with extension cocycle. Together with Theorem 6.1.3. Theorem, we therefore
have the following corollary.

######  Corollary (Isomorphism classes of strongly polarized Lie superalgebras
vs isomorphism classes of flat Lie superalgebras with a cocycle).

The correspondence which associates a strongly polarized Lie superalgebra
$({\mathfrak{g}},\omega,{\mathfrak{a}},N)$, where $\omega$ is orthosymplectic
(resp. periplectic), to its extension triple $({\mathfrak{h}},\nabla,\alpha)$
(resp. $({\mathfrak{h}},\nabla,\beta)$) induces a bijection between
isomorphism classes of strongly polarized Lie superalgebras and isomorphism
classes of ‘even’ (resp. ‘odd’) flat Lie superalgebras with orthosymplectic
(resp. periplectic) extension cocycle.

###  6.3. Change of strong polarization

Let $({\mathfrak{g}},\omega,{\mathfrak{a}},N)$ and
$({\mathfrak{g}},\omega,{\mathfrak{a}},N^{\prime})$ be two strong polariza-
tions belonging to the same Lagrangian extension, where the form $\omega$ is
either orthosymplectic or periplectic. We have showed in Theorem 6.1.4.
Theorem that each of them can be obtained as a $T^{*}$-extension or a $\Pi
T^{*}$-extension by means of a 2-cocycle. We will show that these 2-cocycles
differ by a coboundary. The following Lemma is a superization of a result in
[BC].

######  Lemma ($\alpha$ and $\alpha^{\prime}$ (resp. $\beta$ and
$\beta^{\prime}$) are cohomologous).

(i) If the form $\omega$ is orthosymplectic, then there exists
$\sigma\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{h}},{\mathfrak{h}}^{*})$,
satisfying

(35) $\sigma(u)(v)-(-1)^{p(u)p(v)}\sigma(v)(u)=0,\quad\text{for all
$u,v\in{\mathfrak{h}}$,}$

such that $\alpha^{\prime}=\alpha+\partial_{\rho}\sigma$.

(ii) If the form $\omega$ is periplectic, the there exists
$\mu\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$,
satisfying

(36) $\mu(u)(v)-(-1)^{p(u)p(v)}\mu(v)(u)=0,\quad\text{for all
$u,v\in{\mathfrak{h}}$,}$

such that $\beta^{\prime}=\beta+\partial_{\chi}\mu$.

###### Proof.

We will denote the form by $\omega$ (resp. $\kappa$) if it is orthosymplectic
(resp. periplectic) on ${\mathfrak{g}}$.

As ${\mathfrak{g}}={\mathfrak{a}}\oplus N={\mathfrak{a}}^{\prime}\oplus
N^{\prime}$, let $\pi_{\mathfrak{a}},\pi_{N}$, as well as
$\pi_{\mathfrak{a}},\pi_{N^{\prime}}^{\prime}$ be the corresponding projection
on the summand of ${\mathfrak{g}}$. Let us put
$\tau:=\pi_{\mathfrak{a}}-\pi^{\prime}_{\mathfrak{a}}\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{g}},{\mathfrak{a}})$.
Clearly, $\tau$ is an even map with ${\mathfrak{a}}\subseteq\text{Ker}(\tau)$.
Moreover, since any element $x\in{\mathfrak{g}}$ can be uniquely expressed as
$x=\pi_{a}(x)+\pi_{N}(x)=\pi_{a}^{\prime}(x)+\pi_{N^{\prime}}^{\prime}(x)$, it
follows that $\pi^{\prime}_{N^{\prime}}=\pi_{N}+\tau$.

Since, all $N,N^{\prime}$, and ${\mathfrak{a}}$ are Lagrangian, the
homomorphism $\tau$ satisfies the condition (for all $n,m\in N$)

(37) $\displaystyle\omega(\tau(n),m)+\omega(n,\tau(m))$ $\displaystyle=$
$\displaystyle 0\quad\text{(if the form $\omega$ is orthosymplectic)}$ (38)
$\displaystyle\kappa(\tau(n),m)+\kappa(n,\tau(m))$ $\displaystyle=$
$\displaystyle 0\quad\text{(if the form $\kappa$ is periplectic).}$

Let $u,v\in{\mathfrak{h}}$ and $\tilde{u},\tilde{v}$ their respective lifts to
${\mathfrak{g}}$. We first compute the bracket, and using the fact that
${\mathfrak{a}}$ is abelian, we get

$\displaystyle[\pi^{\prime}_{N^{\prime}}(\pi_{N}(\tilde{u})),\pi^{\prime}_{N^{\prime}}(\pi_{N}(\tilde{v}))]=[\pi_{N}(\tilde{u})+\tau(\pi_{N}(\tilde{u})),\pi_{N}(\tilde{v})+\tau(\pi_{N}(\tilde{v}))]$
$\displaystyle=[\pi_{N}(\tilde{u}),\pi_{N}(\tilde{v})]+[\pi_{N}(\tilde{u}),\tau(\pi_{N}(\tilde{v}))]+[\tau(\pi_{N}(\tilde{u})),\pi_{N}(\tilde{v})].$

Let us first prove Part (i). By Eq. (33), we thus have

$\displaystyle\alpha_{({\mathfrak{g}},\omega,{\mathfrak{a}},N^{\prime})}(u,v)=\omega(\pi_{\mathfrak{a}}^{\prime}([\pi^{\prime}_{N^{\prime}}(\tilde{u}),\pi^{\prime}_{N^{\prime}}(\tilde{v})],\cdot)=\omega(\pi_{\mathfrak{a}}([\pi^{\prime}_{N^{\prime}}(\tilde{u}),\pi^{\prime}_{N^{\prime}}(\tilde{v})])-\tau([\pi^{\prime}_{N^{\prime}}(\tilde{u}),\pi^{\prime}_{N^{\prime}}(\tilde{v})]),\cdot)$
$\displaystyle=\omega(\pi_{\mathfrak{a}}([\pi_{N}(\tilde{u}),\pi_{N}(\tilde{v})]+[\pi_{N}(\tilde{u}),\tau(\pi_{N}(\tilde{v}))]+[\tau(\pi_{N}(\tilde{u})),\pi_{N}(\tilde{v})])-\tau([\pi_{N}(\tilde{u}),\pi_{N}(\tilde{v})]),\cdot)$
$\displaystyle=\alpha_{({\mathfrak{g}},\omega,{\mathfrak{a}},N)}(u,v)+\omega([\pi_{N}(\tilde{u}),\tau(\pi_{N}(\tilde{v}))]+[\tau(\pi_{N}(\tilde{u})),\pi_{N}(\tilde{v})])-\tau([\pi_{N}(\tilde{u}),\pi_{N}(\tilde{v})]),\cdot).$

Since ${\mathfrak{a}}$ is contained in $\text{Ker}(\tau)$, then $\tau$ defines
an element
$\bar{\tau}\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{h}},{\mathfrak{a}})$
given by $\bar{\tau}(u):=\tau(\tilde{u})$ for all $u\in{\mathfrak{h}}$, and
therefore
$\sigma=i_{\omega}\circ\bar{\tau}\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{h}},{\mathfrak{h}}^{*})$.
Using Eq. (33), we deduce from the above that
$\alpha^{\prime}=\alpha+\partial_{\rho}\sigma$. Now, using Eq.(37), we get

$\begin{array}[]{lcl}\sigma(u)(v)-(-1)^{p(u)p(v)}\sigma(v)(u)&=&i_{\omega}\circ\bar{\tau}(u)(v)-(-1)^{p(u)p(v)}i_{\omega}\circ\bar{\tau}(v)(u)\\\\[5.69054pt]
&=&\omega(\bar{\tau}(u),\tilde{v})-(-1)^{p(u)p(v)}\omega(\bar{\tau}(v),\tilde{u})\\\\[5.69054pt]
&=&\omega(\tau(\tilde{u}),\tilde{v})+\omega(\tilde{u},\tau(\tilde{v}))=0.\end{array}$

For Part (ii) the proof is similar. In this case, choose $\mu=\Pi\circ
i_{\kappa}\circ\bar{\tau}\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$.
∎

###  6.4. Equivalence classes of $T^{*}$-extensions and $\Pi T^{*}$-extensions

Let ${\mathfrak{h}}$ be an arbitrary Lie superalgebra. An orthosymplectic
(resp. periplectic) $T^{*}$-extension  (resp. $\Pi T^{*}$-extension)
$({\mathfrak{g}},\omega,{\mathfrak{a}})$ of ${\mathfrak{h}}$ is an
orthosymplectic (resp. a periplectic) quasi-Frobenius Lie superalgebra
$({\mathfrak{g}},\omega)$, together with an extension of Lie superalgebras

$0\rightarrow{\mathfrak{a}}\rightarrow{\mathfrak{g}}\rightarrow{\mathfrak{h}}\rightarrow
0,$

such that the image of ${\mathfrak{a}}$ in ${\mathfrak{g}}$ is a Lagrangian
ideal of $({\mathfrak{g}},\omega)$.

An isomorphism of $T^{*}$-extension or $\Pi T^{*}$-extension of the Lie
superalgebra ${\mathfrak{h}}$ is an isomorphism of Lie superalgebras
$\Phi:({\mathfrak{g}},\omega)\rightarrow({\mathfrak{g}}^{\prime},\omega^{\prime})$
both equipped with orthosymplectic forms (or periplectic forms) such that the
diagram below is commutative:

(39)
${0}$${{\mathfrak{a}}}$${{\mathfrak{g}}}$${{\mathfrak{h}}}$${0}$${0}$${{\mathfrak{a}}^{\prime}}$${{\mathfrak{g}}^{\prime}}$${{\mathfrak{h}}}$${0}$$\scriptstyle{\Phi|_{\mathfrak{a}}}$$\scriptstyle{\Phi}$$\scriptstyle{=}$

######  Proposition (Isomorphic extensions give the same quotient Lie
superalgebra).

Isomorphic orthosymplectic $T^{*}$-extension or periplectic $\Pi
T^{*}$-extensions of ${\mathfrak{h}}$ give rise to the same associated
quotient flat Lie superalgebra $({\mathfrak{h}},\nabla)$.

