text:
\section{Introduction}
The category $\Diff$ of diffeological spaces and smooth maps has
excellent properties such as completeness, cocompleteness, and
cartesian closedness, and provides a framework for combining
analysis with geometry and topology.
But there is one barrier to spread the use of
diffeology. Compared with continuous maps, smooth maps are far
more difficult to handle with and it often occurs that we cannot
find an appropriate smooth map that satisfies the given
requirements.
Such a situation typically occurs in analysis, e.g.\ when solving
partial differential equations.
Distributions (or generalized functions) make it possible to
differentiate functions whose derivatives do not exist in the
classical sense (e.g.\ locally integrable functions).
They are widely used in the theory of partial differential
equations, where it may be easier to establish the existence of
weak solutions than classical ones, or appropriate classical
solutions may not exist.
They are also important in physics and engineering where many
problems naturally lead to differential equations whose solutions
or initial conditions are distributions.
The aim of this paper is to bring in such generalized notion of
maps into the framework of diffeology, and utilize them to
investigate diffeological spaces in a more flexible manner.

We now state the main result of the paper.  For diffeological
spaces $X$ and $Y$, we denote by $\Cinfty(X,Y)$ the set of smooth
maps from $X$ to $Y$ equipped with the functional diffeology
(cf.\ \cite{Zemmour}).  A diffeological space is called a
\emph{smooth differential algebra} if it has a structure of a
differential algebra such that its sum, product and partial
differential operators are smooth.
Given an open subset $U$ of $\mathbb{R}^n$, let $\mathcal{D}'(U)$ denote the
locally convex topological vector space of Schwartz
distributions.  Then the main theorem of the paper can be stated
as follows.

\begin{thm}
  \label{thm:main theorem}
  There is a supercategory\/ $\eDiff$\/ of\/ $\Diff$\/ enjoying
  the properties below.  Let\/ $\mathbb{F}$\/ be either $\mathbb{R}$ or $\mathbb{C}$.
  \begin{enumerate}
  \item $\eDiff$ has the same objects as $\Diff$ and is enriched
    over $\Diff$.
  \item $\eDiff$ has all small limits and colimits, and the
    inclusion $\Diff \to \eDiff$ preserves (hence creates) limits
    and colimits.

  \item The space $\eC(X,Y)$ of morphisms from $X$ to $Y$ in
    $\eDiff$ includes $\Cinfty(X,Y)$ as a subspace, and there is
    a natural diffeomorphism
    \[
      \eC(X \times Y,Z) \cong \Cinfty(X,\eC(Y,Z)).
    \]
  \item Let $\widehat{\mathbb{F}} = \eC(\mathbb{R}^0,\mathbb{F})$.  Then $\widehat{\mathbb{R}}$ is a
    non-archimedean real closed field and $\widehat{\mathbb{C}}$ is an
    algebraically closed field of the form
    $\widehat{\mathbb{C}} = \widehat{\mathbb{R}} + \sqrt{-1}\,\widehat{\mathbb{R}}$.
  \item For every $X \in \Diff$, $\eC(X,\mathbb{F})$ is an
    $\widehat{\mathbb{F}}$-algebra containing
    $\Cinfty(X,\mathbb{F})$ as an $\mathbb{F}$-subalgebra.
  \item If $U$ is an open subset of\/ $\mathbb{R}^n$ then $\eC(U,\mathbb{F})$ is
    a smooth differential algebra admitting a continuous linear
    embedding $\mathcal{D}'(U) \to \eC(U,\mathbb{F})$ preserving partial
    derivatives.
  \end{enumerate}
\end{thm}

A member of $\eC(X,Y)$ is called an \emph{asymptotic map} from
$X$ to $Y$, and is called an \emph{asymptotic function} if $Y$ is
$\mathbb{R}$ or $\mathbb{C}$.

The construction of asymptotic functions is based on Colombeau's
theory of simplified algebra \cite{Colombeau} and its variant by
Todorov-Vernaeve \cite{Todorov-Vernaeve}.
We briefly review in Section~2 the basics of Colombeau's
simplified algebra and its reinterpretation as a smooth
differential algebra given by Giordano-Wu~\cite{Giordano-Wu}.  It
is shown that the embedding $\mathcal{D}'(U) \to \bmc{G}^s(U)$ into the
smooth version of Colombeau's simplified algebra is continuous
with respect to the $D$-topology associated with the diffeology
of $\bmc{G}^s(U)$.

Section~3 provides a step-by-step construction of $\eC(X,Y)$.
For a Euclidean open set $U$, the space $\eC(U,\mathbb{F})$ closely
resembles $\bmc{G}^s(U)$ and there is a natural homomorphism
$\bmc{G}^s(U) \to \eC(U,\mathbb{F})$ of smooth differential algebras.
But there is a significant difference between the two: The set of
scalars of Colombeau's theory is not a field but a ring with
(lots of) zero divisors, while on the other hand, our $\widehat{\mathbb{F}}$ is
isomorphic to the scalar field of Todorov-Vernaeve's theory, and
hence to Robinson's field of asymptotic numbers.  Once
$\eC(U,\mathbb{F})$ is established as a functor of Euclidean open sets
$U$, it is straightforward, by the nature of diffeological
spaces, to construct $\eC(X,Y)$ for arbitrary $X$ and $Y$, and to
show that diffeological spaces and asymptotic maps form a
category $\eDiff$ enriched over $\Diff$.

Section~4 completes the proof of the main theorem by verifying
that the remaining properties (2) and (3) of the main theorem
really hold.

Finally, in Section~5 we prove, as an illustrating example of the
use of asymptotic maps, that any smooth relative cell complex
$(X,A)$ enjoys the homotopy extension property in $\eDiff$, that
is, for any asymptotic map $f \colon X \to Y$ and asymptotic
homotopy $f|A \simeq g$ on $A$, there is an asymptotic homotopy
$f \simeq \tilde{g}$ extending $f|A \simeq g$.
Note that the homotopy extension property plays a key role in the
development of homotopy theory.  But it fails if maps and
homotopies are restricted to smooth ones.

The author wish to thank Dan Christensen for pointing out a wrong
statement in the initial version of the manuscript that $\eDiff$
is cartesian closed.
As the property (4) of the main theorem says, $\widehat{\mathbb{F}} = \eC(\pt,\mathbb{F})$
is not isomorphic to $\mathbb{F}$, meaning that $\eDiff$ is not cartesian
closed.

\section{Colombeau algebra of generalized functions}
We briefly recall the basic notations and properties of
simplified Colombeau algebra of generalized functions.
Henceforth, $\Gamma$ denotes the interval $(0,1]$, $\mathbb{F}$ is either
$\mathbb{R}$ or $\mathbb{C}$, and $\mathbb{N}$ is the set of non-negative integers.  If
$X$ is a topological or diffeological space then its underlying
set is denoted by $\abs{X}$.

\subsection{Simplified Colombeau algebra}
Given an open set $U$ in $\mathbb{R}^n$, let
$\mathcal{E}_{\mathbb{F}}(U) = \Cinfty(U,\mathbb{F})^{\Gamma}$ be the set of
$\Gamma$-nets in the locally convex space of smooth functions
$U \to \mathbb{F}$.  Then the simplified Colombeau algebra on $U$ is
defined to be the quotient
\[
  \mathcal{G}^s(U) = \mathcal{M}(\mathcal{E}_{\mathbb{F}}(U))/\mathcal{N}(\mathcal{E}_{\mathbb{F}}(U))
\]
of the subalgebra of \emph{moderate nets} over that of
\emph{negligible nets}.  Here, we say that
$(f_{\epsilon}) \in \mathcal{E}_{\mathbb{F}}(U)$ is \emph{moderate} if it
satisfies the condition
\[
  \forall K \Subset U,\ \forall \alpha \in \mathbb{N}^n,\ \exists m \in
  \mathbb{N},\ \displaystyle\max_{x \in K}\abs{D^{\alpha}f_{\epsilon}(x)}
  = O(1/\epsilon^m)
\]
and is \emph{negligible} if
\[
  \forall K \Subset U,\ \forall \alpha \in \mathbb{N}^n,\ \forall p \in
  \mathbb{N},\ \displaystyle\max_{x \in K}\abs{D^{\alpha}f_{\epsilon}(x)}
  = O(\epsilon^p)
\]
where the notation $K \Subset U$ indicates that $K$ is a compact
subset of $U$.

