-coloring is
the chromatic number $\chi(G)$ of $G$. For a set $S\subseteq V(G)$ we define
$c(S)=\\{c(u):u\in S\\}$.
Definition 1 For integers $k,r>0$, a conditional $(k,r)$-coloring of $G$ is a
surjective mapping $c\colon V(G)\to\\{1,2,\ldots,k\\}$ such that both the
following conditions (C1) and (C2) hold:

> (C1) if $(u,v)\in E(G)$, then $c(u)\neq c(v)$.
> (C2) for any $v\in V(G)$, $|c(N_{G}(v))|\geq$ min {$d(v),r$}.

Given an integer $r>0$, the smallest integer $k$ such that $G$ has a proper
$(k,r)$-coloring is called the rth-order conditional chromatic number of $G$,
denoted by $\chi_{r}(G)$. It is proved in [4] that the conditional
$(k,r)$-coloring problem of a graph is $NP$-complete. Two of our earlier work
can be found in [5, 6]. We follow commonly used terminology and notations (see
for example [1, 2, 7]).

##  2 Unique conditional colorability

If $\chi(G)=k$ and every $k$-coloring of $G$ induces the same partition of
$V(G)$, then $G$ is called uniquely k-colorable. In a similar way we define
unique $(k,r)$-colorability of graphs.
Definition 2 If $\chi_{r}(G)=k$ and every conditional $(k,r)$-coloring of $G$
induces the same partition of $V(G)$, then $G$ will be called uniquely
$(k,r)$-_colorable_.

######  Proposition 1.

If $G$ is uniquely $n$-colorable and $r\leq n-1$ then $\chi_{r}(G)=n$.

###### Proof.

Since $G$ is uniquely $n$-colorable, let $c\colon V(G)\to\\{1,2,\ldots,n\\}$
be the proper coloring of $G$ and w.l.o.g. for $1\leq i\leq n$ let the color
class $C_{i}$ be defined as $C_{i}=\\{v:c(v)=i\\}$. For all $u\in V(G)$ if
$u\in C_{i}$ then for all $j\in\\{1,2,\ldots,n\\}$ there exists a $v\in
C_{j}\;(j\neq i)$ – this implies that for every $u\in V(G)$, $d(u)\geq n-1$
and $|c(N(u))|=n-1$. Note that $c$ is also a conditional $(n,r)$-coloring of
$G$ because (C2) is also satisfied as for every $u\in V(G),|c(N(u))|=n-1\geq$
min $\\{r,d(u)\\}$. ∎

The definition of conditional $(k,r)$-coloring of $G$ and Proposition 2.1
together imply:

######  Corollary 1.

Every uniquely $n$-colorable graph $G$ is also uniquely $(n,n-1)$-colorable.

######  Proposition 2.

For every $k\geq 3$, there exists a uniquely $(3,2)$-colorable graph $G_{k}$
with $k+2$ vertices.

###### Proof.

We take $G_{1}$ to be $C_{3}$. Suppose that $k\geq 1$ and assume that $G_{k}$
has been obtained. From $G_{k}$, we construct $G_{k+1}$ by introducing a new
vertex $w$. The vertex and edge sets of $G_{k+1}$ are defined thus:

$\displaystyle V(G_{k+1})$
$\displaystyle=V(G_{k})\cup\\{w\\},\text{where}\;w\notin V(G_{k}).$
$\displaystyle E(G_{k+1})$
$\displaystyle=E(G_{k})\cup\\{(u,w),(v,w)\\},\text{where}\ (u,v)\in E(G_{k}).$

Evidently, $|V(G_{k})|=k+2$. We need to show that for any $k$, $G_{k}$ is
uniquely $(3,2)$-colorable. We prove the result by induction on $k$. Clearly
$G_{1}$ is uniquely $(3,2)$-colorable. For some $k>1$, assume $G_{k}$ is
uniquely $(3,2)$-colorable. Then $G_{k+1}$ is also uniquely $(3,2)$-colorable
because the new vertex $w\in G_{k+1}$ is assigned a third color different from
its two neighbors which are adjacent; by the inductive hypothesis the result
follows. ∎

######  Proposition 3.

Every path $P_{n}\;(n\geq 3)$ is uniquely $(3,2)$-colorable.

###### Proof.

From [3] we get $\chi_{2}(P_{n})=3$ . Define a conditional $(3,2)$-coloring
$c\colon V(P_{n})\to\\{1,2,3\\}$ by

$\displaystyle c^{-1}(1)=C_{1}$ $\displaystyle=\\{v_{i}:i\bmod 3=1\\},$
$\displaystyle c^{-1}(2)=C_{2}$ $\displaystyle=\\{v_{i}:i\bmod 3=2\\},$
$\displaystyle c^{-1}(3)=C_{3}$ $\displaystyle=\\{v_{i}:i\bmod 3=0\\},$

where, $C_{1}$,$C_{2}$ and $C_{3}$ are the color classes. In the conditional
$(3,2)$-coloring of $P_{n}$, for any two vertices $v_{i},v_{j}\;(i\neq j)$ in
$V(P_{n})$ if $|i-j|\bmod 3=0$ then $v_{i}$ and $v_{j}$ must be colored same;
otherwise either (C1) will be violated at $v_{min\\{i,j\\}+1}$ or (C2) will be
violated at $v_{max\\{i,j\\}-1}$. Since $c$ is the only coloring wherein for
all $v_{i},v_{j}\in V(P_{n})$, $c(v_{i})=c(v_{j})$, if $|i-j|\bmod 3=0$,
$P_{n}$ is uniquely $(3,2)$-colorable. ∎

######  Proposition 4.

If $T$ ($\neq P_{n}$) is a rooted tree with $n$ verices and $k=\chi_{r}(T)$
then $T$ is not uniquely $(k,r)$-colorable unless $k=n$.

###### Proof.

Let the root of $T$ be a vertex of degree $\Delta(T)$. Let $c\colon
V(T)\to\\{1,2,\ldots,k\\}$ be a conditional $(k,r)$-coloring of $T$. We show
that a new conditional $(k,r)$-coloring of $T$ can be obtained based on $c$ if
$k\neq n$. We know that every tree $T$ has at least two vertices say, $u,v$
with degree less than two. Let $p(v)$ deonte the parent of $v$. We have the
following cases:
Case 1: $p(u)=p(v):$
Case 1.1: $c(u)=c(v):$ Since $r\geq 2$ assigning one of the colors from the
set $c(N_{T}(p(u)))\setminus\\{c(v)\\}$ to $u$ results in a new conditional
$(k,r)$-coloring.
Case 1.2: $c(u)\neq c(v):$
Case 1.2.1: There exists a vertex $w\in V(T)$ such that $c(w)=c(u)$ or
$c(w)=c(v):$ Interchanging the colors of $u$ and $v$ results in a different
induced partition of $V(T)$.
Case 1.2.2: There doesn’t exist a vertex $w\in V(T)$ such that $c(w)=c(u)$ or
$c(w)=c(v):$
Case 1.2.2.1: $r\geq\Delta:$ If $n\neq\Delta+1$ then there exists a vertex
$w^{\prime}\in N_{T}(p(u))$ such that $d(w^{\prime})\geq 2$; then
interchanging the colors of $u$ and $w^{\prime}$ results in a different
induced partition of $V(T)$ because the subtree rooted at $w^{\prime}$ doesn’t
contain any vertex colored $c(u)$. If $n=\Delta+1$ then $k=n$.
Case 1.2.2.2: $r<\Delta:$ There must exist at least two vertices
$w_{1},w_{2}\in N_{T}(p(u))$ such that $c(w_{1})=c(w_{2})$. Interchanging the
colors of $u$ and $w_{1}$ results in a different induced partition of $V(T)$
because the subtree rooted at $w_{1}$ doesn’t contain any vertex colored
$c(u)$.
Case 2: $p(u)\neq p(v):$
Case 2.1: $d(p(u))<$ min $\\{r,\Delta(T)\\}$ or $d(p(v))<$ min
$\\{r,\Delta(T)\\}:$ Assigning to $u$ any color in the set $c(V(T))\setminus
c(N_{T}[p(u)])$ if $d(p(u))<$ min $\\{r,\Delta(T)\\}$ or to $v$ any color in
the set $c(V(T))\setminus c(N_{T}[p(v)])$ if $d(p(v))<$ min
$\\{r,\Delta(T)\\}$ gives a new conditional $(k,r)$-coloring.
Case 2.2: $d(p(u))>$ min $\\{r,\Delta(T)\\}$ or $d(p(v))>$ min
$\\{r,\Delta(T)\\}:$ If $d(p(u))>$ min $\\{r,\Delta(T)\\}$ there must exist a
vertex $u^{\prime}\in N_{T}(p(u))$ such that $c(u)=c(u^{\prime})$ or two
vertices $u_{1},u_{2}\in N_{T}(p(u))$ such that $c(u_{1})=c(u_{2})\neq c(u)$.
Assigning to $u$ any of the color in the set
$c(V(T))\setminus\\{c(u^{\prime}),c(p(u))\\}$ or intechanging the colors of
$u$ and $u_{1}$ and the colors $c(u)$, $c(u_{1})$ in the subtree rooted at
$u_{1}$ gives a new conditional $(k,r)$-coloring in the former and latter
cases respectively. The case $d(p(v))>$ min $\\{r,\Delta(T)\\}$ is similar.
Case 2.3: $d(p(u))=d(p(v))=$ min $\\{r,\Delta(T)\\}:$
Case 2.3.1: min $\\{r,\Delta(T)\\}>2:$ There exists a $w\in V(T)$ such that
$c(u)\neq c(w)$ and $p(u)=p(w)$. Making the color of $u$ as $c(w)$ and in the
subtree rooted at $w$, swapping the colors $c(u)$ and $c(w)$ gives a new
conditional $(k,r)$-coloring.
Case 2.3.2: min $\\{r,\Delta(T)\\}=2:$ Since $T\neq P_{n}$ so $\Delta(T)>2$
and $r=2$. There exists an ancestor of $u$ and $v$ with degree $\geq 2$
because the root of $T$ has maximum degree. Let $w$ with $d(w)\geq 2$ be the
closest ancestor of $u$ and $v$. Then there exists two children of $w$ namely
$w_{1}$ and $w_{2}$ which are ancestors of $u$, $v$ respectively. If $w$ is
the root of $T$ and $c(w_{1})\neq c(w_{2})$, interchanging the colors
$c(w_{1})$ and $c(w_{2})$ in the subtree rooted at $w_{1}$ gives a new
conditional $(k,r)$-coloring. If $w$ is not the root of $T$ and $c(w_{1})\neq
c(w_{2})$, interchanging the colors $c(w_{1})$ and $c(w_{2})$ in the subtree
rooted at $w$ gives a new conditional $(k,r)$-coloring. Otherwise (i.e., if
$c(w_{1})=c(w_{2})$), there exists a $w_{3}\in N_{T}(w)$ such that
$c(w_{3})\neq c(w_{1})$ and interchanging the colors $c(w_{1})$ and $c(w_{2})$
in the subtree rooted at $w_{1}$ gives a new conditional $(k,r)$-coloring. ∎

##  3 Conditional colorability of some graphs

######  Theorem 1.

Let $G_{1}$ and $G_{2}$ be two graphs where $\chi(G_{1})=k_{1}$,
$\chi(G_{2})=k_{2}$ and w.l.o.g. let $k_{1}\leq k_{2}$. Then
$\chi_{r}(G_{1}+G_{2})$ = $\chi(G_{1}+G_{2})$ = $k_{1}+k_{2}$, where $r\leq
k_{1}+1$.

###### Proof.

In the graph $G_{1}+G_{2}$, $V(G_{2})\subset N_{G_{1}+G_{2}}(u)$ if $u\in
V(G_{1})$ or $V(G_{1})\subset N_{G_{1}+G_{2}}(u)$ if $u\in V(G_{2})$.
Therefore $c(V(G_{1}))\cap c(V(G_{2}))=\emptyset$, and in any proper
$k$-coloring of $G_{1}+G_{2}$, for all $u\in V(G_{1}+G_{2})$,
$|c(N_{G_{1}+G_{2}}(u))|\geq\min\\{d(u),r\\}$. This implies that every proper
$k$-coloring of $G_{1}+G_{2}$ is also a proper $(k,r)$-coloring – if not, we
get a contradiction: suppose that $\chi(G_{1}+G_{2})=k\neq k_{1}+k_{2}$; since
$c(V(G_{1}))\cap c(V(G_{2}))=\emptyset$, either $k_{1}\neq\chi(G_{1})$ or
$k_{2}\neq\chi(G_{2})$ which contradicts the given condition. ∎

######  Theorem 2.

Let $G(V_{1},V_{2},E)$ be a bipartite graph, $S_{1}=\bigcap_{u\in
V_{1}}N_{G}(u)$, $S_{2}=\bigcap_{v\in V_{2}}N_{G}(v)$ and w.l.o.g. let
$|S_{1}|\leq|S_{2}|$. Then $\chi_{r}(G)=2r$ where $r\leq|S_{1}|$ .

###### Proof.

In any proper coloring of $G$, from the given conditions $|c(V_{1})|\geq r$
and $|c(V_{2})|\geq r$ as $G$ is bipartite. Since $r\leq|S_{1}|$ and
$c(S_{1})\cap c(S_{2})=\emptyset$ we have $\chi_{r}(G)\geq 2r$. But there
exists a proper $2r$-coloring of $G$ such that $|c(S_{1})|=|c(S_{2})|=r$
because every bipartite graph is bicolorable and $r\geq 2$. This coloring also
satisfies (C2) as $S_{1}\subseteq V_{2}$ and $S_{2}\subseteq V_{1}$. Thus
$\chi_{r}(G)\leq 2r$. Hence $\chi_{r}(G)=2r$. ∎

######  Theorem 3.

Let $T_{1},T_{2}$ be two non trivial trees with $n_{1},n_{2}$ number of
vertices respectively and w.l.o.g. let $n_{1}\leq n_{2}$. Then
$\chi_{r}(T_{1}+T_{2})=2(r-1)$, where $4\leq r\leq\ n_{1}+1$.