###### Proof.

An immediate consequence of Lemma 6.2.1. Lemma. ∎

We now construct for any flat Lie superalgebra $({\mathfrak{h}},\nabla)$ a
cohomology, which describes all ortho-symplectic $T^{*}$-extensions or
periplectic $\Pi T^{*}$-extensions of ${\mathfrak{h}}$ with associated flat
Lie superalgebra $({\mathfrak{h}},\nabla)$. First, we define Lagrangian $1$\-
and $2$-cochains on ${\mathfrak{h}}$ as

$\begin{array}[]{lcl}C^{1}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})&:=&\\{\phi\in
C^{1}({\mathfrak{h}},{\mathfrak{h}}^{*})\mid\phi(u)(v)-(-1)^{p(u)p(v)}\phi(v)(u)=0\quad\text{for
all }u,v\in{\mathfrak{h}}\\},\\\\[5.69054pt]
C^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})&:=&\\{\alpha\in
C^{2}({\mathfrak{h}},{\mathfrak{h}}^{*})\mid\alpha\text{ satisfies
(\ref{eq5.4})}\\}.\end{array}$
$\begin{array}[]{lcl}C^{1}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))&:=&\\{\phi\in
C^{1}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))\mid\phi(u)(v)-(-1)^{p(u)p(v)}\phi(v)(u)=0\quad\text{for
all }u,v\in{\mathfrak{h}}\\},\\\\[5.69054pt]
C^{2}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))&:=&\\{\beta\in
C^{2}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))\mid\alpha\text{ satisfies
(\ref{eq5.4b})}\\}.\end{array}$

Furthermore, let $\rho=\rho^{\nabla}$ (resp. $\chi=\chi^{\nabla}$) be the
representation of ${\mathfrak{h}}$ on ${\mathfrak{h}}^{*}$ (resp.
$\Pi({\mathfrak{h}}^{*})$) associated to $\nabla$, as defined in (22) (resp.
(23)). Denote by $\partial_{\rho}=\partial^{i}_{\rho}$ (resp.
$\partial_{\chi}=\partial^{i}_{\chi}$) the corresponding coboundary operators
for cohomology with $\rho$-coefficients (resp. $\chi$-coefficients).

######  Lemma (The coboundary image of Lagrangian cochains).

(i) The coboundary operator
$\partial_{\rho}:C^{1}({\mathfrak{h}},{\mathfrak{h}}^{*})\rightarrow
C^{2}({\mathfrak{h}},{\mathfrak{h}}^{*})$ maps the subspace
$C^{1}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})$ to
$C^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})\cap
Z^{2}({\mathfrak{h}},{\mathfrak{h}}^{*})$.

(ii) The coboundary operator
$\partial_{\chi}:C^{1}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))\rightarrow
C^{2}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$ maps the subspace
$C^{1}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$ to
$C^{2}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))\cap
Z^{2}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$.

###### Proof.

Let us prove Part (i). Let $\phi\in C^{1}({\mathfrak{h}},{\mathfrak{h}}^{*})$,
and let $u,v,w\in{\mathfrak{h}}$. We have

$\begin{array}[]{lcl}(\partial_{\rho}\phi)(u,v)&=&(-1)^{p(u)p(\phi)}\rho(u)\cdot\phi(v)-(-1)^{p(v)(p(u)+p(\phi))}\rho(v)\cdot\phi(u)-\phi([u,v])\\\\[5.69054pt]
&=&-(-1)^{p(u)p(v)}\phi(v)\circ\nabla_{u}+\phi(u)\circ\nabla_{v}-\phi([u,v]).\end{array}$

Let us check that condition (26) is satisfied. Indeed, for
$u,v,w\in{\mathfrak{h}}$ we have

$\displaystyle(-1)^{p(u)p(w)}(\partial_{\rho}\phi)(u,v)(w)+\circlearrowleft(u,v,w)$
$\displaystyle=(-1)^{p(u)p(w)}(-(-1)^{p(u)p(v)}\phi(v)\circ\nabla_{u}(w)+\phi(u)\circ\nabla_{v}(w)-\phi([u,v])(w))+\circlearrowleft(u,v,w).$

Now, using the fact that the connection is flat, the equation above becomes

$\displaystyle(-1)^{p(u)p(w)}(\partial_{\rho}\phi)(u,v)(w)+\circlearrowleft(u,v,w)$
$\displaystyle=(-1)^{p(u)p(v)}(\phi(v)([w,u])-(-1)^{p(v)(p(w)+p(u))}\phi([w,u])(v))+\circlearrowleft(u,v,w)=0$

This follows that $\partial_{\rho}\phi\in
C^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})$. ∎

Let
$Z^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*}):=C^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})\cap
Z^{2}({\mathfrak{h}},{\mathfrak{h}}^{*})$ and
$Z^{2}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*})):=C^{2}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))\cap
Z^{2}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$ denote the space of Lagrangian
cocycles. We define the Lagrangian extension cohomology of the flat Lie
algebra $({\mathfrak{h}},\nabla)$ by

$\begin{array}[]{rcl}H^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})&:=&\displaystyle\frac{Z^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})}{\partial_{\rho}C^{1}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})},\\\\[11.38109pt]
H^{2}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))&:=&\displaystyle\frac{Z^{2}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))}{\partial_{\chi}C^{1}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))}.\end{array}$

######  Theorem (A link between extensions and cohomology classes).

Every orthosymplectic $T^{*}$-extension or periplectic $\Pi T^{*}$-extensions
$({\mathfrak{g}},\omega,{\mathfrak{a}})$ of the flat Lie algebra
$({\mathfrak{h}},\nabla)$ gives rise to a characteristic extension class

$\begin{array}[]{ll}[\alpha]\in
H^{2}_{{\bar{0}},L}({\mathfrak{h}},{\mathfrak{h}}^{*})&\text{{(}if $\omega$ is
orthosymplectic{)}},\\\\[5.69054pt] [\beta]\in
H^{2}_{{\bar{0}},L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))&\text{{(}if
$\omega$ is periplectic{)}}.\end{array}$

Two extensions of $({\mathfrak{h}},\nabla)$ are isomorphic if and only if they
have the same extension class in either
$H^{2}_{L}({\mathfrak{h}},{\mathfrak{h}}^{*})$ or
$H^{2}_{L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$.

###### Proof.

Let $({\mathfrak{g}},\omega,{\mathfrak{a}})$ be a Lagrangian extension of
$({\mathfrak{h}},\nabla)$, where $\omega$ is orthosymplectic (resp.
periplectic). Choose a strong polarization
$({\mathfrak{g}},\omega,{\mathfrak{a}},N)$. By virtue of Theorem 6.1.3.
Theorem, $({\mathfrak{g}},\omega,{\mathfrak{a}},N)$ is a Lagrangian extension
by means of $\alpha\in Z^{2}_{{\bar{0}},L}({\mathfrak{h}},{\mathfrak{h}}^{*})$
(resp. $\beta\in Z^{2}_{{\bar{0}},L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$)
if $p(\omega)={\bar{0}}$ (resp. $p(\omega)={\bar{1}}$). Its cohomology class
$[\alpha]$ (resp. $[\beta]$) is independent of $N$ by Lemma 6.3.1. Lemma.

Now suppose
$\Phi:({\mathfrak{g}},\omega,{\mathfrak{a}})\rightarrow({\mathfrak{g}}^{\prime},\omega^{\prime},{\mathfrak{a}}^{\prime})$
is an isomorphism of ${\mathfrak{h}}$ (see, diagram (39)). Choose a
complementary Lagrangian subspace $N$ for
$({\mathfrak{g}},\omega,{\mathfrak{a}})$. Then,
$\Phi:({\mathfrak{g}},\omega,{\mathfrak{a}},N)\rightarrow({\mathfrak{g}}^{\prime},\omega^{\prime},{\mathfrak{a}}^{\prime},N^{\prime}=\Phi(N))$
is an isomorphism of strongly polarized orthosymplectic (resp. periplectic)
Lie superalgebras. By Lemma 6.2.2. Lemma, we have
$\alpha_{{\mathfrak{g}}^{\prime},\omega^{\prime},{\mathfrak{a}}^{\prime},N^{\prime}}=(\Phi_{\mathfrak{h}})_{*}\alpha_{{\mathfrak{g}},\omega,{\mathfrak{a}},N}$
(resp.
$\beta_{{\mathfrak{g}}^{\prime},\omega^{\prime},{\mathfrak{a}}^{\prime},N^{\prime}}=(\Phi_{\mathfrak{h}})_{*}\beta_{{\mathfrak{g}},\omega,{\mathfrak{a}},N}$),
since by assumption
$\Phi|_{\mathfrak{h}}=\mathop{\mathrm{id}}\nolimits_{\mathfrak{h}}$. This
shows that isomorphic extensions over ${\mathfrak{h}}$ have the same
cohomology class.

It remains to show that two extensions of ${\mathfrak{h}}$ with the same class
are isomorphic. By Theorem 6.1.3. Theorem, it is enough to show that any two
strongly polarized orthosymplectic (resp. periplectic) Lie superalgebras give
rise to isomorphic extensions of ${\mathfrak{h}}$ if the corresponding classes
$\alpha$ and $\alpha^{\prime}$ if $p(\omega)={\bar{0}}$ (resp. $\beta$ and
$\beta^{\prime}$ if $p(\omega)={\bar{1}}$) satisfy
$[\alpha]=[\alpha^{\prime}]\in
H^{2}_{{\bar{0}},L}({\mathfrak{h}},{\mathfrak{h}}^{*})$ (resp.
$[\beta]=[\beta^{\prime}]\in
H^{2}_{{\bar{0}},L}({\mathfrak{h}},\Pi({\mathfrak{h}}^{*}))$). That is
$\alpha^{\prime}=\alpha-\partial_{\rho}\sigma$ (resp.
$\beta^{\prime}=\beta-\partial_{\chi}\mu$). The map
$(u,\xi)\mapsto(u,\xi+\sigma(u))$ (resp. $(u,\xi)\mapsto(u,\xi+\mu(u))$) is an
ismorphism. ∎

######  Remark (A link between ordinary cohomology and Lagrangian cohomology).

The kernel of the map $H^{2}_{L}({\mathfrak{g}},{\mathfrak{g}}^{*})\rightarrow
H^{2}_{\rho}({\mathfrak{g}},{\mathfrak{g}}^{*})$ (resp.
$H^{2}_{L}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))\rightarrow
H^{2}_{\chi}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))$) is given by

$\kappa_{L}=\frac{B^{2}_{\rho}({\mathfrak{g}},{\mathfrak{g}}^{*})\cap
Z^{2}_{L}({\mathfrak{g}},{\mathfrak{g}}^{*})}{B^{2}_{L}({\mathfrak{g}},{\mathfrak{g}}^{*})},\left(\text{resp.
}\kappa_{L}=\frac{B^{2}_{\chi}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))\cap
Z^{2}_{L}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))}{B^{2}_{L}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))}\right),$

where

$B^{2}_{\rho}({\mathfrak{g}},{\mathfrak{g}}^{*})=\\{\partial^{1}_{\rho}\lambda\mid\lambda\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{g}},{\mathfrak{g}}^{*})\\}\text{
(resp.
}B^{2}_{\rho}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))=\\{\partial^{1}_{\rho}\lambda\mid\lambda\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))\\})$

is the set of ordinary two-coboundaries with $\rho$-coefficients (resp.
$\chi$-coefficient) and

$B^{2}_{L}({\mathfrak{g}},{\mathfrak{g}}^{*})=\\{\partial^{1}_{\rho}\lambda\mid\lambda\in
C^{1}_{L}({\mathfrak{g}},{\mathfrak{g}}^{*})\\}\text{ (resp.
}B^{2}_{L}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))=\\{\partial^{1}_{\rho}\lambda\mid\lambda\in
C^{1}_{L}({\mathfrak{g}},\Pi({\mathfrak{g}}^{*}))\\})$

is the set of two-coboundaries for Lagrangian extension cohomology.

