# Schwarz-Pick Lemma for Harmonic and Hyperbolic Harmonic Functions

Adel Khalfallah  A. Khalfallah, Department of Mathematics, King Fahd
University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
<EMAIL_ADDRESS>,  Bojana Purtić  B. Purtić, Faculty of mathematics,
University of Belgrade, Studentski Trg 16, Belgrade, Republic of Serbia
<EMAIL_ADDRESS>and  Miodrag Mateljević  M. Mateljević, Faculty of
mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Republic of
Serbia<EMAIL_ADDRESS>
###### Abstract.

We establish some inequalities of Schwarz–Pick type for harmonic and
hyperbolic harmonic functions on the unit ball of $\mathbb{R}^{n}$ and we
disprove a recent conjecture of Liu [26].

###### 2020 Mathematics Subject Classification:

31B05,31C05, 30C80

##  1\. Introduction

By $\omega_{n}$ or $V({\mathbb{B}}^{n})$ we denote the $n$-volume of the unit
ball $\mathbb{B}^{n}$ in $\mathbb{R}^{n}$, and by $\sigma_{n}$ the
$(n-1)$-volume of the unit sphere ${\mathbb{S}^{n-1}}$; note that
$\sigma_{n}=n\omega_{n}$. Next, $\sigma$ denotes the rotation invariant Borel
measure on ${\mathbb{S}^{n-1}}$, $\sigma^{0}=\sigma/\sigma_{n}$ and $|.|$ is
the Euclidean norm. Thus $\sigma^{0}$ is the unique rotation invariant
normalized Borel measure on ${\mathbb{S}^{n-1}}$ such that
$\sigma^{0}({\mathbb{S}^{n-1}})=1$. In this paper, the expressions
$\omega_{n-1}/\omega_{n}$ and $\sigma_{n-1}/\sigma_{n}$ often appear so it is
convenient to denote them by $\omega_{*}(n)$ and $\sigma_{*}(n)$ respectively,
that is,

$\omega_{*}(n):=\frac{\omega_{n-1}}{\omega_{n}},\quad\mbox{ and
}\quad\sigma_{*}(n):=\frac{\sigma_{n-1}}{\sigma_{n}}.$

Recall that a mapping $u\in\mathcal{C}^{2}(\mathbb{B}^{n},\mathbb{R})$ is said
to be hyperbolic harmonic if $\Delta_{h}u=0,$ where $\Delta_{h}$ is the
hyperbolic Laplacian operator defined by

$\Delta_{h}u(x)=(1-|x|^{2})^{2}\Delta
u+2(n-2)(1-|x|^{2})\sum_{i=1}^{n}x_{i}\frac{\partial u}{\partial x_{i}}(x),$

here $\Delta$ denotes the Laplacian on $\mathbb{R}^{n}$. Clearly for $n=2$,
hyperbolic harmonic and harmonic functions coincide.

In [25], Liu proved the Khavinson conjecture, which says for bounded harmonic
functions on the unit ball of $\mathbb{R}^{n}$ the sharp constants in the
estimates for their radial derivatives and for their gradients coincide.

######  Theorem A ([25]).

For $n\geq 3$, if $u$ is a bounded harmonic function on $\mathbb{B}^{n}$ into
$\mathbb{R}$, then we have the following sharp inequality

$|\nabla
u(x)|\leq\frac{c_{n}}{1-|x|^{2}}\Phi_{n}(|x|)|u|_{\infty},\,x\in\mathbb{B}^{n},$

with $c_{n}=(n-1)\omega_{*}(n),$ and

$\Phi_{n}(r)=\int_{-1}^{1}\frac{\left|t-\frac{n-2}{2}r\right|(1-t^{2})^{\frac{n-3}{2}}}{(1-2tr+r^{2})^{\frac{n-2}{2}}}\,dt.$

For more details and development regarding the Khavinson conjecture for
harmonic functions, see [15, 16, 17, 18, 19, 21, 22].

In [26], the author further proved that, when $n\geq 4$, the function
$\Phi_{n}$ is decreasing on [0,1], thus

$\max_{r\in[0,1]}\Phi_{n}(r)=\Phi_{n}(0)=\frac{2}{n-1}.$

In contrast, if $n=3$, then

$\Phi_{3}(r)=\frac{2}{3}\frac{(1+\frac{1}{3}r^{2})^{3/2}-1+r^{2}}{r^{2}}$

is strictly increasing on $[0,1]$ and attains its maximum at $r=1$, thus

$\max_{r\in[0,1]}\Phi_{3}(r)=\Phi_{3}(1)=\frac{16}{9\sqrt{3}}.$

######  Theorem B ([26], Schwarz-Pick lemma for harmonic functions).

Let $u$ be a real-valued bounded harmonic function on the unit ball
$\mathbb{B}^{n}$ of $\mathbb{R}^{n}$.

  1. (1)

When $n=2$ or $n\geq 4$, the following sharp inequality holds:

$|\nabla u(x)|\leq 2\omega_{*}(n)\frac{|u|_{\infty}}{1-|x|^{2}},\quad
x\in{\mathbb{B}}^{n}.$ (1.1)

Equality holds if and only if $x=0$ and $u=U\circ T$ for some orthogonal
transformation $T$, where $U$ is the Poisson integral of the function that
equals 1 on a hemisphere and $-1$ on the remaining hemisphere.

  2. (2)

When $n=3$, we have

$|\nabla u(x)|\leq\frac{8}{3\sqrt{3}}\frac{|u|_{\infty}}{1-|x|^{2}},\quad
x\in{\mathbb{B}}^{3}.$ (1.2)

The constant $\frac{8}{3\sqrt{3}}$ here is the best possible.

######  Remark 1.1.

  1. (1)

Note that the inequality (1.1) holds when $n=3$ at $x=0$. Curiously, the
inequality (1.1) fails when $n=3$ in general. Note that
$\frac{8}{3\sqrt{3}}\approx 1.5396,$ while the constant
$\frac{2V({\mathbb{B}}^{n-1})}{V({\mathbb{B}}^{n})}$ in (1.1) equals to
$\frac{3}{2}$ when $n=3$.

  2. (2)

The inequality (1.1) at $x=0$ was previously proved in [5, Theorem 6.26] and
in [6, Corollary 1] for harmonic functions fixing the origin.

The classical Schwarz-Pick lemma states that an analytic function of
$\mathbb{D}$ into itself satisfies

$|f^{\prime}(z)|\leq\frac{1-|f(z)|^{2}}{1-|z|^{2}},z\in\mathbb{D},$

where $\mathbb{D}$ denotes the unit disc of the complex plane $\mathbb{C}$.
For complex-valued harmonic function of $\mathbb{D}$ into itself, Colonna [11]
proved the following sharp Schwarz-Pick lemma:

$|Df(z)|\leq\frac{4}{\pi}\frac{1}{1-|z|^{2}},z\in\mathbb{D},$

where $|Df(z)|=\left|\frac{\partial f(z)}{\partial
z}\right|+\left|\frac{\partial f(z)}{\partial\overline{z}}\right|$.

In the planar case, Kalaj and Vuorinen [12] obtained the following inequality
for real harmonic functions with values in $(-1,1)$.

$|\nabla u(z)|\leq\frac{4}{\pi}\frac{1-|u(z)|^{2}}{1-|z|^{2}},\quad|z|<1.$
(1.3)

Based on (1.1) and (1.3), Liu suggested the following conjecture.

######  Conjecture 1 ([26]).

If $n\geq 4$ and $u:\mathbb{B}^{n}\rightarrow(-1,1)$ is a harmonic function,
then

$|\nabla u(x)|\leq 2\omega_{*}(n)\frac{1-u^{2}(x)}{1-|x|^{2}},\quad|x|<1.$
(1.4)

First, by providing a counter-example, we disprove Conjecture 1 for $n\geq 4$.
Our main tool is Theorem C giving a sharp estimate of the norm of the gradient
at zero of functions having generalized Poisson transformations; such estimate
is based on the Burgeth’s method, see [6, 7].

Let us introduce some notations. If $x\in\mathbb{B}^{n}\setminus\\{0\\}$,
define $\hat{x}=\frac{x}{|x|}\in{\mathbb{S}^{n-1}}$ and
$\hat{0}=e_{n}=(0,\ldots,1)$, the north pole. $S(\hat{x},\gamma)$ denotes the
hyperspherical cap with center $\hat{x}$ and contact angle $\gamma\in[0,\pi]$:

$S(\hat{x},\gamma)=\\{y\in{\mathbb{S}^{n-1}}\,:\,\,\langle\hat{x},y\rangle>\cos\gamma\\}.$

For $\alpha,\beta\in\mathbb{R}$, $\beta>0$, the generalized Poisson kernel is
defined by

$P_{\alpha,\beta}(x,y)=\frac{(1-|x|^{2})^{\alpha}}{|x-y|^{2\beta}},\,x\in\mathbb{B}^{n}\mbox{
and }y\in{\mathbb{S}^{n-1}}.$

For $f\in L^{1}({\mathbb{S}^{n-1}})$, set

$P_{\alpha,\beta}[f](x)=\int_{{\mathbb{S}^{n-1}}}P_{\alpha,\beta}(x,y)f(y)\,d\sigma^{0}(y).$

By $A=A_{n}^{cap}(\gamma)$ we denote the normalized $(n-1)$-dimensional volume
of the spherical cap with contact angle $\gamma$.