###### Proof.

Every nontrivial tree has at least two vertices with degree one [2]. Therefore
there exist vertices $u,v$ where $u\in V(T_{1})$ and $v\in V(T_{2})$, such
that $d(u)=d(v)=1$. Therefore, $d_{u}(T_{1}+T_{2})=1+n_{2}$ and
$d_{v}(T_{1}+T_{2})=1+n_{1}$. If $\chi_{r}(T_{1}+T_{2})<2(r-1)$, then either
$|c(V(T_{1}))|<r-1$ or $|c(V(T_{2}))|<r-1$ or both because $c(V(T_{1}))\cap
c(V(T_{2}))=\emptyset$. Hence (C2) is violated at $u$ or $v$ or both.
Therefore, $\chi_{r}(T_{1}+T_{2})\geq 2(r-1)$. Since every tree is
$2$-colorable and $r\geq 4$, properly color $V(T_{1})$, $V(T_{2})$ in
$T_{1}+T_{2}$ using $r-1$ colors each such that $|c(V(T_{1}+T_{2}))|=2(r-1)$.
The resulting coloring is a conditional $(2(r-1),r)$-coloring of
$T_{1}+T_{2}$, as (C1) is satisfied because $c(V(T_{1}))\cap
c(V(T_{2}))=\emptyset$, and (C2) is satisfied because $c(V(T_{1}))\cap\
c(V(T_{2}))=\emptyset$ and for all $w\in V(T_{1}+T_{2})$,
$|c(N_{T_{1}+T_{2}}(w))|\geq(r-1)+1\geq$ min $\\{d(w),r\\}$. Hence the result.
∎

######  Theorem 4.

Given any two graphs $G_{1}$ and $G_{2}$, let $r_{1}$ and $r_{2}$ be such that
$r_{1}\geq\delta(G_{1})$ and $r_{2}\geq\delta(G_{2})$. Then $\chi_{r}(G_{1}\
\Box\ G_{2})\leq\chi_{r_{1}}(G_{1}).\chi_{r_{2}}(G_{2})$ where
$r\leq\delta(G_{1})+\delta(G_{2})$.

###### Proof.

Let $\chi_{r_{1}}(G_{1})=g_{1}$ and $\chi_{r_{2}}(G_{2})=g_{2}$. Let
$c_{G_{1}}$ (resp. $c_{G_{2}}$) be a proper $(g_{1},r_{1})$\- (resp.
$(g_{2},r_{2})$-) coloring of $G_{1}$ (resp. $G_{2}$). Then let $c_{G_{1}\Box
G_{2}}$ be a coloring of $G_{1}\Box G_{2}$ wherein we assign to any vertex
$(u_{1},u_{2})\in V(G_{1}\ \Box\ G_{2})$ the color denoted by the ordered pair
$(c_{g_{1}}(u_{1}),c_{g_{2}}(u_{2}))$. This coloring uses $g_{1}.g_{2}$ colors
and it defines a proper coloring of $G_{1}\ \Box\ G_{2}$. Therefore
$c_{G_{1}\Box G_{2}}$ satisfies (C1). Let $(u_{1},u_{2})\in V(G_{1}\ \Box\
G_{2})$ such that $u_{1}\in V(G_{1})$ and $u_{2}\in V(G_{2})$. Since
$c_{G_{1}}$ and $c_{G_{2}}$ satisfy (C2), by the definition of $G_{1}\Box
G_{2}$, a vertex $(u_{1},u_{2})$ has at least
$\min\\{r_{1},\delta(G_{1})\\}=\delta(G_{1})$ distinctly colored neighbors of
the form $(u^{\prime},u_{2})$ because $|c(N_{G_{1}}(u_{1}))|\geq\delta(G_{1})$
and at least $\min\\{r_{2},\delta(G_{2})\\}=\delta(G_{2})$ distinctly colored
neighbors of the form $(u_{1},u^{\prime\prime})$ because
$|c(N_{G_{2}}(u_{2}))|\geq\delta(G_{2})$. Therefore $|c(N_{G_{1}\ \Box\
G_{2}}((u_{1},u_{2}))|\geq\delta(G_{1})+\delta(G_{2})\geq r$. Hence
$c_{G_{1}\Box G_{2}}$ satisfies (C2) and the result follows. ∎

######  Theorem 5.

Let $L(T)$ be the line graph of complete $k$-ary tree $T$ with height $h\geq
2$. Then

$\chi_{r}(L(T))=\left\\{\begin{array}[]{l l}k+1,&\quad\mbox{if $r\leq k$.
{}}\\\ 2k+1,&\quad\mbox{if $r=\Delta$. {}}\\\ \end{array}\right.$

###### Proof.

Let $V(L(T))=\left\\{v_{1},v_{2},\dotsc,v_{e(h)}\right\\}$, where
$e(h)=\frac{k^{h+1}-1}{k-1}-1$. In $T$ we assume that the root is at level $0$
and for each $l$ ($1\leq l\leq h$), $v_{e(l-1)+1}$ to $v_{e(l)}$ represent the
edges between levels $l-1$ and $l$, numbered from ‘left’ to ‘right’. It can be
seen that $\Delta(L(T))=2k$ and $\omega(L(T))=k+1$. In the ordering
$v_{e(h)},\dotsc,v_{1}$ of the vertices of $L(T)$, for each $i$ ($1\leq i\leq
e(h)$), $v_{i}$ is a simplicial vertex in the subgraph induced by
$\\{v_{i},\dotsc,v_{1}\\}$. Hence the ordering is a p.e.o. and $L(T)$ is
chordal. As every chordal graph is perfect, $\chi(L(T))=\omega(L(T))=k+1$.
Since every vertex of $L(T)$ is in a $K_{k+1}$, we also have
$\chi_{r}(L(T))=k+1$ if $r\leq k$. Thus $\chi_{r}(L(T))=k+1$, if $r\leq k$.
From [3] we know $\chi_{r}(G)\geq\min\\{r,\Delta\\}+1$. Taking $G=L(T)$ we
have $\chi_{r}(L(T))\geq\min\\{r,\Delta\\}+1=2k+1$ if $r=\Delta$. Similar to
the greedy (vertex) coloring, color the vertices in the order
$v_{1},\dotsc,v_{e(h)}$ by assigning to each vertex the first available color
not already used for any of the lower indexed vertices within distance two. In
the assumed order, each vertex has at most $\Delta$ lower indexed vertices
within distance two; therefore $\chi_{\Delta}(L(T))\leq\Delta+1=2k+1$. Hence
$\chi_{r}(L(T))=2k+1$ if $r=\Delta$. ∎

Definition 3 The wheel graph $W_{n}$ consists of a $C_{n-1}$ together with a
center vertex $s$ that is adjacent to all the $n-1$ vertices in the $C_{n-1}$.

######  Theorem 6.

If $W_{n}$ denotes the wheel graph, for any $r\geq 3$

$\chi_{r}(W_{n})=\left\\{\begin{array}[]{l
l}\chi_{r}(C_{n-1})+1,&\quad\mbox{if $r\leq\chi_{r}(C_{n-1}).$ {}}\\\
\min\\{r,n-1\\}+1,&\quad\mbox{if $r>\chi_{r}(C_{n-1}).${}}\\\
\end{array}\right.$

###### Proof.

By definition $d(s)=n-1$. For any other $u\;(\neq s)\in V(W_{n}),\;d(u)=3$.
Since $r\geq 3$, every conditional $(k,r)$-coloring of $W_{n}$ is also a
conditional $(k,r)$-coloring of its induced subgraph $C_{n-1}$. There exists
conditional $(k,r)$-coloring of $G$ iff $k\geq\chi_{r}(G)$. Hence conditional
$(\chi_{r}(C_{n-1}),r)$-coloring of $C_{n-1}$ with a new color to $s$
satisfies (C1) and gives $|c(N_{W_{n}}(v))|\geq\min\\{r,d(v)\\}$, for all
$v\in V(W_{n})\backslash s$. If $r\leq\chi_{r}(C_{n-1})$, then
$|c(N_{W_{n}}(s))|\geq r$. Thus for each $v\in V(W_{n})$,
$|c(N_{W_{n}}(v))|\geq\min\\{r,d(v)\\}$ satisfying (C2) – in total
$\chi_{r}(C_{n-1})+1$ colors are used. If $r>\chi_{r}(C_{n-1})$, then
$|c(N_{W_{n}}(s))|<r$, and we need to give new colors to
$\min\\{r,n-1\\}-\chi_{r}(C_{n-1})$ vertices adjacent to $s$. In total
$\chi_{r}(C_{n-1})+1+\min\\{r,n-1\\}-\chi_{r}(C_{n-1})=\min\\{r,n-1\\}+1$
colors are used. Hence the result. ∎

Definition 4 The $n$-gear $G_{n}$ consists of a cycle $C_{2n}$ on $2n$
vertices where every other vertex on the cycle is adjacent to a $(2n+1)th$
center vertex labeled $v_{0}$. The vertices in the $C_{2n}$ are labeled
sequentially $v_{1},\ldots,v_{2n}$ such that for $1\leq i\leq 2n-1$, $v_{i}$
is adjacent to $v_{i+1}$, $v_{1}$ is adjacent to $v_{2n}$, and every vertex in
$V_{oddi}=\\{v_{i}|\;\text{i is odd and}\;1\leq i\leq 2n-1\\}$ is adjacent to
$v_{0}$.

######  Theorem 7.

If $G_{n}$ is the $n$-gear then for any $n\geq 3$

$\chi_{r}(G_{n})=\left\\{\begin{array}[]{l l}4,&\quad\mbox{if $r=2$. {}}\\\
\chi_{2}(C_{2n})+1,&\quad\mbox{if $r=3$.{}}\\\
min\\{r,\Delta\\}+1,&\quad\mbox{if $r\geq 4$. {}}\\\ \end{array}\right.$

###### Proof.

Let $k=\chi_{r}(G_{n})$. From [3] we know $\chi_{r}(G)\geq
min\\{r,\Delta\\}+1$. Taking $G=G_{n}$ we have $\chi_{r}(G_{n})\geq
min\\{r,\Delta\\}+1$. Let $k=\chi_{r}(G_{n})$.
Case 1: $r=2:$ We have $k\geq 3$. We assume that $k=3$. Let $c\colon
V(G_{n})\to\\{1,2,3\\}$ be a conditional $(3,2)$-coloring where $c(v_{i})=i$
for $i=1,2,3$. Then across $v_{1},v_{2},v_{3}$ (C1) must be true and in
particular (C2) must hold at $v_{2}$. To satisfy (C1) at $v_{3},c(v_{4})\neq
3$. We branch into two cases. If $c(v_{4})=1$, then by (C2) at $v_{4}$ we must
have $c(v_{5})\notin\\{1,3\\}$. Therefore we must have $c(v_{5})=2$. To
satisfy (C1), $c(v_{0})\notin\\{1,2,3\\}$. On the other hand if $c(v_{4})=2$
then to satisfy (C2) at $v_{4}$ we must have $c(v_{5})\notin\\{2,3\\}$. Hence
$c(v_{5})=1$. To satisfy (C2) at $v_{3}$ while preserving proper coloring,
$c(v_{0})\notin\\{1,2,3\\}$. Therefore $k\geq 4$. To show that $k=4$, it
suffices to construct a conditional $(4,2)$-coloring of $G_{n}$. Define
$c\colon V(G_{n})\to\\{1,2,3,4\\}$ as follows:
Case 1.1: $n\equiv 0\pmod{3}:$ Set $c(v_{0})=4$ and

$c(v_{i})=\left\\{\begin{array}[]{l l}1,&\quad\mbox{if \ $i\bmod 3=1$. {}}\\\
2,&\quad\mbox{if \ $i\bmod 3=2$.{}}\\\ 3,&\quad\mbox{if \ $i\bmod 3=0$. {}}\\\
\end{array}\right.$

Case 1.2: $n\equiv 2\pmod{3}:$ Modify $c$ in case (1.1) by the assignment
$c(v_{2n})=2$.
Case 1.3: $n\equiv 1\pmod{3}:$ Modify $c$ in case (1.1) by the assignments
$c(v_{2n-4})=2,c(v_{2n-3})=1,c(v_{2n-2})=3,c(v_{2n-1})=2$ and $c(v_{2n})=3$.
It can be verified that $c$ is a conditional $(4,2)$-coloring of $G_{n}$. Thus
$k\leq 4$, and hence $\chi_{2}(G_{n})=4$.
Case 2: $r=3:$ We have $k\geq 4$. By (C1) $c(v_{i})\neq c(v_{0})$ and by (C2)
at $v_{i}$, $c(v_{i+1})\neq c(v_{0})$ for all odd $i$ in the range $1\leq
i\leq 2n-1$. Hence $c(v_{i})\neq c(v_{0})$ for all $i\;(i\neq 0)$. Since
$d(v_{i})\leq 3$ for $1\leq i\leq 2n$ and $r=3$, a conditional
$(\chi_{3}(G_{n}),3)$-coloring of $G_{n}$ gives a conditional
$(\chi_{2}(C_{2n}),2)$-coloring of $C_{2n}$. In turn, a conditional
$(\chi_{2}(C_{2n}),2)$-coloring of $C_{2n}$ with an additional color to
$v_{0}$ gives a conditional $(\chi_{3}(G_{n}),3)$-coloring of $G_{n}$.
Case 3: $r\geq 4:$ Let $l=\min\\{r,\Delta(G_{n})\\}$. We know
$\chi_{r}(G_{n})\geq l+1$. Since $v_{0}$ is the only vertex with $d(v_{0})\geq
r$ and $\chi_{r}(C_{2n})\leq l$, conditional $(l,r)$-coloring of
$V(G_{n})\setminus\\{v_{0}\\}$ such that $|c(V_{oddi})|=l$, with $(l+1)$th
color assigned to $v_{0}$ results in a conditional $(l+1,r)$-coloring of
$V(G_{n})$. Thus $\chi_{r}(G_{n})\leq l+1$, and hence
$\chi_{r}(G_{n})=\min\\{r,\Delta\\}+1$. ∎

## References

  * [1] M. C. Golumbic, Algorithmic graph theory and perfect graphs, Elsevier, North Holland, 2004.
  * [2] F. Harary, Graph theory, Addison-Wesley, MA, 1969.
  * [3] H. J. Lai, J. Lin, B. Montgomery, T. Shui, and S. Fan, Conditional colorings of graphs, Discrete Math. 306 (2006), pp. 1997-2004.
  * [4] X. Li, X. Yao, W. Zhou, and H. Broersma, Complexity of conditional colorability of graphs, Appl. Math. Lett. 22 (2009), pp. 320-324.
  * [5] P.V.S. Reddy, K.V.Iyer, “On conditional coloring of some graphs”, 76th Ann. Conf. Ind. Math. Soc.,Surat, Gujarat, Dec. 2010. (arxiv.org/abs/1011.5289)
  * [6] P.V.S.Reddy, K.V.Iyer, “Conditional coloring of some parameterized graphs”, Submitted to Util. Math., May. 2010. (arxiv.org/abs/1012:2251)
  * [7] D. B. West, Introduction to graph theory, Prentice Hall of India, 2003.


aainstitutetext: New York University Abu Dhabi
PO Box 129188, Saadiyat Island, Abu Dhabi, United Arab
Emiratesbbinstitutetext: Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, Franceccinstitutetext: PRISMA+ Cluster of Excellence and Mainz
Institute for Theoretical Physics
Johannes Gutenberg University, 55099 Mainz, Germany

# Probing Reheating with Graviton Bremsstrahlung

Nicolás Bernal b Simon Cléry b Yann Mambrini c and Yong Xu111Corresponding
author<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the stochastic gravitational wave (GW) spectrum resulting from
graviton bremsstrahlung during inflationary reheating. We focus on an inflaton
$\phi$ oscillating around a generic monomial potential
$V(\phi)\propto\phi^{n}$, considering two different reheating scenarios: $i)$
inflaton decay and $ii)$ inflaton annihilation. We show that in the case of a
quadratic potential, the scattering of the inflatons can give rise to larger
GW amplitude than the decay channel. On the other hand, the GW spectrum
exhibits distinct features and redshifts in each scenario, which makes it
possible to distinguish them in the event of a discovery. Specifically, in the
case of annihilation, the GW frequency can be shifted to values higher than
those of decay, whereas the GW amplitude generated by annihilation turns out
to be smaller than that in the decay case for $n\geq 4$, due to the different
scaling of radiation during reheating. We also show that the differences in
the GW spectrum become more prominent with increasing $n$. Finally, we
highlight the potential of future high-frequency GW detectors to distinguish
between the different reheating scenarios.