##  7\. Examples of double, $\Pi T^{*}$\- and $T^{*}$-extensions

######  Proposition (Examples of ${\mathscr{D}}_{\bar{1}}$-extensions).

(i) The Lie superalgebra $C^{1}_{1/2}+A$ – see Table 2 – cannot be obtained as
a double extension even though it has a center.

(ii) The Lie superalgebra $(2A_{1,1}+2A)^{3}_{p}$ (for $p=1/2$) – see Table 2
– can be obtained as a double ${\mathscr{D}}_{\bar{1}}$-extension of the
Abelian Lie superalgebra ${\mathfrak{a}}:=\mathbb{R}^{1|1}$ spanned by
$\\{e_{2},e_{3}\\}$ with (for the notations, see Theorem 4.2.1. Theorem):

$\omega_{\mathfrak{a}}=e_{2}\wedge e_{3},\quad x=e_{1},\quad e=e_{4},\quad
a_{0}=\frac{1}{2}e_{2},\quad{\mathscr{D}}(e_{3})=\frac{1}{2}e_{2},\quad{\mathscr{D}}(e_{2})=0,\quad{\mathscr{D}}^{*}\equiv{\mathscr{D}}.$

###### Proof.

Let us prove Part (i). A direct computation shows that the center of
${\mathfrak{g}}:=C^{1}_{1/2}+A$ is
$\mathop{\mathrm{{Span}}}\nolimits\\{e_{4}\\}$. Observe that
$\omega_{\mathfrak{g}}(e_{4},e_{4})\not=0$. Now, the dual to the center
$x=e_{4}$ with respect to the form $\omega_{\mathfrak{g}}$ is
$x^{*}=e_{3}+e_{4}$. If the Lie superalgebra is a double extension of a Lie
sub-superalgebra ${\mathfrak{a}}$, then ${\mathfrak{a}}$ would be spanned by
$e_{1}$ and $e_{2}$. Now the condition $[x^{*},x^{*}]_{\mathfrak{g}}=2a_{0}$
implies that $a_{0}=\frac{1}{2}e_{2}$. On the other hand, the equation
$[x^{*},a]_{\mathfrak{g}}={\mathscr{D}}(a)+\omega_{\mathfrak{a}}(a,a_{0})x$
implies that
${\mathscr{D}}(e_{1})=-\frac{1}{2}(e_{3}+e_{4})\not\in{\mathfrak{a}}$ which is
a contradiction.

Let us prove Part (ii). For short, let us write
${\mathfrak{g}}:=(2A_{1,1}+2A)^{3}_{p}$, where $p=1/2$. A direct computation
shows that ${\mathscr{D}}^{2}=\mathop{\mathrm{ad}}\nolimits_{a_{0}}=0$. The
map ${\mathscr{D}}$ is obviously a derivation since the Lie superalgebra
${\mathfrak{a}}$ is abelian. Besides, the two conditions
${\mathscr{D}}(a_{0})={\mathscr{D}}^{*}(a_{0})=0$ are obviously satisfied, by
construction of ${\mathscr{D}}$ and ${\mathscr{D}}^{*}$. Now, the map

$(a,b)\mapsto\omega_{\mathfrak{a}}({\mathscr{D}}\circ{\mathscr{D}}(a)+(1+(-1)^{p(a)}){\mathscr{D}}^{*}\circ{\mathscr{D}}(a)-{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a),b)$

is not only a coboundary but actually identically zero. Moreover,
$\omega_{\mathfrak{a}}(a_{0},[a,b]_{\mathfrak{a}})=0$ for all
$a,b\in{\mathfrak{a}}$ since ${\mathfrak{a}}$ is abelian. It remains to check
the Lie brackets. We have

$[e_{3},e_{3}]_{\mathfrak{g}}=[e_{3},e_{3}]_{\mathfrak{a}}+(\omega_{\mathfrak{a}}(\mathscr{D}(e_{3}),e_{3})-\omega_{\mathfrak{a}}(e_{3},\mathscr{D}(e_{3})))x=\left(\frac{1}{2}\omega_{\mathfrak{a}}(e_{2},e_{3})-\frac{1}{2}\omega_{\mathfrak{a}}(e_{3},e_{2})\right)e_{1}=e_{1},$
$[e_{3},e_{4}]_{\mathfrak{g}}={\mathscr{D}}(e_{3})+\omega_{\mathfrak{a}}(a_{0},e_{3})x=\frac{1}{2}e_{2}+\frac{1}{2}\omega_{\mathfrak{a}}(e_{2},e_{3})e_{1}=\frac{1}{2}(e_{1}+e_{2}).\qed$

######  Proposition (Examples of $T^{*}$\- and $\Pi T^{*}$-extensions).

(i) The Lie superalgebra $D^{5}$ – see Table 1 – is a $\Pi T^{*}$-extension of
the Lie superalgebra ${\mathfrak{h}}$ spanned by $e_{1}\mid e_{4}$ with the
bracket $[e_{1},e_{4}]=e_{4}$, and where $\Pi({\mathfrak{h}}^{*})$ is spanned
by $e_{2}=\Pi(e_{4}^{*})\mid e_{3}=\Pi(e_{1}^{*})$. The $2$-form $\beta\equiv
0$ and the connection $\nabla$ on ${\mathfrak{h}}$ is given as follows:

$\nabla_{e_{4}}(e_{1})=e_{4},\quad\nabla_{e_{1}}(e_{1})=e_{1},\quad\nabla_{e_{1}}(e_{4})=2e_{4},\quad\nabla_{e_{4}}(e_{4})=0.$

(ii) The Lie superalgebra $C_{1}^{1}+A$ – see Table 2 for its structure – is a
$T^{*}$-extension of the abelian Lie superalgebra ${\mathfrak{h}}$ spanned by
$e_{1}\mid X:=e_{3}-2e_{4}$ where the $2$-form $\alpha$ is given by

$\alpha=e_{1}^{*}\otimes X\wedge X+X^{*}\otimes e_{1}\wedge X.$

The dual space ${\mathfrak{h}}^{*}$ is spanned by
$e_{2}=\frac{1}{2}e_{1}^{*}\mid e_{3}=X^{*}$ and the connection $\nabla$ on
${\mathfrak{h}}$ is given by

$\nabla_{e_{1}}(e_{1})=-e_{1},\quad\nabla_{e_{1}}(X)=-X,\quad\nabla_{X}(e_{1})=-X,\quad\nabla_{X}(X)=0.$

(iii) The Lie superalgebra $D^{7}_{-1,q}$ ($q\leq-1$) – see Table 1 – can be
obtained as a $\Pi T^{*}$-extension of the Lie superalgebra ${\mathfrak{h}}$
spanned by $e_{1},e_{2}$ with the bracket $[e_{1},e_{2}]=e_{2}$, and where the
dual space $\Pi({\mathfrak{h}}^{*})$ is spanned by
$e_{3}=\Pi(e_{1}^{*})+\Pi(e_{2}^{*}),e_{4}=\Pi(e_{1}^{*})$. The $2$-form
$\beta\equiv 0$ and the connection $\nabla$ on ${\mathfrak{h}}$ is given by

$\nabla_{e_{1}}(e_{2})=e_{2},\quad\nabla_{e_{1}}(e_{1})=-qe_{1}+(q+1)e_{2},\quad\nabla_{e_{2}}(e_{1})=\nabla_{e_{2}}(e_{2})=0.$

###### Proof.

Let us just prove Part (ii) and skip Part (i) and (iii) as the computation is
similar. For short, we put
${\mathfrak{k}}={\mathfrak{h}}\oplus\Pi({\mathfrak{h}}^{*})$. Let us first
show that the connection is torsion-free. Indeed,

$\nabla_{e_{1}}(X)-\nabla_{X}(e_{1})-[e_{1},X]=-X-(-X)-0=0.$

Let us check the flatness. Indeed,

$\nabla_{e_{1}}\circ\nabla_{X}-\nabla_{X}\circ\nabla_{e_{1}}-\nabla_{[e_{1},X]}=\nabla_{e_{1}}\circ\nabla_{X}-\nabla_{X}\circ\nabla_{e_{1}}.$

Evaluating this expression at $X$ gives zero. Let us evaluate it at $e_{1}$,
we get

$\nabla_{e_{1}}\circ\nabla_{X}(e_{1})-\nabla_{X}\circ\nabla_{e_{1}}(e_{1})=\nabla_{e_{1}}(-X)-\nabla_{X}(-e_{1})=X-X=0.$

Let us check the Lie bracket. We have

$[e_{4},e_{3}]_{\mathfrak{k}}=[\frac{1}{2}(e_{3}-X),e_{3}]_{\mathfrak{k}}=-\frac{1}{2}[X,X^{*}]_{\mathfrak{k}}=-\frac{1}{2}X^{*}\circ\nabla_{X}=\frac{1}{2}e_{1}^{*}=e_{2},$
$[e_{1},e_{3}]_{\mathfrak{k}}=[e_{1},X^{*}]_{\mathfrak{k}}=-X^{*}\circ\nabla_{e_{1}}=X^{*}=e_{3},\quad[e_{1},e_{2}]_{\mathfrak{k}}=[e_{1},\frac{1}{2}e_{1}^{*}]_{\mathfrak{k}}=-\frac{1}{2}e_{1}^{*}\circ\nabla_{e_{1}}=\frac{1}{2}e_{1}^{*}=e_{2},$
$\displaystyle[e_{4},e_{4}]_{\mathfrak{k}}=[\frac{1}{2}(e_{3}-X),\frac{1}{2}(e_{3}-X)]_{\mathfrak{k}}=\frac{1}{4}\left([X^{*}-X,X^{*}-X]_{\mathfrak{k}}\right)=\frac{1}{4}(-2[X,X^{*}]_{\mathfrak{k}}+[X,X]_{\mathfrak{k}})$
$\displaystyle=\frac{1}{4}(-2X^{*}\circ\nabla_{X}-2e_{1}^{*})=\frac{1}{4}(2e_{1}^{*}-2e_{1}^{*})=0.\qed$

######  Proposition (Filiforms as double and $\Pi T^{*}$-extensions).

(i) The filiform Lie superalgebra $L^{n,n}$ can be obtained as a $\Pi
T^{*}$-extension of the filiform Lie algebra $L^{n}$ where the connection
$\nabla$ on $L^{n}$ is given by

$\nabla_{X_{1}}(X_{i})=X_{i+1}\quad\text{ for $1\leq i\leq n-1$ \quad and
\quad$\nabla_{X_{j}}(X_{i})=0$ \quad for all $j>1$}.$

The odd part $\Pi({(L^{n}})^{*})$ is given by

$Y_{i}=(-1)^{i}\sum_{j=1}^{n-i+1}\Pi X_{j}^{*}\quad\text{ for
$i=1,\ldots,n$.}$

(ii) For $n$ even and $m$ odd, the filiform Lie superalgebra $L^{n,m}$ can be
obtained as a ${\mathscr{D}}_{\bar{0}}$-extension of the abelian Lie
superalgebra ${\mathbb{R}}^{n-2,1}$, followed by successive
${\mathscr{D}}_{\bar{1}}$-extensions.