As described in \cite[Theorem 1.1]{Giordano-Wu}, the
correspondence $U \mapsto \mathcal{G}^s(U)$ is a fine and supple sheaf
of differential algebras, and there is a sheaf embedding of
vector spaces $\iota_U \colon \mathcal{D}'(U) \to \mathcal{G}^s(U)$ preserving
partial derivatives.  Note, however, that the construction is not
canonical because it depends on particular choice of a
$\delta$-net.
\subsection{Colombeau algebra as a diffeological space}
\label{subsec:smooth Colombeau algebra}
Giordano and Wu introduced in \cite[\S{5}]{Giordano-Wu} a
diffeological space structure for $\mathcal{G}^s(U)$ defined as
follows.  Instead of the locally convex topology, let us endow
$\Cinfty(U,\mathbb{F})$ with the functional diffeology (cf.\
\cite{Zemmour}); so that a map $\rho \colon V \to \Cinfty(U,\mathbb{F})$
is a plot for $\Cinfty(U,\mathbb{F})$ if and only if the composition
below is smooth.
\[
  V \times U \xrightarrow{\rho \times 1} \Cinfty(U,\mathbb{F}) \times U
  \xrightarrow{\ev} \mathbb{F}, \quad (x,y) \mapsto \rho(x)(y)
\]
Let $\bmc{E}_{\mathbb{F}}(U) = \Cinfty(U,\mathbb{F})^{\Gamma}$, where $\Gamma$ is
regarded as a discrete diffeological space.  Then moderate nets
and negligible nets form diffeological subspaces
$\mathcal{M}(\bmc{E}_{\mathbb{F}}(U))$ and $\mathcal{N}(\bmc{E}_{\mathbb{F}}(U))$),
respectively, and hence a diffeological quotient
\[
  \bmc{G}^s(U) = \mathcal{M}(\bmc{E}_{\mathbb{F}}(U))/\mathcal{N}(\bmc{E}_{\mathbb{F}}(U)).
\]

\begin{thm}[Giordano-Wu]
  \label{thm:smooth Colombeau algebra}
  The space $\bmc{G}^s(U)$ is a smooth differential algebra, and
  there is a linear embedding
  $\iota_U \colon \mathcal{D}'(U) \to \bmc{G}^s(U)$ enjoying the following
  properties:
  \begin{enumerate}
  \item $\iota_U$ is continuous with respect to the weak dual
    topology on $\mathcal{D}'(U)$ and the $D$-topology on $\bmc{G}^s(U)$.
  \item $\iota_U$ preserves partial derivatives, i.e.\
    $D^{\alpha}(\iota_U(u)) = \iota_U(D^{\alpha}u)$ holds for all
    $u \in \mathcal{D}'(U)$ and $\alpha \in \mathbb{N}^n$.

  \item $\iota_U$ restricts to the embedding of smooth
    differential algebra $\Cinfty(U) \to \bmc{G}^s(U)$ which
    takes $f$ to the class of the constant net with value $f$.
  \end{enumerate}
\end{thm}

\begin{proof}
  That $\bmc{G}^s(U)$ is a smooth differential algebra is proved
  in \cite[Theorem 5.1]{Giordano-Wu}.  The properties (2) and (3)
  follows, respectively, from (vii) and (viii) of \cite[Theorem
  1.1]{Giordano-Wu}.
  %
  To prove (1), let $\bmc{D}(U)'$ be the smooth dual of the
  vector space $\bmc{D}(U)$ of test functions equipped with the
  canonical diffeology (cf.\ \cite[Definition 4.1]{Giordano-Wu}),
  and $(T\bmc{D}(U))'$ be the continuous dual of $T\bmc{D}(U)$.
  Then we have
  \(
    \abs{\mathcal{D}'(U)} = \abs{(T\bmc{D}(U))'} = \abs{\bmc{D}(U)'}
  \)
  by \cite[Lemma 4.7]{Giordano-Wu}, and the inclusion
  \[
    \mathcal{D}'(U) \to (T\bmc{D}(U))'
  \]
  is continuous because the $D$-topology of $\bmc{D}(U)$ is finer
  than the locally convex topology of $\mathcal{D}(U)$ by \cite[Corollary
  4.12 (i)]{Giordano-Wu}.
  %
  Also, the embedding
  $\iota_U \colon \bmc{D}(U)' \to \bmc{G}^s(U)$ is smooth by
  \cite[Theorem 5.1]{Giordano-Wu}, and hence
  \[
    T\iota_U \colon T(\bmc{D}(U)') \to T\bmc{G}^s(U)
  \]
  is continuous.
  %
  Thus, to prove (1) it suffices to show that the inclusion
  \[
    (T\bmc{D}(U))' \to T(\bmc{D}(U)')
  \]
  is continuous.  By definition, $(T\bmc{D}(U))'$ has the
  coresest topology such that the evaluation map
  $(T\bmc{D}(U))' \times T\bmc{D}(U) \to \mathbb{F}$ is continuous.  But
  this is equivalent to say that it has the final topology with
  respect to those parametrizations
  $\sigma \colon V \to \abs{(T\bmc{D}(U))'}$ such that for every
  plot $\rho \colon W \to \bmc{D}(U)$ the composition below is
  continuous.
  \begin{equation}
    \label{eq:condition for continuous dual}
    V \times W \xrightarrow{\sigma \times \rho}
    \abs{(T\bmc{D}(U))'} \times \bmc{D}(U) \xrightarrow{\ev} \mathbb{F}
  \end{equation}
  %
  On the other hand, $T(\bmc{D}(U)')$ has the final topology with
  respect to those
  $\sigma \colon V \to \abs{\bmc{D}(U)'} = \abs{T\bmc{D}(U)'}$
  such that for every plot $\rho \colon W \to \bmc{D}(U)$ the
  composition below is smooth.
  \begin{equation}
    \label{eq:plot for smooth dual}
    V \times W \xrightarrow{\sigma \times \rho}
    \abs{\bmc{D}(U)'} \times \bmc{D}(U) \xrightarrow{\ev} \mathbb{F}
  \end{equation}
  Since smooth maps are continuous, we see that $(T\bmc{D}(U))'$
  has finer topology than $T(\bmc{D}(U)')$.  Hence the inclusion
  $(T\bmc{D}(U))' \to (\bmc{D}(U)')$ is continuous.
\end{proof}

\begin{note}
  The embedding $\iota_U$ preserves the products of smooth maps,
  but does not preserve the products of continuous maps because
  if otherwise, it contradicts to the Schwartz impossibility
  result \cite{Schwartz}.
\end{note}

\section{The space of asymptotic maps}
We assign to any pair of diffeological spaces $(X,Y)$ a
diffeological space $\eC(X,Y)$ consisting of certain generalized
maps which we call \emph{asymptotic maps}.  The construction
consists of three steps: First, we introduce asymptotic functions
on Euclidean open sets, then extend them to the ones on general
diffeological spaces, and finally, construct asymptotic maps
between arbitrary diffeological spaces.

\subsection{Asymptotic functions on Euclidean open sets}
We now modify the definition of $\bmc{G}^s(U)$ so that the
scalars form a true field.  For this purpose, we introduce, as in
the definition of Todorov-Vernaeve's algebra
\cite{Todorov-Vernaeve}, a $\{0,1\}$-valued finitely additive
measure on $\Gamma$ given by an ultrafilter $\mathcal{U}$ generated by the
collection $\{(0,\epsilon) \mid 0 < \epsilon \leq 1\}$.
Observe that $\mathcal{U}$ contains those subsets $A$ for which there is a
subset $B$ such that $B$ has Lebesgue outer measure $0$ and
$A \cup B \supset (0,\epsilon]$ for some $0 < \epsilon \leq 1$.
We say that a predicate $P(\epsilon)$ defined on $\Gamma$ holds
\emph{almost everywhere as $\epsilon \to 0$} (``a.e.'' for short)
if $\{\epsilon \in \Gamma \mid P(\epsilon)\} \in \mathcal{U}$.