MITP-23-065

##  1 Introduction

Cosmic inflation is an elegant paradigm that solves the horizon, flatness, and
monopole problems of the (old) standard cosmology Lyth:2009zz . A successful
inflationary model must incorporate an efficient reheating mechanism to align
with observational data. In the conventional scenario, after inflation, the
inflaton field descends to the potential minimum, starting to oscillate and
transfer energy to light degrees of freedom of the Standard Model (SM), which
eventually undergo thermalization, forming the SM bath. Energy transfer can
proceed through decay or annihilation of the inflaton, facilitated by the
introduction of couplings between the inflaton and the daughter fields
Kofman:1997yn . Due to the inevitable coupling between the metric and the
energy-momentum tensor, the emission of graviton degrees of freedom is also
anticipated through radiative bremsstrahlung processes Weinberg:1965nx ;
Barker:1969jk . However, within the framework of general relativity, the
graviton production rate is suppressed by the squared Planck mass $M_{P}^{2}$,
where $M_{P}=1/\sqrt{8\pi\,G_{N}}\simeq 2.4\times 10^{18}$ GeV. To achieve
efficient graviton production, a high-energy scale and consequently a large
energy-momentum tensor become imperative. This precise scenario unfolds during
the reheating phase.

The physics that takes place during reheating is difficult to test
experimentally. One of the very few observables (if not the only one) is
gravitational waves (GWs). It is also known that graviton bremsstrahlung
during inflationary reheating gives rise to an interesting stochastic GW
background (SGWB)222We notice that fluctuations in the thermal plasma could
also give rise to SGWB Ghiglieri:2015nfa ; Ghiglieri:2020mhm ;
Ringwald:2020ist ; Klose:2022knn ; Ringwald:2022xif ; Ghiglieri:2022rfp .
Nakayama:2018ptw ; Huang:2019lgd ; Ghoshal:2022kqp ; Barman:2023ymn ;
Barman:2023rpg ; Kanemura:2023pnv . The spectrum typically peaks at ultrahigh
frequencies depending on the detailed dynamics of reheating, such as the shape
of the inflaton potential or the inflaton-matter coupling Barman:2023ymn ;
Barman:2023rpg . It was also recently shown that the GW spectrum can be
boosted during bosonic reheating through inflaton decay, once the inflaton
oscillates around the minimum of a potential steeper than quadratic,
$V(\phi)\propto\phi^{n}$ with $n>2$ Barman:2023rpg . In this case, the
inflaton decay rate has a time dependence proportional to $1/\phi_{0}^{n-2}$
and $\phi_{0}^{n-2}$ for bosonic and fermionic reheating, respectively
Garcia:2020wiy , where we have assumed that after inflation,
$\phi(t)=\phi_{0}(t)\,\mathcal{P}(t)$, with $\mathcal{P}(t)$ a quasiperiodic
function encoding the (an)harmonicity of short-timescale oscillations in the
potential and $\phi_{0}(t)$ is the envelope subject to the effect of redshift
and decay. Note that $\phi_{0}(t)$ is a decreasing function of time, leading
to the radiation or entropy released in bosonic reheating being smaller
compared to that during fermionic reheating. Consequently, the GW amplitude
generated in bosonic reheating suffers less dilution effects, and hence tends
to feature larger amplitudes.333Note that the GW amplitude can be written as
$\Omega_{\text{GW}}\propto 1/\rho_{R}$, where $\rho_{R}$ correspond to the
energy densities stored in radiation. If $\rho_{R}$ is smaller,
$\Omega_{\text{GW}}$ can be larger.

In the existing literature, the analysis for graviton bremsstrahlung is
limited to reheating scenarios considering inflaton decays Nakayama:2018ptw ;
Huang:2019lgd ; Ghoshal:2022kqp ; Barman:2023ymn ; Barman:2023rpg ;
Kanemura:2023pnv . In this work, we generalize previous studies by including
inflaton annihilation, which is also a viable reheating scenario
Garcia:2020eof ; Garcia:2020wiy ; Barman:2021ugy , and compare their
observational signatures in the form of GW emission. In particular, we assume
that the inflaton transfers its energy predominantly to the bosonic sector,
which could account for the degrees of freedom of the SM Higgs doublet.444Note
that for fermionic reheating, extra vector-like fermionic degree of freedoms
beyond the SM of particle physics have to be introduced; in that sense, the
bosonic reheating scenario is more minimal. Although reheating is not possible
through inflaton scattering for $n=2$, SM radiation and therefore SM bath
temperature feature distinct behaviors depending on the reheating channel for
$n>2$. More specifically, in the decay scenario, the temperature evolves as
$T(a)\propto a^{-\frac{3}{2n+4}}$, while for annihilation $T(a)\propto
a^{-\frac{9}{2n+4}}$, where $a$ denotes the cosmic scale factor Garcia:2020wiy
. We show that the temperature dependence in the redshift is transmitted to
the spectrum and amplitude of the GWs generated by the graviton-bremsstrahlung
process through the dilution effect, which makes it possible to distinguish
between the two reheating channels. In particular, we aim to compare the
different features of the GW spectrum in inflaton decays and annihilations.
This is particularly interesting because it offers the potential to utilize
the GW features to pinpoint reheating scenarios with inflaton decays and
annihilations. To this end, we first calculate the graviton emission rate from
inflaton annihilation. Then we compute the bremsstrahlung-induced GW spectrum
by solving a set of coupled Boltzmann equations.

The paper is organized as follows. In Section 2, we study the two reheating
setups together with the graviton emission rates in each scenario. In Section
3, the solution for the differential GW spectrum is presented for the
inflaton-annihilation scenario. Furthermore, we investigate the GW spectrum,
paying particular attention to the difference between inflaton bosonic decay
and annihilation. Finally, we summarize our findings in Section 4.

##  2 Reheating and Graviton Emission

###  2.1 Generalities

After inflation ends, the inflaton starts to coherently oscillate around the
potential minimum, transferring its energy to the SM bath: a process called
inflationary reheating Kofman:1997yn . The energy-transfer efficiency is
controlled by the type and strength of the inflaton-matter coupling. In this
paper, we focus on a bosonic reheating scenario, considering that the inflaton
$\phi$ decays or annihilates into a real scalar $\varphi$, which could account
for the degrees of freedom of the SM Higgs doublet.

The relevant interactions are described by the Lagrangian density

$\mathcal{L}\supset\mu\,\phi\,\varphi^{2}+\sigma\,\phi^{2}\,\varphi^{2},$ (1)

where $\mu$ and $\sigma$ are couplings with mass dimensions 1 and 0,
respectively. Expanding the space-time metric $g_{\mu\nu}$ around the flat
Minkowski background $g_{\mu\nu}\simeq\eta_{\mu\nu}+(2/M_{P})\,h_{\mu\nu}$,
one can write the effective coupling Choi:1994ax

$\sqrt{-g}\,\mathcal{L}\supset-\frac{1}{M_{P}}\,h_{\mu\nu}\,T^{\mu\nu},$ (2)

where $g$ denotes the determinant of $g_{\mu\nu}$ and $T^{\mu\nu}$ corresponds
to the energy-momentum tensor.

During reheating, we assume that the inflaton oscillates around a generic
monomial potential of the form Turner:1983he ; Garcia:2020wiy ; Bernal:2022wck

$V(\phi)=\lambda\,M_{P}^{4}\left(\frac{\phi}{M_{P}}\right)^{n},$ (3)

where $\lambda$ is a dimensionless parameter. Such potential could originate,
for example, from Starobinsky inflation Starobinsky:1980te , polynomial
inflation Drees:2021wgd ; Bernal:2021qrl ; Drees:2022aea  or the
$\alpha$-attractor inflationary $T$\- or $E$-models Kallosh:2013hoa ;
Kallosh:2013yoa . The effective mass $m_{\phi}$ of the inflaton can be defined
as the second derivative of its potential and is Garcia:2020wiy ;
Bernal:2022wck

$m_{\phi}(a)^{2}\simeq
n\,(n-1)\,\lambda^{\frac{2}{n}}\,M_{P}^{\frac{2\,(4-n)}{n}}\rho_{\phi}(a)^{\frac{n-2}{n}},$
(4)

which is a time-dependent quantity for $n\neq 2$ given as a function of the
inflaton energy density

$\rho_{\phi}=\frac{1}{2}\,\dot{\phi}^{2}+V(\phi)\,,$ (5)

where dots $(\dot{\phantom{.}})$ correspond to derivatives with respect to
time.

The evolution of the energy densities for the inflaton $\rho_{\phi}$ and the
SM radiation $\rho_{R}$ can be tracked by the following Boltzmann equations
Turner:1983he ; Bernal:2022wck ; Garcia:2020wiy

$\displaystyle\frac{d\rho_{\phi}}{dt}+3H\,(1+w_{\phi})\rho_{\phi}$
$\displaystyle=-(1+w_{\phi})\,\gamma\,,$ (6)
$\displaystyle\frac{d\rho_{R}}{dt}+4\,H\,\rho_{R}$
$\displaystyle=+(1+w_{\phi})\,\gamma\,,$ (7)

where $w_{\phi}=(n-2)/(n+2)$ denotes the equation of state Bernal:2022wck ;
Garcia:2020wiy  and $\gamma\equiv\Gamma\,\rho_{\phi}$ is the interaction rate
density for the production of SM states from inflatons. Additionally, $H$
corresponds to the Hubble expansion rate

$H^{2}=\frac{\rho_{R}+\rho_{\phi}}{3\,M_{P}^{2}}\,,$ (8)

with

$\rho_{R}(T)=\frac{\pi^{2}}{30}\,g_{\star}(T)\,T^{4},$ (9)

and $g_{\star}(T)$ being the number of relativistic degrees of freedom
contributing to the SM energy density.555Note that during our study, we
assumed instantaneous thermalization, for details of the effects of non-
instantaneous thermalization, see Refs. Mukaida:2015ria ; Garcia:2018wtq . The
end of the reheating period is defined as the moment at which the SM radiation
starts to dominate the total energy density of the universe and corresponds to
a reheating temperature $T_{\text{RH}}$ (or equivalently, a scale factor
$a_{\text{RH}}\equiv a(T_{\text{RH}})$) given by
$\rho_{R}(T_{\text{RH}})=\rho_{\phi}(T_{\text{RH}})=3\,M_{P}^{2}\,H_{\text{RH}}^{2}$,
where $H_{\text{RH}}\equiv H(T_{\text{RH}})$.

During reheating, that is, when $\Gamma\ll H$, Eq. (6) admits an approximate
analytical solution given by

$\rho_{\phi}(a)\simeq\rho_{\phi}(a_{\text{RH}})\left(\frac{a_{\text{RH}}}{a}\right)^{\frac{6\,n}{n+2}},$
(10)

and therefore, the effective inflaton mass can be rewritten as

$m_{\phi}(a)=m_{\phi}^{\text{RH}}\left(\frac{a_{\text{RH}}}{a}\right)^{\frac{3\,(n-2)}{n+2}},$
(11)

with

$m_{\phi}^{\text{RH}}\equiv
m_{\phi}(a_{\text{RH}})\simeq\sqrt{n\,(n-1)}\,\lambda^{\frac{1}{n}}\,M_{P}\left(\sqrt{3}\,\frac{H_{\text{RH}}}{M_{P}}\right)^{\frac{n-2}{n}}.$
(12)

Depending on the form of $\gamma$, the solution for $\rho_{R}(a)$ and
therefore the temperature $T(a)$ during reheating exhibits distinct
characteristics, which would consequently give rise to different dilution
effects for the generated GW spectrum. In the following sections, we will
present the scenarios with inflaton bosonic decay and annihilation.

However, before proceeding, it is important to comment on several points. We
have assumed that the effects of non-perturbative preheating Amin:2014eta ;
Garcia:2021iag , inflaton fragmentation Lozanov:2016hid ; Garcia:2023eol ;
Garcia:2023dyf , and gravitational reheating Bernal:2020qyu ; Clery:2021bwz ;
Haque:2022kez ; Barman:2023opy ; Haque:2023zhb  are subdominant compared to
perturbative decay. Note that even if non-perturbative phenomena during
reheating could lead to the equation of state approaching $1/3$ during
reheating, perturbative decay is still necessary to fully deplete the inflaton
energy Dufaux:2006ee ; Maity:2018qhi . Furthermore, for the bosonic decay
scenario, the inefficiency of preheating has been highlighted as the result of
self-interaction of the generated daughter field Dufaux:2006ee . In
particular, the trilinear coupling $\mu\,\phi\,\varphi^{2}$ gives rise to a
tachyonic squared mass $m^{2}_{\varphi}\sim\mu\,\phi$ for the field $\varphi$
once $\phi$ crosses zero and becomes negative. However, the self-interaction
of the $\varphi$ fields $\lambda_{\varphi}\,\varphi^{4}$ gives rise to a
positive squared mass term
$m^{2}_{\varphi}\sim\lambda_{\varphi}\left\langle\varphi^{2}\right\rangle$
where $\left\langle\varphi^{2}\right\rangle$ corresponds to the variance of
$\varphi$. This backreaction counteracts tachyonic effects and quickly
terminates preheating, making it inefficient Dufaux:2006ee . Additionally,
when the inflaton features self-interactions or, equivalently, when the
inflaton potential is steeper than quadratic, the effects of inflaton
fragmentation are relevant. Indeed, in the case of decays into fermions,
fragmentation is particularly important, making perturbative decays
subdominant Garcia:2023dyf . However, in the case of decay or scattering of
the inflaton to scalars, the effect is milder.