######  Theorem C ([13]).

Let $h^{*}:{\mathbb{S}^{n-1}}\to[-1,1]$ be a bounded function on
${\mathbb{S}^{n-1}}$ with values in $[-1,1]$ and $h=P_{\alpha,\beta}[h^{*}]$.
Then,

$|\nabla h(0)|\leq
D_{n}(\gamma,\beta):=\frac{4\beta}{n}\,\frac{\omega_{n-1}}{\omega_{n}}(\sin\gamma)^{n-1},$
(1.5)

where $\gamma$ is the unique angle in $[0,\pi]$ such that

$A=A_{n}^{cap}(\gamma)=\frac{1+h(0)}{2}.$ (1.6)

The estimate (1.5) is sharp and

$h^{0}_{\alpha,\beta}:=P_{\alpha,\beta}\left[\mathds{1}_{S(e_{n},\gamma)}-\mathds{1}_{S^{c}(e_{n},\gamma)}\right]$
(1.7)

is an extremal function, where
$S^{c}(e_{n},\gamma)={\mathbb{S}^{n-1}}\setminus S(e_{n},\gamma)$.

It is readable that

$D_{n}(\gamma,\beta)=\sup|\nabla h(0)|,$

where the supremum is taken over all functions $h$ which satisfy the
assumptions of Theorem C with the constraint $h(0)=a$, where $\gamma$ is
determined by $A_{n}^{cap}(\gamma)=\frac{1+a}{2}$.

As a corollary, we obtain an estimate of the gradient at zero for harmonic
functions obtained for $(\alpha,\beta)=(1,\frac{n}{2})$ and hyperbolic-
harmonic functions obtained for $(\alpha,\beta)=(n-1,n-1)$ in terms of their
values at the origin.

######  Corollary 1 ([13]).

Let $h:\mathbb{B}^{n}\to(-1,1)$ be a harmonic or hyperbolic harmonic function.
Then the following sharp estimates hold:

$|\nabla h(0)|\leq\left.\begin{cases}2\omega_{*}(n)(\sin\gamma)^{n-1}&\text{
if }h\text{ is harmonic,}\\\ \displaystyle
4\sigma_{*}(n)(\sin\gamma)^{n-1}&\text{ if }h\text{ is hyperbolic-
harmonic,}\end{cases}\right.$

where $\gamma$ is the unique angle in $[0,\pi]$ such that

$A_{n}^{cap}(\gamma)=\frac{1+h(0)}{2}.$

These estimates are sharp and $h^{0}_{1,\frac{n}{2}}$ (resp.,
$h^{0}_{n-1,n-1}$) is an extremal harmonic (resp., hyperbolic-harmonic)
function on $\mathbb{B}^{n}$, see (1.7).

##  2\. Main results

We are now in a position to establish our main results. Our first result is
the following.

######  Theorem 2.1.

For $n\geq 4$, the harmonic function $h^{0}_{1,\frac{n}{2}}$ defined in (1.7)
provides a counter-example to Conjecture 1.

The proof is based on Corollary 1 and some basic properties of the volume of
the unit ball in $\mathbb{R}^{n}$. Recently, several authors presented
interesting monotonicity properties of the $\omega_{n}$, the volume of the
unit ball in $\mathbb{R}^{n}$. The sequence itself is not monotonic and
attains its maximum at $n=5$. It is worth noting that
$\omega_{*}(n)=\frac{\omega_{n-1}}{\omega_{n}}$ has very interesting
properties and there are remarkable upper and lower bounds, see for instance
Borgwardt [8, p. 253 ] and Alzer [3]. Using these estimates, we prove the
following.

######  Theorem 2.2.

For $n\geq 4$ and $a\in[-1,1]$. Then the following inequalities hold

$(\sin\gamma_{a})^{n-1}\geq 1-a^{2}.$ (2.1)

Moreover, the equality holds for $a=-1,0,1$.

$(\sin\gamma_{a})^{n-1}\leq\frac{n-1}{4\sigma_{*}(n)}(1-a^{2}).$ (2.2)

Moreover, the equality holds for $a=-1,1$.
$\gamma_{a}$ is the unique angle in $[0,\pi]$ such that
$A_{n}^{cap}(\gamma_{a})=\frac{1+a}{2}.$

Thus, combining Corollary 1 and the inequality (2.1), we disprove Liu’s
conjecture.

######  Remark 2.3.

  * (i)

For $n=3$, we have $(\sin\gamma_{a})^{n-1}=1-a^{2}$.

  * (ii)

For $n=2$, $(\sin\gamma_{a})^{n-1}\leq 1-a^{2}$.

Indeed, in the planar case, $\displaystyle
A_{2}^{cap}(\gamma_{a})=\frac{\gamma_{a}}{\pi}=\frac{1+a}{2}$, thus
$\displaystyle\gamma_{a}=\frac{\pi}{2}+\frac{\pi a}{2}$ and
$\sin\gamma_{a}=\cos(\frac{\pi a}{2})\leq 1-a^{2}$.

As for $n=3$, we have $\frac{4}{n}\frac{\omega_{n-1}}{\omega_{n}}=1$ and
combining Theorem C and the inequality (2.2) in Theorem 2.2, it yields the
following.

######  Theorem 2.4.

Let $n\geq 3$ and $h^{*}:{\mathbb{S}^{n-1}}\to[-1,1]$ be a function on
${\mathbb{S}^{n-1}}$ with values in $[-1,1]$ and $h=P_{\alpha,\beta}[h^{*}]$.
Then,

$|\nabla h(0)|\leq\beta(1-|h(0)|^{2}).$ (2.3)

The constant $\beta$ is sharp in (2.3).

In particular, we get the following estimate of the gradient at zero for
harmonic and hyperbolic harmonic functions.

######  Corollary 2.

Let $n\geq 3$ and $h:\mathbb{B}^{n}\to(-1,1)$ be a harmonic or hyperbolic
harmonic function. Then the following sharp estimates hold:

$|\nabla
h(0)|\leq\left.\begin{cases}\displaystyle\frac{n}{2}(1-|h(0)|^{2})&\text{ if
}h\text{ is harmonic,}\\\ (n-1)(1-|h(0)|^{2})&\text{ if }h\text{ is
hyperbolic-harmonic}.\end{cases}\right.$

Furthermore, this inequality is strict for $n\geq 4$.

Using the ball of center $x$ and radius $1-|x|$ in the harmonic case and
Möbius transformations in the hyperbolic-harmonic case, we obtain the
following.

######  Theorem 2.5.

Let $n\geq 3$ and $h:\mathbb{B}^{n}\to(-1,1)$ be a harmonic function, then

$|\nabla h(x)|\leq\frac{n}{2}\frac{1-|h(x)|^{2}}{1-|x|}.$

In addition, this inequality is strict for $n\geq 4$.

######  Theorem 2.6.

Let $n\geq 3$ and let $h:\mathbb{B}^{n}\to(-1,1)$ be a hyperbolic harmonic
function, then

$|\nabla h(x)|\leq(n-1)\frac{1-|h(x)|^{2}}{1-|x|^{2}}.$

Therefore,

$d_{h_{2}}(h(x_{1}),h(x_{2}))\leq(n-1)\,d_{h_{n}}(x_{1},x_{2}),\quad
x_{1},x_{2}\in\mathbb{B}^{n},$

where $d_{h_{n}}$ denotes the hyperbolic distance of the unit ball
$\mathbb{B}^{n}$. For $n=2$, $d_{h_{2}}$ is simply the Poincaré distance of
the unit disc.

This result is connected to the Khavinson conjecture for hyperbolic harmonic
functions, see [14].

######  Theorem D.

[14, Theorem 2] Let $n\geq 3$ and let $h:\mathbb{B}^{n}\to(-1,1)$ be a
hyperbolic harmonic function, then

$|\nabla h(x)|\leq 4\sigma_{*}(n)\frac{1}{1-|x|^{2}}.$

For vector valued functions, we prove the following.

######  Theorem 2.7.

Let $n\geq 3$, $m\geq 1$ and $h:\mathbb{B}^{n}\to\mathbb{B}^{m}$ be a harmonic
or hyperbolic harmonic function. Then the following estimates hold.

$|Dh(x)|\leq\left.\begin{cases}\displaystyle\frac{n}{2}\frac{1}{1-|x|}&\text{
if }h\text{ is harmonic,}\\\ \displaystyle(n-1)\frac{1}{1-|x|^{2}}&\text{ if
}h\text{ is hyperbolic-harmonic},\end{cases}\right.$

where

$|Dh(x)|=\sup_{|v|=1}|Dh(x)v|.$

In addition, it yields the following.

######  Theorem 2.8.

Let $n\geq 3$, $m\geq 1$ and $h:\mathbb{B}^{n}\to\mathbb{B}^{m}$ be a harmonic
or hyperbolic harmonic function. Then the following estimates hold:

$|\nabla\,|h|(x)|\leq\left.\begin{cases}\displaystyle\frac{n}{2}\frac{1-|h(x)|^{2}}{1-|x|}&\text{
if }h\text{ is harmonic,}\\\
\displaystyle(n-1)\frac{1-|h(x)|^{2}}{1-|x|^{2}}&\text{ if }h\text{ is
hyperbolic-harmonic}.\end{cases}\right.$

Furthermore, this inequality is strict for $n\geq 4$.