###### Proof.

For Part (i), let us first check that the connection is torsion-free. Indeed,

$\nabla_{X_{1}}X_{i}-\nabla_{X_{i}}X_{1}-[X_{1},X_{i}]=X_{i+1}-X_{i+1}=0.$

Now, the connection is flat since $\nabla_{X_{i}}\circ\nabla_{X_{j}}=0$ and
$\nabla_{[X_{i},X_{j}]}(X_{k})=0$ for all $i,j,k=1,\ldots,n$. Let us now check
the Lie brackets. Indeed, for all $i=1,\ldots,n-1$, we have

$[X_{1},Y_{i}]=[X_{1},(-1)^{i}\sum_{j=1}^{n-i+1}\Pi
X_{j}^{*}]=-(-1)^{i}\sum_{j=1}^{n-i+1}\Pi
X_{j}^{*}\circ\nabla_{X_{1}}=(-1)^{i+1}\sum_{j=1}^{n-i}\Pi X_{j}^{*}=Y_{i+1}.$

Also,

$[X_{1},Y_{n}]=[X_{1},(-1)^{n}\Pi(X_{1}^{*})]=-(-1)^{n}\Pi(X_{1}^{*})\circ\nabla_{X_{1}}=0.$

Let us prove Part (ii). First, let us observe that $L^{n,m}$ is a
${\mathscr{D}}_{\bar{1}}$-extension of $L^{n,m-2}$ as follows (for the
notations, see Theorem 4.2.1. Theorem):

$a_{0}=0,\quad{\mathscr{D}}(X_{1})=-(-1)^{\frac{m+1}{2}}Y_{2},\quad{\mathscr{D}}(a)=0\text{
for all $a\not=X_{1}$},\quad x^{*}=(-1)^{\frac{m+1}{2}}Y_{1},\quad x=Y_{m}.$

Now, all the requirements are satisfied: ${\mathscr{D}}^{2}=0$ and the map
$(a,b)\mapsto\omega({\mathscr{D}}\circ{\mathscr{D}}(a)-{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a),b)$
is not only a coboundary but is identically zero.

By induction, we deduce that $L^{n,m}$ is a successive
${\mathscr{D}}_{\bar{1}}$-extensions of the filiform Lie superalgebra
$L^{n,1}$.

Let us now show that $L^{n,1}$ is a ${\mathscr{D}}_{\bar{0}}$-extension of the
Abelian Lie superalgebra ${\mathbb{R}}^{n-2|1}$. Denote by
$X_{1},\ldots,X_{n}|Y_{1}$ the generators of $L^{n,1}$. Let us choose (for the
notations, see Theorem 4.1.1. Theorem):

$\lambda=Z_{\Omega}=0,\quad x=X_{n},\quad
x^{*}=X_{1},\quad{\mathscr{D}}(X_{i})=X_{i+1}\text{ for
$i=2,\ldots,n-2$},\quad{\mathscr{D}}(Y_{1})={\mathscr{D}}(X_{n-1})=0.$

Now, the map

$(a,b)\mapsto\omega_{\mathfrak{a}}({\mathscr{D}}\circ{\mathscr{D}}(a)+2{\mathscr{D}}^{*}\circ{\mathscr{D}}(a)-{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a)+\lambda({\mathscr{D}}+{\mathscr{D}^{*}})(a),b)$

is not only a coboundary, but it is identically zero. ∎

Acknowledgments. We would like to thank S. Benayadi for several stimulating
and fruitful discussions.

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# Probing the environments surrounding ultrahigh energy cosmic ray
accelerators
and their implications for astrophysical neutrinos

Marco Stein Muzio<EMAIL_ADDRESS>Center for Cosmology and Particle Physics,
Department of Physics, New York University, 726 Broadway, New York, New York
10003, USA  Glennys R. Farrar<EMAIL_ADDRESS>Center for Cosmology and Particle
Physics, Department of Physics, New York University, 726 Broadway, New York,
New York 10003, USA  Michael Unger<EMAIL_ADDRESS>Institut für
Astroteilchenphysik, Karlsruher Institut für Technologie, Karlsruhe 76344,
Germany

###### Abstract

We explore inferences on ultrahigh energy cosmic ray (UHECR) source
environments – constrained by the spectrum and composition of UHECRs and non-
observation of extremely high energy neutrinos – and their implications for
the observed high energy astrophysical neutrino spectrum. We find acceleration
mechanisms producing power-law CR spectra $\propto E^{-2}$ are compatible with
UHECR data, if CRs at high rigidities are in the quasi-ballistic diffusion
regime as they escape their source environment. Both gas-dominated and photon-
dominated source environments are able to account for UHECR observations,
however photon-dominated sources give a better fit. Additionally, gas-
dominated sources are in tension with current neutrino constraints. Accurate
measurement of the neutrino flux at $\sim 10$ PeV will provide crucial
information on the viability of gas-dominated sources, as well as whether
diffusive shock acceleration is consistent with UHECR observations. We also
show that UHECR sources are able to give a good fit to the high energy portion
of the astrophysical neutrino spectrum, above $\sim$ PeV. This common origin
of UHECRs and high energy astrophysical neutrinos is natural if air shower
data is interpreted with the Sibyll2.3c hadronic interaction model, which
gives the best-fit to UHECRs and astrophysical neutrinos in the same part of
parameter space, but not for EPOS-LHC.

##  I Introduction

Discovering the origin of ultrahigh energy cosmic rays (UHECRs), $E\gtrsim
10^{18}$ eV ($1$ EeV), has been a longstanding challenge due to their
incredibly low flux, their uncertain composition, and their deflection in both
Galactic and extragalactic magnetic fields [1]. However high energy and UHE
neutrinos, which do not suffer magnetic deflections or significant energy
losses, are natural signals of interactions of UHECRs (and their secondary
CRs) with photon fields and gas during their propagation from the accelerator
to Earth. The potential of neutrinos to provide multimessenger observations to
help discover UHECR origins has been underscored by the recent coincident
observations of a high energy neutrino (IC-170922A) with the flaring blazar
TXS 0506+056 [2] and another with the tidal disruption event TDE AT2019dsg
[3].

In this paper, we build on the approach of Unger, Farrar, and Anchordoqui (UFA
2015, [4]) and its elaboration in [5], to investigate constraints on UHECR
sources which can be inferred from the latest multimessenger data, using a
realistic but flexible UHECR source model with minimal reliance on any
specific astrophysical scenario. UFA15 proposed such a model for UHECR
sources, taking into account photohadronic interactions of UHECRs propagating
through the source environment after acceleration (potentially delayed in
their escape due to diffusion in a turbulent magnetic field) and before
extragalactic propagation. This model provides a natural explanation for the
observed UHECR spectrum and its composition, and enabled multimessenger
constraints on UHECR sources to be investigated in [5]. We consider here the
effects of gas in the environment surrounding the source and the ability of
UHECR sources to explain the highest energy astrophysical neutrinos.111Earlier
studies have also considered that UHECRs and astrophysical neutrinos may have
a common origin. Qualitative fits to the astrophysical neutrino spectrum
consistent with UHECR observations in the context of photodisintegration
models are given in [6, 7, 8] while for specific astrophysical scenarios see
[9, 10, 11, 12, 13, 14]. Other studies have explored neutrino backgrounds at
higher energies consistent with UHECR data [15, 16, 17, 18] and have set
upper-bounds on the flux of neutrinos produced by UHECRs [19]. Like [4, 5],
our description is agnostic to the acceleration process or specific
astrophysical source type.

Figure 1: Predictions of the UHECR source model producing the best description
of the astrophysical neutrino flux for Sibyll2.3c. Left: The CR predictions
for spectrum (top) and composition (bottom) compared to shifted Auger
observations [20, 21, 22, 23], as detailed in Section III. The red and blue
solid lines show the $\langle X_{\mathrm{max}}\rangle$ and
$\sigma(X_{\mathrm{max}})$ predictions of Sibyll2.3c for pure proton and iron
models. Right: The neutrino and gamma-ray predictions for this model (solid
and dashed lines, respectively). The total neutrino flux due to UHECRs (“UHECR
$\nu$’s”, solid magenta) is broken down by origin: UHECR interactions during
extragalactic propagation (“Propagation $\nu$’s”, dotted dark magenta), UHECR
photohadronic and hadronic interactions in the source (“Source $\nu$’s”, dot-
dashed and dashed dark magenta, respectively). Neutrinos originating from a
source other than UHECRs (“Non-UHECR $\nu$’s”, dot-dot-dashed dark magenta)
are also shown. The observed and inferred values of the extragalactic gamma-
ray flux [24], astrophysical neutrino fluxes [25, 26], flux measurements from
the Glashow event [27], and upper-bounds on the EHE cosmic neutrino flux from
IceCube [28, 27] (black) and Auger [29] (grey) are shown. Data points are as
detailed in the text.

We focus here on the following questions:

  1. (a)

How is the original “UFA15” picture impacted by the presence of gas in the
surroundings of the accelerator, in addition to – or instead of – a photon
field?

  2. (b)

Is the diffusive shock acceleration prediction for the power-law index of the
spectrum of UHECRs emerging from their accelerator consistent with
multimessenger data?

  3. (c)

Can the highest energy neutrinos so far detected be produced by interactions
of UHECRs with photons and/or gas in their source environment, given the
strongly constrained UHECR spectrum and composition?

This paper is organized as follows. In Section II we review the model of UFA15
and detail the elaborations made for this study. Section III gives an overview
of the relevant multimessenger data and the constraints we have adopted.
Finally, in Section IV we report our findings, including the degree to which
gas- and photon-dominated source environments can be distinguished and the
ability of UHECR sources to explain the high energy astrophysical neutrino
flux. For the reader’s orientation, Fig. 1 shows the UHECR model giving the
best-fit to the astrophysical neutrino flux, among those giving a good fit to
UHECR spectrum and composition data using the Sibyll2.3c hadronic interaction
model, as discussed below.