As in the previous section, we denote
$\bmc{E}_{\mathbb{F}}(U) = \Cinfty(U,\mathbb{F})^{\Gamma}$ and define
\[
  \eC(U,\mathbb{F}) = \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{F}}(U))/\widehat{\mathcal{N}}(\bmc{E}_{\mathbb{F}}(U))
\]
where $\widehat{\mathcal{M}}(\bmc{E}_{\mathbb{F}}(U))$ consists of \emph{moderate nets}
$(f_{\epsilon}) \in \bmc{E}_{\mathbb{F}}(U)$ satisfying
\[
  \forall K \Subset U,\ %
  \forall \alpha \in \mathbb{N}^n,\ %
  \exists m \in \mathbb{N},\ %
  \displaystyle\max_{x \in K}\abs{D^{\alpha}f_{\epsilon}(x)} =
  O(1/\epsilon^m) \text{ a.e.}
\]
and $\widehat{\mathcal{N}}(\bmc{E}_{\mathbb{F}}(U))$ consists of \emph{negligible nets}
$(f_{\epsilon}) \in \bmc{E}_{\mathbb{F}}(U)$ satisfying
\[
  \forall K \Subset U,\ %
  \forall \alpha \in \mathbb{N}^n,\ %
  \forall p \in \mathbb{N},\ %
  \displaystyle\max_{x \in K}\abs{D^{\alpha}f_{\epsilon}(x)} =
  O(\epsilon^p) \text{ a.e.}
\]
Clearly, $\eC(U,\mathbb{F})$ inherits ring operations from
$\Cinfty(U,\mathbb{F})$ and every smooth map $F \colon W \to U$ between
Euclidean open sets induces a functorial homomorphism of
$\mathbb{F}$-algebras
\[
  F^* \colon \eC(U,\mathbb{F}) \to \eC(W,\mathbb{F}), \quad %
  F^*([f_{\epsilon}]) = [f_{\epsilon} \circ F]
\]
Moreover, by arguing as in \cite[Theorem~5.1]{Giordano-Wu} we see
that $\eC(U,\mathbb{F})$ is a smooth differential algebra with respect to
partial differential operators
\[
  D^{\alpha} \colon \eC(U,\mathbb{F}) \to \eC(U,\mathbb{F}), \quad %
  D^{\alpha}([f_{\epsilon}]) = [D^{\alpha}f_{\epsilon}] \quad
  (\alpha \in \mathbb{N}^n)
\]
and there is a natural inclusion of smooth differential algebras
\[
  i_U \colon \Cinfty(U,\mathbb{F}) \to \eC(U,\mathbb{F})
\]
which takes a smooth map $f$ to the class of the constant net
with value $f$.

In particular, let $\widehat{\mathbb{F}} = \eC(\mathbb{R}^0,\mathbb{F})$ for $\mathbb{F} = \mathbb{R},\,\mathbb{C}$.
The following result is essentially the same as
\cite[Theorem~7.3]{Todorov-Vernaeve}.

\begin{thm}[Todorov-Vernaeve]
  $\widehat{\mathbb{C}}$ is an algebraically closed field of the form
  $\widehat{\mathbb{R}} + \sqrt{-1}\,\widehat{\mathbb{R}}$.
\end{thm}

\begin{proof}
  It is clear that $\widehat{\mathbb{C}}$ is a ring and that we have
  $\widehat{\mathbb{C}} = \widehat{\mathbb{R}} + \sqrt{-1}\,\widehat{\mathbb{R}}$.
  %
  To see that $\widehat{\mathbb{C}}$ is a field, suppose
  $(a_{\epsilon}) \in \mathcal{M}(\mathbb{C}^{\Gamma})$ represents a non-zero
  class in $\widehat{\mathbb{C}}$.  Then there exist $m,\,p \in \mathbb{N}$ such that
  $\Phi = \{ \epsilon \mid \epsilon^p \leq \abs{a_{\epsilon}}
  \leq 1/\epsilon^m \} \in \mathcal{U}$.
  %
  Let us define $(b_{\epsilon}) \in \mathbb{C}^{\Gamma}$ by
  $b_{\epsilon} = 1/a_{\epsilon}$ if $\epsilon \in \Phi$ and
  $b_{\epsilon} = 1$ if otherwise.  Then we have
  $[b_{\epsilon}] = [a_{\epsilon}]^{-1}$ in $\widehat{\mathbb{C}}$ because
  $a_{\epsilon}b_{\epsilon} = 1$ holds almost everywhere.  Hence
  $\widehat{\mathbb{C}}$ is a field.
  %
  To see that $\widehat{\mathbb{C}}$ is algebraically closed, suppose
  \[
    P(x) = x^p + a_1x^{p-1} + \cdots + a_p
  \]
  is a polynomial with coefficients in $\widehat{\mathbb{C}}$, and choose a
  representative $(a_{k,\epsilon}) \in \mathcal{M}(\mathbb{C}^{\Gamma})$ for
  each $a_k \in \widehat{\mathbb{C}}$.  For $0 < \epsilon \leq 1$, let
  \[
    P_{\epsilon}(x) = x^p + a_{1,\epsilon}x^{p-1} + \cdots +
    a_{p,\epsilon}
  \]
  and take $x_{\epsilon} \in \mathbb{C}$ satisfying
  $P_{\epsilon}(x_{\epsilon}) = 0$.  Then
  $(x_{\epsilon}) \in \mathbb{C}^{\Gamma}$ is moderate because
  $\abs{x_{\epsilon}} \leq 1 + \abs{a_{1,\epsilon}} + \cdots +
  \abs{a_{p,\epsilon}}$ holds, and hence we have
  $P([x_{\epsilon}]) = 0$.
\end{proof}

\begin{crl}
  $\widehat{\mathbb{R}}$ is a real closed field and both $\widehat{\mathbb{R}}$ and $\widehat{\mathbb{C}}$ are
  non-archimedean fields in the sense that they contain non-zero
  infinitesimals.
\end{crl}

\begin{proof}
  Since we have $\widehat{\mathbb{C}} = \widehat{\mathbb{R}} + \sqrt{-1}\,\widehat{\mathbb{R}}$ and $\widehat{\mathbb{C}}$ is
  algebraically closed, $\widehat{\mathbb{R}}$ is a real closed field and is
  totally ordered in such a way that $a \geq 0$ if and only if
  $a = b^2$ holds for some $b \in \widehat{\mathbb{R}}$.
  %
  It is non-archimedean because the asymptotic number $\rho$
  represented by the net $(\epsilon)$ is a non-zero infinitesimal
  in $\widehat{\mathbb{R}}$.
\end{proof}

\begin{note}
  The constant $\rho$ is called the \emph{canonical
    infinitesimal} in $\widehat{\mathbb{R}}$.  If we regard $\rho$ as a positive
  infinitesimal in the nonstandard reals
  ${}^{*}\mathbb{R} = \mathbb{R}^{\Gamma}/\!\sim_{\mathcal{U}}$ then $\widehat{\mathbb{C}}$ and $\widehat{\mathbb{R}}$
  are isomorphic to Robinson's fields of $\rho$-asymptotic
  numbers ${}^{\rho}\mathbb{C}$ and ${}^{\rho}\mathbb{R}$, respectively.  For
  details, see Todorov-Vernaeve
  \cite[Section~7]{Todorov-Vernaeve}.
  \end{note}

By the definition, $\mathcal{M}(\bmc{E}_{\mathbb{F}}(U))$ and
$\mathcal{N}(\bmc{E}_{\mathbb{F}}(U))$ are subalgebras of
$\widehat{\mathcal{M}}(\bmc{E}_{\mathbb{F}}(U))$ and $\widehat{\mathcal{N}}(\bmc{E}_{\mathbb{F}}(U))$, respectively.
Hence we have

\begin{prp}
  \label{prp:asymptotic functions vs distributions}
  The natural map $\bmc{G}^s(U) \to \eC(U,\mathbb{F})$ induced by the
  inclusions $\mathcal{M}(\bmc{E}_{\mathbb{F}}(U)) \to \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{F}}(U))$
  and $\mathcal{N}(\bmc{E}_{\mathbb{F}}(U)) \to \widehat{\mathcal{N}}(\bmc{E}_{\mathbb{F}}(U))$ is a
  homomorphism of smooth differential algebras over\/ $\mathbb{F}$.
\end{prp}

By the definition, the composition
$\mathcal{D}'(U) \to \bmc{G}^s(U) \to \eC(U,\mathbb{F})$ is a linear monomorphism.
Thus we have the following by Theorem~\ref{thm:smooth Colombeau
  algebra}.