Lastly, to be dominant, the gravitational reheating mechanism requires
$w_{\phi}\gtrsim 0.65$, corresponding to $n>9$ Clery:2021bwz ; Haque:2022kez ;
Co:2022bgh ; Barman:2022qgt . This bound can be relaxed to $n>4$ if one
introduces a non-minimal coupling between gravity and a pair of inflatons
Clery:2022wib , or even to $n\geq 2$ if gravity couples non-minimally to a
single inflaton Barman:2023opy . In the current work, we focus on scenarios
with $2\leq n\leq 6$ minimally coupled to gravity with high-temperature
reheating, where the effects of inflaton fragmentation and gravitational
reheating are expected to be subdominant.

###  2.2 Graviton emission from inflaton decay

In this section, we briefly review the case of inflaton bosonic decays
Barman:2023rpg ; Barman:2023ymn . The rate of inflaton decay into a pair of
real scalars $\varphi$ with mass $m$ is given by

$\Gamma^{1\to
2}\simeq\frac{m_{\phi}}{8\pi}\left(\frac{\mu_{\text{eff}}}{m_{\phi}}\right)^{2},$
(13)

where

$\mu_{\text{eff}}^{2}(n)\simeq(n+2)(n-1)\,\mu^{2}\,\frac{\varpi}{m_{\phi}}\sum_{j=1}^{\infty}j\,|\mathcal{P}_{j}|^{2}\left\langle\left(1-\frac{4y^{2}}{j^{2}}\mathcal{P}\right)^{1/2}\right\rangle$
(14)

corresponds to the effective coupling after averaging over inflaton
oscillations Garcia:2020wiy , and $y\equiv m/\varpi$. We introduced the
Fourier decomposition of the oscillating field

$\phi(t)=\phi_{0}(t)\,\mathcal{P}(t)=\phi_{0}(t)\sum_{j=1}^{\infty}\mathcal{P}_{j}(t)\,e^{-i\,j\,\varpi\,t}.$
(15)

The $j^{\text{th}}$ oscillating mode is associated with an energy
$E_{j}=j\,\varpi$ where $\varpi$ is the frequency of the oscillating
background field. It is a time-dependent quantity related to the effective
inflaton mass and amplitude via

$\varpi=\sqrt{\frac{n\,\pi}{2(n-1)}}\,\frac{\Gamma(\frac{1}{2}+\frac{1}{n})}{\Gamma(\frac{1}{n})}\,m_{\phi}=\alpha_{n}\,\sqrt{\lambda}\,M_{P}\left(\frac{\phi_{0}}{M_{P}}\right)^{\frac{n-2}{2}}.$
(16)

We note that the energy in each mode is time dependent for $n>2$ and that each
mode contributes to the production rate, as can be seen from the sum in the
expression above. In what follows, we neglect the contribution of higher-order
modes $j>1$ and look at the leading contribution $j=1$.

In the case of decays, the rate density $\gamma=\rho_{\phi}\,\Gamma^{1\to 2}$,
and therefore Eq. (7) admits the analytical solution Xu:2023lxw

$\rho_{R}(a)\simeq\frac{n}{1+2n}\rho_{\phi}(a_{\text{end}})\left(\frac{a_{\text{end}}}{a}\right)^{\frac{6}{n+2}}\left[1-\left(\frac{a_{\text{end}}}{a}\right)^{\frac{2+4n}{2+n}}\right]\left(\frac{a_{\text{end}}}{a_{\text{RH}}}\right)^{\frac{6(n-1)}{2+n}},$
(17)

where $a_{\text{end}}$ and $a_{\text{RH}}$ denote the scale factor at the end
of inflation and of reheating, respectively, which implies that during
reheating

$T(a)=T_{\text{RH}}\left(\frac{a_{\text{RH}}}{a}\right)^{\alpha},$ (18)

with

$\alpha\equiv\frac{3}{2n+4}\,.$ (19)

It is important to note that away from the instantaneous decay approximation
of the inflaton, the SM bath could reach temperatures higher than
$T_{\text{RH}}$, up to $T_{\text{max}}$ Giudice:2000ex . This can be observed
in Fig. 1, where we show the evolution of the energy densities of the inflaton
and SM radiation (left) and the temperature of the SM bath (right) as a
function of the scale factor for the cases of inflaton annihilations and
decays. Reheating occurs in the range $a_{\text{end}}<a<a_{\text{RH}}$.
Earlier times correspond to the cosmic inflationary epoch, while later times
to the SM radiation-dominated isentropic era, where $T(a)\propto a^{-1}$. We
note that to solve Eqs. (6) and (7), one has to fix the interaction rate
density $\gamma$ of the inflaton (and, therefore, its mass and couplings to
the daughter particles), its initial energy density
$\rho_{\phi}(a_{\text{end}})$ and $n$. We recall that at the end of inflation
$\rho_{R}(a_{\text{end}})=0$ is expected. However, it is more convenient to
choose, instead of $\gamma$ and $\rho_{\phi}(a_{\text{end}})$ as free
parameters, $T_{\text{RH}}$ and $T_{\text{max}}$. The latter set of parameters
is used in the following analysis. In the upper panels of Fig. 1, we have
chosen $n=4$, $T_{\text{RH}}=10^{13}$ GeV with $T_{\text{max}}=3.6\times
10^{13}$ GeV, while in the lower panels $n=6$, $T_{\text{RH}}=10^{13}$ GeV
with $T_{\text{max}}=2.4\times 10^{13}$ GeV. We consider $T_{\text{RH}}$ as a
free parameter and then compute $T_{\text{max}}$ in the $\alpha$-attractor
$T$-model considering the central values of the amplitude of the primordial
spectrum of curvature perturbations and its associated spectral tilt:
$n_{s}=0.9659$ and $A_{s}=2.1\times 10^{-9}$ Planck:2018vyg .

Figure 1: Evolution for energy densities (left) and temperature (right), for
$T_{\text{RH}}=10^{13}$ GeV. In the upper panels, we have chosen $n=4$ and
$T_{\text{max}}=3.6\times 10^{13}$ GeV for decays ($T_{\text{max}}=4.3\times
10^{14}$ GeV for annihilations), while in the lower panels $n=6$ and
$T_{\text{max}}=2.4\times 10^{13}$ GeV for decays ($T_{\text{max}}=1.2\times
10^{14}$ GeV for annihilations).

Figure 2: Inflaton decays into a pair of $\varphi$ and a graviton
$h_{\mu\nu}$.

In addition to the 2-body decay, there is an inexorably associated graviton
emission by bremsstrahlung, as shown in Fig. 2. It is interesting to note that
from the four possible diagrams, two of the matrix elements (the first and the
last) vanish because the inflaton decays with vanishing three momenta and the
graviton polarization tensor is traceless Barman:2023ymn . The differential
decay width for the inflaton condensate with respect to the graviton energy
$E_{\omega}$ for this 3-body decay process is given by Garcia:2020wiy

$\frac{d\Gamma^{1\to
3}}{dE_{\omega}}=\frac{m_{\phi}}{\rho_{\phi}(1+w_{\phi})}\sum\limits_{j=1}^{\infty}\int\left|\mathcal{M}^{1\to
3}_{j}\right|^{2}\frac{d\rm PS^{(3)}}{dE_{\omega}}\,,$ (20)

where $d\rm PS^{(3)}$ is the 3-body Lorentz-invariant phase space, leading
to666Note that this is consistent with the result presented in Ref.
Barman:2023ymn  where it was assumed that the final state contains different
particles and that there are two decay channels. The prefactor in Ref.
Barman:2023ymn  is $1/32$, which recovers to $1/16$ after multiplying
$2^{2}/2$.

$\frac{d\Gamma^{1\to
3}}{dE_{\omega}}\simeq\frac{1}{16\pi^{3}}\left(\frac{\mu_{\rm
eff}}{M_{P}}\right)^{2}\left[\frac{(1-2x)\,(1-2x+2y^{2})}{4\,x\,\zeta^{-1}}+\frac{y^{2}\,(y^{2}+2x-1)}{x}\ln\left(\frac{1+\zeta}{1-\zeta}\right)\right],$
(21)

with $x\equiv E_{\omega}/\varpi$, and

$\zeta\equiv\sqrt{1-\frac{4\,y^{2}}{1-2\,x}}\,.$ (22)

In the above expression, we have considered only the contribution of the first
mode $j=1$ in $\mu_{\rm eff}$. The graviton energy spans the range

$0<E_{\omega}\leq m_{\phi}\left(\frac{1}{2}-2\,y^{2}\right).$ (23)

We see that when $E_{\omega}\to m_{\phi}\left(\frac{1}{2}-2\,y^{2}\right)$,
the decay becomes kinematically forbidden,777It was shown in Ref.
Mambrini:2022uol  that graviton bremsstrahlung can also be the source of dark
matter production. and therefore $d\Gamma^{1\to 3}/dE_{\omega}\to 0$. Note
that here $E_{\omega}$ corresponds to the graviton energy at emission, which
can be at most half of the inflaton mass in the limit $y\to 0$.

###  2.3 Graviton emission from inflaton annihilation

Instead of decay, the transfer of energy to the SM bath during reheating could
be dominated by inflaton annihilation, see e.g. Refs. Hooper:2018buz ;
Garcia:2021gsy  for motivated scenarios. The $2\to 2$ annihilation cross
section of inflatons in a pair of real scalars $\varphi$, in the non-
relativistic limit is given by

$\Gamma^{2\to 2}\simeq\frac{\sigma^{2}_{\rm
eff}\,\rho_{\phi}}{8\pi\,m_{\phi}^{3}}\,,$ (24)

where

$\sigma_{\text{eff}}^{2}\simeq\sigma^{2}\,n\,(n+2)(n-1)^{2}\,\frac{\varpi}{m_{\phi}}\,\sum_{j=1}^{\infty}j\,|\mathcal{P}_{j}|^{2}\left\langle\left(1-\frac{4\,y^{2}}{j^{2}}\,\mathcal{P}\right)^{1/2}\right\rangle$
(25)

denotes the effective coupling after averaging inflaton oscillations
Garcia:2020wiy . We have introduced the Fourier decomposition of the
oscillating background field square

$\phi(t)^{2}=\phi_{0}^{2}(t)\,\mathcal{P}^{(2)}(t)=\phi_{0}^{2}(t)\,\sum\limits_{j=1}^{\infty}\mathcal{P}^{(2)}_{j}(t)\,e^{-i\,j\,\varpi\,t}.$
(26)

The corresponding interaction rate density is $\gamma\equiv\Gamma^{2\to
2}\,\rho_{\phi}$, and therefore Eq. (7) admits the analytical solution
Xu:2023lxw

$\rho_{R}(a)\simeq\frac{n}{2n-5}\,\rho_{\phi}(a_{\text{end}})\left(\frac{a_{\text{end}}}{a}\right)^{\frac{18}{n+2}}\left[1-\left(\frac{a_{\text{end}}}{a}\right)^{\frac{2(2n-5)}{2+n}}\right]\left(\frac{a_{\text{end}}}{a_{\text{RH}}}\right)^{\frac{6(n-3)}{n+2}},$
(27)

which in turn implies that during reheating

$\alpha\equiv\frac{9}{2\,n+4}\,,$ (28)

for $n\geq 5/2$. Interestingly, for $n=2$ (and, in general, for $n<5/2$), the
radiation energy density dilutes faster than nonrelativistic matter (i.e. the
inflaton), which implies that it cannot overcome the inflaton energy density,
and, therefore, the universe cannot become radiation dominated. In the case of
annihilation, Fig. 1 also shows the evolution of the energy densities (left)
and the temperature of the SM (right) as a function of the scale factor, for
$T_{\text{RH}}=10^{13}$ GeV, and $T_{\text{max}}=4.3\times 10^{14}$ GeV for
$n=4$ (top), or $T_{\text{max}}=1.2\times 10^{14}$ GeV for $n=6$ (bottom). It
is clear that in the annihilation case, the evolution of radiation or
temperature is steeper than that in the decay case, which has important
implications for the GW spectrum, as will be shown in the next section.

There are some self-consistency conditions that must be met. First, one has to
ensure that the inflaton mass satisfies $m_{\phi}(a)<M_{P}$ Barman:2023rpg ,
which leads to upper bounds on the reheating energy scale:
$m_{\phi}^{\text{RH}}<M_{P}\,(T_{\text{max}}/T_{\text{RH}})^{2(2-n)/3}$ for
annihilations and
$m_{\phi}^{\text{RH}}<M_{P}\,(T_{\text{max}}/T_{\text{RH}})^{2(2-n)}$ for
decays. Alternatively, if one treats $m_{\phi}^{\text{RH}}$ as a free
parameter, the condition $m_{\phi}(a)<M_{P}$ also gives upper bounds on the
temperature ratio, which reads
$(T_{\text{max}}/T_{\text{RH}})<(m_{\phi}^{\text{RH}}/M_{P})^{3/(4-2n)}$ for
annihilations and
$(T_{\text{max}}/T_{\text{RH}})<(m_{\phi}^{\text{RH}}/M_{P})^{1/(4-2n)}$ for
decays. Furthermore, we can take advantage of the recent constraint on the
inflationary tensor-to-scalar ratio $r<0.035$, obtained from BICEP/Keck 2018
BICEP:2021xfz , to derive an upper bound on the inflationary scale, denoted by
$H_{I}<2.0\times 10^{-5}~{}M_{P}$. This upper bound on the inflationary scale,
in turn, allows us to establish upper bounds on the reheating temperature.
Consequently, for the scenario of annihilation, the upper bounds on the
reheating temperature are given by $T_{\text{RH}}<3.3\times
10^{15}~{}\text{GeV}\,(T_{\text{max}}/T_{\text{RH}})^{-4/3}$ with $n=4$ and
$T_{\text{RH}}<3.1\times 10^{15}$ GeV $(T_{\text{max}}/T_{\text{RH}})^{-2}$
with $n=6$. On the other hand, for the inflaton decay scenario, the bounds on
the reheating temperature are expressed as $T_{\text{RH}}<2.8\times
10^{15}~{}\text{GeV}\,(T_{\text{max}}/T_{\text{RH}})^{-4}$ for $n=4$ and
$T_{\text{RH}}<2.5\times
10^{15}~{}\text{GeV}\,(T_{\text{max}}/T_{\text{RH}})^{-6}$ for $n=6$. We have
checked that the values of $T_{\text{max}}$ and $T_{\text{RH}}$ shown in Fig.
1 satisfy these constraints.