Recall that

$\left|\nabla\,|h|(x)\right|=\sup_{\beta\in{\mathbb{S}^{n-1}}}\lim_{t\to
0^{+}}\frac{\left|\,|h(x+t\beta)|-|h(x)|\,\right|}{t}.$

Thus $\left|\nabla\,|h|(x)\right|$ coincides with the gradient of $|h|$ at
$x$, if $h(x)\not=0$. Moreover, if $h(x)=0$, then
$\left|\nabla\,|h|(x)\right|=|Dh(x)|$. Therefore, Theorem 2.8 can be seen as
an extension of Theorem 2.7.

We mention that in [12], the authors considered the corresponding theorem for
vector harmonic functions defined on the unit disc, see [12, Theorem 1.10]. A
Schwarz lemma for the modulus of a vector-valued analytic functions was
previously considered in [24].

In [7], Burgeth introduced $R_{\alpha,\beta}(|x|,h(x))$ and provides estimates
for the radial derivative in terms of $|x|$ and $h(x)$. In the three-
dimensional case, we get explicit formula for $R_{\alpha,\beta}(|x|,h(x))$. In
particular, we can modify [7, Corollary 3] in the following way.

######  Theorem 2.9.

If $u:\mathbb{B}^{3}\rightarrow(-1,1)$ is a harmonic function, then for each
$x\in\mathbb{B}^{3}$,

$\frac{-3-|x|u(x)}{1-|x|^{2}}(1-u^{2}(x))/2\leq
D_{r}u(x)\leq\frac{3-|x|u(x)}{1-|x|^{2}}(1-u^{2}(x))/2.$ (2.4)

Hence

$|\nabla u(x)|\leq 2\frac{1-u^{2}(x)}{1-|x|^{2}},$ (2.5)

and therefore

$d_{h_{2}}(u(x_{1}),u(x_{2}))\leq 2d_{h_{3}}(x_{1},x_{2}),\quad
x_{1},x_{2}\in\mathbb{B}^{3}.$ (2.6)

We should mention that, recently, several versions of Schwarz lemma for
harmonic and pluriharmonic mappings were established, see [9, 10, 20, 27, 28,
29].

We close our paper, by the following question as an adjustment to Liu’s
conjecture.

######  Question 1.

Let $n\geq 3$ and $h:\mathbb{B}^{n}\to(-1,1)$ be a harmonic function. Is it
true that

$|\nabla h(x)|\leq\frac{n}{2}\frac{1-|h(x)|^{2}}{1-|x|^{2}},\quad
x\in\mathbb{B}^{n}?$

##  3\. Proofs of the main results

### Proof of Theorem 2.1

To simplify notations, let us denote

$A=A_{n}^{cap}(\gamma).$

Using spherical coordinates, the formula for the normalized area of the
spherical cap with contact angle $\gamma\in[0,\pi]$ is given by

$A=\sigma_{*}(n)A_{0}(\gamma),$ (3.1)

where

$A_{0}(\gamma)=\int_{0}^{\gamma}(\sin\theta)^{n-2}d\theta,$

see for example [1].

######  Lemma 3.1.

Using the notations of Theorem C, for $\gamma\in(0,\pi)$, we have

$D_{n}(\gamma,\beta)=\beta C_{n}(\gamma)(1-|h(0)|^{2}),$ (3.2)

where

$C_{n}(\gamma)=\dfrac{1}{(n-1)}\dfrac{(\sin\gamma)^{n-1}}{A_{0}(\gamma)(1-A_{n}^{cap}(\gamma))},$
(3.3)

and

$C_{n}(\gamma)\to 1\mbox{ as }\gamma\to 0^{+}.$

###### Proof.

By the constraint condition (1.6), we have $1+h(0)=2A$ and $1-h(0)=2(1-A)$.
Therefore

$1-|h(0)|^{2}=4A(1-A).$ (3.4)

Recall that
$D_{n}(\beta,\gamma)=\dfrac{4\beta\omega_{*}(n)}{n}(\sin\gamma)^{n-1}=\dfrac{4\beta\sigma_{*}(n)}{n-1}(\sin\gamma)^{n-1}$.

By (3.4), we obtain
$D_{n}(\beta,\gamma)=\dfrac{\beta}{n-1}\dfrac{\sigma_{*}(n)(\sin\gamma)^{n-1}}{A(1-A)}(1-|h(0)|^{2}).$
Therefore, by (3.1), we get (3.3). The limit of $C_{n}(\gamma)$ as $\gamma$
goes to $0$ equals to $1$ is a direct consequence of L’Hop̂ital’s rule.
∎

First, we disprove Conjecture 1. Assume that the inequality (1.4) is true.
Then, in particular for $x=0$, we have

$|\nabla u(0)|\leq 2\omega_{*}(n)(1-a^{2}),$ (3.5)

where $a=u(0)$. Since the estimate (1.5) in Theorem C under the constrain
$a=u(0)$ is sharp, then there is extremal function $u^{0}$, such that
$D_{n}(\gamma,\beta)=|\nabla u^{0}(0)|$.

Thus

$D_{n}(\gamma,\beta)=\beta C_{n}(\gamma)(1-a^{2})\leq
2\omega_{*}(n)(1-a^{2}).$ (3.6)

Therefore $\beta C_{n}(\gamma)\leq 2\omega_{*}(n)$. In particular, in harmonic
case where $\beta=n/2$, we have

$nC_{n}(\gamma)\leq 4\omega_{*}(n).$

As the limit of $C_{n}(\gamma)$ is $1$ as $\gamma\to 0$, it yields

$n\leq 4\omega_{*}(n).$ (3.7)

Since $\omega_{n}$ assumes its maximal value when $n=5$, $\omega_{*}(5)<1$, we
disprove Liu’s conjecture.

We leave the reader to disprove Conjecture 1 for $n\geq 4$ using Alzer’s
estimate below.

###  3.1. Proof of Theorem 2.2

A remarkable upper and lower bounds for the ratio
$\frac{\omega_{n-1}}{\omega_{n}}$ are proved by Borgwardt[7, p.253]. He showed
that for $n\geq 2$

$\sqrt{\frac{n}{2\pi}}\leq\omega_{*}(n)\leq\sqrt{\frac{n+1}{2\pi}}.$ (3.8)

More refinements of these estimates are established by Alzer [3].

######  Theorem 3.1.

[3] For $n\geq 2$, we have

$\sqrt{\frac{n+A}{2\pi}}\leq\omega_{*}(n)\leq\sqrt{\frac{n+B}{2\pi}},$ (3.9)

with the best possible constants

$A=\frac{1}{2}\mbox{ and }B=\frac{\pi}{2}-1.$

For more properties of the volume of the unit ball in $\mathbb{R}^{n}$, see
[4, 23].

As $\sigma_{*}(n)=\frac{n-1}{n}\,\omega_{*}(n)$ and using (3.8) or (3.9), one
can easily check the following lemma.

######  Lemma 3.2.

For $n\geq 4$, we have

$\frac{1}{2}<\sqrt{\frac{n-1}{8}}<\sigma_{*}(n)<\frac{n-1}{4}.$ (3.10)

Recall that the area of the spherical cap of contact angle $\gamma\in[0,\pi]$
is given by

$A(\gamma)=\sigma_{*}(n)\int_{0}^{\gamma}\sin^{n-2}\theta d\theta.$ (3.11)

Let $a\in[-1,1]$, then there exists a unique angle $\gamma(a)\in[0,\pi]$ such
that

$A(\gamma(a))=\frac{1+a}{2}.$ (3.12)

Clearly, the mapping $a\mapsto\gamma(a)$ is strictly increasing from $[-1,1]$
to $[0,\pi]$.

Differentiating the equation (3.12) with respect to $a$, we obtain

$\sigma_{*}(n)\gamma^{\prime}(a)\sin^{n-2}\gamma(a)=\frac{1}{2},\quad\mbox{
for }a\in(-1,1).$ (3.13)

Let us consider the function $h$ defined on $[-1,1]$ by

$h(a):=\sin^{n-1}\gamma(a)-1+a^{2}.$ (3.14)

As $h$ is even, it is enough to study the function $h$ on $[0,1].$

For each $a\in[0,1)$, we have

$h^{\prime}(a)=(n-1)\gamma^{\prime}(a)\sin^{n-2}\gamma(a)\cos\gamma(a)+2a.$

In view of the equation (3.13), we get

$h^{\prime}(a)=\frac{(n-1)\cos\gamma(a)}{2\sigma_{*}(n)}+2a.$

The second derivative of $h$ is given by

$h^{\prime\prime}(a)=-\frac{n-1}{2\sigma_{*}(n)}\sin\gamma(a)\gamma^{\prime}(a)+2.$

Again using (3.13), we deduce that

$h^{\prime\prime}(a)=-\frac{n-1}{4\sigma_{*}(n)^{2}}\sin^{3-n}\gamma(a)+2.$

For $n\geq 4$, we conclude that $h^{\prime\prime}$ is strictly decreasing on
$[0,1]$ because $\gamma$ is increasing with values in $[\pi/2,\pi]$. Moreover,
we have

$h^{\prime\prime}(0)=-\frac{n-1}{4\sigma_{*}(n)^{2}}+2,\mbox{ and
}\quad\lim_{a\to 1}h^{\prime\prime}(a)=-\infty.$ (3.15)

Using the inequality (3.10), it yields

$h^{\prime\prime}(0)>0,\mbox{ for }n\geq 4.$ (3.16)

Therefore, there exists $a_{n}\in(0,1)$ such that $h^{\prime\prime}(a)>0$ on
$(0,a_{n})$ and $h^{\prime\prime}(a)<0$ on $(a_{n},1)$. Thus $h^{\prime}$ is
increasing on $[0,a_{n})$ and decreasing on $(a_{n},1)$. Moreover,

$h^{\prime}(0)=0\mbox{ and }h^{\prime}(1)=-\frac{n-1}{2\sigma_{*}(n)}+2.$

Now, using (3.10), we deduce that

$h^{\prime}(1)<0.$

Therefore there exists $b_{n}\in(0,1)$ such that $h$ is increasing on
$(0,b_{n})$ and decreasing on $(b_{n},1)$. As $h(0)=h(1)=0$, we conclude that
the $\min_{a\in[0,1]}h(a)=h(0)=0$. Finally, we get $h(a)\geq 0\mbox{ for all
}a\in[0,1]$, that is, $\sin^{n-1}\gamma(a)\geq 1-a^{2},$ for $n\geq 4$.