\begin{crl}
  \label{crl:embedding of distributions}
  There exists a (non-canonical) continuous linear embedding
  $\mathcal{D}'(U) \to \eC(U,\mathbb{F})$ preserving partial derivatives.
\end{crl}

\subsection{Asymptotic functions on general diffeological spaces}
Given an arbitrary diffeological space $X$ with diffeology
$\mathcal{D}_X$, we define
\[
  \eC(X,\mathbb{F}) = \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{F}}(X))/ \widehat{\mathcal{N}}(\bmc{E}_{\mathbb{F}}(X))
\]
where $\bmc{E}_{\mathbb{F}}(X) = \Cinfty(X,\mathbb{F})^{\Gamma}$ and for
$\mathcal{L} = \mathcal{M},\; \mathcal{N}$ we put
\[
  \widehat{\mathcal{L}}(\bmc{E}_{\mathbb{F}}(X)) = \{ (f_{\epsilon}) \in
  \bmc{E}_{\mathbb{F}}(X) \mid (f_{\epsilon} \circ \rho) \in
  \widehat{\mathcal{L}}(\bmc{E}_{\mathbb{F}}(U))\ \text{for

    $\forall\rho \colon U \to X \in \mathcal{D}_X$} \}
\]
One easily observes that if $X$ is an open set in $\mathbb{R}^n$ then
$\eC(X,\mathbb{F})$ is identical with the algebra constructed in the
previous step.
Also, there is a natural inclusion
$i_X \colon \Cinfty(X,\mathbb{F}) \to \eC(X,\mathbb{F})$ which takes
$f \in \Cinfty(X,\mathbb{F})$ to the class of the constant net with value
$f$.  Evidently, we have the following.

\begin{prp} For every $X \in \Diff$, $\eC(X,\mathbb{F})$ is an algebra
  over the field\/ $\widehat{\mathbb{F}}$, and the following hold.
  \begin{enumerate}
  \item Any smooth map $F \colon X \to Y$ induces a functorial
    homomorphism of\/ $\widehat{\mathbb{F}}$-algebras
    $F^* \colon \eC(Y,\mathbb{F}) \to \eC(X,\mathbb{F})$.
  \item The natural inclusion
    $i_X \colon \Cinfty(X,\mathbb{F}) \to \eC(X,\mathbb{F})$ is a homomorphism
    of\/ $\mathbb{F}$-algebras.
  \end{enumerate}
\end{prp}

\subsection{Asymptotic maps between diffeological spaces}
Given diffeological spaces $X$ and $Y$, a net
$(f_{\epsilon}) \in \Cinfty(X,Y)^{\Gamma}$ is said to be
\emph{moderate} if it satisfies the condition:
\[
  (u_{\epsilon} \circ f_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(X))
  \quad (\,\forall (u_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(Y))\,)
\]
Let $\widehat{\mathcal{M}}(\Cinfty(X,Y)^{\Gamma})$ be the subspace consisting of
moderate nets in $\Cinfty(X,Y)$.

\begin{lmm}
  \label{lmm:composition of nets}
  If $(f_{\epsilon}) \in \widehat{\mathcal{M}}(\Cinfty(X,Y)^{\Gamma})$ and
  $(g_{\epsilon}) \in \widehat{\mathcal{M}}(\Cinfty(Y,Z)^{\Gamma})$ then we have
  $(g_{\epsilon} \circ f_{\epsilon}) \in
  \widehat{\mathcal{M}}(\Cinfty(X,Z)^{\Gamma})$.
\end{lmm}

\begin{proof}
  Because $(g_{\epsilon})$ is moderate we have
  $(u_{\epsilon} \circ g_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(Y))$
  for every $(u_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(Z))$.  But then
  we have
  $(u_{\epsilon} \circ g_{\epsilon} \circ f_{\epsilon}) \in
  \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(X))$, and hence
  $(g_{\epsilon} \circ f_{\epsilon}) \in
  \widehat{\mathcal{M}}(\Cinfty(X,Z)^{\Gamma})$.
\end{proof}

We define the space of \emph{asymptotic maps} from $X$ to $Y$ as
the quotient
\[
  \eC(X,Y) = \widehat{\mathcal{M}}(\Cinfty(X,Y)^{\Gamma})/\sim
\]
where $(f_{\epsilon})$ and $(f'_{\epsilon})$ are equivalent in
$\widehat{\mathcal{M}}(\Cinfty(X,Y)^{\Gamma})$ if
\[
  ((u_{\epsilon} \circ f'_{\epsilon} - u_{\epsilon} \circ
  f_{\epsilon}) \circ \rho_{\epsilon}) \in \widehat{\mathcal{N}}(\bmc{E}_{\mathbb{R}}(U))
\]
holds for every $(u_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(Y))$ and
$(\rho_{\epsilon}) \in \widehat{\mathcal{M}}(\Cinfty(U,X)^{\Gamma})$.

\begin{note}
  Even if we replace $\bmc{E}_{\mathbb{R}}(-)$ in the definition above
  with $\bmc{E}_{\mathbb{C}}(-)$, the resulting quotient space is the
  same as $\eC(X,Y)$.
\end{note}

\begin{prp}
  There exists a supercategory\/ $\eDiff$ of\/ $\Diff$ which has
  the same objects as $\Diff$ and is enriched over $\Diff$ with
  $\eC(X,Y)$ as the space of morphisms from $X$ to $Y$.
\end{prp}

\begin{proof}
  For given $X,\; Y,\;Z \in \Diff$ consider the smooth map
  \[
    \Cinfty(Y,Z)^{\Gamma} \times \Cinfty(X,Y)^{\Gamma} \to
    \Cinfty(X,Z)^{\Gamma}
  \]
  which takes a pair $((g_{\epsilon}),(f_{\epsilon}))$ to the
  levelwise composition $(g_{\epsilon} \circ f_{\epsilon})$.
  %
  By Lemma~\ref{lmm:composition of nets}, the map above restricts
  to a smooth map
  \[
    c_{X,Y,Z} \colon \widehat{\mathcal{M}}(\Cinfty(Y,Z)^{\Gamma}) \times
    \widehat{\mathcal{M}}(\Cinfty(X,Y)^{\Gamma}) \to \widehat{\mathcal{M}}(\Cinfty(X,Z)^{\Gamma}).
  \]
  Moreover, if $(f_{\epsilon}) \sim (f'_{\epsilon})$ and
  $(g_{\epsilon}) \sim (g'_{\epsilon})$ hold in
  $\widehat{\mathcal{M}}(\Cinfty(X,Y)^{\Gamma})$ and $\widehat{\mathcal{M}}(\Cinfty(Y,Z)^{\Gamma})$,
  respectively, then we have
  $(g_{\epsilon} \circ f_{\epsilon}) \sim (g'_{\epsilon} \circ
  f'_{\epsilon})$ because
  \begin{multline*}
    ((u_{\epsilon} \circ (g'_{\epsilon} \circ f'_{\epsilon}) -
    u_{\epsilon} \circ (g_{\epsilon} \circ f_{\epsilon})) \circ
    \rho_{\epsilon})
    \\
    = ((u_{\epsilon} \circ (g'_{\epsilon} \circ f'_{\epsilon}) -
    u_{\epsilon} \circ (g_{\epsilon} \circ f'_{\epsilon}) +
    u_{\epsilon} \circ (g_{\epsilon} \circ f'_{\epsilon}) -
    u_{\epsilon} \circ (g_{\epsilon} \circ f_{\epsilon})) \circ
    \rho_{\epsilon})
    \\
    = ((u_{\epsilon} \circ g'_{\epsilon} - u_{\epsilon} \circ
    g_{\epsilon}) \circ (f'_{\epsilon} \circ \rho_{\epsilon})) +
    (((u_{\epsilon} \circ g_{\epsilon}) \circ f'_{\epsilon} -
    (u_{\epsilon} \circ g_{\epsilon}) \circ f_{\epsilon}) \circ
    \rho_{\epsilon})
  \end{multline*}
  is negligible for every
  $(u_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(Z))$ and
  $(\rho_{\epsilon}) \in \widehat{\mathcal{M}}(\Cinfty(U,X)^{\Gamma})$.
  %
  Hence $c_{X,Y,Z}$ induces a smooth composition
  \[
    \eC(Y,Z) \times \eC(X,Y) \to \eC(X,Z).
  \]
  It is now evident that we can define $\eDiff$ to be the
  category enriched over $\Diff$ with diffeological spaces as
  objects and with $\eC(X,Y)$ as the space of morphisms from $X$
  to $Y$.
\end{proof}

\section{Proof of the main theorem}
We have already shown that $\eDiff$ satisfies all the properties
stated in Theorem~\ref{thm:main theorem} except for (2) and (3).