Figure 3: Inflaton annihilation into a pair of $\varphi$ and a graviton.

Similarly to the case of decay, there is a $2\to 3$ inflaton annihilation
process containing the emission of a graviton, as shown in Fig. 3. Using the
Feynman rules and the formalism as presented in Ref. Barman:2023ymn , we
obtain the following amplitudes

$\displaystyle i\mathcal{M}^{1}_{j}$
$\displaystyle=-\,\frac{i\,\sigma\phi_{0}^{2}}{M_{P}}\,\frac{l_{1\,\mu}\,l_{1\,\nu}\,\epsilon^{\star\mu\nu}}{l_{1}\cdot\omega}\,\mathcal{P}_{j}^{(2)},$
(29) $\displaystyle i\mathcal{M}^{2}_{j}$
$\displaystyle=-\frac{i\,\sigma\phi_{0}^{2}}{M_{P}}\,\frac{l_{2\,\mu}\,l_{2\,\nu}\epsilon^{\star\mu\nu}}{l_{2}\cdot\omega}\,\mathcal{P}_{j}^{(2)},$
(30) $\displaystyle i\mathcal{M}^{3}_{j}$
$\displaystyle=\frac{i\,\sigma\phi_{0}^{2}}{M_{P}}\,\frac{p_{\mu}\,p_{\nu}\,\epsilon^{\star\mu\nu}}{p\cdot\omega}\,\mathcal{P}_{j}^{(2)},$
(31) $\displaystyle i\mathcal{M}^{4}_{j}$
$\displaystyle=\frac{i\,\sigma\phi_{0}^{2}}{M_{P}}\,\frac{q_{\mu}\,q_{\nu}\,\epsilon^{\star\mu\nu}}{q\cdot\omega}\,\mathcal{P}_{j}^{(2)},$
(32) $\displaystyle i\mathcal{M}_{5}$
$\displaystyle\propto\eta_{\mu\nu}\,\epsilon^{\star\mu\nu},$ (33)

where

$\displaystyle p\cdot\omega$
$\displaystyle=2m_{\phi}\left(E_{\omega}+E_{p}-m_{\phi}\right),$ (34)
$\displaystyle q\cdot\omega$
$\displaystyle=(l_{1}\,+l_{2}-\omega-p)\cdot\omega=2m_{\phi}\,E_{\omega}-p\cdot\omega\,,$
(35)

and $\epsilon^{\mu\nu}$ are the polarisation tensors of the graviton. Note
that $\mathcal{M}_{5}$ vanishes due to the traceless condition of the massless
graviton, while $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ vanish since we assume
that the inflaton condensate annihilates with a vanishing three-momentum
Barman:2023ymn .

The $2\to 3$ differential interaction rate reads888Note that there we have
$2\,m_{\phi}$ since the initial energy for annihilation is twice larger
compared to that from the decay, which is defined in Eq. (20).

$\frac{d\Gamma^{2\to
3}}{dE_{\omega}}=\frac{2\,m_{\phi}}{\rho_{\phi}(1+w_{\phi})}\sum\limits_{j=1}^{\infty}\int|\mathcal{M}^{2\to
3}_{j}|^{2}\frac{d\rm PS^{(3)}}{dE_{\omega}}\,,$ (36)

leading to

$\frac{d\Gamma^{2\to 3}}{dE_{\omega}}\simeq\frac{\sigma^{2}_{\rm
eff}}{16\pi^{3}}\frac{\rho_{\phi}}{m_{\phi}^{2}M_{P}^{2}}\left[\frac{(1-x)\,(2-2x+y^{2})}{x\,\beta^{-1}}+\frac{y^{2}(y^{2}+4x-4)}{2x}\log\left(\frac{1+\beta}{1-\beta}\right)\right],$
(37)

where we have considered only the contribution of the first mode in
$\sigma_{\rm eff}$, with

$\beta\equiv\sqrt{1-\frac{y^{2}}{1-x}}\,,$ (38)

and the graviton energy in the range

$0<E_{\omega}\leq m_{\phi}\left(1-y^{2}\right).$ (39)

As expected, in the limit $E_{\omega}\to m_{\phi}(1-y^{2})$, there is no
graviton emission as $2\to 3$ annihilations become kinematically forbidden.
Note that for annihilation, the graviton energy $E_{\omega}$ can be as large
as the inflaton mass in the limit $y\to 0$, which is twice larger than that
from the decay (cf. Eq. (23)). We will see in the next section that such
distinctions have important implications for the range of frequency for the
emitted GWs in the two processes.

##  3 Gravitational Wave Spectrum

###  3.1 Overview

Before analyzing in more detail the spectrum and constraints of GWs generated
by bremsstrahlung of gravitons, in this section we summarize the procedure in
the case of inflaton decay and scattering, in the simplest case of a quadratic
potential $V(\phi)=\frac{1}{2}m_{\phi}^{2}\,\phi^{2}$. We refer the reader to
Appendix A for the complete treatment, which takes into account the sum over
all modes in the case of generic potentials. Although we know that complete
reheating is not possible in this scenario for pure scattering Garcia:2020eof
; Garcia:2020wiy , this example will help us better understand the context and
expected results. Moreover, it is always possible to imagine mixed scenarios,
where graviton emission by scattering is independent of the process
responsible for reheating, such as gravitational reheating.

The differential Boltzmann equation for the energy density $\rho_{\text{GW}}$
stored in the form of graviton radiation generated by inflaton decay is

$\frac{d}{dt}\left(\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right)+4\,H\,\frac{d\rho_{\text{GW}}}{dE_{\omega}}=\frac{\rho_{\phi}}{m_{\phi}}\,\frac{d\Gamma^{1\rightarrow
3}}{dE_{\omega}}\times E_{\omega}\,,$ (40)

where $d\Gamma^{1\to 3}/dE_{\omega}$ is given by Eq. (21). This equation can
be rewritten as

$\frac{d}{da}\left(a^{4}\,\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right)=\frac{a^{3}}{H}\,\frac{\rho_{\phi}}{m_{\phi}}\,\frac{d\Gamma^{1\rightarrow
3}}{dE_{\omega}}\times E_{\omega}\,,$ (41)

which, in the limit $m$, $E_{\omega}\ll m_{\phi}$, admits the approximate
analytical solution

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}\simeq\frac{\sqrt{3}}{160\pi^{3}}\,\frac{\mu^{2}\sqrt{\rho_{\rm
RH}}}{M_{P}}\,,$ (42)

where we used $\rho_{\phi}(a)=\rho_{\rm
RH}\left(\frac{a_{\text{RH}}}{a}\right)^{3}$ in the case of a quadratic
potential. To obtain the gravitational energy density today
$\Omega_{\text{GW}}^{0}$, at energy $E_{\omega}$, we need to apply the
redshift $(a_{\text{RH}}/a_{0})^{4}$ for the energy density and
$a_{\text{RH}}/a_{0}$ for the graviton energy. We then have

$\Omega_{\text{GW}}^{0}=\frac{1}{\rho_{c}^{0}}\,\left.\frac{d\rho_{\text{GW}}}{d\ln
E_{\omega}}\right|_{a_{0}}=\frac{\sqrt{3}}{160\pi^{3}}\,\frac{\mu^{2}\,\rho_{R}^{0}}{M_{P}\,\rho_{c}^{0}\,\sqrt{\rho_{\rm
RH}}}\times E_{\omega}\,,$ (43)

where $\Omega_{R}^{0}=\frac{\rho_{R}^{0}}{\rho_{c}^{0}}$ is the relative
radiation density today, $\rho_{c}^{0}\simeq 1.05\times 10^{-5}~{}h^{2}$
GeV/cm3 the critical density at present, and we have used
$a_{\text{RH}}/a_{0}=(\rho_{R}^{0}/\rho_{\rm RH})^{1/4}$. Alternatively,
expressing Eq. (43) as function of the frequency defined by
$E_{\omega}(a)=2\pi\,f\,\frac{a_{0}}{a}$, we obtain

$\Omega_{\text{GW}}^{1\rightarrow 3}h^{2}\simeq
10^{-20}\left(\frac{10^{10}~{}\text{GeV}}{T_{\text{RH}}}\right)\left(\frac{\mu}{10^{10}~{}\text{GeV}}\right)^{2}\left(\frac{f}{10^{8}~{}\rm{Hz}}\right),$
(44)

with $h\simeq 0.67$ the present dimensionless Hubble parameter. Now, if one
considers that the reheating is also given by the same decaying channel, we
can express the GW spectrum completely as a function of $T_{\text{RH}}$ (or
$\mu$) considering $\Gamma^{1\to 2}=\frac{\mu^{2}}{8\pi\,m_{\phi}}$ from Eq.
(13), and $\rho_{\rm RH}\simeq\Gamma_{\phi}^{2}\,M_{P}^{2}$. In this case,
$\mu=10^{10}$ GeV gives $T_{\text{RH}}\sim 10^{12}$ GeV with $m_{\phi}\simeq
10^{13}~{}\text{GeV}$. We note that Eq. (44) agrees with the earlier result
presented in Ref. Barman:2023ymn .

For scatterings, the procedure is exactly the same, except that
$\Gamma^{1\rightarrow 3}$ must be replaced by $\Gamma^{2\rightarrow 3}$ and
$m_{\phi}\to 2m_{\phi}$ for annihilation in Eq. (40). We then obtain

$\Omega_{\text{GW}}^{2\rightarrow 3}h^{2}\simeq
10^{-19}\,\sigma^{2}\left(\frac{T_{\text{RH}}}{10^{10}~{}\text{GeV}}\right)^{\frac{7}{3}}\left(\frac{\rho_{\rm
end}}{6.25\times
10^{62}~{}\text{GeV}^{4}}\right)^{\frac{1}{6}}\left(\frac{10^{13}~{}\text{GeV}}{m_{\phi}}\right)^{2}\left(\frac{f}{10^{8}~{}\rm{Hz}}\right),$
(45)

where $\rho_{\rm end}$ represents the energy density stored in the inflaton
field at the end of inflation. We normalized Eq. (45) with parameters typical
of the $T$-model $\alpha$-attractors, that is, $\rho_{\rm end}\sim(5\times
10^{15})^{4}$ GeV4 and $m_{\phi}\sim 10^{13}$ GeV. For the sake of
completeness, we report in Appendix A the complete expression summed over all
the inflaton oscillatory modes for a generic potential
$V(\phi)\propto\phi^{n}$.

Figure 4: GW spectrum (top) and GW strain (bottom) for the quadratic inflaton
potential $n=2$ with $T_{\text{RH}}=10^{10}$ GeV (left) and
$T_{\text{RH}}=10^{13}$ GeV (right). Solid black lines represent the signal
for inflaton decays with $\mu=10^{10}$ GeV whereas dashed black lines
represent the signal for inflaton scattering with $\sigma=1$. Sensitivities of
future GW detectors are represented by solid orange lines.

We illustrate in the top left panel of Fig. 4 the GW spectrum predicted for
inflaton decay (solid black line) and scattering (dotted black line) assuming
$\mu=10^{10}$ GeV, $\sigma=1$, $T_{\text{RH}}=10^{10}$ GeV and $n=2$. We note
that the spectra (in the low-frequency regime) nicely match the approximate
expressions in Eqs. (44) and (45), respectively. As mentioned in the previous
section, the emitted graviton energy from inflaton annihilation can be twice
as large as that from the decay, and consequently, the upper limit of the GW
frequency can be twice as large. This explains why the dotted curve
(annihilation) peaks at a frequency twice as large as the solid line (decay).

In addition, we overlay several proposed high-frequency GW detectors999See
Ref. Aggarwal:2020olq  for a recent review. with high sensitivity in the
figure. These include LISA LISA:2017pwj , the Einstein Telescope (ET)
Punturo:2010zz ; Hild:2010id ; Sathyaprakash:2012jk ; Maggiore:2019uih , the
Big Bang Observer (BBO) Crowder:2005nr ; Corbin:2005ny ; Harry:2006fi ,
ultimate DECIGO (uDECIGO) Seto:2001qf ; Kudoh:2005as , GW-electromagnetic wave
conversion in vacuum (solid) and in a Gaussian beam (GB) (dotted) Li:2003tv ;
Ringwald:2020ist , resonant cavities Berlin:2021txa ; Berlin:2023grv ;
Herman:2022fau , and the International Axion Observatory (IAXO)
Armengaud:2014gea ; IAXO:2019mpb . Sensitivity curves for these detectors were
adapted from Refs. Ringwald:2020ist ; Ringwald:2022xif . The energy stored in
GWs exhibits characteristics similar to those of dark radiation, which, in
turn, contributes to the effective number of neutrino species, denoted
$N_{\text{eff}}$. In this paper, we present a compilation of various
constraints related to this phenomenon. The Planck 2018 mission provides
valuable information, establishing a 95% CL result of $N_{\text{eff}}=2.99\pm
0.34$ Planck:2018vyg . Future experiments, such as COrE COrE:2011bfs  and
Euclid EUCLID:2011zbd , are expected to significantly enhance these
constraints at the $2\sigma$ level, resulting in $\Delta
N_{\text{eff}}\lesssim 0.013$. It is also interesting to mention a bound of
$\Delta N_{\text{eff}}\lesssim 3\times 10^{-6}$ reported in Ref. Ben-
Dayan:2019gll  based on a hypothetical cosmic-variance-limited (CVL) CMB
polarization experiment. We note that for a reheating temperature as low as
$10^{10}$ GeV, it is not possible to detect graviton bremsstrahlung, even in
future experiments.

However, the top right panel of Fig. 4 shows an equivalent result, except for
$T_{\text{RH}}=10^{13}$ GeV, which corresponds roughly to an instantaneous
reheating. We then clearly see that the spectrum derived from the scattering,
in the case $n=2$, is larger by orders of magnitude than that generated by the
inflaton decay. It is also interesting to note that there is a weak dependence
of our results on the parameter $n$. We have made this explicit in Eq. (70) in
the case of scattering. Furthermore, in our analysis above, we distinguish
between the reheating process, which involves $T_{\text{RH}}$, and the
bremsstrahlung process. We can also ask whether it is possible to test what
the reheating process is (decay or scattering) by measuring the graviton
bremsstrahlung generated during the reheating period. But for that, one needs
to look for $n\geq 4$, as reheating is not possible by scattering for $n=2$,
which is the purpose of the following section.