Next, in order to prove the second estimate in Theorem 2.2, we need the
following lemma.

######  Lemma 3.3.

Let $a\in[0,1]$ and $n\geq 4$. Then

$-\cos\gamma(a)\leq a,$

Moreover, the equality holds at $a=0$ or $a=1$.

###### Proof.

Consider the function $G$ on $[0,1]$ defined by

$G(a)=a+\cos\gamma(a).$

$G$ is differentiable on $(0,1)$ and

$G^{\prime}(a)=1-\sin\gamma(a)\gamma^{\prime}(a)=1-\frac{1}{2\sigma_{*}(n)}\sin^{3-n}\gamma(a).$

Thus, the function $G^{\prime}$ is strictly decreasing as $\gamma(a)$ belongs
to $[\pi/2,\pi]$. Moreover, we have

$G^{\prime}(0)=1-\frac{1}{2\sigma_{*}(n)}>0\mbox{ and}\lim_{a\to
1}G^{\prime}(a)=-\infty.$

Therefore, $G$ is strictly increasing on $[0,c_{n}]$ and decreasing on
$(c_{n},1)$, where $c_{n}\in(0,1)$ is the unique zero of $G^{\prime}$. As
$G(0)=G(1)=0$, we get $G(a)>0$ on $(0,1)$.
∎

It remains to prove the following:
$(\sin\gamma)^{n-1}\leq\frac{n-1}{4\sigma_{*}(n)}(1-a^{2})$, $n\geq 4$.

Consider

$g(a)=1-a^{2}-\frac{4\sigma_{*}(n)}{n-1}\sin^{n-1}\gamma(a).$

Then

$g^{\prime}(a)=-2a-2\cos\gamma(a).$

By Lemma 3.3, we have $g^{\prime}(a)\leq 0$ and $g$ is decreasing. Hence
$g(a)\geq g(1)=0$ on $[0,1]$ and the conclusion follows.

### Proof of Theorem 2.4

Theorem 2.4 is a direct consequence of Theorem C and Theorem 2.2. In
particular, in Corollary 2, we obtain the corresponding inequality for
harmonic and hyperbolic harmonic functions, by considering the corresponding
values of $\beta$.

### Proof of Theorem 2.5 and 2.6

Let $h:\mathbb{B}^{n}\to(-1,1)$ be a harmonic function and let
$x\in\mathbb{B}^{n}$. Consider $g$ the harmonic function defined on
$\mathbb{B}^{n}$ by

$g(y)=h(x+y(1-|x|)).$

Clearly, we have $g(0)=h(x)$ and $|\nabla g(0)|=(1-|x|)|\nabla h(x)|$. By
applying Corollary 2 to the function $g$, we get the desired inequality.

In the hyperbolic harmonic case, we compose with a Möbius transformation
sending $0$ to $x$. More precisely, let $x\in\mathbb{B}^{n}$ be fixed. By the
Möbius invariance of $\Delta_{h}$, the function $h\circ\varphi_{x}$ is also a
bounded hyperbolic harmonic function, where

$\varphi_{x}(y):=\frac{|y-x|^{2}x-(1-|x|^{2})(y-x)}{1-2\langle
y,x\rangle+|y|^{2}|x|^{2}},$

which is a Möbius transformation of $\mathbb{B}^{n}$. Theorem 2.6 follows by
replacing $h$ by $h\circ\varphi_{x}$ in Corollary 2 and noting that
$\nabla(h\circ\varphi_{x})(0)=-(1-|x|^{2})\nabla h(x)$, see [30, p. 18].

###  3.2. Proof of Theorem 2.7

As the proofs for the harmonic and the hyperbolic harmonic case are similar,
we will provide only the proof in the harmonic setting. Let
$h:\mathbb{B}^{n}\to\mathbb{B}^{m}$ be a harmonic vector-function and $\theta$
be a unit vector in $\mathbb{R}^{m}$. Consider $h_{\theta}$ the function
defined by

$h_{\theta}(x)=\langle h(x),\theta\rangle.$

Then $h_{\theta}$ is a harmonic function with values in $(-1,1)$.
Consequently, by Theorem 2.5, we get

$|\langle
Dh(x)v,\theta\rangle|=|Dh_{\theta}(x)v|\leq\frac{n}{2}\frac{1}{1-|x|},\quad\mbox{for
all }v\in\mathbb{R}^{n},\,|v|=1.$

Therefore, $|Dh(x)|\leq\frac{n}{2}\frac{1}{1-|x|}.$

###  3.3. Proof of Theorem 2.8

Let $x_{0}\in\mathbb{B}^{n}$ and $h:\mathbb{B}^{n}\to\mathbb{B}^{m}$ be a
harmonic function. By Theorem 2.7, we have to consider only the case where
$h(x_{0})\not=0$. Define

$g(x):=\langle h(x),\frac{h(x_{0})}{|h(x_{0})|}\rangle.$

Then $g$ is a harmonic function on $\mathbb{B}^{n}$ with codomain $(-1,1)$
with $g(x_{0})=|h(x_{0})|.$ It follows from Theorem 2.5, that

$|\nabla g(x_{0})|\leq\frac{n}{2}\frac{1-|g(x_{0})|^{2}}{1-|x_{0}|}.$

Indeed, easy computations show that $|\nabla g(x_{0})|=|\nabla|h|(x_{0})|$ as
$g_{x_{i}}(x_{0})=|h|_{x_{i}}(x_{0})$, where $g_{x_{i}}(x_{0})$ denotes the
partial derivative of $g$ with respect to the variable $x_{i}$ at $x_{0}$.

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# Gamma-rays flashes from dark photons in neutron star mergers

Melissa D. Diamond<EMAIL_ADDRESS>Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, MD 21218, U.S.A.  Gustavo Marques-Tavares
<EMAIL_ADDRESS>Maryland Center for Fundamental Physics, Department of Physics,
University of Maryland, College Park, MD 20742, U.S.A.

###### Abstract

In this letter we begin the study of visible dark sector signals coming from
binary neutron star mergers. We focus on dark photons emitted in the 10 ms - 1
s after the merger, and show how they can lead to bright transient gamma-ray
signals. The signal will be approximately isotropic, and for much of the
interesting parameter space will be close to thermal, with an apparent
temperature of $\sim 100$ keV. These features can distinguish the dark photon
signal from the expected short gamma-ray bursts produced in neutron star
mergers, which are beamed in a small angle and non-thermal. We calculate the
expected signal strength and show that for dark photon masses in the $1-100$
MeV range it can easily lead to total luminosities larger than $10^{46}$ ergs
for much of the unconstrained parameter space. This signal can be used to
probe a large fraction of the unconstrained parameter space motivated by
freeze-in dark matter scenarios with interactions mediated by a dark photon in
that mass range. We also comment on future improvements when proposed
telescopes and mid-band gravitational detectors become operational.

##  I Introduction

The detection of gravitational waves (GW) from binary neutron star (BNS)
mergers, and the observation of their electromagnetic counterparts has
inaugurated a new era in multi-messenger astronomy Abbott et al. (2017a);
Abbott et al. (2017b, c). Upcoming observations will shed light on the physics
of neutron stars, the origin of heavy elements and models of stellar evolution
Cowan et al. (2021); Meszaros (2002); Pian (2020); Mészáros et al. (2019).
This new window into the Universe offers great potential as a probe of physics
beyond the standard model (BSM), and in particular of scenarios involving very
weakly interacting new states.

When two neutron stars merge, a meta-stable remnant with nuclear densities and
temperatures in the 10s of MeV forms, similar to the proto-neutron stars
formed in core-collapse supernovae Camelio et al. (2021); Endrizzi et al.
(2020); Radice (2020); Perego et al. (2019). These hot remnants are a
promising source of new weakly coupled particles. It has already been shown
that such events can produce a large flux of neutrinos Radice et al. (2018).
Nonetheless, the low merger rate implies that if we are interested in seeing
at least $\mathcal{O}(1)$ merger/year, their typical distance will be $\sim
100$ Mpc, making direct observation of the new particles challenging due to
their small flux at large distances and small couplings. However, if the new
particles are unstable and can decay to visible particles, such as leptons or
photons, they can produce very bright signals making BNS mergers a powerful
probe of new physics.