##  II Model

###  II.1 Overview

To perform this analysis we extend the UFA15 framework by adding interactions
with gas in the source environment (as described in Section II.2), as well as
making a variety of technical improvements to the analysis in [4]. Based on
the results of [5], we adopt a source evolution following the star-formation
rate (SFR, [30]) and take a single-mass injection of CRs into the source
environment.222Source evolutions which are stronger at high redshift generally
result in a larger neutrino flux at Earth, but the quality of the fit to UHECR
observations degrades and requires an extremely hard CR spectrum escaping the
source [5, 31]. Source evolutions with lower source densities at high redshift
produce smaller neutrino fluxes at Earth and also give a worse fit to UHECR
data when considering photodisintegration models, so are not explored in this
study. However, UHECR data alone does not strongly constrain the source
evolution. A more complex injection composition was found not to be needed
[5], so for simplicity we adopt single-mass injection. We approximate the gas
to be pure hydrogen since other components make up less than $10\%$ by number.
We also introduce an alternate treatment of the energy dependence of the
escape time, based on the behavior of diffusion coefficients and reflecting
the finite size of sources. Details of our treatment of systematic
uncertainties are given in Section III.

A CR nucleus of energy $E$, mass $A$, and charge $Z$ has interactions with
photons and gas at a rate $\tau^{-1}_{\gamma}(E,A)$ and $\tau^{-1}_{g}(E,A)$
when propagating in the source environment. These rates are specified by their
cross sections, the photon spectral density distribution, and the gas density.
Thus we can fully characterize UHECR interactions with photons and gas in a
given source environment by knowing the parameters specifying the photon
spectrum, and $\tau_{\gamma}$ and $\tau_{g}$ for a reference nucleus and
energy. Following [4], we adopt 10 EeV 56Fe as this reference.

Only the total number of interactions prior to escape matters in the
processing of nuclei injected by the accelerator, so only the ratio of the
interaction and escape times is relevant for fixing the composition and energy
spectra of CRs emerging from the source environment, given the injected
composition and spectrum.333The overall normalization of the energy spectrum
is set by the product of the UHECR luminosity per source and the number
density of UHECR sources. However, this is unrelated to the interaction and
escape times within the source environment and so can be fit independently to
obtain the best-fit. We denote these ratios for the reference nucleus by the
model parameters
$r_{\mathrm{esc}}\equiv\tau_{\mathrm{esc}}^{\mathrm{ref}}/\tau_{\mathrm{int}}^{\mathrm{ref}}=\langle
N_{\mathrm{int}}^{\mathrm{ref}}\rangle$, the ratio of the escape and total
interaction times, and
$r_{g\gamma}\equiv\tau_{g}^{\mathrm{ref}}/\tau_{\gamma}^{\mathrm{ref}}=\langle
N_{\gamma}^{\mathrm{ref}}\rangle/\langle N_{g}^{\mathrm{ref}}\rangle$, the
ratio of the hadronic and photohadronic interaction times.

In previous work [4, 5], the escape time was taken to be a power law in
rigidity, $R\equiv E/Z$, so that $\tau_{\rm esc}=\tau^{\mathrm{ref}}_{\rm
esc}(R/R_{\mathrm{ref}})^{\delta_{\rm esc}}$ with $\delta_{\mathrm{esc}}$ a
free parameter of the model limited to be within $[-1,-1/3]$, covering the
expected range from Kolmogorov to Bohm diffusion. Here we adopt a more
detailed parameterization (described in Section II.3) based on simulations of
CR propagation in turbulent magnetic fields, e.g. [32]:
$\tau_{\mathrm{esc}}(R)=\frac{L^{2}}{6D(R)}+\frac{L}{c}$. This implies faster
escape at high rigidities by capturing the quasi-ballistic regime.

The interplay between the energy dependence of the interaction time,
$\tau_{\mathrm{int}}(E)$, and the rigidity dependence of the escape time from
the environment, $\tau_{\rm esc}(R)$, governs the energy dependence of the
number of interactions before escape, as was noted in equation (4) of [4].
There, the focus was on how the high-pass filter mechanism creates the ankle
in the UHECR spectrum and produces the observed light extragalactic UHECRs
below the ankle. As noted in [4], this mechanism leads to a hardening of the
escaping CR spectrum. Thus the Auger combined fit to the spectrum and
composition [31] favoring an $\sim E^{-1}$ or harder spectrum escaping the
source, does not necessarily mean that the accelerator produces such a hard
spectrum. In principle, the fundamental spectrum from the accelerator could be
much softer, e.g., $\sim E^{-2}$ as predicted by diffusive shock acceleration,
with the low-energy component of any given $A$ being more depleted through
interactions because of their longer residence time in the environment – thus
hardening the escaping spectrum for each composition.444An alternative
mechanism to harden the accelerated spectrum is through magnetic suppression
of the spectrum at low-rigidities due to the horizon induced by the
extragalactic magnetic field [33, 34].

In this work we find that UHECR data is compatible with an accelerator
spectrum $E^{-2}$, if the peak of that spectrum falls in the quasi-ballistic
regime where the rapid change of escape time with energy leads to a strong
hardening of the escaping flux. This requires source conditions such that high
rigidity CRs are able to enter the quasi-ballistic diffusion regime. In this
case, diffusive shock acceleration remains a possible mechanism for UHECR
acceleration. As we shall show below, the viability of this scenario can be
tested with accurate measurement of the neutrino flux at $\sim 10$ PeV.

A description of how our model’s parameters can be related to astrophysical
quantities will be given in a forthcoming paper.

###  II.2 Incorporation of hadronic interactions in the source environment

Photohadronic interactions in the source environment are accounted for as
described in Appendix B and Appendix C of [4], with the adjustment of the
definition of the total interaction time to include hadronic interactions, so
that

$\displaystyle\tau^{-1}_{\mathrm{int}}(E,A)=\tau^{-1}_{g}(E,A)+\tau^{-1}_{\gamma}(E,A)$
(1)

is the total interaction time for a CR of mass $A$ and energy $E$, where
$\tau_{g}$ is its hadronic interaction time and $\tau_{\gamma}$ is its
photohadronic interaction time.

The photohadronic interaction time is the total interaction time due to both
photopion (PP) and photodisintegration (PD) interactions with the ambient
photon field,

$\displaystyle\tau_{\gamma}^{-1}(E,A)=\tau_{\mathrm{PP}}^{-1}(E,A)+\tau_{\mathrm{PD}}^{-1}(E,A).$
(2)

(a)  (b)

Figure 2: The escape (solid lines), photohadronic interaction (dashed lines),
and hadronic interaction (dashed-dotted lines) lengths for different nuclear
masses (colors) in the UHECR source model best reproducing the astrophysical
neutrino flux for Sibyll2.3c (left) and EPOS-LHC (right). Hadronic interaction
times were calculated under the respective hadronic interaction models. In
particular while the interaction times are similar, the escape times in the
two panels differ significantly at high energies, reflecting significantly
different best-fit values of $r_{\mathrm{size}}=L/\lambda_{c}$. The
corresponding fits are pictured in Figs. 1 and 6, respectively.

The relation between the interaction times for each of these interaction
channels and the ambient photon field are given below.

Throughout the present work we consider either a pure or generalized black-
body photon spectral density distribution555 In [5], we considered both black-
body and broken power-law parametrizations of the photon spectral density
distribution in the source environment and found that both gave comparable
fits to the UHECR data. Black-body spectra, though, produced fewer extremely
high energy cosmic neutrinos, allowing more conservative constraints to be
set. The qualitative conclusions are expected to be similar for black-body and
broken power-law parametrizations [4, 35]. of the form

$\displaystyle n(\varepsilon)=n_{0}f_{\mathrm{BB}}(\varepsilon)$ (3)

where $n_{0}$ is the photon density in units of
$I_{\mathrm{BB}}\equiv\frac{16\pi\zeta(3)}{(hc)^{3}}\left(kT\right)^{3}$, the
black-body photon number density, and $f_{\mathrm{BB}}$ is the spectral
density distribution of a black-body (BB) photon field with temperature $T$
given by

$\displaystyle f_{\mathrm{BB}}(\varepsilon)$
$\displaystyle=\frac{8\pi}{(hc)^{3}}\frac{\varepsilon^{2}}{\mathrm{e}^{\frac{\varepsilon}{kT}}-1}.$
(4)

The generalized black-body spectral density (3) corresponds to the spectral
density of a grey-body for $n_{0}<1$. With this definition, the physical
number density of the photon spectral density distribution $n(\varepsilon)$ is
given by $n_{\gamma}=n_{0}I_{\mathrm{BB}}$.

The interaction time for each of the photohadronic interaction channels, $i=$
PD or PP, is given by

$\displaystyle\tau_{i}^{-1}(E,A)$
$\displaystyle=\frac{c}{2}\int_{0}^{\infty}\mathrm{d}\varepsilon\frac{n(\varepsilon)}{\gamma^{2}\varepsilon^{2}}\int_{0}^{2\gamma\varepsilon}\mathrm{d}\varepsilon^{\prime}\varepsilon^{\prime}\sigma_{A\gamma,i}(\varepsilon^{\prime})$
(5)
$\displaystyle=n_{0}\frac{c}{2}\int_{0}^{\infty}\mathrm{d}\varepsilon\frac{f_{\mathrm{BB}}(\varepsilon)}{\gamma^{2}\varepsilon^{2}}\int_{0}^{2\gamma\varepsilon}\mathrm{d}\varepsilon^{\prime}\varepsilon^{\prime}\sigma_{A\gamma,i}(\varepsilon^{\prime})$
(6) $\displaystyle=n_{0}\tau_{\mathrm{BB},i}^{-1}(E,A,T)$ (7)

where $\sigma_{A\gamma,i}$ is the photonuclear cross section for photohadronic
interaction channel $i$ (provided by CRPropa [36] based on TALYS and SOPHIA to
appropriately handle the different photon energy regimes), $\gamma$ is the
Lorentz factor, and $\tau_{\mathrm{BB},i}$ is the interaction time for a
black-body distribution with temperature $T$.