To prove that $\eDiff$ has all limits, it suffices to show that
it has all small products and equalizers.
Given a family of diffeological spaces $\{X_j\}_{j \in J}$, let
$X = \prod_{j \in J}X_j$ be their product in $\Diff$.  Then $X$
is a product even in the supercategory $\eDiff$.  To see this,
suppose $\{f_j \colon L \to X_j\}$ is a family of asymptotic maps
and let $(f_{j,\epsilon})$ be a representative for each $f_j$.
Then there exist unique smooth maps $u_{\epsilon} \colon L \to X$
satisfying $f_{j,\epsilon} = p_j \circ u_{\epsilon}$, where
$p_j \colon X \to X_j$ is the projection.  But then,
$(u_{\epsilon}) \in \Cinfty(L,X)^{\Gamma}$ is moderate and
induces a unique asymptotic map
$u = [u_{\epsilon}] \colon L \to X$ satisfying
$f_j = p_j \circ u$.  Hence $\eDiff$ has all small products.
To see that equalizers exist in $\eDiff$, suppose $(f,g)$ is a
pair of asymptotic maps from $X$ to $Y$.  Let $(f_{\epsilon})$
and $(g_{\epsilon})$ be representatives for $f$ and $g$,
respectively, and put
\[
  \Eq(f,g) = \{ x \in X \mid f_{\epsilon}(x) = g_{\epsilon}(x) \
  \text{a.e.} \}.
\]
Clearly, this definition does not depend on the choice of
representatives for $f$ and $g$, and the inclusion
$\Eq(f,g) \to X$ is an equalizer for $(f,g)$ because any
asymptotic map $k \colon Z \to X$ satisfying
$f \circ k = g \circ k$ factors through $\Eq(f,g)$.
Therefore, $\eDiff$ has all small limits and the inclusion
$\Diff \to \eDiff$ preserves limits because it preserves products
and equalizers.

Similarly, we can prove that $\eDiff$ has all colimits by showing
that it has all coproducts and coequalizers.  Given a family of
diffeological spaces $\{X_j\}_{j \in J}$, their coproduct
$\coprod_{j \in J}X_j$ in $\Diff$ is a coproduct in the
supercategory $\eDiff$, and any pair of asymptotic maps $(f,g)$
from $X$ to $Y$ has a coequalizer $Y \to \Coeq(f,g)$ defined as
the projection of $Y$ onto its quotient space by the least
equivalence relation such that $y \sim y'$ if both
$y = f_{\epsilon}(x)$ a.e.\ and $y' = g_{\epsilon}(x)$ a.e.\ hold
for some $x \in X$.
Again, the inclusion $\Diff \to \eDiff$ preserves colimits
because it preserves coproducts and coequalizers.  Thus we see
that the property (2) holds.

To prove (3), consider the natural isomorphism
\[
  \gamma \colon \Cinfty(X \times Y,Z)^{\Gamma} \to
  \Cinfty(X,\Cinfty(Y,Z)^{\Gamma})
\]
We first show that $\gamma$ restricts to an isomorphism
\begin{equation}
  \label{eq:iso between moderate nets}
  \widehat{\mathcal{M}}(\Cinfty(X \times Y,Z)^{\Gamma}) \cong
  \Cinfty(X,\widehat{\mathcal{M}}(\Cinfty(Y,Z)^{\Gamma}))
\end{equation}
Let $(f_{\epsilon}) \in \Cinfty(X \times Y,Z)^{\Gamma}$ and
$(g_{\epsilon}) = \gamma((f_{\epsilon})) \in
\Cinfty(X,\Cinfty(Y,Z)^{\Gamma})$.  Then for any
$(u_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(Z))$ and
$\rho = (\rho',\rho'') \in \Cinfty(U,X \times Y)$, we have
\[
  u_{\epsilon}(f_{\epsilon}(\rho(t))) =
  u_{\epsilon}(g_{\epsilon}(\rho'(t))(\rho''(t))) \quad (t \in U)
\]
In particular, by taking $\rho'$ to be the constant map with
value $x$, we see that if $(f_{\epsilon})$ is moderate then so is
$(g_{\epsilon}(x))$ for every $x \in X$.  Hence $\gamma$ maps
$\widehat{\mathcal{M}}(\Cinfty(X \times Y,Z)^{\Gamma})$ into
$\Cinfty(X,\widehat{\mathcal{M}}(\Cinfty(Y,Z)^{\Gamma}))$.
Conversely, we can show that
$(g_{\epsilon}) \in \Cinfty(X,\widehat{\mathcal{M}}(\Cinfty(Y,Z)^{\Gamma})$ implies
$(f_{\epsilon}) \in \widehat{\mathcal{M}}(\Cinfty(X \times Y,Z)^{\Gamma})$ as
follows.  Let
\[
  G_{\epsilon}(s,t) =
  u_{\epsilon}(g_{\epsilon}(\rho'(s))(\rho''(t))) \quad (s,\,t
  \in U)
\]
Then, for any $K \Subset U$, $\alpha \in \mathbb{N}^{\dim U}$, and
$s_0 \in K$ there exists an $m \in \mathbb{N}$ such that the following
holds for $s = s_0$:
\begin{equation}
  \label{eq:estimation for DG}
  \max_{t \in K}\abs{D^{\alpha}G_{\epsilon}(s,t)} =
  O(1/\epsilon^m) \text{ a.e.}
\end{equation}
But then, (\ref{eq:estimation for DG}) holds for all $s$
sufficiently near $s_0$ because $(D^{\alpha}G_{\epsilon}(s,t))$
is smooth with respect to $s$.  It follows, by the compactness of
$K$, that (\ref{eq:estimation for DG}) holds for all $s \in K$
with sufficiently large $m$.
In particular, as we have
$u_{\epsilon}(f_{\epsilon}(\rho(t))) = G_{\epsilon}(t,t)$, the
composite $u_{\epsilon} \circ f_{\epsilon} \circ \rho$ is
moderate for all $u_{\epsilon}$ and $\rho$.  Hence
$(f_{\epsilon})$ is moderate and $\gamma$ restricts to an
isomorphism (\ref{eq:iso between moderate nets}).

Now, let
$(f_{\epsilon}),\, (f'_{\epsilon}) \in \widehat{\mathcal{M}}(\Cinfty(X \times
Y,Z)^{\Gamma})$ and
$(g_{\epsilon}) = \gamma((f_{\epsilon}))$,
$(g'_{\epsilon}) = \gamma((f'_{\epsilon}))$.  Then we have
\[

  u_{\epsilon}(f'_{\epsilon}(\rho_{\epsilon}(t))) -
  u_{\epsilon}(f_{\epsilon}(\rho_{\epsilon}(t))) =
  u_{\epsilon}(g'_{\epsilon}(\rho'_{\epsilon}(t))(\rho''_{\epsilon}(t)))
  -
  u_{\epsilon}(g_{\epsilon}(\rho'_{\epsilon}(t))(\rho''_{\epsilon}(t)))
\]
for every $(u_{\epsilon}) \in \widehat{\mathcal{M}}(\bmc{E}_{\mathbb{R}}(Z))$ and
$(\rho_{\epsilon}) = ((\rho'_{\epsilon},\rho''_{\epsilon})) \in
\widehat{\mathcal{M}}(\Cinfty(U,X \times Y)^{\Gamma})$.
From this, we see that $(f_{\epsilon}) \sim (f'_{\epsilon})$
holds if and only if $(g_{\epsilon}(x)) \sim (g'_{\epsilon}(x))$
holds for every $x \in X$.  Therefore, $\gamma$ induces a natural
isomorphism
\[
  \eC(X \times Y,Z) \cong \Cinfty(X,\eC(Y,Z))
\]

\begin{note}
  Let $\widehat{X} = \eC(\mathbb{R}^0,X)$ for any $X \in \Diff$.  Then, by
  taking $Y = \mathbb{R}^0$ in the isomorphism above, we obtain
  $\eC(X,Z) \cong \Cinfty(X,\widehat{Z})$.  In particular, we have
  $\eC(X,\mathbb{F}) \cong \Cinfty(X,\widehat{\mathbb{F}})$, meaning that
  \emph{$\mathbb{F}$-valued asymptotic function} is the same thing as
  \emph{$\widehat{\mathbb{F}}$-valued smooth function}.
\end{note}

\section{Smooth cell complexes and asymptotic homotopy}
Asymptotic maps can be expected to provide ``weak solutions'' to
such problems that are hard to solve within the usual framework
of smooth maps.  In this section, we present a modest application
related to homotopy theory of diffeological spaces.