There have been a number of proposals to search for new physics affecting the
GW signal from binary black-holes or neutron star mergers Endlich et al.
(2017); Hook and Huang (2018); Sagunski et al. (2018); Croon et al. (2018);
Huang et al. (2019); Bezares et al. (2019); Dror et al. (2020); Sennett et al.
(2020), in addition to potential cooling constraints due to emission of weakly
coupled BSM particles from the remnant Dietrich and Clough (2019); Harris et
al. (2020). In this letter, we initiate the study of dark sector
electromagnetic signals in BNS mergers. For concreteness, we focus exclusively
on dark photons, though this framework is applicable to other dark sector
models. The dark photon is a new massive vector field that kinetically mixes
with the photon, and through this mixing interacts with charged standard model
matter Holdom (1986); Fabbrichesi et al. (2020). It corresponds to one of the
three renormalizable portals between the Standard Model (SM) and dark sectors,
which are sectors not charged under the SM gauge group Essig et al. (2013);
Alexander et al. (2016). Dark matter might be part of such a dark sector, and
its interactions with the standard model mediated by the dark photon may
account for the observed dark matter density. The relic abundance can be
obtained either through standard freeze-out, corresponding to scenarios in
which dark matter reaches thermal equilibrium with the visible sector and
requires larger couplings, or through freeze-in in which the interactions are
so weak that dark matter never reaches thermal equilibrium with the SM Hall et
al. (2010); Chu et al. (2012).

The cross-sections suggested by the freeze-out scenario motivated a large
experimental program intended to cover the parameter space corresponding to
the observed relic density. In contrast, the freeze-in scenario points to tiny
couplings, with the kinetic mixing parameter as small as $10^{-11}$, making it
more difficult to probe experimentally. There are a number of recent direct
detection proposals aimed at probing the freeze-in parameters for very light
dark photons, which take advantage of the cross-section enhancement at low
velocities characteristic of interactions mediated by light particles (e.g.
Essig et al. (2016); Knapen et al. (2018); Abramoff et al. (2019); Aralis et
al. (2020)). Probing the freeze-in scenario for larger dark photon masses,
when there is no significant enhancement of the cross-section, is much more
challenging. Most constraints on this scenario come from searches sensitive to
the dark photon directly, generally using cosmological or astrophysical
signals Redondo and Postma (2009); An et al. (2013); Fradette et al. (2014);
Kazanas et al. (2014); Chang et al. (2017); Hardy and Lasenby (2017); DeRocco
et al. (2019); Hong et al. (2020). We will show that dark photons in the mass
range $\sim 1-100$ MeV can be produced copiously in the remnant of a neutron
star merger. Their decays lead to a transient bright gamma-ray signal that can
be used to search for dark photons in much of the remaining viable parameter
space for freeze-in with a mediator in that mass range.

##  II Dark Photon production and decay

We will concentrate on dark photon production and assume that any other new
particles are heavy and irrelevant for BNS mergers. The relevant terms in the
lagrangian are

$\mathcal{L}\supset\frac{1}{2}m^{\prime}A_{\mu}^{\prime}A^{\prime\mu}-\frac{1}{4}F_{\mu\nu}^{\prime}F^{\prime\mu\nu}-\frac{\epsilon}{2}F_{\mu\nu}^{\prime}F^{\mu\nu}\,,$
(1)

where $A^{\prime\mu}$ is the dark photon, $m^{\prime}$ its mass, $F^{\mu\nu}$
the photon field strength, and $\epsilon$ the kinetic mixing.

The production of dark photons in the proto-neutron star is dominated by
nucleon-nucleon bremsstrahlung, as in the supernova case. The flux of dark
photons can be calculated following Ref. Chang et al. (2017),

$\frac{dN}{dVdt}=\int\frac{d\omega\omega^{2}v}{2\pi^{2}}e^{-\omega/T}(\Gamma_{\text{abs},L}+2\Gamma_{\text{abs},T}),$
(2)

where $\omega$ is the frequency of the dark photon, $v$ its velocity, $T$ the
local temperature and $\Gamma_{\text{abs},T/L}$ the absorption width for the
transverse/longitudinal dark photon modes. Integrating this over the
production volume gives the total number of dark photons produced. The
absorption width can be computed using the soft-radiation approximation, as
discussed in Ref. Rrapaj and Reddy (2016). Ignoring Pauli blocking (which is
at most an $\mathcal{O}(1)$ effect), we find

$\displaystyle\Gamma_{\text{abs, }X}=$ $\displaystyle\,\frac{32\alpha
n_{\text{n}}n_{\text{p}}\epsilon^{2}\,m^{\prime\,4}}{3\pi\omega^{3}\left((m^{\prime\,2}-\text{Re}\Pi_{X})^{2}+\text{Im}\Pi_{X}^{2}\right)}\left(\frac{\pi
T}{m_{\text{N}}}\right)^{3/2}$ (3) $\displaystyle\times$
$\displaystyle\langle\sigma_{\text{np}}^{(2)}(T)\rangle\,\times\left\\{\begin{array}[]{ll}1\,,&X=T\\\
(m^{\prime}/\omega)^{2}\,,&X=L\end{array}\right.\,,$

where $X=T(L)$ refers to the transverse (longitudinal) polarization of the
dark photon, $m_{N}$ is the nucleon mass, $n_{n/p}$ is the neutron/proton
density, $\Pi_{X}$ is the in medium polarization tensor of the photon (see
Supplemental Materials sup  for their explicit form) and
$\langle\sigma_{\text{np}}^{(2)}\rangle$ is the weighted proton neutron
scattering cross-setion taken from Ref. Rrapaj and Reddy (2016). From Eq. 2,
we calculate the dark photon luminosity

$\frac{dE}{dVdt}=\int\frac{d\omega\omega^{3}v}{2\pi^{2}}e^{-\omega/T}(\Gamma_{\text{abs},L}+2\Gamma_{\text{abs},T})\,.$
(4)

Based on BNS merger simulations in Ref. Perego et al. (2019); Radice (2020);
Bernuzzi et al. (2020); Endrizzi et al. (2020), we assume a simplified,
spherically symmetric description of the merger remnant, with a constant
temperature, density and electron fraction: $T=30$ MeV, $\rho=4\times 10^{14}$
g cm-3, $Y_{e}=0.1$ in the region 5 km-10 km from the center of the remnant.
We consider only the dark photons produced in this hot region and ignore
contributions from other colder regions. In Fig. 1, we show the luminosity of
dark photons as a function of mass and how it varies with the remnant’s
temperature.

Figure 1: Luminosity of dark photons with $\epsilon=10^{-10}$, produced in BNS
mergers using the simplified profile. We use $T=30$ MeV (blue curve) for
calculations in this work. The dashed and dotted lines show luminosity for
other values of temperature. The enhancement near $10$ MeV comes from resonant
mixing between the dark photon and the photon.

The time evolution of the remnant depends on the mass of the original neutron
stars and the equation of state of nuclear matter. Different initial
conditions can produce a variety of different remnants that persist for
between $1-1000$ ms before collapsing to a black hole Perego et al. (2019);
Camelio et al. (2021); Endrizzi et al. (2020); Bernuzzi et al. (2020); Radice
(2020). Light remnants might not collapse at all, instead forming a heavier
neutron star Fujibayashi et al. (2020). Nonetheless, cooling will decrease
their temperature from $T\sim 30$ MeV within a few seconds Dietrich and Clough
(2019); Radice et al. (2020). To illustrate this range of possibilities, we
present results assuming two different scenarios for dark photon emissions:
emissions lasting $10$ ms after the merger and emissions lasting $1$ s after
the merger. Note that most analyses of GW170817 conclude that the remnant must
have lasted for at least $10$ ms Murguia-Berthier et al. (2021); Shibata et
al. (2019, 2017); Ruiz et al. (2018). For simplicity, we ignore gravitational
redshift, which depends strongly on the density distribution in the central
region of the merger (we estimate its impact in the Supplemental Material sup
).

After production, the dark photons decay to electron-positron pairs, forming
an expanding plasma shell. The dark photon decay width at rest is

$\Gamma=\frac{1}{3}\alpha\epsilon^{2}m^{\prime}\sqrt{1-\frac{4m_{\text{e}}^{2}}{m^{\prime
2}}}\left(1+\frac{2m_{\text{e}}^{2}}{m^{\prime 2}}\right)\,,$ (5)

where $m_{e}$ is the electron mass. The initial Lorentz factor of the
expanding shell is approximately the average Lorentz factor of the dark photon
flux,

$\gamma_{0}=\frac{\langle\omega\rangle}{m^{\prime}}\,,$ (6)

where $\langle\omega\rangle$ is the average dark photon energy. In the merger
frame, the dark photon decay length is

$d=\frac{\gamma_{0}v}{\Gamma}\,.$ (7)

The width of the plasma shell immediately after the decay is

$\delta=\frac{1}{\gamma_{0}\Gamma}\,,\qquad\delta^{\prime}=\frac{1}{\Gamma}$
(8)

in the star frame and plasma frame respectively.

The photon signal depends on the evolution of this plasma. For simplicity, we
focus on parameters such that the dark photons decay at least 1000 km from the
center of the merger, where ambient baryon density and magnetic fields can be
safely neglected. 111 Magnetic fields do not directly affect the dark photons,
but would lead to non-trivial dynamics for the electron-positron plasma formed
after the decay.  When calculating the number of leptons that results from
dark photon decays, we only include the fraction coming from decays at least
1000 km away from the merger. Dark photons in most of the viable parameter
space have long enough decay lengths that this requirement does not cause a
significant change in sensitivity.