For the sake of completeness, and in order to project sensitivity curves from
different GW experiments, it is also convenient to define the dimensionless
strain $h_{c}$ in terms of the GW spectral energy density as Maggiore:1999vm

$h_{c}(f)\equiv\frac{H_{0}}{f}\,\sqrt{\frac{3}{2\pi^{2}}\,\Omega_{\text{GW}}(f)}\simeq
1.26\times
10^{-18}\left(\frac{\rm{Hz}}{f}\right)\sqrt{h^{2}\,\Omega_{\text{GW}}(f)}\,,$
(46)

where $H_{0}\equiv H(T_{0})\simeq 1.44\times 10^{-42}$ GeV is the present-day
Hubble parameter Planck:2018vyg . As an illustration, in Fig. 4 (bottom) we
depict the strain parameter $h_{c}$ as a function of frequency $f$ for the
same set of parameters as the one used in Fig. 4 (top).

###  3.2 Combining graviton bremsstrahlung and reheating

In the previous section, the system of Boltzmann equations for the background
was solved in the general case, without directly linking the reheating channel
with the graviton emission channel. However, to compute the GW spectrum one
has to keep track of the evolution of the differential GW energy density
$\rho_{\text{GW}}$ and radiation $\rho_{R}$, and therefore instead of Eqs. (6)
and (7) one has to solve Barman:2023rpg

$\displaystyle\frac{d\rho_{\phi}}{dt}+\frac{6\,n}{2+n}\,H\,\rho_{\phi}=-\frac{2\,n}{2+n}\,\left(\gamma^{(0)}+\gamma^{(1)}\right),$
(47)
$\displaystyle\frac{d\rho_{R}}{dt}+4\,H\,\rho_{R}=+\frac{2\,n}{2+n}\,\gamma^{(0)}+\frac{2\,n}{2+n}\,\int\frac{d\gamma^{(1)}}{dE_{\omega}}\,\frac{E_{\phi}-E_{\omega}}{E_{\phi}}\,dE_{\omega}\,,$
(48)
$\displaystyle\frac{d\rho_{\text{GW}}}{dt}+4\,H\,\rho_{\text{GW}}=+\frac{2\,n}{2+n}\,\int\frac{d\gamma^{(1)}}{dE_{\omega}}\,\frac{E_{\omega}}{E_{\phi}}\,dE_{\omega}\,,$
(49)

where $E_{\omega}$ corresponds to the graviton energy at the moment of
production, and $E_{\phi}$ corresponds to the initial total inflaton energy,
either $E_{\phi}=m_{\phi}$ for decays or $E_{\phi}=2\,m_{\phi}$ for
annihilations. The term $(E_{\phi}-E_{\omega})/E_{\phi}$ describes the energy
fraction that goes to radiation. The density rates $\gamma^{(0)}$ and
$\gamma^{(1)}$ correspond to the cases without and with graviton emission,
respectively. More specifically, for inflaton decays, we have
$\gamma^{(0)}=\Gamma^{1\to 2}\rho_{\phi}$ and $\gamma^{(1)}=\Gamma^{1\to
3}\rho_{\phi}$, and $\gamma^{(0)}=\Gamma^{2\to 2}\rho_{\phi}$ and
$\gamma^{(1)}=\Gamma^{2\to 3}\rho_{\phi}$ for inflaton
annihilations.101010Note that if the decay final states have $N$ degrees of
freedom, one shall multiply a factor $N$ in the rates.

For the GW spectrum, what we need is the differential GW energy density;
therefore, it is convenient to write Eq. (49) as

$\frac{d}{da}\frac{d(\rho_{\text{GW}}/\rho_{R})}{dE_{\omega}}\simeq\frac{2n}{2+n}\frac{1}{a\,H(T_{\text{RH}})}\left(\frac{a_{\text{RH}}}{a}\right)^{\frac{3n}{2+n}-4\alpha}\left[\frac{d\Gamma^{(1)}}{dE_{\omega}}\frac{E_{\omega}}{m_{\phi}}-\frac{d(\rho_{\text{GW}}/\rho_{R})}{dE_{\omega}}\Gamma^{(0)}\right].$
(50)

Note that $E_{\omega}$ corresponds to the energy at the moment when gravitons
are emitted. However, to compute the GW spectrum, we need to consider the
redshift and sum over different (redshifted) energies. Therefore, in Eq. (50)
the change of variable
$E_{\omega}(E_{\omega}^{\prime},\,a)=E_{\omega}^{\prime}\,\frac{a_{\text{RH}}}{a}$,
with $E_{\omega}^{\prime}$ being the energy at $a=a_{\text{RH}}$, is required,
leading to Barman:2023rpg

$\frac{d}{da}\frac{d(\rho_{\text{GW}}/\rho_{R})}{dE_{\omega}^{\prime}}\simeq\frac{a_{\text{RH}}}{a}\frac{2n}{2+n}\frac{1}{a\,H(T_{\text{RH}})}\left(\frac{a_{\text{RH}}}{a}\right)^{\frac{3n}{2+n}-4\alpha}\left[\frac{d\Gamma^{(1)}}{dE_{\omega}^{\prime}}\frac{E_{\omega}^{\prime}}{m_{\phi}}-\frac{a}{a_{\text{RH}}}\frac{d(\rho_{\text{GW}}/\rho_{R})}{dE_{\omega}^{\prime}}\Gamma^{(0)}\right],$
(51)

where the extra overall factor $a_{\text{RH}}/a$ arises from the Jacobian
related to the changing of variables.

The solution for Eq. (51) in the case of inflaton decays was recently reported
in Ref. Barman:2023rpg . In this work, we focus on the yet unexplored case of
inflaton annihilation. In that case, Eq. (51) admits analytical solutions for
different values of $n$. Recalling that reheating by inflaton annihilation is
not viable in the case $n=2$, we explore the scenarios with $n=4$ and $n=6$.

  * •

For $n=4$, it follows that

$\frac{d(\rho_{\text{GW}}/\rho_{R})}{dE_{\omega}^{\prime}}\simeq\frac{2\,m_{\phi}^{\text{RH}}}{\,M_{P}^{2}\,\pi^{2}}\left(1-\frac{E_{\omega}^{\prime}}{m_{\phi}^{\text{RH}}}\right)^{2}\left[\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{8/9}-1\right],$
(52)

for $0\leq E_{\omega}^{\prime}\leq m_{\phi}^{\text{RH}}$.

  * •

For $n=6$, one has

$\displaystyle\frac{d(\rho_{\text{GW}}/\rho_{R})}{dE_{\omega}^{\prime}}\simeq\frac{3\,m_{\phi}^{\text{RH}}}{2\pi^{2}M_{P}^{2}}$
$\displaystyle\Bigg{[}\left(\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{16/9}-1\right)-4\frac{E_{\omega}^{\prime}}{m_{\phi}^{\text{RH}}}\left(\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{8/9}-1\right)$
$\displaystyle\quad+\frac{16}{9}\left(\frac{E_{\omega}^{\prime}}{m_{\phi}^{\text{RH}}}\right)^{2}\log\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)\Bigg{]}$
(53)

for $0\leq E_{\omega}^{\prime}\leq m_{\phi}^{\text{RH}}$, while for
$m_{\phi}^{\text{RH}}\leq E_{\omega}^{\prime}\leq
m_{\phi}^{\text{RH}}\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{8/9}$
it becomes

$\displaystyle\frac{d(\rho_{\text{GW}}/\rho_{R})}{dE_{\omega}^{\prime}}\simeq\frac{3\,m_{\phi}^{\text{RH}}}{2\pi^{2}\,M_{P}^{2}}$
$\displaystyle\Bigg{[}\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{16/9}-4\frac{E_{\omega}^{\prime}}{m_{\phi}^{\text{RH}}}\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{8/9}$
$\displaystyle\quad+\left(\frac{E_{\omega}^{\prime}}{m_{\phi}^{\text{RH}}}\right)^{2}\left(3+2\log\left[\frac{m_{\phi}^{\text{RH}}}{E_{\omega}^{\prime}}\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{8/9}\right]\right)\Bigg{]}.$
(54)

Several comments on the regimes for $E_{\omega}^{\prime}$ shown above are in
order. Note that the graviton energy at emission always lies in the range
$0<E_{\omega}\leq m_{\phi}(a)$, which together with
$E_{\omega}^{\prime}=E_{\omega}\,a/a_{\text{RH}}$ implies
$0<E_{\omega}^{\prime}\leq
m_{\phi}^{\text{RH}}\,(a/a_{\text{RH}})^{2(n-4)/(n+2)}$. Subsequently, for
$n=4$, one has $0<E_{\omega}^{\prime}\leq m_{\phi}^{\text{RH}}$. However, for
$n=6$, there are two energy regimes: low-energy gravitons in the range
$0<E_{\omega}^{\prime}\leq m_{\phi}^{\text{RH}}$ can be produced during the
entire reheating period, while high-energy gravitons with
$m_{\phi}^{\text{RH}}\leq E_{\omega}^{\prime}\leq
m_{\phi}^{\text{RH}}\,\left(T_{\text{max}}/T_{\text{RH}}\right)^{8/9}$ can
only be produced in the last stages of reheating. Additionally, we notice that
the GW spectrum features a smaller boost proportional to powers of the ratio
$T_{\text{max}}/T_{\text{RH}}$, compared to the decay case, as for
annihilations the entropy dilution is more prominent, as shown in Fig. 1.

The primordial GW spectrum $\Omega_{\text{GW}}(f)$ at present per logarithmic
frequency $f$ is

$\Omega_{\text{GW}}(f)=\Omega_{\gamma}^{(0)}\,\frac{g_{\star}(T_{\text{RH}})}{g_{\star}(T_{0})}\left[\frac{g_{\star
s}(T_{0})}{g_{\star
s}(T_{\text{RH}})}\right]^{4/3}\,\frac{d(\rho_{\text{GW}}(T_{\text{RH}})/\rho_{R}(T_{\text{RH}}))}{d\ln
E_{\omega}^{\prime}}\,,$ (55)

where $\Omega_{\gamma}^{(0)}h^{2}\simeq 2.47\times 10^{-5}$ is the current
photon density, $T_{0}\simeq 2.73$ K is the CMB temperature Planck:2018vyg ,
and $g_{\star s}(T)$ is the number of relativistic degrees of freedom that
contribute to the SM entropy. The present GW frequency is associated with the
graviton energy $E_{\omega}^{\prime}$ at the end of reheating via

$f=\frac{E_{\omega}^{\prime}}{2\pi}\,\frac{a_{\text{RH}}}{a_{0}}=\frac{E_{\omega}^{\prime}}{2\pi}\,\frac{T_{0}}{T_{\text{RH}}}\left[\frac{g_{\star
s}(T_{0})}{g_{\star s}(T_{\text{RH}})}\right]^{1/3},$ (56)

considering the redshift of the graviton energy from the end of reheating and
the present epoch. Note that the frequency is bounded from above since the
graviton at production could carry at most half of the total inflaton energy,
namely $E_{\omega}(a)\leq m_{\phi}(a)$ for inflaton annihilation, which
translates into

$f\leq\frac{m_{\phi}^{\text{RH}}}{2\pi}\,\frac{a_{\text{RH}}}{a_{0}}\left(\frac{a_{\text{RH}}}{a}\right)^{\frac{2(n-4)}{n+2}}\leq\frac{m_{\phi}^{\text{RH}}}{2\pi}\,\frac{T_{0}}{T_{\text{RH}}}\left[\frac{g_{\star
s}(T_{0})}{g_{\star
s}(T_{\text{RH}})}\right]^{1/3}\times\begin{dcases}1&\text{ for }n\leq 4\,,\\\
\left(\frac{T_{\text{max}}}{T_{\text{RH}}}\right)^{\frac{2(n-4)}{\alpha(n+2)}}&\text{
for }n>4\,.\end{dcases}$ (57)

In the case of an inflaton decay, $E_{\omega}(a)\leq m_{\phi}(a)/2$, leading
to an additional $1/2$ on the right-hand side of Eq. (57).

In Fig. 5, we present the GW spectrum for the benchmark parameters: ①
$T_{\text{RH}}=10^{13}$ GeV, $m_{\phi}^{\text{RH}}=1.1\times 10^{11}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=43$ (for $n=4$) and $T_{\text{RH}}=10^{13}$ GeV,
$m_{\phi}^{\text{RH}}=2.6\times 10^{10}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=12$ (for $n=6$), and ②
$m_{\phi}^{\text{RH}}=5\times 10^{16}$ GeV, $T_{\text{RH}}=5\times 10^{13}$
GeV and $T_{\text{max}}/T_{\text{RH}}=4$ for both $n=4$ and $n=6$. Note that
the parameters in ① are derived from the $\alpha$-attractor $T$-model (cf.
Fig. 1). In addition, the model parameters we have considered satisfy the
limits detailed in Section 2.3. GW signals are depicted for two different
cases: $n=6$ (solid black line) and $n=4$ (dotted black line). For the case
with $n=6$, the signal shows a boost factor
$\propto\left(T_{\text{max}}/T_{\text{RH}}\right)^{16/9}$ (cf. Eq. (• ‣ 3.2)),
which exceeds the boost factor
$\propto\left(T_{\text{max}}/T_{\text{RH}}\right)^{8/9}$ observed in the case
$n=4$ (cf. Eq. (52)), and therefore the strain of the GW spectrum
corresponding to $n=6$ is expected to be higher. (Note that this conclusion is
based on the same model parameters, that is, ②.) Furthermore, it should be
noted that the upper limit of the GW frequency is higher for $n=6$ compared to
the case with $n=4$, as explained by Eq. (57). The wider range of frequencies
for $n=6$ adds to the distinction between the two cases in the spectrum. These
results suggest that the parameter space of the inflaton annihilation
reheating scenario might be probed by next-generation GW detectors as
gravitational cavities and $\Delta N_{\text{eff}}$ experiments.