Let $I$ be the interval $[0,1]$ equipped with standard
diffeology.  As in the case of $\Top$ or $\Diff$, there is a
notion of homotopy in the category $\eDiff$: a \emph{homotopy}
between two asymptotic maps $f,\, g \colon X \to Y$ is an
asymptotic map $H \colon X \times I \to Y$ satisfying
$H \circ i_0 = f$ and $H \circ i_1 = g$, where $i_{\alpha}$ is
the inclusion $X \to X \times \{\alpha\} \subset X \times I$ for
$\alpha = 0,\,1$.
Denote by $L^n$ the subspace
$\partial I^n \times I \cup I^n \times \{0\}$ of $I^{n+1}$.

\begin{lmm}
  \label{lmm:asymptotic deformation retraction}
  $L^n$ is a deformation retract of $I^{n+1}$ in $\eDiff$.
\end{lmm}

\begin{proof}
  It is easy to construct a piecewise linear retraction
  $r \colon I^{n+1} \to L^n$ by using suitable polyhedral
  decomposition of $I^{n+1}$.
  %
  Let $r_i$ be the $i$-th component function of $r$.  Then there
  exists for each $i$ a moderate net $(R_{i,\epsilon})$ which
  represents the image of $r_i$ under the embedding of continuous
  functions into asymptotic functions.
  %
  Now, let
  $R_{\epsilon} = (R_{1,\epsilon},\cdots,R_{n+1,\epsilon})$.
  Then the net $(R_{\epsilon})$ is moderate, and represents an
  asymptotic retraction $R \colon I^{n+1} \to L^n$.
  %
  Moreover, there is a retracting homotopy
  $H \colon I^{n+1} \times I \to I^{n+1}$ between $R$ and the
  identity represented by a net $(h_{\epsilon})$, where
  \[
    h_{\epsilon}(u,t) = (1-t)R(u) + tu, \ (u,t) \in I^{n+1}
    \times I
  \]
  showing that $L^n$ is a deformation retract of $I^{n+1}$.
\end{proof}

\begin{crl}
  \label{crl:extendability of asymptotic maps}
  Any asymptotic map $L^n \to X$ can be extended to an asymptotic
  map $I^{n+1} \to X$.
\end{crl}

\begin{dfn}
  \label{dfn:relative cell complex}

  Let $(X,A)$ be a pair of diffeological spaces.  We say that
  $(X,A)$ is a \emph{smooth relative cell complex} if there is an
  ordinal $\delta$ and a $\delta$-sequence
  $Z \colon \delta \to \Diff$ such that the inclusion
  $i \colon A \to X$ coincides with the composition
  $Z_{0} \to \mbox{colim}Z$ and for each successor ordinal
  $\beta < \delta$, there exists a smooth map
  $\phi_{\beta} \colon \partial I^n \to Z_{\beta-1}$ (called an
  \emph{attaching map}) such that $Z_{\beta}$ is diffeomorphic to
  the adjunction space $Z_{\beta-1} \cup_{\phi_{\beta}} I^n$
  i.e.\ a pushout of
  \[
    Z_{\beta-1} \xleftarrow{\phi_{\beta}} \partial I^n \hookrightarrow
    I^n
  \]
  In particular, if $A = \emptyset$ then $X$ is called a
  \emph{smooth cell complex}.
\end{dfn}

The next theorem says that every smooth relative cell complex
enjoys homotopy extension property in the extended category
$\eDiff$.

\begin{thm}
  \label{thm:HEP}
  Let $(X,A)$ be a smooth relative cell complex, and
  $f \colon X \to Y$ be an asymptotic map.  Suppose there is an
  asymptotic homotopy $h \colon A \times I \to Y$ satisfying
  $h_0 = f|A$.  Then there exists an asymptotic homotopy
  $H \colon X \times I \to Y$ satisfying $H_0 = f$ and
  $H|A \times I = h$.
\end{thm}

To prove this, we need the lemma below (cf.\
\cite[Lemma~4.8]{model_cat}).

\begin{lmm}
  \label{lmm:subduction of pushout}
  Let $\phi \colon \partial I^n \to Y$ be a smooth map, and
  $Z = Y \cup_\phi I^n$ be the adjunction space given by $\phi$
  and the inclusion $\partial I^n \to I^n$.  Then the map
  \[
    \textstyle i \times 1 \bigcup \Phi \times 1 \colon Y \times I
    \coprod I^n \times I \to Z \times I
  \]
  induced by the natural maps $i \colon Y \to Z$ and
  $\Phi \colon I^n \to Z$ is a subduction.
\end{lmm}

\begin{proof}
  Let $P \colon U \to Z \times I$ be a plot of $Z \times I$ given
  by $P(r) = (\sigma(r),\sigma'(r))$, and let $r \in U$.  Since
  $\sigma$ is a plot of $Z$, there exists a plot
  $Q_1 \colon V \to Y$ such that $\sigma |V = i \circ Q_1$ holds,
  or a plot $Q_2 \colon V \to I^n$ such that
  $\sigma|V = \Phi \circ Q_2$ holds.  In either case, we have
  \[
    \textstyle P|V = (i \times 1 \bigcup \Phi \times 1) \circ
    (Q_{\alpha},\sigma')|V,
  \]
  where $\alpha$ is either $1$ or $2$.  This means that
  $i \times 1 \bigcup \Phi \times 1$ is a subduction.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:HEP}]
  Let $Z \colon \delta \to \Diff$ be a $\delta$-sequence such
  that the composition $Z_0 \to \colim Z$ is the inclusion
  $A \to X$.  We construct an asymptotic homotopy
  $H \colon X \times I \to Y$ by transfinite induction on
  $\beta < \delta$.
  %
  Suppose $\beta$ is a successor ordinal, and there is an
  asymptotic homotopy
  $H_{\beta-1} \colon Z_{\beta-1} \times I \to Y$ such that the
  following hold.
  \[
    H_{\beta-1}|X \times \{0\} = f|Z_{\beta-1}, \quad
    H_{\beta-1}|A \times I = h
  \]
  Let $i_{\beta} \colon Z_{\beta-1} \to Z_{\beta}$ be the
  inclusion.  Then we have a commutative diagram
  \[
    \xymatrix@C=46pt@R=36pt{%
      \partial I^n \times I \cup I^n \times \{0\} \ar[r]^-{(\phi_{\beta}
        \times 1) \cup \Phi_{\beta}} \ar[d] & Z_{\beta-1} \times
      I \cup \Phi_{\beta}(I^n) \times \{0\} \ar[r]^-{H_{\beta-1}
        \cup f} \ar[d] & Y
      \\
      I^n \times I \ar[r]^-{\Phi_{\beta} \times 1} \ar@{.>}[urr]
      & Z_{\beta} \times I \ar@{.>}[ur] }%
  \]
  By Corollary~\ref{crl:extendability of asymptotic maps} the
  composition of upper arrows can be extended to an asymptotic
  map $K_{\beta} \colon I^n \times I \to Y$.
  %
  In order to further extend $K_{\beta}$ over
  $Z_{\beta} \times I$, let us take representatives
  $(H_{\beta-1,\epsilon})$ for $H_{\beta-1}$ and
  $(K_{\beta,\epsilon})$ for $K_{\beta}$.
  %
  Then for each $\epsilon > 0$, there is a unique map
  $H_{\beta,\epsilon} \colon Z_{\beta} \times I \to Y$ such that
  \[
    \xymatrix@C=50pt{%
      Z_{\beta-1} \times I \coprod I^n \times I
      \ar[r]^-{H_{\beta-1,\epsilon} \bigcup K_{\beta,\epsilon}}
      \ar[d]_-{i_{\beta} \times 1 \bigcup \Phi_{\beta} \times 1}
      & Y
      \\
      Z_{\beta} \times I \ar[ru]_-{H_{\beta,\epsilon}} }%
  \]
  commutes.  But then $H_{\beta,\epsilon}$ is smooth because
  $i_{\beta} \times 1 \bigcup \Phi_{\beta} \times 1$ is a
  subduction by Lemma~\ref{lmm:subduction of pushout}.  The
  resulting net $(H_{\beta,\epsilon})$ is moderate and defines an
  asymptotic map $H_{\beta} \colon Z_{\beta} \times I \to Y$
  satisfying
  $H_{\beta} \circ (\Phi_{\beta} \times 1) = K_{\beta}$.
  %
  Thus, by applying transfinite induction on the
  $\delta$-sequence $Z$, we obtain an asymptotic homotopy
  $H \colon X \times I \to Y$ extending
  $h \colon A \times I \to Y$.
\end{proof}