##  III Gamma-ray signal

We are interested in the photon signal arising from the the electrons and
positrons produced by dark photon decays. With our simplified approximation
for the remnant, the signal will be isotropic222Realistically the remnant will
not be spherically symmetric and will have $\mathcal{O}(1)$ variations in the
luminosity depending on the inclination angle with respect to the plane of the
merger., peaked between 100 keV and 10s of MeV and visible within about a
second of the merger. The isotropic nature of the signal, its duration and its
spectral information can be used to distinguish it from the short gamma-ray
burst (GRB) expected to be produced from relativistic jets after the remnant
collapses to a black hole Abbott et al. (2017c); Meszaros (2006).

We track the dynamics of the plasma starting from a radius equal to the decay
length of the dark photon in the star frame, given by Eq. 7. We describe the
plasma using quantities defined in the co-expanding frame, in which the lepton
momenta distribution is isotropic and which is initially related to the merger
frame by a Lorentz factor given by Eq. 6. In this frame, the initial
temperature is directly related to the dark photon mass, $T\approx
m^{\prime}/6$, since in that frame each electron and positron generated by the
dark photon has approximately $m^{\prime}/2$ energy, with a small spread
related to the dark photon boost distribution. The initial number density is
given by

$n_{\text{e}}\approx\frac{N_{\text{tot}}\gamma_{0}}{4\pi d^{3}}\,,$ (9)

where we take

$N_{\text{tot}}=\frac{dN}{dVdt}\times
V_{\text{emit}}t_{\text{emit}}\left(e^{-1000\text{km}/d}-e^{-1}\right)$ (10)

as the number of dark photons which decay between 1000 km and one decay length
from the merger remnant. $V_{\text{emit}}$ and $t_{\text{emit}}$ are the
volume and time over which dark photons are emitted, respectively. As in the
standard fireball model Piran (1999); Meszaros (2006), when the plasma energy
is dominated by relativistic particles, the density evolves as $\rho\propto
r^{-4}$, where $r$ is the distance of the plasma shell from the remnant (in
the remnant’s rest frame), the Lorentz factor of the shell increases as
$\gamma\propto r$, and the width in the merger frame, which also determines
the duration of the signal, remains constant. If expansion is the only
processes which changes the total number density then the total number of
particles is conserved, the total number density scales as $n\propto r^{-3}$,
and the temperature goes as $T\propto r^{-1}$.

The electron/positron densities in the plasma can be large enough for pair
annihilation, $e^{+}e^{-}\rightarrow\gamma\gamma$, to be very efficient. In
this case, the pair creation and annihilation processes quickly leads the
number densities of electrons and photons to be related by detailed balance,

$\frac{n_{e}}{n_{\gamma}}=\frac{n_{e}^{\text{eq}}}{n_{\gamma}^{\text{eq}}}\,.$
(11)

The RHS of Eq.11 refers to the equilibrium number densities for leptons and
photons. In this regime, the photon mean free path will be short and the
plasma is optically thick. An observable signal emerges once the lepton number
density becomes low enough for the plasma to become optically thin, allowing
photons to escape. Photon bremsstrahlung from electron and positron scattering
can also play an important role in the dynamics of the plasma. While pair
creation/annihilation preserves the total number of particles, bremsstrahlung
increases the total number of particles and consequently the total number
density. Energy conservation implies that the temperature decreases as the
number density goes up, thus, bremsstrahlung can lower the peak energy of the
photon signal.

To determine if pair annihilation and bremsstrahlung are important effects, we
compare their rates in the plasma frame to the number density dilution rate
coming from the expansion,

$\Gamma_{\text{exp}}=-3\gamma/r\,.$ (12)

The rate for pair annihilation is given by Svensson (1982)

$\Gamma_{\text{annih}}\approx\frac{\pi
n_{e}\alpha^{2}}{m_{e}^{2}}\left(1+\frac{2(T/m_{e})^{2}}{1+\log\left(\frac{2T}{m_{e}e^{\gamma_{E}}}+1.3\right)}\right)^{-1}\\!,$
(13)

where $\gamma_{E}$ is the Euler-Mascheroni constant. The photon production
rate from $e^{+}e^{-}$ (or $e^{-}e^{-}$) bremsstrahlung in a relativistic gas
($T\gg m_{e}^{2})$ is ALEXANIAN (1968)

$\displaystyle\Gamma_{\text{brem}}$
$\displaystyle\approx\frac{2n_{e}\alpha^{3}\log\left(e^{\gamma_{E}}m_{e}^{2}/T^{2}\right)}{9m_{e}^{2}}\left[12\log\left(e^{\gamma_{E}}m_{e}^{2}/T^{2}\right)\right.$
(14)
$\displaystyle\left.-84+48\log\left(e^{\gamma_{E}}m_{e}/T\right)\right]\,,$

where we imposed an infrared cutoff on the photon energy
$\omega_{\gamma}>m_{e}^{2}/T$, corresponding to photons that can exchange
$\mathcal{O}(T)$ of energy in a single scattering and thus quickly thermalize.
For $T\lesssim 1\,{\rm MeV}$, we switch to the non-relativistic rate (for the
dominant $e^{+}e^{-}$ case) Haug (1985)

$\Gamma_{\text{brem}}\approx\frac{64}{3\sqrt{\pi}}\frac{n_{e}\alpha^{3}}{\sqrt{Tm_{e}^{3}}}\,,$
(15)

where we imposed an infrared cutoff $\omega_{\gamma}>T$, again corresponding
to photons that can exchange $\mathcal{O}(T)$ energy in a single scattering.
If the annihilation and bremsstrahlung rates are both greater than the
expansion rate at the beginning of the plasma evolution, right after most dark
photons have decayed, then the plasma thermalizes quickly. Since
$n_{\gamma}\ll T^{3}$, these processes increase the total number density and
decrease the temperature before any significant expansion until $T\leq m_{e}$,
at which point both annihilation and bremsstrahlung rates decay exponentially
due to the loss of leptons. Dark photons with masses above $\sim 10$ MeV, have
another mechanism for thermalizing rapidly. Even if the pair annihilation rate
is initially slower than the expansion, the energy loss rate from
bremsstrahlung ALEXANIAN (1968)

$\frac{d\log\rho_{e}}{dt}\Big{|}_{\text{brem}}=-\frac{8n_{e}\alpha^{3}}{m_{e}^{2}}\left(\text{Log}\left(\frac{2T}{e^{\gamma_{E}}m_{e}}\right)+\frac{5}{4}\right)$
(16)

can be faster than that due to expansion

$\frac{d\log\rho_{e}}{dt}\Big{|}_{\text{exp}}=-\frac{4\gamma}{r}\,,$ (17)

where $\rho_{e}=3Tn_{e}$ is the energy density of the shell in the shell frame
immediately after the dark photons decay. This energy loss decreases the
plasma temperature (with insignificant change to the lepton number density),
increasing the pair annihilation rate sufficiently to make it faster than the
expansion. Afterwards, the fireball evolves as if both pair annihilation and
bremsstrahlung were already efficient when the plasma first formed.

The resulting photon spectrum in the observer frame is approximately thermal,
with an apparent temperature given by $\gamma_{*}T_{*}$, where $\gamma_{*}$
and $T_{*}$ are respectively the plasma’s Lorentz factor and temperature when
it becomes optically thin for photons. The photon interaction rate drops
exponentially once the plasma temperature is below $m_{e}$ because the
electron density becomes Boltzmann suppressed, leading to $T_{*}\sim m_{e}/10$
with only a mild logarithmic sensitivity to the initial conditions of the
plasma shell. In the thermalized scenario, the temperature drops before the
plasma expands significantly, and thus $\gamma_{*}=\gamma_{0}\lesssim 10$, and
effectively all of the energy radiated in dark photons gets converted into
photons with energies in the $10-1000$ keV range. The photon signal duration
is set by light crossing time of the plasma shell, about 0.1-100 s depending
on the dark photon parameters. The luminosity of the signal can be estimated
by dividing the total energy output by the width of the plasma shell as shown
in Eq. 8.

The parameter space region in which this thermal spectrum occurs is shown in
Fig. 2 for two different assumptions about the duration of the remnant. It
also shows two curves which mark the total energy emitted in dark photons. One
curve shows where the signal would be detectable at the Fermi gamma-ray burst
monitor (GBM), assuming all of the energy gets converted to photons in the
sensitivity range of the instrument ($100-2000$ keV) and a merger distance of
$100$ Mpc. The other curve shows the region where the total energy emitted in
dark photons is $10^{44}$ ergs as a potential target for future detectors. If
the “fireball” forms, the dark photon energy would get converted to the GBM
range because the initial boost $\gamma_{0}$ is $\mathcal{O}(1)$ for all
relevant masses. This shows that one could probe most of the remaining
parameter space above $\epsilon\approx 10^{-11}$, motivated by freeze-in dark
matter scenarios, for $m^{\prime}\approx 1-100$ MeV by searching for gamma-ray
signals which coincide with a neutron star merger.

Figure 2: The left (right) plot shows the conditions produced when the merger
emits dark photons for 10 ms (1 s). The red region marks where the dark
photons form a fireball which produces a thermal spectrum, below it dark
photons generate a dimmer non-thermal signal. The dark gray region shows where
the dark photon decay length is less than 1000 km (close enough to the merger
remnant that baryons could affect the signal). The purple line shows the
minimum $\epsilon$ needed to produce a signal visible to Fermi GBM for a
merger 100 Mpc away ($\sim 2\times 10^{46}$ ergs), and the blue line shows
parameters for a $10^{44}$ erg signal. The light gray regions marked SN
Cooling and SN Emission show the parameter space already ruled out by
astrophysical observations DeRocco et al. (2019); Chang et al. (2017). The
light blue regions labeled $N_{eff}$ and BBN mark space excluded by
cosmological observations Ellis et al. (1992); Zhang et al. (2007); Fradette
et al. (2014); Ibe et al. (2020). For potential additional complementary
constraints see also DeRocco et al. (2019); Sung et al. (2019); Caputo et al.
(2022).