In order to account for hadronic interactions we must assume a hadronic
interaction model (HIM), as hadronic interactions within the source
environment occur at similar center-of-mass energies as in extensive air
showers initiated by UHECRs on Earth. Once we have assumed a particular HIM,
labeled $m$, the hadronic interaction time is determined by the density of gas
$n_{g}$ in the source environment and the $pA$ cross section
$\sigma_{g}^{m}(E,A)$, for a collision between a proton at rest and CR of
energy $E$ in that model:

$\displaystyle\tau_{g}^{-1}(E,A)=n_{g}\sigma^{m}_{g}(E,A)c.$ (8)

Particle production in these interactions also depends upon the assumed HIM.
Using CRMC v1.6.0 [37], we built interaction matrices for the EPOS-LHC [38]
and Sibyll2.3c [39] HIMs.666Throughout this work, the HIM assumed for
calculation of interactions in the source environment always matched the HIM
used to interpret air shower observations. Simulating the collision between a
CR of nucleus $A$ and energy $E_{j}$ with a proton at rest, we generated an
interaction matrix $I_{ij}^{sA,m}$ by tabulating all the secondaries of type
$s$ (e.g. $\nu$, $\pi$, $\gamma$, $A^{\prime}$) and energy $E_{i}$ predicted
by HIM $m$. Pions and neutrons were treated as stable for this purpose,777Here
we assume the energy loss time due to synchrotron radiation exceeds either the
decay time of the $\pi\rightarrow\mu\rightarrow\nu$ chain or the escape time
of pions and muons. See e.g. [40] for a discussion of cooling-damped sources.
In particular, our results here are robust to cooling as will be shown in a
forthcoming paper. in order to treat these particles in a consistent way with
those produced in photohadronic interactions (as described in [4]), and all
other particles were decayed before being tabulated. The final interaction
matrices give the average number of secondaries of energy $E_{i}$ produced in
a collision. With the interaction matrices in hand, one can easily compute the
total number of secondaries $s$ produced in logarithmic interval
$[\lg{E_{i}},\lg{E_{i}}+\mathrm{d}\lg{E_{i}})$ by a flux of CRs as

$\displaystyle\frac{\mathrm{d}N^{s}}{\mathrm{d}\lg{E_{i}}}=\sum_{\begin{subarray}{c}j>i,A\end{subarray}}I^{sA,m}_{ij}f_{g}(A,E_{j})f_{\mathrm{int}}(A,E_{j})\frac{\mathrm{d}N^{A}}{\mathrm{d}\lg{E_{j}}}\left|\frac{\mathrm{d}\lg{E_{j}}}{\mathrm{d}\lg{E_{i}}}\right|$
(9)

where $f_{g}\equiv\left(1+\tau_{g}/\tau_{\gamma}\right)^{-1}$ is the fraction
of interactions which are hadronic, and
$f_{\mathrm{int}}\equiv\left(1+\tau_{\mathrm{int}}/\tau_{\mathrm{esc}}\right)^{-1}$
is the fraction of CRs which interact. In the above formula, we have neglected
reinteraction of secondary CRs for simplicity and clarity, but in our full
calculation reinteractions are accounted for.

It is important to note that hadronic interactions always dominate at low
enough energies such that the $\gamma$-CR center-of mass-energy is too low to
photopion produce or excite the giant dipole resonance. This is reflected in
the energy dependence of the interaction times, as Fig. 2 shows. Thus even in
photon-dominated environments (meaning environments where UHECRs primarily
interact photohadronically), astrophysical neutrinos may still be the product
of hadronic interactions with gas, given a sufficient gas density and number
of interactions. On the other hand, CRs in gas-dominated sources primarily
interact hadronically at both high and low energies.

###  II.3 CR diffusion and escape from the source environment

In previous analyses [4] and [5], the rigidity dependence of the diffusion
coefficient was modeled as a single power law in rigidity: $D(R)\propto
R^{-\delta_{\mathrm{esc}}}$ and $\tau_{\mathrm{esc}}\propto
R^{\delta_{\mathrm{esc}}}$. This is a good approximation for CRs that are in
the diffusive regime for all the relevant rigidities. For a more realistic
treatment we use a fit to the diffusion simulation of [32] with the functional
form suggested in [41] (leading to very similar numerical results as found
therein):

$\displaystyle D(R)$
$\displaystyle=\frac{c\lambda_{c}}{6\pi}\left[\left(\frac{R}{R_{\mathrm{diff}}}\right)^{1/3}+\frac{1}{2}\left(\frac{R}{R_{\mathrm{diff}}}\right)+\frac{2}{3}\left(\frac{R}{R_{\mathrm{diff}}}\right)^{2}\right],$
(10)

where $r_{L}(R)\equiv 1.1\text{
kpc}\left(\frac{\text{$\mu$G}}{B}\right)\left(\frac{R}{\text{EV}}\right)$ is
the Larmor radius, $\lambda_{c}$ is the coherence length of the magnetic
field, $B$ is its RMS field strength, and $R_{\mathrm{diff}}$ is the
characteristic rigidity scale of diffusion, defined by $2\pi
r_{L}(R_{\mathrm{diff}})\equiv\lambda_{c}$. For convenience, we define the
dimensionless diffusion coefficient,
$d(R)\equiv\left(R/R_{\mathrm{diff}}\right)^{1/3}+\frac{1}{2}\left(R/R_{\mathrm{diff}}\right)+\frac{2}{3}\left(R/R_{\mathrm{diff}}\right)^{2}$,
so that $D(R)=\frac{c\lambda_{c}}{6\pi}d(R)$.

CRs of rigidity $R$ that enter the diffusion regime have an escape time given
by $\tau_{\mathrm{esc}}=\frac{L^{2}}{6D(R)}$. However, in reality the source
may not be large enough for all CRs to enter the diffusion regime before
escaping the source, especially those at high rigidities with large Larmor
radii. The source’s physical size $L$ then imprints a minimum escape time of
$L/c$. To account for this effect we adopt the following parametrization for
the escape time, which has been verified against simulation data shown in
[32]:

$\displaystyle\tau_{\mathrm{esc}}(R)=\frac{L^{2}}{6D(R)}+\frac{L}{c}~{}.$ (11)

This treatment imposes that the escape time be at least as large as the
crossing time.

The escape time for a nucleus of rigidity $R$ can then be calculated in terms
of the escape time of the reference nucleus as

$\displaystyle\tau_{\mathrm{esc}}(R)$
$\displaystyle=\tau_{\mathrm{esc}}^{\mathrm{ref}}\frac{\frac{L^{2}}{6D(R)}+\frac{L}{c}}{\frac{L^{2}}{6D(R_{\mathrm{ref}})}+\frac{L}{c}}$
$\displaystyle=\tau_{\mathrm{esc}}^{\mathrm{ref}}\frac{\frac{\pi}{d(R)}\frac{L}{\lambda_{c}}+1}{\frac{\pi}{d(R_{\mathrm{ref}})}\frac{L}{\lambda_{c}}+1}$
$\displaystyle=\tau_{\mathrm{esc}}^{\mathrm{ref}}\frac{\frac{\pi
r_{\mathrm{size}}}{d(R)}+1}{\frac{\pi
r_{\mathrm{size}}}{d(R_{\mathrm{ref}})}+1},$ (12)

where $R_{\mathrm{ref}}=10/26\text{ EV}\simeq 0.38\text{ EV}$ is the rigidity
of the reference nucleus, $\tau_{\mathrm{esc}}^{\mathrm{ref}}$ is its escape
time, and we have introduced the model parameter $r_{\mathrm{size}}\equiv
L/\lambda_{c}$.

We find that in order for a soft $E^{-2}$ acceleration spectrum to be
adequately hardened to account for UHECR data, high rigidity CRs must be in
the $\tau_{\mathrm{esc}}\sim E^{-2}$ regime. This implies that
$r_{\mathrm{size}}\gg 1$ in order for conventional acceleration mechanisms to
explain UHECR observations.

(a)  (b)

Figure 3: $N_{\sigma}^{\prime}$, the number of standard deviations improvement
of the global UHECR best-fit relative to the local UHECR best-fit, as a
function of model parameters
$r_{\mathrm{esc}}\equiv\tau_{\mathrm{esc}}^{\mathrm{ref}}/\tau_{\mathrm{int}}^{\mathrm{ref}}$
and $r_{g\gamma}\equiv\tau_{g}^{\mathrm{ref}}/\tau_{\gamma}^{\mathrm{ref}}$
for Sibyll2.3c (left) and EPOS-LHC (right). Contours mark $99\%$ confidence
level (CL) exclusion regions based on UHECR data (white contour and dotted
region), non-observation of neutrinos above $10^{15.9}$ eV (grey contour and
hatched region), and upper-bounds on the extragalactic gamma-ray flux (cyan
contour and hatched region). The global best-fit points
($N^{\prime}_{\sigma}=0$) correspond to $\chi^{2}/ndf$’s of $58.5/58$ and
$106.2/58$ for Sibyll2.3c and EPOS-LHC, respectively.

##  III Data & Methodology

Our model is fit to the UHECR spectrum and composition data of Auger [20, 21,
22, 42, 43, 44, 23], applying a $+0.8$-bin shift of the Auger energy scale
($+20\%$, slightly larger than the quoted systematic uncertainty of $14\%$
[45]). We also shift $\langle X_{\mathrm{max}}\rangle$ by $-10$ g/cm2 on
average, following the energy dependence quoted in [43]. These shifts were
determined via a preliminary study by scanning over the energy and $\langle
X_{\mathrm{max}}\rangle$ shifts and fixing their values to the combination
which enabled the best-fit to the data. To assess the goodness-of-fit we
compute a combined $\chi^{2}$ to the spectrum data and $\langle
X_{\mathrm{max}}\rangle$ and $\sigma\left(X_{\mathrm{max}}\right)$ data mapped
into $\langle\ln{A}\rangle$ and $\mathrm{V}(\ln{A})$ using the parametrization
of [42],

$\displaystyle\chi^{2}=$
$\displaystyle\displaystyle\sum_{i}^{N_{\mathrm{spec}}}\frac{(J_{m,i}-J_{i})^{2}}{\sigma_{J,i}^{2}}+\displaystyle\sum_{j}^{N_{\mathrm{comp}}}\frac{(\langle\ln{A}\rangle_{m,j}-\langle\ln{A}\rangle_{j})^{2}}{\sigma_{\langle\ln{A}\rangle,j}^{2}}$
$\displaystyle+\displaystyle\sum_{j}^{N_{\mathrm{comp}}}\frac{(\mathrm{V}(\ln{A})_{m,j}-\mathrm{V}(\ln{A})_{j})^{2}}{\sigma_{\mathrm{V}(\ln{A}),j}^{2}},$
(13)

where $N_{\mathrm{spec}}$ and $N_{\mathrm{comp}}$ are the number of data
points in the spectrum and composition, respectively, $x_{m,i}$ denotes the
model prediction of quantity $x$ (flux $J$, mean logarithmic mass
$\langle\ln{A}\rangle$ or its variance $\mathrm{V}$) at energy bin $i$, and
errors include both systematic and statistical errors for the spectrum [20,
21, 22] and statistical errors for the mean and variance on $\ln{A}$ [42, 43,
44, 23]. For spectral energy bins above the highest-energy data point, we
follow [46] by adding an additional $2n_{i}$ to the $\chi^{2}$, where $n_{i}$
is the expected number of observed events predicted by the model in energy bin
$i$ given the exposure of the dataset. Our final figure of merit is
$\chi^{2}_{\mathrm{CR}}=\chi^{2}+2\sum_{i}n_{i}$, where $i$ runs over energy
bins above the highest-energy data point in the spectrum.