Figure 5: GW signature from graviton bremsstrahlung via inflaton annihilation
during reheating, assuming the following parameters: ① $T_{\text{RH}}=10^{13}$
GeV, $m_{\phi}^{\text{RH}}=1.1\times 10^{11}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=43$ (for $n=4$) and $T_{\text{RH}}=10^{13}$ GeV,
$m_{\phi}^{\text{RH}}=2.6\times 10^{10}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=12$ (for $n=6$); ② $m_{\phi}^{\text{RH}}=5\times
10^{16}$ GeV, $T_{\text{RH}}=5\times 10^{13}$ GeV and
$T_{\text{max}}/T_{\text{RH}}=4$ for both $n=4$ and $n=6$.

To facilitate comparison of our results on inflaton annihilation with inflaton
decays, in Fig. 6, we show the GW spectrum of several benchmark parameters. In
the left panel with $n=4$, the parameters considered are: ①
$T_{\text{RH}}=10^{13}$ GeV, $m_{\phi}^{\text{RH}}=1.1\times 10^{11}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=3.6$ (for decay) and $T_{\text{RH}}=10^{13}$
GeV, $m_{\phi}^{\text{RH}}=1.1\times 10^{11}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=43$ (for annihilation); ②
$T_{\text{RH}}=10^{12}$ GeV, $T_{\text{max}}/T_{\text{RH}}=7$,
$m_{\phi}^{\text{RH}}=8\times 10^{14}$ GeV for both decay and annihilation. In
the right frame with $n=6$: ① $T_{\text{RH}}=10^{13}$ GeV,
$m_{\phi}^{\text{RH}}=2.6\times 10^{10}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=2.4$ (for decay) and $T_{\text{RH}}=10^{13}$
GeV, $m_{\phi}^{\text{RH}}=2.6\times 10^{10}$ GeV,
$T_{\text{max}}/T_{\text{RH}}=12$ (for annihilation); ② $T_{\text{RH}}=3\times
10^{12}$ GeV, $T_{\text{max}}/T_{\text{RH}}=3$, $m_{\phi}^{\text{RH}}=3\times
10^{14}$ GeV for both decay and annihilation.

Solid lines correspond to the case where the inflaton annihilates, while
dotted curves correspond to the case of decay. For $n=4$, we notice that the
GW frequency in the annihilation scenario can be shifted to a higher value
compared to the decay model; this is because the maximum energy of the emitted
graviton can be twice higher in the annihilation scenario. In the context of
the same model parameters, e.g., ②, we find that the GW amplitude in the case
of annihilation is smaller than that in the case of decay. This is because for
annihilation, the spectrum has an enhancement during reheating
$\propto\left(T_{\text{max}}/T_{\text{RH}}\right)^{8/9}$ (cf. Eq. (52)) that
is lower than that in the case of decays, where the enhancement is
$\propto\left(T_{\text{max}}/T_{\text{RH}}\right)^{8/3}$ Barman:2023rpg . This
distinctive enhancement originates from the evolution of the temperature and
hence the dilution effect during reheating. Due to the steeper slope of
temperature evolution in the annihilation scenario (cf. Fig. 1), it is
expected that for larger $n$, the dilution effect becomes more prominent.
Consequently, the spectrum in the annihilation model becomes suppressed
compared to that in the decay case, as shown in the figure. With a progressive
increase in the parameter $n$, the distinction between the GW spectra in the
two scenarios considered becomes increasingly pronounced, as illustrated in
the right panel.

Figure 6: Comparison for GW signature from graviton bremsstrahlung via
inflaton decay and annihilation during reheating. The left and right frames
correspond to the spectrum with $n=4$ and $n=6$, respectively.

For the aforementioned benchmark parameters, we notice that the GW signature
might be within the sensitivity of future cavity experiments. Specifically,
for $n=6$, we observe that the GW spectrum exhibits a discernible signal that
may be detectable by resonant cavity detectors. On the contrary, the amplitude
of the GW spectrum in the annihilation scenario is considerably lower and lies
beyond the range of sensitivity of the cavity detectors.

In the event that a null signal is observed in future high-frequency GW
measurements, it would enable the exclusion of certain regions of the
parameter space. Consequently, our proposed approach, utilizing
bremsstrahlung-induced GWs, could give useful information on the reheating
dynamics.

##  4 Conclusions

In this study, we have revisited graviton bremsstrahlung during the phase of
inflationary reheating in the presence of an inflaton field $\phi$ oscillating
around a generic monomial potential $\phi^{n}$. Unlike previous analyses, we
have examined for the first time the gravitational wave (GW) spectrum within
the context of a reheating scenario involving inflaton annihilations. We
calculated the corresponding graviton emission rate, which is presented in Eq.
(37) and the corresponding GW spectra are illustrated in Fig. 4. This is
complementary to previous studies where only inflaton decays were assumed
Nakayama:2018ptw ; Huang:2019lgd ; Ghoshal:2022kqp ; Barman:2023ymn ;
Barman:2023rpg ; Kanemura:2023pnv .

We showed first that, depending on the reheating temperature, in the case of a
quadratic potential with $n=2$, the scattering of the inflaton can give a much
larger amount of graviton through bremsstrahlung than the decay channel, as
one can see in the right panel of Fig. 4. Interestingly, these spectra could
be in the region probed by next-generation GW observatories.

We then concentrated our study on the assumption that the channel responsible
for the bremsstrahlung is also responsible for the reheating process. The
reheating temperature is then directly related to the GW spectrum. For $n>2$
we found that the GW spectrum features a power-law enhancement on the ratio
$T_{\text{max}}/T_{\text{RH}}$, whose effect becomes more prominent with
increasing $n$, as shown in Fig. 5. Compared to the inflaton decay scenario,
we found that the GW spectrum exhibits distinct characteristics, depicted in
Fig. 6, which could potentially allow us to distinguish between decays and
annihilations. These distinctions arise from the disparate scaling behaviors
exhibited by the radiation released during the reheating process (cf. Fig. 1),
which behaves as a dilution effect for the GW spectrum. Particularly
noteworthy is the fact that, for certain model parameters, the GW spectrum in
the annihilation scenario is smaller than in the decay case, where the latter
could fall within the detectability range of future cavity detectors. Our
results thus suggest a new potential avenue for probing the parameter space of
reheating by utilizing bremsstrahlung-induced GWs.

## Acknowledgments

The authors want to thank Basabendu Barman, Mathias Pierre and Óscar Zapata
for very fruitful discussions. N.B. received funding from the Spanish FEDER /
MCIU-AEI under the grant FPA2017-84543-P and thanks the ICTP staff for their
hospitality during his stay when this study was completed. S.C. and Y.M.
acknowledge the support of the Institut Pascal at Université Paris-Saclay
during the Paris-Saclay Astroparticle Symposium 2023, with the support of the
P2IO Laboratory of Excellence (program “Investissements d’avenir”
ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis of the
Graduate School of Physics of Université Paris-Saclay, as well as IJCLab, CEA,
IAS, OSUPS, and the IN2P3 master project UCMN. Y.X. has received support from
the Cluster of Excellence “Precision Physics, Fundamental Interactions, and
Structure of Matter” (PRISMA+ EXC 2118/1) funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) within the German
Excellence Strategy (Project No. 39083149). This project has received funding
and support from the European Union’s Horizon 2020 research and innovation
program under the Marie Sklodowska-Curie grant agreement No. 860881-HIDDeN.

##  Appendix A Graviton Bremsstrahlung for a Monomial Potential

We can solve Eq. (41) for a generic potential of the inflaton during reheating
$V(\phi)=\lambda\,M_{P}^{4}\left(\frac{\phi}{M_{P}}\right)^{n}$. We obtain the
generic solution for the $1\rightarrow 3$ process

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}^{1\rightarrow
3}=\frac{\sqrt{3}\mu^{2}M_{P}}{32\pi^{3}}\alpha_{n}^{2}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)^{\frac{1}{2}}\int_{a_{\text{end}}}^{a_{\text{RH}}}\frac{da}{a_{\text{RH}}}\left(\frac{a}{a_{\text{RH}}}\right)^{\frac{6}{n+2}}\sum\limits_{j=1}^{\infty}j^{2}|\mathcal{P}_{j}|^{2}\left(1-2x_{j}(a)\right)^{2},$
(58)

for $y\to 0$. For the case $n=2$, the frequency of the oscillation is fixed as
$\varpi=m_{\phi}$, and there is only one mode in the Fourier decomposition
corresponding to $j=1$ associated with the coefficient
$\mathcal{P}_{1}=\frac{1}{2}$. Hence, ignoring the redshift dependence of
$x_{j}$,111111For a complete treatment of the energy redshift during reheating
we refer the reader to Ref. Barman:2023rpg . we recover

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}^{1\rightarrow
3}=\frac{\sqrt{3}\mu^{2}M_{P}}{160\pi^{3}}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)^{1/2}\left[1-\left(\frac{a_{\text{end}}}{a_{\text{RH}}}\right)^{5/2}\right]\left(1-2\frac{E_{w}}{m_{\phi}}\right)^{2}.$
(59)

The relic density of GWs is obtained by

$\Omega^{0}_{\rm
GW}h^{2}=\frac{h^{2}}{\rho_{c}^{0}}\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}\left(\frac{a_{\text{RH}}}{a_{0}}\right)^{4}\times
E_{\omega}(a_{\text{RH}})\,,$ (60)

where we can relate the present energy of the gravitons with the current
frequency of the GW,
$E_{\omega}(a_{\text{RH}})=2\pi\,f\,(a_{0}/a_{\text{RH}})$. This leads in case
$n=2$ to the following relic density for $a_{\text{end}}\ll a_{\text{RH}}$ and
$E_{\omega}(a_{\text{RH}})\ll m_{\phi}$

$\Omega^{0}_{\rm
GW}h^{2}\simeq\frac{h^{2}\mu^{2}\sqrt{3}}{80\pi^{2}}\times\frac{f}{M_{P}}\times\frac{\Omega_{R}^{0}}{(\rho_{\rm
RH}\rho_{R}^{0})^{1/4}}\,.$ (61)

For the generic case $n>2$, the integration of the energy density spectrum is
generally more involved, as the energy in each mode is time dependent, leading
to mode-dependent boundaries of the integral. We give an approximate solution
for a part of the spectrum where
$E_{\omega}(a_{\text{RH}})\ll\omega(a_{\text{RH}})$. In this case, all the
modes contribute similarly to the production of gravitons until the end of
reheating, with a contribution proportional to $j^{2}|\mathcal{P}_{j}|^{2}$
for each mode $j$

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}^{1\rightarrow
3}\simeq\frac{\sqrt{3}\mu^{2}M_{P}}{32\pi^{3}}\alpha_{n}^{2}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)^{1/2}\left(\frac{n+2}{n+8}\right)\left[1-\left(\frac{a_{\text{end}}}{a_{\text{RH}}}\right)^{\frac{n+8}{n+2}}\right]\sum\limits_{j=1}^{\infty}j^{2}|\mathcal{P}_{j}|^{2}$
(62)

Hence, we see that in the low-frequency limit, the influence of the shape of
the inflaton potential is only a change in the amplitude of the spectrum

$\frac{\Omega_{\rm GW}^{n>2}}{\Omega_{\rm
GW}^{n=2}}\simeq\frac{n+2}{n+8}\frac{\alpha_{n}^{2}}{5}\sum\limits_{j=1}^{\infty}j^{2}|\mathcal{P}_{j}|^{2}\,.$
(63)

Computing the integral with all modes provides a peak structure in the GW
spectrum that we did not consider in our analysis.

Now, we consider the scattering channel. We can obtain similar expressions for
the $2\rightarrow 3$ process

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}^{2\to
3}=\frac{\sqrt{3}\sigma^{2}M_{P}^{3}}{8\pi^{3}\lambda^{2/n}}\alpha_{n}^{2}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)^{\frac{n+4}{2n}}\int_{a_{\text{end}}}^{a_{\text{RH}}}\frac{da}{a_{\text{RH}}}\left(\frac{a}{a_{\text{RH}}}\right)^{-\frac{6}{n+2}}\sum\limits_{j=1}^{\infty}j^{2}|\mathcal{P}_{j}^{(2)}|^{2}\left(1-x_{j}(a)\right)^{2}.$
(64)

In the case $n=2$, we recover

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}^{2\rightarrow
3}=\frac{\sqrt{3}\sigma^{2}M_{P}^{3}}{16\pi^{3}\lambda}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)^{3/2}\left[\left(\frac{a_{\text{RH}}}{a_{\text{end}}}\right)^{1/2}-1\right]\left(1-\frac{E_{\omega}}{m_{\phi}}\right)^{2},$
(65)

leading, in the limit $a_{\text{end}}\ll a_{\text{RH}}$ and
$E_{\omega}(a_{\text{RH}})\ll m_{\phi}$, to

$\Omega^{0}_{\rm
GW}h^{2}\simeq\frac{h^{2}\sigma^{2}\sqrt{3}}{4\pi^{2}}\times\frac{f}{M_{P}}\times\frac{\rho_{\rm
end}^{1/6}\rho_{\rm
RH}^{7/12}\Omega_{R}^{0}}{m_{\phi}^{2}(\rho_{R}^{0})^{1/4}}\,,$ (66)

where we have used the relation $\lambda=m_{\phi}^{2}/(2M_{P}^{2})$
considering the quadratic potential of the inflaton.