\providecommand{\bysame}{\leavevmode\hbox
  to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}


text:
\section{Introduction}
\label{sec:intro}

The detection of MeV neutrinos from the Sun enabled direct observations of nuclear reactions in the core of the Sun, as well as studies of fundamental neutrino properties (Bahcall 2005). Existing MeV neutrino "telescopes" are also capable of detecting neutrinos from supernova explosions in our local Galactic neighborhood, at distances $<100$~kpc, such as supernova 1987A. The detection of neutrinos emitted by SN1987A provided a direct observation of the core collapse process and constraints on neutrino properties (Raffelt 2005).

MeV neutrino detectors are unlikely to enable the detection of an extra-Galactic neutrino signal. In order to extend the distance accessible to neutrino astronomy to extra-Galactic scales, high energy, $>1$~TeV, neutrino telescopes are being constructed (Halzen 2005). The advantage in going to high energy is twofold: The neutrino interaction cross section increases with energy, and the construction of larger detectors is possible at higher energy.

The target sources of high energy neutrino telescopes are different than those of low energy neutrino telescopes, since, unlike MeV neutrinos, TeV neutrinos are not produced in thermonuclear interactions in stars. $>1$~TeV neutrino telescopes are aiming at detecting "cosmic accelerators," which produce high energy particles
(Waxman 2005). The existence of extra-Galactic high-energy neutrino sources is implied by the existence of cosmic-rays with energies extending to $>10^{20}$~eV: The $>10^{19}$~eV cosmic-ray flux is likely dominated by extra-Galactic sources of protons (Nagano \& Watson 2000) (see, however, Watson 2004), and the interaction of such high energy protons with radiation fields or gas (either near or far from the source) leads to the production of charged pions which decay to produce neutrinos (e.g. $p+\gamma\rightarrow n+\pi^+$, $\pi^+\rightarrow\mu^++\nu_\mu\rightarrow e^++\nu_\mu+\bar{\nu}_\mu+\nu_e$).

The origin of the highest energy, $>10^{19}$~eV, cosmic rays (UHECRs) is a mystery: The identity of the UHECR sources and the mechanism(s) by which particles are accelerated to such extreme energies are unknown. Identifying the sources and understanding the acceleration mechanism are among the most interesting and important challenges of high energy astrophysics (Bhattacharjee \& Sigl 2000, Waxman 2004a). Identifying the UHECR sources and probing the physical mechanisms driving them are among the main science targets of high energy neutrino detectors (Waxman 2005). These issues would not be discussed, however, in detail in this review, which is focused on the relation between neutrino astronomy and gamma-ray bursts (GRBs).

A phenomenological, model independent discussion of extra-Galactic high energy neutrino sources is presented in \S~\ref{sec:WB}. It is shown that the UHECR observations set an upper bound to the extra-Galactic high energy neutrino flux. The experimental implications of this bound and the current state of the experimental effort are briefly reviewed in \S~\ref{sec:nu_tel}. High energy neutrino production in GRBs is discussed in \S~\ref{sec:GRBnu}. Testing for fundamental neutrino properties using GRB neutrinos is discussed in \S~\ref{sec:nu_phys}. A summary and a brief discussion of future prospects is given in \S~\ref{sec:discussion}.

\section{Cosmic accelerators and the neutrino background}
\label{sec:WB}

The cosmic-ray spectrum extends to energies $>10^{20}$~eV. The flux at the highest observed energies is rather low,
\begin{equation}\label{eq:jCR}
    j_{CR}(>10^{20}{\rm eV})\approx 1/{\rm km^2}/100{\rm yr}/(2\pi\,{\rm sr}).
\end{equation}
The flattening of the spectrum at $\sim10^{19}$~eV and the isotropic distribution of arrival directions suggest that the cosmic-ray flux above $\sim10^{19}$~eV is dominated by extra-Galactic sources (Nagano \& Watson 2000). There is some evidence that the highest energy particles are light nucleons (presumably protons), but the identity of the cosmic-ray primaries is still debated (e.g. Watson 2004). Under the assumption that the UHECRs are extra-Galactic protons, the observed flux and spectrum of UHECRs determines the rate (per unit time and volume) and the spectrum with which high energy protons are produced by extra-Galactic sources to be (Waxman 1995b, Bahcall \& Waxman 2003)
\begin{equation}
    E_p^2\frac{d\dot{N}_p}{dE_p}=0.65\times 10^{44} {\rm erg~Mpc^{-3}~yr^{-1}}\phi(z).
\label{eq:energyrate}
\end{equation}
Here, $\phi(z)$ accounts for redshift evolution and $\phi(z=0)=1$. The flux and spectrum above $10^{19}$~eV are only weakly dependent on $\phi(z)$ since proton energy loss limits the protons' propagation distance.
Protons of energy exceeding the threshold for pion production in interaction with the cosmic microwave background photons, $\sim5\times10^{19}$~eV, lose most of their energy over a time short compared to the age of the universe, or, equivalently, over a propagation distance $d_{\rm GZK}\simeq100$~Mpc, which is short compared to the Hubble distance (the "GZK effect"; Greisen 1966, Zatsepin \& Kuzmin 1966).

The identity of the UHECR sources as well as the acceleration mechanism are not yet known. The stringent constraints, which are imposed on the properties of possible UHECR sources by the high energies observed, rule out almost all source candidates, and suggest that GRBs and active galactic nuclei (AGN) are the most plausible sources (Bhattacharjee \& Sigl 2000, Waxman 2004a). Active galactic nuclei are disfavored as sources of UHECRs, since no AGN powerful enough to allow acceleration to $>10^{20}$~eV is present out to a distance $\sim d_{\rm GZK}$ from earth. The suggested association between GRB and UHECR sources is based on two arguments (Waxman 1995a, Waxman 2004b): (i) The constraints that UHECR sources must satisfy to allow proton acceleration to $>10^{20}$~eV are similar to those inferred for GRB sources from $\gamma$-ray observations, and (ii) The average energy generation rate of UHECR's is similar to the $\gamma$-ray generation rate of GRBs (Associations based on different arguments were suggested in Milgrom \& Usov 1995 and in Vietri 1995).

Adopting the hypothesis that UHECRs are extra-Galactic protons, an upper bound to the extra-Galactic high energy neutrino intensity may be obtained by the following argument. Let us consider first the neutrino intensity that would have been produced had the high energy cosmic-ray protons lost all their energy to pion production in $p\gamma$ or $pp(n)$ interactions. In this case, the resulting intensity of high energy muon neutrinos (including muon and anti-muon neutrinos, neglecting mixing) would have been (Waxman \& Bahcall 1999, Bahcall \& Waxman 2001)
\begin{equation}
E_\nu^2\Phi_\nu=E_\nu^2\Phi_\nu^{WB}=
\frac{1}{4}\frac{c}{4\pi}E_p^2\left(\frac{d\dot{N}_p}{dE_p}\right)_{z=0}\xi_z t_H.
\label{eq:WB}
\end{equation}
Here $t_H$ is the Hubble time and the factor of $1/4$ accounts for the fact that $\approx1/4$ of the pions' energy is converted to muon neutrino energy (since charged pions and neutral pions are produced with roughly equal probability, and $\approx1/2$ of the charged pion energy is channelled into muon neutrinos). $\xi_z$ is (a dimensionless parameter) of order unity, which depends on the redshift evolution of $E_p^2d\dot{N}_p/dE_p$ (see eq.~\ref{eq:energyrate}). $\Phi_\nu^{WB}$ is an upper bound to the neutrino intensity produced by sources which, like GRBs and AGN jets, are optically thin for high-energy nucleons to $p\gamma$ and $pp(n)$ interactions (since in such sources protons lose only a small fraction of their energy to pion production before escaping the source). For sources of this type, therefore,
\begin{equation}
\label{eq:WB_bound}
E_\nu^2\Phi_\nu<E_\nu^2\Phi_\nu^{WB}
=2\times10^{-8}\xi_z\left[\frac{(E_p^2d\dot{N}_p/dE_p)_{z=0}}{10^{44}{\rm erg/Mpc^3yr}}\right]
{\rm GeV\,cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}.
\end{equation}
In order to obtain a conservative upper bound, we adopt $(E_p^2d\dot{N}_p/dE_p)_{z=0}=10^{44}{\rm erg/Mpc^3yr}$ and a rapid redshift evolution, $\phi(z)=(1+z)^3$ up to $z=2$, following the evolution of star formation rate. This evolution yields $\xi_z\approx3$.