##  IV Discussion and future prospects

The LIGO/Virgo collaboration is currently sensitive to neutron star mergers
within $\sim 100$ Mpc of Earth Instrument Science List et al. (2020). The
first BNS merger detection, GW170817, was about half this distance, unusually
close given the estimated rate for BNS mergers. There was an associated gamma-
ray signal observed shortly after by Fermi-GBM and Integral Goldstein et al.
(2017), which has largely confirmed the expectation that BNS mergers are
responsible for short gamma-ray bursts (sGRB). Making a discovery of dark
photons using the gamma-ray signal discussed in this paper would require
distinguishing the dark photon signal from usual sGRBs. This can, in
principle, be done using the fact that sGRBs are expected to be highly beamed
since they are produced by a relativistic jet, leading to large variations of
the observed luminosity from mergers at similar distances. However, potential
off-axis gamma-ray emission is still not fully understood, and might be an
important background for our proposal Lazzati et al. (2017); Gottlieb et al.
(2018); Lazzati (2020). Even if there is a significant background from off-
axis emission, one can use information about the spectrum, signal arrival time
and duration as important features to distinguish our signal from sGRBs (in
the Supplementary Materials sup  we show how timing can be used to put
constraints on the dark photon model).

The LIGO/Virgo collaboration will be sensitive to neutron star mergers as far
as $\sim 200$ Mpc when it reaches design sensitivity in the near future Aasi
et al. (2015), at which point it will detect multiple mergers per year. Having
large merger statistics, and utilizing realistic remnant profiles from
simulations, would allow one to probe the parameter space in Fig. 2 above the
purple line and within the fireball region assuming better understanding of
the backgrounds from sGRBs. This would cover a large portion of the remaining
parameter space motivated by dark matter freeze-in, $\epsilon\gtrsim 10^{-11}$
Chu et al. (2012).

There are two challenges to probing the parameter space in which a fireball
never forms. One is modeling the multiple processes that produce photons from
a dilute electron positron plasma. We leave the required detailed analysis of
this non-thermal emission to future work, but we expect that for part of the
parameter space, the main signal will be in $\sim 10$ MeV photons, for which
we currently do not have very good coverage. There are a number of proposals
such as e-ASTROGRAM De Angelis et al. (2017), AMEGO Caputo et al. (2019), and
MeVCube Lucchetta et al. (2020) that target the low MeV range which would
significantly increase our reach to dark photons from BNS. The other challenge
is that, for smaller couplings the lepton density after decay is smaller and
only a reduced fraction of the energy gets converted to photons, making the
signal dimmer. One promising direction to compensate the low photon luminosity
is to use detectors with better angular resolution, which decreases the
background to the signal. Future proposed GW mid-band detectors such as AMIGO
Ni et al. (2020), MAGIS Abe et al. (2021), AION Badurina et al. (2020), ELGAR
Canuel et al. (2020) should be able to detect BNS merger events in advance and
with better localization, allowing future x-ray and gamma-ray telescopes with
narrower fields of view to observe the merger.

We have shown that neutron star mergers can be used to search for unstable
dark sector particles. We demonstrated this by studying the gamma-ray
signatures that arise from dark photon decays, and showed that this can probe
a large portion of unconstrained parameter space, including much of the
remaining parameter space motivated by dark matter freeze-in. Utilizing the
full potential of BNS as probes of dark sectors requires a better
understanding of the remnant dynamics and further investigation of how to
distinguish the signal arising from dark sectors from the that expected from
sGRBs associated with BNS mergers.

###### Acknowledgements.

The authors thank Tim Dietrich, Julian Krolik, and David Radice for useful
discussions and Daniel Egaña-Ugrinovic for comments on the draft. GMT was
supported in part by the NSF grants PHY-1914480, PHY-1914731, by the Maryland
Center for Fundamental Physics (MCFP) and by the US-Israeli BSF Grant 2018236.
MD was supported in part by NSF grant PHY-1818899.

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Visible dark photon flashes from neutron star mergers

Supplementary Material
Melissa Diamond and Gustavo Marques-Tavares

##  I Dark photon production calculation

The differential number flux of dark photons is given by:

$\frac{dN}{dVdt}=\int
d\omega\frac{dN}{dVdtd\omega}=\int\frac{d\omega\omega^{2}v}{2\pi^{2}}e^{-\omega/T}(\Gamma_{\text{abs},L}^{{}^{\prime}}+2\Gamma_{\text{abs},T}^{{}^{\prime}})$
(S1)

Where $\omega$ is the frequency of the dark photon, $v$ is its velocity and
$\Gamma_{\text{abs},T/L}^{{}^{\prime}}$ is the absorptive width of the
transverse/longitudinal mode. Inverse proton-neutron bremsstrahlung is the
dominant absorption process in the remnant, so that will be the one considered
here. Using the soft radiation approximation the absorptive width is given by
Rrapaj and Reddy (2016); Chang et al. (2017)

$\Gamma^{{}^{\prime}}_{\text{ibr},L|T}=\frac{32}{3\pi}\frac{\alpha(\epsilon_{m})^{2}n_{\text{n}}n_{\text{p}}}{\omega^{3}}\left(\frac{\pi
T}{m_{\text{N}}}\right)^{3/2}\langle\sigma_{\text{np}}^{(2)}(T)\rangle\left(\frac{m^{\prime
2}}{\omega^{2}}\right)_{L}\,,$ (S2)

where $n_{\text{n}}$ is the number density of neutrons, $n_{\text{p}}$ is the
number density of protons, $m_{\text{N}}$ is the neutron mass, $m^{\prime}$ is
the dark photon mass, $T$ is the background temperature,
$\langle\sigma_{\text{np}}^{(2)}(T)\rangle$ is the thermal-averaged proton
neutron dipole scattering cross section, $(\epsilon_{m})^{2}$ is the in medium
mixing angle and $\left(\frac{m^{\prime 2}}{\omega^{2}}\right)_{L}$ only
applies to the longitudinal term and is $1$ for the transverse polarizations.
The cross section $\langle\sigma_{np}^{(2)}(T)\rangle$ is taken from Ref.
Rrapaj and Reddy (2016).

Plasma effects change the mixing between the photon and the dark photon. This
can be taken into account by using an effective mixing parameter Chang et al.
(2017)

$(\epsilon_{m})^{2}_{L|T}=\frac{\epsilon^{2}}{\left(1-\text{Re}\Pi_{L|T}/m^{\prime
2}\right)^{2}+\left(\text{Im}\Pi_{L|T}/m^{\prime 2}\right)^{2}}\,,$ (S3)

where $\Pi$ is the polarization tensor. Its real component is given by:

$\text{Re}\Pi_{L}=\frac{3\omega_{p}^{2}}{v^{2}}\left(1-v^{2}\right)\left[\frac{1}{2v}ln\left(\frac{1+v}{1-v}\right)-1\right]$
(S4)
$\text{Re}\Pi_{T}=\frac{3\omega_{p}^{2}}{2v^{2}}\left[1-\frac{1-v^{2}}{2v}ln\left(\frac{1+v}{1-v}\right)\right]\,.$
(S5)

Here the velocity of the dark photon is determined by
$v=\sqrt{1-\frac{m^{\prime 2}}{\omega^{2}}}$, and $\omega_{p}$ is the plasma
frequency, which for the remnant is well approximated by the one for a gas of
degenerate electrons given by

$\omega_{p}^{2}=\frac{4\pi\alpha
n_{e}}{\sqrt{m_{e}^{2}+\left(3\pi^{2}n_{e}\right)^{2/3}}}\,,$ (S6)

where $m_{e}$ and $n_{e}$ denote the electron mass and number density
respectively.