(a)  (b)

Figure 4: Spectral index $\gamma_{\mathrm{inj}}$ of CRs injected into the
source environment, $E^{\gamma_{\mathrm{inj}}}$, as a function of model
parameters
$r_{\mathrm{esc}}\equiv\tau_{\mathrm{esc}}^{\mathrm{ref}}/\tau_{\mathrm{int}}^{\mathrm{ref}}$
and $r_{g\gamma}\equiv\tau_{g}^{\mathrm{ref}}/\tau_{\gamma}^{\mathrm{ref}}$
for Sibyll2.3c (left) and EPOS-LHC (right). Contours mark $99\%$ CL exclusion
regions based on UHECR data (white contour and dotted region) and non-
observation of neutrinos above $10^{15.9}$ eV (grey contour and hatched
region).

We follow the PDG [47, 48] defining the number of sigma from the best-fit as
$N_{\sigma}^{\prime}=S^{-1}\sqrt{\chi^{2}_{\mathrm{model}}-\chi^{2}_{\mathrm{min}}}$,
where $S=\sqrt{\chi^{2}_{\mathrm{min}}/N_{\mathrm{dof}}}$ is the scale factor
introduced in [48] to enlarge the uncertainties to account for a
$\chi^{2}_{\mathrm{min}}/N_{\mathrm{dof}}>1$, $\chi^{2}_{\mathrm{model}}$ is
the $\chi^{2}$ for a given model, $\chi^{2}_{\mathrm{min}}$ is the $\chi^{2}$
of the best-fit model (under the assumption of that HIM), and
$N_{\mathrm{dof}}$ is the number of degrees of freedom. We adopt the condition
$N_{\sigma}^{\prime}<2.58$, corresponding to $99\%$ CL, when placing
constraints based on UHECR data.

We also place constraints using bounds on extremely high energy (EHE)
neutrinos ($E_{\nu}>10^{15.9}$ eV) from IceCube [28], updated to reflect the
newly observed Glashow event [27]. Models which predict $N_{\nu}^{\rm
EHE}>4.74$ are excluded at the $99\%$ CL, since IceCube has observed no
neutrinos at these energies and this energy range is expected to be background
free [49]. We allow fits that are within $2.58\sigma$ of the best-fit to CRs
to stray from the local best-fit parameters, as long as they respect
$N_{\nu}^{\rm EHE}<4.74$, so as to determine if any choice of parameters is
compatible with both sets of data. This process can introduce apparent
sampling artifacts into the results, e.g. along the grey contour in Fig. 3b.

Finally, we consider the ability of current extragalactic gamma-ray data to
constrain our model’s parameters. For this we consider the extragalactic
gamma-ray background (EGB) measured by the Fermi-Large Area Telescope (LAT)
[24]. In order to place conservative bounds we adopt the published EGB
spectrum under the assumption of LAT’s Galactic foreground model B, the
foreground model which attributes the least Galactic emission to the measured
gamma-ray flux. We consider a model to be excluded if its predicted gamma-ray
flux exceeds the EGB model B measurement by more than the error bar in any
energy bin. In plots we also give an estimate of the truly diffuse gamma-ray
background (TDGRB) – as described in Section 2 of [5] – and the flux measured
in the highest energy bin of the isotropic gamma-ray background (IGRB) model B
by LAT [24]. As is shown in Fig. 3a, gamma-ray data is weakly constraining,
only ruling out models also excluded by bounds on EHE neutrinos. Thus we set
aside the gamma-ray constraints for the rest of our analysis.

##  IV Results

###  IV.1 Implications of gas in the source environment

To a good approximation hadronic interactions partially disintegrating a
nucleus preserve the energy-per-nucleon of the primary CR. This feature allows
CR interactions with gas to realize the mechanism explored in [4], and
expanded upon in [5], for explaining the observed UHECR spectrum and
composition. Fits in Fig. 3 show that both gas- and photon-dominated
environments can give an adequate accounting of UHECR data. This figure shows
how the quality of fit to UHECR data changes with $r_{\mathrm{esc}}$,
controlling the average number of interactions CRs undergo before escape, and
$r_{g\gamma}$, controlling whether the source environment is gas-dominated
(small values) or photon-dominated (large values). Contours in this figure
show the regions excluded by CR (white dotted region), EHE neutrino (grey
hatched region), and gamma-ray (cyan hatched region) data. Figure 3a shows
that while Sibyll2.3c is able to describe the UHECR data even for gas-
dominated sources, it prefers photon-dominated sources. EPOS-LHC, though,
prefers sources where photon interactions are more comparable to gas
interactions, as can be seen in Fig. 3b. We also note that, in all cases,
small values of $r_{esc}$ provide a poor fit to UHECR data since the injected
UHECR spectrum must undergo a sufficient number of photodissociation
interactions in order to account for the composition observed at Earth, in
particular the position of the protonic component. Otherwise the composition
of injected UHECRs must be fine tuned.

(a)  (b)

Figure 5: Reduced $\chi^{2}_{\nu}$ of UHECR model predictions of the
astrophysical neutrino flux as a function of model parameters
$r_{\mathrm{esc}}\equiv\tau_{\mathrm{esc}}^{\mathrm{ref}}/\tau_{\mathrm{int}}^{\mathrm{ref}}$
and $r_{g\gamma}\equiv\tau_{g}^{\mathrm{ref}}/\tau_{\gamma}^{\mathrm{ref}}$
for Sibyll2.3c (left) and EPOS-LHC (right). Contours mark best-fitting region
to CR data (dashed white contour) and $99\%$ CL exclusion regions based on
UHECR data (solid white contour and dotted region) and non-observation of
neutrinos above $10^{15.9}$ eV (grey contour and hatched region).

An interesting feature of gas-dominated source environments is that they are
able to achieve a good fit to Auger observations even for spectral indices as
soft as $\gamma_{\mathrm{inj}}\sim-2$, as is predicted for diffusive shock
acceleration; see Fig. 4, which shows the best-fit accelerator spectral index
for CRs. This is due to the fact that the hadronic interaction time depends
very weakly on energy (see Fig. 2), so that it is well approximated by a
single power law $\tau_{g}\sim E^{\zeta}$ with $\zeta\sim 0$. Following the
analysis of [4] (see Section 2), this implies that quasi-ballistic-regime CRs
(i.e. $\tau_{\mathrm{esc}}\sim E^{-2}$) interacting dominantly with gas, can
produce a spectrum escaping the source environment with hard spectral index
$\gamma_{\mathrm{esc}}\gtrsim-1$ even when the injected spectral index is
$\gamma_{\mathrm{inj}}\sim-2$ to $-3$.888This mechanism can also be achieved
by photohadronic interactions, since at high energies $\tau_{\gamma}\sim
E^{0}$ as well. But in order for the peak of the CR spectrum to be in this
regime the photon field needs a very high temperature and CRs require a large
number of interactions on average. These two requirements cause the
description of the UHECR spectrum and composition to degrade, as well as
producing an excess of EHE neutrinos which violates the IceCube bounds. This
clearly demonstrates that processing of the CR spectrum by the source
environment can substantially modify the escaping spectral index to be as hard
as is required by the Auger spectrum [31], without relying on exotic
acceleration mechanisms.

Since a soft injected spectral index requires hadronic interactions to
dominate to get a good fit to UHECR data, a substantial flux of neutrinos is
produced. This creates a potential tension with the IceCube bound on EHE
neutrinos, as shown in Fig. 4. The viability of a soft injected spectral index
will be decisively tested by accurate measurement of the neutrino flux at
$\sim 10$ PeV. It is also worth noting that in the gas-dominated regime, where
such soft spectral indices can be achieved, EPOS-LHC suffers from tension with
both EHE neutrinos and the poor quality of fit it produces to UHECR data.

###  IV.2 High energy astrophysical neutrino flux

The astrophysical neutrino flux has been studied using several different
IceCube datasets [50, 26, 25, 51] with some tension in the various derived
neutrino spectra [51]. We compare the neutrino flux predicted by our UHECR
model to the differential flux measurement presented in the Glashow event
paper [27] and to the IceCube Cascades dataset [25], because this dataset has
the least tension with other IceCube measurements of the astrophysical
neutrino flux [25, 51]. However, we stress that until the neutrino spectrum is
better determined the results of our analysis must be taken as provisional.

There is no reason to assume that all astrophysical neutrinos are of UHECR
origin. Therefore we allow for a component of neutrinos originating from some
other source. This non-UHECR component is parametrized by a single power law
with an exponential cutoff: $\phi_{\nu}=\phi_{\nu
0}(E/E_{0})^{\gamma_{\nu}}e^{-E/E_{\mathrm{max},\nu}}$, where $E_{0}=10^{13}$
eV. Given an UHECR model prediction of the neutrino flux, the normalization,
spectral index, and cutoff energy of the non-UHECR neutrinos are tuned so that
the total neutrino flux gives the best-fit to the IceCube data.

We measure our model’s description of the astrophysical neutrino flux using a
$\chi^{2}$ to the data points,

$\displaystyle\chi^{2}=\displaystyle\sum_{i}\frac{(\phi_{m,i}-\phi_{i})^{2}}{\sigma_{\phi,i}^{2}}~{}.$
(14)

For energy bins with upper-bounds, we follow the same procedure as for
$\chi^{2}_{\mathrm{CR}}$ discussed in Section III, adding an additional
$2n_{i}$ to the $\chi^{2}$. Our final figure of merit is
$\chi^{2}_{\nu}=\chi^{2}+2\sum_{i}n_{i}$, where $i$ runs over energy bins with
upper-bounds. We calculate the figure of merit including the Glashow event
data point and the portion of the IceCube Cascades dataset within the
sensitive energy range, from $16$ TeV to $2.6$ PeV, as determined by IceCube
[25].

Fig. 5 shows the ability of UHECR models to accurately describe the
astrophysical neutrino flux. The best-fitting source environments are photon-
dominated with $\langle N_{\mathrm{int}}\rangle\sim 100-1000$ interactions on
average for the reference nucleus. Remarkably, for Sibyll2.3c this region also
corresponds to the region best-fitting UHECR data. By contrast these regions
are disconnected for EPOS-LHC, making a common origin possible but less
natural than for Sibyll2.3c. Given that the region best-fitting the
astrophysical neutrino flux corresponds to photon-dominated sources,
uncertainties in the modeling of hadronic interactions in the source
environment are not relevant to this result.

We note that at low values of $r_{\mathrm{esc}}$ very few interactions occur
before escape so that UHECRs do not produce a substantial neutrino flux. In
this case the goodness-of-fit to the astrophysical neutrino flux approaches a
constant value because the spectrum is dominated by the non-UHECR component of
neutrinos.