In the general case $n>2$, again the situation is more complicated as the
frequency of the background oscillations evolves with time. However, we give
an approximate solution for a part of the spectrum where
$E_{\omega}(a_{\text{RH}})\ll\omega(a_{\text{RH}})$

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}^{2\rightarrow
3}\simeq\frac{\sqrt{3}\sigma^{2}M_{P}^{3}}{8\pi^{3}\lambda^{1/2}}\alpha_{4}^{2}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)\ln{\left(\frac{\rho_{\rm end}}{\rho_{\rm
RH}}\right)}\sum\limits_{j=1}^{\infty}j^{2}\left|\mathcal{P}_{j}^{(2)}\right|^{2}$
(67)

for $n=4$, or

$\left.\frac{d\rho_{\text{GW}}}{dE_{\omega}}\right|_{a_{\text{RH}}}^{2\rightarrow
3}\simeq\frac{n+2}{n-4}\frac{\sqrt{3}\sigma^{2}M_{P}^{3}}{8\pi^{3}\lambda^{2/n}}\alpha_{n}^{2}\left(\frac{\rho_{\rm
RH}}{M_{P}^{4}}\right)^{\frac{n+4}{2n}}\left[1-\left(\frac{a_{\text{end}}}{a_{\text{RH}}}\right)^{\frac{n-4}{n+2}}\right]\sum\limits_{j=1}^{\infty}j^{2}\left|\mathcal{P}_{j}^{(2)}\right|^{2}$
(68)

for $n>4$. In the low-frequency limit, the influence of the shape of the
inflaton potential for the $2\to 3$ process again is only a change in the
amplitude of the spectrum

$\displaystyle\frac{\Omega_{\rm GW}^{n=4}}{\Omega_{\rm GW}^{n=2}}$
$\displaystyle\simeq
2\alpha_{4}^{2}\lambda^{\frac{1}{2}}\sum\limits_{j=1}^{\infty}j^{2}\left|\mathcal{P}^{(2)}_{j}\right|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(n=4),$
(69) $\displaystyle\frac{\Omega_{\rm GW}^{n>2}}{\Omega_{\rm GW}^{n=2}}$
$\displaystyle\simeq\frac{n+2}{n-4}\frac{2\alpha_{n}^{2}}{\lambda^{\frac{2-n}{n}}}\sum\limits_{j=1}^{\infty}j^{2}\left|\mathcal{P}^{(2)}_{j}\right|^{2}~{}~{}~{}~{}~{}~{}(n>4).$
(70)

## References

  * (1) D.H. Lyth and A.R. Liddle, _The primordial density perturbation: Cosmology, inflation and the origin of structure_ (2009).
  * (2) L. Kofman, A.D. Linde and A.A. Starobinsky, _Towards the theory of reheating after inflation_ , _Phys. Rev. D_ 56 (1997) 3258 [hep-ph/9704452].
  * (3) S. Weinberg, _Infrared photons and gravitons_ , _Phys. Rev._ 140 (1965) B516.
  * (4) B.M. Barker, S.N. Gupta and J. Kaskas, _Graviton bremsstrahlung and infrared divergence_ , _Phys. Rev._ 182 (1969) 1391.
  * (5) J. Ghiglieri and M. Laine, _Gravitational wave background from Standard Model physics: Qualitative features_ , _JCAP_ 07 (2015) 022 [1504.02569].
  * (6) J. Ghiglieri, G. Jackson, M. Laine and Y. Zhu, _Gravitational wave background from Standard Model physics: Complete leading order_ , _JHEP_ 07 (2020) 092 [2004.11392].
  * (7) A. Ringwald, J. Schütte-Engel and C. Tamarit, _Gravitational Waves as a Big Bang Thermometer_ , _JCAP_ 03 (2021) 054 [2011.04731].
  * (8) P. Klose, M. Laine and S. Procacci, _Gravitational wave background from non-Abelian reheating after axion-like inflation_ , _JCAP_ 05 (2022) 021 [2201.02317].
  * (9) A. Ringwald and C. Tamarit, _Revealing the cosmic history with gravitational waves_ , _Phys. Rev. D_ 106 (2022) 063027 [2203.00621].
  * (10) J. Ghiglieri, J. Schütte-Engel and E. Speranza, _Freezing-In Gravitational Waves_ , 2211.16513.
  * (11) K. Nakayama and Y. Tang, _Stochastic Gravitational Waves from Particle Origin_ , _Phys. Lett. B_ 788 (2019) 341 [1810.04975].
  * (12) D. Huang and L. Yin, _Stochastic Gravitational Waves from Inflaton Decays_ , _Phys. Rev. D_ 100 (2019) 043538 [1905.08510].
  * (13) A. Ghoshal, R. Samanta and G. White, _Bremsstrahlung high-frequency gravitational wave signatures of high-scale nonthermal leptogenesis_ , _Phys. Rev. D_ 108 (2023) 035019 [2211.10433].
  * (14) B. Barman, N. Bernal, Y. Xu and Ó. Zapata, _Gravitational wave from graviton Bremsstrahlung during reheating_ , _JCAP_ 05 (2023) 019 [2301.11345].
  * (15) B. Barman, N. Bernal, Y. Xu and Ó. Zapata, _Bremsstrahlung-induced gravitational waves in monomial potentials during reheating_ , _Phys. Rev. D_ 108 (2023) 083524 [2305.16388].
  * (16) S. Kanemura and K. Kaneta, _Gravitational Waves from Particle Decays during Reheating_ , 2310.12023.
  * (17) M.A.G. Garcia, K. Kaneta, Y. Mambrini and K.A. Olive, _Inflaton Oscillations and Post-Inflationary Reheating_ , _JCAP_ 04 (2021) 012 [2012.10756].
  * (18) M.A.G. Garcia, K. Kaneta, Y. Mambrini and K.A. Olive, _Reheating and Post-inflationary Production of Dark Matter_ , _Phys. Rev. D_ 101 (2020) 123507 [2004.08404].
  * (19) B. Barman and N. Bernal, _Gravitational SIMPs_ , _JCAP_ 06 (2021) 011 [2104.10699].
  * (20) S.Y. Choi, J.S. Shim and H.S. Song, _Factorization and polarization in linearized gravity_ , _Phys. Rev. D_ 51 (1995) 2751 [hep-th/9411092].
  * (21) M.S. Turner, _Coherent Scalar Field Oscillations in an Expanding Universe_ , _Phys. Rev. D_ 28 (1983) 1243.
  * (22) N. Bernal and Y. Xu, _WIMPs during reheating_ , _JCAP_ 12 (2022) 017 [2209.07546].
  * (23) A.A. Starobinsky, _A New Type of Isotropic Cosmological Models Without Singularity_ , _Phys. Lett. B_ 91 (1980) 99.
  * (24) M. Drees and Y. Xu, _Small field polynomial inflation: reheating, radiative stability and lower bound_ , _JCAP_ 09 (2021) 012 [2104.03977].
  * (25) N. Bernal and Y. Xu, _Polynomial inflation and dark matter_ , _Eur. Phys. J. C_ 81 (2021) 877 [2106.03950].
  * (26) M. Drees and Y. Xu, _Large field polynomial inflation: parameter space, predictions and (double) eternal nature_ , _JCAP_ 12 (2022) 005 [2209.07545].
  * (27) R. Kallosh and A. Linde, _Universality Class in Conformal Inflation_ , _JCAP_ 07 (2013) 002 [1306.5220].
  * (28) R. Kallosh, A. Linde and D. Roest, _Superconformal Inflationary $\alpha$-Attractors_, _JHEP_ 11 (2013) 198 [1311.0472].
  * (29) K. Mukaida and M. Yamada, _Thermalization Process after Inflation and Effective Potential of Scalar Field_ , _JCAP_ 02 (2016) 003 [1506.07661].
  * (30) M.A.G. Garcia and M.A. Amin, _Prethermalization production of dark matter_ , _Phys. Rev. D_ 98 (2018) 103504 [1806.01865].
  * (31) M.A. Amin, M.P. Hertzberg, D.I. Kaiser and J. Karouby, _Nonperturbative Dynamics Of Reheating After Inflation: A Review_ , _Int. J. Mod. Phys. D_ 24 (2014) 1530003 [1410.3808].
  * (32) M.A.G. Garcia, K. Kaneta, Y. Mambrini, K.A. Olive and S. Verner, _Freeze-in from preheating_ , _JCAP_ 03 (2022) 016 [2109.13280].
  * (33) K.D. Lozanov and M.A. Amin, _Equation of State and Duration to Radiation Domination after Inflation_ , _Phys. Rev. Lett._ 119 (2017) 061301 [1608.01213].
  * (34) M.A.G. Garcia and M. Pierre, _Reheating after Inflaton Fragmentation_ , 2306.08038.
  * (35) M.A.G. Garcia, M. Gross, Y. Mambrini, K.A. Olive, M. Pierre and J.-H. Yoon, _Effects of Fragmentation on Post-Inflationary Reheating_ , 2308.16231.
  * (36) N. Bernal, J. Rubio and H. Veermäe, _UV Freeze-in in Starobinsky Inflation_ , _JCAP_ 10 (2020) 021 [2006.02442].
  * (37) S. Clery, Y. Mambrini, K.A. Olive and S. Verner, _Gravitational portals in the early Universe_ , _Phys. Rev. D_ 105 (2022) 075005 [2112.15214].
  * (38) M.R. Haque and D. Maity, _Gravitational reheating_ , _Phys. Rev. D_ 107 (2023) 043531 [2201.02348].
  * (39) B. Barman, N. Bernal and J. Rubio, _Rescuing Gravitational-Reheating in Chaotic Inflation_ , 2310.06039.
  * (40) M.R. Haque, D. Maity and R. Mondal, _$\nu$ GRe: Gravitational $\nu$trino Reheating_, 2311.07684.
  * (41) J.F. Dufaux, G.N. Felder, L. Kofman, M. Peloso and D. Podolsky, _Preheating with trilinear interactions: Tachyonic resonance_ , _JCAP_ 07 (2006) 006 [hep-ph/0602144].
  * (42) D. Maity and P. Saha, _(P)reheating after minimal Plateau Inflation and constraints from CMB_ , _JCAP_ 07 (2019) 018 [1811.11173].
  * (43) R.T. Co, Y. Mambrini and K.A. Olive, _Inflationary gravitational leptogenesis_ , _Phys. Rev. D_ 106 (2022) 075006 [2205.01689].
  * (44) B. Barman, S. Cléry, R.T. Co, Y. Mambrini and K.A. Olive, _Gravity as a portal to reheating, leptogenesis and dark matter_ , _JHEP_ 12 (2022) 072 [2210.05716].
  * (45) S. Clery, Y. Mambrini, K.A. Olive, A. Shkerin and S. Verner, _Gravitational portals with nonminimal couplings_ , _Phys. Rev. D_ 105 (2022) 095042 [2203.02004].
  * (46) Y. Xu, _Constraining axion and ALP dark matter from misalignment during reheating_ , _Phys. Rev. D_ 108 (2023) 083536 [2308.15322].
  * (47) G.F. Giudice, E.W. Kolb and A. Riotto, _Largest temperature of the radiation era and its cosmological implications_ , _Phys. Rev. D_ 64 (2001) 023508 [hep-ph/0005123].
  * (48) Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * (49) Y. Mambrini, K.A. Olive and J. Zheng, _Post-inflationary dark matter bremsstrahlung_ , _JCAP_ 10 (2022) 055 [2208.05859].
  * (50) D. Hooper, G. Krnjaic, A.J. Long and S.D. Mcdermott, _Can the Inflaton Also Be a Weakly Interacting Massive Particle?_ , _Phys. Rev. Lett._ 122 (2019) 091802 [1807.03308].
  * (51) M.A.G. Garcia, Y. Mambrini, K.A. Olive and S. Verner, _On the Realization of WIMPflation_ , _JCAP_ 10 (2021) 061 [2107.07472].
  * (52) BICEP, Keck collaboration, _Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season_ , _Phys. Rev. Lett._ 127 (2021) 151301 [2110.00483].
  * (53) N. Aggarwal et al., _Challenges and opportunities of gravitational-wave searches at MHz to GHz frequencies_ , _Living Rev. Rel._ 24 (2021) 4 [2011.12414].
  * (54) LISA collaboration, _Laser Interferometer Space Antenna_ , 1702.00786.
  * (55) M. Punturo et al., _The Einstein Telescope: A third-generation gravitational wave observatory_ , _Class. Quant. Grav._ 27 (2010) 194002.
  * (56) S. Hild et al., _Sensitivity Studies for Third-Generation Gravitational Wave Observatories_ , _Class. Quant. Grav._ 28 (2011) 094013 [1012.0908].
  * (57) B. Sathyaprakash et al., _Scientific Objectives of Einstein Telescope_ , _Class. Quant. Grav._ 29 (2012) 124013 [1206.0331].
  * (58) M. Maggiore et al., _Science Case for the Einstein Telescope_ , _JCAP_ 03 (2020) 050 [1912.02622].
  * (59) J. Crowder and N.J. Cornish, _Beyond LISA: Exploring future gravitational wave missions_ , _Phys. Rev. D_ 72 (2005) 083005 [gr-qc/0506015].
  * (60) V. Corbin and N.J. Cornish, _Detecting the cosmic gravitational wave background with the big bang observer_ , _Class. Quant. Grav._ 23 (2006) 2435 [gr-qc/0512039].
  * (61) G.M. Harry, P. Fritschel, D.A. Shaddock, W. Folkner and E.S. Phinney, _Laser interferometry for the big bang observer_ , _Class. Quant. Grav._ 23 (2006) 4887.
  * (62) N. Seto, S. Kawamura and T. Nakamura, _Possibility of direct measurement of the acceleration of the universe using 0.1-Hz band laser interferometer gravitational wave antenna in space_ , _Phys. Rev. Lett._ 87 (2001) 221103 [astro-ph/0108011].
  * (63) H. Kudoh, A. Taruya, T. Hiramatsu and Y. Himemoto, _Detecting a gravitational-wave background with next-generation space interferometers_ , _Phys. Rev. D_ 73 (2006) 064006 [gr-qc/0511145].
  * (64) F.-Y. Li, M.-X. Tang and D.-P. Shi, _Electromagnetic response of a Gaussian beam to high frequency relic gravitational waves in quintessential inflationary models_ , _Phys. Rev. D_ 67 (2003) 104008 [gr-qc/0306092].
  * (65) A. Berlin, D. Blas, R. Tito D’Agnolo, S.A.R. Ellis, R. Harnik, Y. Kahn et al., _Detecting high-frequency gravitational waves with microwave cavities_ , _Phys. Rev. D_ 105 (2022) 116011 [2112.11465].
  * (66) A. Berlin, D. Blas, R. Tito D’Agnolo, S.A.R. Ellis, R. Harnik, Y. Kahn et al., _MAGO 2.0: Electromagnetic Cavities as Mechanical Bars for Gravitational Waves_ , 2303.01518.
  * (67) N. Herman, L. Lehoucq and A. Fúzfa, _Electromagnetic Antennas for the Resonant Detection of the Stochastic Gravitational Wave Background_ , 2203.15668.
  * (68) E. Armengaud et al., _Conceptual Design of the International Axion Observatory (IAXO)_ , _JINST_ 9 (2014) T05002 [1401.3233].
  * (69) IAXO collaboration, _Physics potential of the International Axion Observatory (IAXO)_ , _JCAP_ 06 (2019) 047 [1904.09155].
  * (70) COrE collaboration, _COrE (Cosmic Origins Explorer) A White Paper_ , 1102.2181.
  * (71) EUCLID collaboration, _Euclid Definition Study Report_ , 1110.3193.
  * (72) I. Ben-Dayan, B. Keating, D. Leon and I. Wolfson, _Constraints on scalar and tensor spectra from $N_{eff}$_, _JCAP_ 06 (2019) 007 [1903.11843].
  * (73) M. Maggiore, _Gravitational wave experiments and early universe cosmology_ , _Phys. Rept._ 331 (2000) 283 [gr-qc/9909001].