Within the merger, standard model photons are in local thermal equilibrium.
Thus, the imaginary part of the polarization becomes:

$\text{Im}\Pi_{L|T}=-\omega\left(1-e^{-\omega/T}\right)\Gamma_{abs,L|T}\,,$
(S7)

where $\Gamma_{abs,L|T}$ is the absorptive width of the standard model photon,
taken to be

$\Gamma_{ibr,L|T}=\frac{32\alpha}{3\pi}\frac{n_{n}n_{p}}{\omega^{3}}\left(\frac{\pi
T}{m_{N}}\right)^{3/2}\langle\sigma_{np}^{(2)}(T)\rangle\left(\frac{m^{\prime
2}}{\omega^{2}}\right)_{L}\,,$ (S8)
$\Gamma^{\prime}_{abs}=(\epsilon_{m})^{2}_{L|T}\Gamma_{abs}\,.$ (S9)

##  II Gravitational Redshift

The number of dark photons emitted, their Lorentz factor upon decaying and the
total energy stored in the expanding fireball shell will be altered by
redshifting effects as the dark photons climb out of the gravitational well
created by the merger remnant. To compute the exact impact of gravitational
redshift on the signal requires a more accurate picture for the density
profile of the remnant than the one we used to compute the signal. Redshifting
will reduce the number of dark photons that escape the merger, and will reduce
the total energy of the photos that do. Still assuming spherical symmetry, the
number of dark photons that escape would now be found by taking

$\frac{dN}{dt}=4\pi\int_{r_{min}}^{r_{max}}r^{2}dr\int_{\omega_{m}}^{\infty}\frac{d\omega\omega^{2}v}{2\pi^{2}}e^{-\omega/T}(\Gamma_{\text{abs},L}+2\Gamma_{\text{abs},T}),$
(S10)

Where $\omega_{m}$ is the minimal energy a dark photon needs to escape the
merger

$\omega_{m}=\frac{m^{\prime}}{\sqrt{1-2\left(\frac{GM_{int}(r)}{r}\right)}}\,,$
(S11)

where $M_{int}(r)$ is the total mass contained within the radius $r$ of the
merger remnant. The total energy carried away by the dark photons can be found
using

$\frac{dE}{dt}=4\pi\int_{r_{min}}^{r_{max}}r^{2}dr\int_{\omega_{m}}^{\infty}\frac{d\omega\omega^{3}\sqrt{1-2\left(\frac{GM_{int}(r)}{r}\right)}v}{2\pi^{2}}e^{-\omega/T}(\Gamma_{\text{abs},L}+2\Gamma_{\text{abs},T})\,.$
(S12)

Figure S1: Range of possible ratios between dark photon luminosity when
redshifting effects are included and when such effects are neglected for a
$2.7M_{\odot}$ merger remnant. The upper limit for this ratio comes from a
model which assumes the remnant has a constant density throughout of $4\times
10^{14}$ g cm3. The lower limit on this ratio comes from a model where the
remnant has density of $4\times 10^{14}$ g cm3 in the dark photon emission
region (5 km-10 km from the center), with the remaining mass of the remnant
stored in the inner 5 km. This strongly suppresses the dark photon luminosity
because the amount of mass in the innermost region would be enough to form a
black hole, preventing many dark photons from escaping.

The average energy of the dark photons, $E_{ave}$, is found by dividing the
total energy output after accounting for redshifting by the total number of
dark photons that successfully escape the merger. In Fig. S1 we show the
approximate range of the ratio between the dark photon luminosity when
redshifting effects are considered and when such effects are neglected for a
$2.7M_{\odot}$ merger remnant, $r_{E}$. The upper bound of this ratio is for a
model that assumes the remnant has a constant density of $4\times 10^{14}$ g
cm-3 throughout, extended out to a radius such that the total mass is
$2.7M_{\odot}$. The lower bound is set by a model where the remnant has a
density of $4\times 10^{14}$ g cm-3 in the emission region (between 5 km and
10 km from the merger center), no density in the region beyond 10 km from the
core, and the remaining mass of the 2.7 $M_{\odot}$ remnant residing in the
inner 5 km of the remnant. Most models of merger remnants have similar
densities to those used here in the emission region, and become more dense
closer to the core Perego et al. (2019); Radice (2020); Bernuzzi et al.
(2020); Endrizzi et al. (2020), however the density in the lower curve is so
large that it would already be a black-hole which is not the case for the
simulations. The first model underestimates the central region’s density and
thus leads to a shallower gravitational potential. The second model, on the
other hand, significantly overestimates the density in the central region,
leading to a large escape velocity and thus a significant decrease in
luminosity due to gravitational redshifting. The figure shows how sensitive to
the density modeling this effect is, specially when considering dark photon
masses larger than the remnants temperature. The real effect, using a more
realistic density profile, will be somewhere in between the two limiting
cases, but would primarily affect the large mass end of the plot.

##  III Ejecta from Binary Neutron Star Merger

Figure S2: The above plot shows the parts of dark photon parameter space which
may be affected by fast moving ejecta. The red lines (solid for 10 ms, dashed
for 1 s) mark the minimum $\epsilon$ where fast moving ejecta begin to affect
the observed gamma ray signal. For reference the original region where dark
photons decay within 1000 km of the merger remnant and produce no distinct
signal is marked in dark gray and the minimum couplings needed for the
produced gamma-ray signal to be visible by Fermi GBM from 100 Mpc away are
shown in purple (solid for 10 ms and dashed for 1 s). Ejecta will only impact
signals from dark photons with the highest masses.

In the moments after a Binary Neutron Star Merger, a small fraction of the
remnant mass can be ejected with large velocities of around 0.1-0.3 Hajela et
al. (2019); Radice et al. (2018). Even though dark photons can pass through
this ejecta without interacting, it may affect the fireball dynamics and
photon signal if dark photons decay before they have passed through the
ejecta. The average dark photon produced in the merger moves faster than 0.3
for all of the relevant parameter space, and so some fraction of the dark
photons will always outrun the ejecta. Depending on how long the dark photon
emission lasts, dark photons emitted at late times may not be able to outrun
the ejecta by the time they decay, and our approximation of ignoring baryon
matter in the fireball would not be realistic.

In order to investigate the regions of parameter space that could be affected
by the presence of the fast ejecta, we found the parameters for which a dark
photon with average velocity would outrun the fast ejecta before decaying
considering the case when the dark photon is emitted 10 ms and 1 s after the
merger (which correspond to the last dark photons emitted in the two scenarios
considered in the paper). The results are shown in Fig. S2. Even for remnants
that can emit dark photons for a total period of $\Delta t$, we can break up
the emission in smaller time intervals $\delta t$, and because the average
dark photon is faster than the ejecta, there will always be some small
interval for which the average dark photon will outrun the ejecta and our
analysis ignoring baryons holds (due to the spread in velocity of the dark
photon, shells emitted at different times can reach one another, but for the
purposes of our analysis, it only matters whether most of the particles in
that shell outrun the ejecta or not). One can see from Fig. S2 that for the 10
ms emission case, the requirement that the decay occurs at least 1000 km away
from the remnant is almost enough to guarantee that most dark photons will
outrun the ejecta, since the small region in between the two curves
corresponds to the largest couplings we are sensitive to, considering a
smaller emission time to guarantee that all dark photons outrun the baryons
would lead to a significant change in sensitivity. For the 1 s emission
scenario, note that we only need to worry about the region of parameter space
which would be covered by a 1 s emission but not for a 10 ms emission, since
from our previous argument we could focus on the signal coming from the first
10 ms of emission of the more long lived remnant. This region corresponds to
weaker couplings, which lead to longer decay lengths and thus almost all of it
is below the line corresponding to outrunning the ejecta in Fig. S2, except
for a very small region at corresponding to masses larger than 100 MeV.

##  IV Constraining Power

There are a variety of properties of the predicted gamma-ray signal produced
by dark photons that can differ from the observed gamma-ray signals in binary
neutron star mergers. This can be used to place constraints on the dark photon
parameter space. In this section we explore two such properties, the timing of
the signal and its duration.

The gamma-ray burst associated with GW170817, GRB 170817A, is the only GRB
observed in association with a BNS merger, and we will use it as a study case
to show how one can use specific observations to constrain dark photon
parameter space. As discussed in the main text, dark photons can be produced
in the BNM remnant during the time that it is hot, and has not yet collapsed
into a black hole. For the remnant associated with GRB 170817A, this is
believed to be at least 10ms Margalit and Metzger (2017), and therefore we
will use a 10ms emission for our analysis. The dark photons would stream out
from remnant at an average velocity

$v=\sqrt{1-\left(\frac{m^{\prime}}{\langle\omega\rangle}\right)^{2}}\,,$ (S13)

where $\langle\omega\rangle$ is their average energy. The much lighter
electrons and positrons produced by their decay, would stream at near the
speed of light. The time delay between the gravitational wave signal and the
gamma-ray signal, is then the difference between the times it takes the
gravitational waves, travelling at the speed of light, and a dark photon,
travelling with velocity given by Eq. S13, to traverse the dark photon decay
length,

$\delta t=\left(\frac{1-v}{v}\right)d\,.$ (S14)

The gamma-ray burst associated with GW170817 appeared about 1.7 s after the
merger Abbott et al. (2017d). Additional studies of the burst limited
precursor gamma-ray emission to $11\times 10^{-7}$ ergs s-1 cm-2 for bursts
lasting up to 1s and $3.7\times 10^{-6}$ ergs s-1 cm-2 for bursts lasting up
to 0.1s in the 200 s leading up to GRB 170817A Goldstein et al. (2017). Using
this, we show an example of how one could rule out dark photons which produce
a gamma-ray signal detectable to Fermi GBM but which arrives earlier than the
observed signal did in Figure S3.

A complimentary way to constrain dark photon parameter space is to consider
the duration of the observed signal. The gamma-ray signal from dark photons
should persist for the light crossing time of the fireball. The thickness of
the fireball shell, $\delta$, is defined in Eq.(8) of the main text. Follow-up
analysis of GRB 170817A limited the extended emission succeeding the burst
averaged over a 10s interval to $6.6\times 10^{-8}$ ergs s-1 cm-2 for the 100s
after the burst Goldstein et al. (2017). We present the regions of dark photon
parameter space which would produce excess emission beyond the end of GRB
170817A in Figure S3.

Figure S3: Example parameter space that could be excluded by observed emission
from GRB 170817A assuming 10ms emission and the simplified remnant model
discussed in the main text. The gray region is ruled out by dark photons
producing excess emission spread out over the time after GRB 170817A. The
green region is ruled out by excess emission which would reach earth between
the gravitational wave signal of GW170817 and the gamma-ray signal GRB
170817A.