The predicted neutrino fluxes with the smallest $\chi^{2}_{\nu}$ allowed by
multimessenger data are shown in Figs. 1 and 6 for Sibyll2.3c and EPOS-LHC,
respectively.999We note that our model results in a neutrino spectrum which is
photohadronically-produced in the source environment and has a lower peak
energy than some other models, e.g. [17], because we account for CR
interactions with the source environment’s hot (relative to the CMB) photon
field. Neutrinos produced during extragalactic UHECR propagation also have a
lower peak energy than other models due to the lower average energy-per-
nucleon of escaping protons imprinted by interactions in the source
environment. These protons primarily produce neutrinos by interacting with the
cosmic optical background (COB) which has a peak energy of $\sim 1$ eV. The
parameters for each of these models can be found in Table 1. These predictions
make clear that UHECR sources may be responsible for the astrophysical
neutrino flux at high energies, most importantly producing a sufficient flux
to account for the observed Glashow event without conflicting with the bounds
on EHE neutrinos. The best-fitting neutrino predictions have a characteristic
dip in the neutrino spectrum at $\sim 500$ TeV. This is very similar to the
Hypothesis E model investigated by IceCube in [25], which was shown to be
favored over a single power-law spectrum by more than $1\sigma$.

Parameter | Sibyll2.3c | EPOS-LHC
---|---|---
$\gamma_{\mathrm{inj}}$ | $-1.16$ | $-0.68$
$\log_{10}(R_{\mathrm{max}}/\mathrm{V})$ | $18.59$ | $18.47$
$\log_{10}{r_{\mathrm{esc}}}$ | $2.65$ | $2.65$
$\log_{10}{r_{g\gamma}}$ | $4.35$ | $4.45$
$\log_{10}(R_{\mathrm{diff}}/\mathrm{V})$ | $17.79$ | $18.14$
$\log_{10}{r_{\mathrm{size}}}$ | $0.98$ | $4.93$
$T/\mathrm{K}$ | $8987$ | $9000$
$A_{\mathrm{inj}}$ | $33.5$ | $28.0$
$f_{\mathrm{gal}}$ | $0.82$ | $0.79$
$\gamma_{\mathrm{gal}}$ | $-3.46$ | $-3.60$
$\log_{10}(E^{\mathrm{galFe}}_{\mathrm{max}}/\mathrm{eV})$ | $19.00$ | $18.94$
$A_{\mathrm{gal}}$ | $26.6$ | $30.5$
$\phi_{\nu 0}$ [$10^{-15}$/(GeV cm2 s sr)] | $1.98$ | $1.98$
$\gamma_{\nu}$ | $-2.06$ | $-2.06$
$\log_{10}(E_{\mathrm{max,\nu}}/\mathrm{eV})$ | $13.89$ | $13.89$
Table 1:  Parameters corresponding to the models presented in Figs. 1 and 6
for Sibyll2.3c and EPOS-LHC, respectively. $R_{\mathrm{max}}$ is the maximum
rigidity of the injected CR spectrum, where the spectrum is cutoff
exponentially; $A_{\mathrm{inj}}$ is the mass number of the CRs injected into
the source environment (non-integers represent the average mass due to a
mixture of two consecutive mass numbers); $f_{\mathrm{gal}}$ is the fraction
of the observed flux at $10^{17.55}$ eV which is Galactic;
$\gamma_{\mathrm{gal}}$ is the spectral index, $E^{\gamma_{\mathrm{gal}}}$, of
the Galactic spectrum; $E^{\mathrm{galFe}}_{\mathrm{max}}$ is the maximum
energy of Galactic iron, where the Galactic component is cutoff exponentially
(this parameter sets the maximum rigidity of the Galactic component);
$A_{\mathrm{gal}}$ is the mass number of the Galactic component (this
component is also approximated as having a single mass). All other parameters
are defined in the text.

The parameters in Table 1 show that this description of the astrophysical
neutrino flux requires a hard acceleration index
$\gamma_{\mathrm{inj}}\gtrsim-1$ and hot photon field $T\sim 9000$ K
(corresponding to a peak photon energy of $\sim 1$ eV), independent of the
HIM. In particular, the fit assuming EPOS-LHC additionally relies on a large
ratio of source size-to-magnetic coherence length,
$r_{\mathrm{size}}=L/\lambda_{c}\sim 10^{5}$, so that CRs at the highest
energies enter the quasi-ballistic diffusion regime. Such a large separation
of scales may not be possible for real astrophysical sources, so this
requirement could be viewed as an additional strain on EPOS-LHC in the case of
a common origin scenario. If the common origin scenario could be validated,
multimessenger observations would be difficult to reconcile with the source
properties required by EPOS-LHC. By contrast, the fit assuming Sibyll2.3c
requires only a ratio of source size-to-magnetic coherence length of $\sim
10$. Such a moderate ratio of scales could be accommodated by realistic
astrophysical sources and allows Sibyll2.3c to comfortably explain both UHECR
data and astrophysical neutrinos. This HIM-dependence of astrophysical
parameters is indirect and occurs due to differences in the inferred-
composition at Earth. Future work will more systematically investigate the
HIM-dependence of astrophysical parameters to determine the robustness of
these differences.

Figure 6: Predictions of the UHECR source model producing the best description
of the astrophysical neutrino flux for EPOS-LHC. Left: Same as in Fig. 1,
except the red and blue solid lines show the $\langle X_{\mathrm{max}}\rangle$
and $\sigma(X_{\mathrm{max}})$ predictions of EPOS-LHC for pure proton and
iron models. Right: Same as in Fig. 1.

##  V Summary

The main findings of this work can be summarized as follows:

  1. (i)

Both photon- and gas-dominated source environments can explain UHECR
observations, but gas-dominated environments do so less precisely and are in
tension with EHE neutrino constraints (see Fig. 3).

  2. (ii)

Gas-dominated environments can sufficiently harden CR spectra to fit UHECR
data with a soft accelerator spectral index, as in diffusive shock
acceleration, if high-rigidity CRs enter the quasi-ballistic diffusion regime
before escaping the source environment (see Fig. 4 and Section IV.1).

  3. (iii)

Neutrinos will probe whether UHECR interactions in the source environment are
predominantly hadronic or photohadronic, and provide information on the
spectral index of the accelerator (see Figs. 3 and 4 and Section IV.1).

  4. (iv)

UHECR sources may account for high energy astrophysical neutrinos, above $\sim
1$ PeV. If so, it will be possible constrain hadronic interaction models (see
Fig. 5, Figs. 1 and 6, and Section IV.2).

These results directly address the questions presented in Section I. In
particular, result (i) directly addresses question (a); results (ii) and (iii)
address question (b); and, result (iv) addresses question (c).

###### Acknowledgements.

We would like to acknowledge F. Oikonomou for useful feedback on our analysis
and A. Fedynitch, N. Globus, K. Murase, W. Winter, and the anonymous referee
for helpful comments on the manuscript. The research of MSM was supported in
part by the NYU James Arthur Graduate Award and the Ted Keusseff Fellowship.
The research of MSM and GRF was supported in part by NSF-2013199. This work
was supported in part through the NYU IT High Performance Computing resources,
services, and staff expertise.

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##  Appendix A Single-component astrophysical neutrino flux

Figs. 7 and 8 show the best-fitting astrophysical neutrino flux produced by
UHECR sources alone (i.e. not including a non-UHECR component of neutrinos as
was explored in Section IV.2). In particular, these fits cannot reproduce the
low-energy part of the astrophysical neutrino flux observed by IceCube. This
makes clear that UHECR sources alone cannot explain the entire flux of
astrophysical neutrinos while also accurately reproducing the UHECR spectrum
and composition and remaining consistent with bounds on EHE neutrinos.

##  Appendix B Softest spectral index $\gamma_{\mathrm{inj}}\lesssim-2$
compatible with multimessenger data

Fig. 9 shows the predictions of the UHECR model with the softest accelerator
spectral index $\gamma_{\mathrm{inj}}\lesssim-2$ compatible with
multimessenger data. This fit assumes the Sibyll2.3c HIM. (No corresponding
fit is presented for EPOS-LHC as such soft spectral indices are entirely ruled
out by UHECR and EHE neutrino constraints.) Table 2 gives the parameters for
this particular fit which has spectral index $\gamma_{\mathrm{inj}}=-2.1$,
similar to the spectral index predicted by diffusive shock acceleration. To
achieve this fit a source size-to-magnetic coherence length ratio of $\sim
3000$ is required so that CRs at the highest rigidities enter the
quasiballistic regime, allowing for a sufficient hardening of the spectrum.

The right panel of Fig. 9 shows that a soft spectral index
$\gamma_{\mathrm{inj}}\sim-2$ can account for the general level of flux
observed in astrophysical neutrinos at all energies, but not the shape. The
observed astrophysical neutrino flux has an inferred spectral index of
$\gamma\sim-2.5$ using a single power-law spectrum. However since the spectral
index of neutrinos is inherited from the spectral index of the accelerated
CRs, this is too soft to be compatible with multimessenger data.

Parameter |  Sibyll2.3c
---|---
$\gamma_{\mathrm{inj}}$ |  $-2.1$
$\log_{10}(R_{\mathrm{max}}/\mathrm{V})$ |  $18.8$
$\log_{10}{r_{\mathrm{esc}}}$ |  $2.75$
$\log_{10}{r_{g\gamma}}$ |  $0.45$
$\log_{10}(R_{\mathrm{diff}}/\mathrm{V})$ |  $16.97$
$\log_{10}{r_{\mathrm{size}}}$ |  $3.51$
$T/\mathrm{K}$ |  $200$
$A_{\mathrm{inj}}$ |  $30.1$
$f_{\mathrm{gal}}$ |  $0.82$
$\gamma_{\mathrm{gal}}$ |  $-3.42$
$\log_{10}(E^{\mathrm{galFe}}_{\mathrm{max}}/\mathrm{eV})$ |  $18.88$
$A_{\mathrm{gal}}$ |  $26.3$
Table 2:  Parameters corresponding to the model in Fig. 9, having the softest
spectral index $\gamma_{\mathrm{inj}}\lesssim-2$ of the accelerator compatible
with multimessenger data. See the caption of Table 1 for parameter
definitions. Figure 7: Predictions of the UHECR source model producing the
best description of the astrophysical neutrino flux, when omitting a non-UHECR
neutrino component; otherwise as in Fig. 1. Figure 8: As in Fig. 7 but for
EPOS-LHC. Figure 9: Predictions of the UHECR source model with the softest
accelerator spectral index $\gamma_{\mathrm{inj}}\lesssim-2$ compatible with
multimessenger data. This fit assumes Sibyll2.3c. A non-UHECR component of
astrophysical neutrinos has been omitted. Left: Same as in Fig. 1. Right: Same
as in Fig. 1.