\section{Neutrino telescopes}
\label{sec:nu_tel}

The UHECR upper bound on the extra-Galactic neutrino intensity, eq.~(\ref{eq:WB_bound}), is compared in fig.~\ref{fig:WBbound} with current experimental limits, and with the expected sensitivity of planned neutrino telescopes. The figure indicates that km-scale (i.e. giga-ton) neutrino telescopes are needed to detect the expected extra-Galactic flux in the energy range of $\sim1$~TeV to $\sim1$~PeV. Much larger effective volume is required to detect the flux at higher energy.
\begin{figure}[htbp]
\epsfxsize=12cm
\centerline{\epsfbox{fig1.eps}}
\caption{The upper bound imposed by UHECR observations on the extra-Galactic high energy muon neutrino intensity (lower-curve: no evolution of the energy production rate, upper curve: assuming evolution following star formation rate), compared with the atmospheric neutrino background and with the experimental upper bounds (solid lines) of optical Cerenkov experiments, BAIKAL (Balkanov et al. 2002) and AMANDA (Ahrens et al. 2003; Halzen 2005), and of coherent Cerenkov radio experiments (RICE, Kravchenko et al. 2003; GLUE, Gorham et al. 2004, Silvestri et al. 2004). The dotted curve is the maximum contribution due to possible extra-Galactic component of lower-energy, $<10^{17}$~eV, protons, that may be "hidden under"  the Galactic heavy nuclei flux (Waxman \& Bahcall 1999, Bahcall \& Waxman 2001). The curve labelled "GZK" shows the intensity due to interaction with micro-wave background photons. Dashed curves show the expected sensitivity of 0.1~Gton (AMANDA, ANTARES, NESTOR) and 1~Gton (IceCube, km3net) optical Cerenkov detectors (Halzen 2005), of the coherent radio Cerenkov (balloon) experiment ANITA (Silvestri et al. 2004) and of the Auger air-shower detector (sensitivity to $\nu_\tau$) (Saltzberg 2005). Space air-shower detectors (OWL-AIRWATCH) may also achieve the sensitivity required to detect fluxes lower than the WB bound at energies $>10^{18}$~eV (Saltzberg 2005).}
\label{fig:WBbound}
\end{figure}

The AMANDA detector operating in the south pole has already achieved an effective detector mass of $\sim0.1$~Gton. It detects neutrinos mainly by measuring the optical Cerenkov light emitted by high energy muons produced by muon neutrino interactions in the ice. The IceCube extension of AMANDA, planned to reach a $\sim1$~Gton effective detector mass, is well under way: The AMANDA detector consists of 660 optical modules, and during the 05/06 (south pole) summer season alone 600 additional IceCube optical modules were successfully deployed. The full IceCube detector, consisting of 4800 modules, is expected to be completed in 4~yrs. A $\sim0.1$~Gton optical Cerenkov experiment, ANTARES, is currently under construction in the Mediterranean sea.  The ANTARES experiment has been successfully operating a single line of 75 optical modules since March 06, and is fully funded to be expanded in the coming few years to reach a $\sim0.1$~Gton detector mass. In parallel, R\&D has begun on a $\sim1$~Gton Mediterranean optical Cerenkov detector within the km3net collaboration. For more details on these optical Cerenkov neutrino telescopes see Halzen (2005) and references therein, and http://icecube.wisc.edu/, http://antares.in2p3.fr/, http://www.nestor.org.gr/hena/intro/intro.htm, http://www.km3net.org/.

At ultra high energies, $\gg1$~PeV, much larger detectors, $\gg1$~Gton, are required. The construction of sufficiently large detectors may be possible by taking advantage of the large coherent Cerenkov radio signals emitted by the particle showers produced by the interaction of ultra high energy neutrinos in the ice. The RICE detector (Kravchenko et al. 2003) is aiming at detecting these radio signals using antennas buried in the ice, while the ANITA experiment (Silvestri et al. 2004) will be using balloon born antennas over the south pole (a full engineering flight is planned for ANITA at the end of 06). The large effective mass required at ultra high energies may also be achieved by large cosmic ray detectors (such as Auger) searching for neutrino induced air showers. For discussion of these experimental techniques see Saltzberg (2005) and references therein, and http://www.ps.uci.edu/~anita/, http://www.bartol.udel.edu/~spiczak/rice/rice.html.

\section{GRB neutrinos}
\label{sec:GRBnu}

Gamma-ray bursts are short, typically tens of seconds long, flashes of gamma-rays, carrying most of their energy in $>1$~MeV photons. The detection in the past few years of "afterglows," delayed X-ray, optical and radio emission from GRB sources, proved that the sources lie at cosmological distances, and provided strong support for the following scenario of GRB production (for recent reviews see Piran 2005, M\'esz\'aros 2006). The energy source is believed to be rapid mass accretion on a newly formed solar-mass black hole. Observations suggest that the formation of the central compact object is associated, at least in some cases, with type Ib/c supernovae. The energy release drives an ultra-relativistic, $\Gamma\sim10^{2.5}$, plasma outflow. The emission of $\gamma$-rays is assumed to be due to internal collisionless shocks within the relativistic wind, the "fireball," which occur at a large distance from the central black-hole due to variability in the wind emitted from the central "engine." It is commonly assumed that electrons are accelerated to high energy within the collisionless shocks, and that synchrotron emission from these shock accelerated electrons produces the observed $\gamma$-rays.

If protons are present in the wind, as assumed in the fireball model, they would also be accelerated to high energy in the region were electrons are accelerated. As mentioned in \S~\ref{sec:WB}, the high energy to which protons may be accelerated in GRB winds and the rate (per unit time and volume) at which GRBs produce energy make GRBs likely sources of UHECR protons (see Waxman 2001 for a pedagogical review). If protons are indeed accelerated to high energy in GRBs, then high energy neutrino emission is also expected.

\subsection{100 TeV fireball neutrinos}
\label{sec:fireball_nu}

Protons accelerated in the fireball internal shocks, where GRB $\gamma$-rays are expected to be produced, lose energy through photo-production of pions in interactions with fireball photons. The decay of charged pions produced in this interaction results in the production of high energy neutrinos. The key relation is between the observed photon energy, $E_\gamma$, and the accelerated proton's energy, $E_p$, at the threshold of the
$\Delta$-resonance. In the observer frame,
\begin{equation}
E_\gamma \,E_{p} = 0.2 \, {\rm GeV^2} \, \Gamma^2\,.
\label{eq:keyrelation}
\end{equation}
For $\Gamma\approx10^{2.5}$ and $E_\gamma=1$~MeV, we see that characteristic proton energies $\sim 10^{16}$~eV are required to produce pions. Since neutrinos produced by pion decay typically carry $5\%$ of the proton energy, production of $\sim 10^{14}$~eV neutrinos is expected (Waxman \& Bahcall 1997).

The fraction of energy lost by protons to pions, $f_\pi$, is $f_\pi\approx0.2$ (Waxman \& Bahcall 1997, Waxman 2001). Assuming that GRBs generate the observed UHECRs, the expected GRB muon and anti-muon neutrino flux may be estimated using eq.~(\ref{eq:WB}) (Waxman \& Bahcall 1997, 1999),
\begin{equation}
E_\nu^2\Phi_{\nu}\approx 0.8\times10^{-8}{f_\pi\over0.2}{\rm GeV\,cm}^{-2}{\rm s}^{-1}{\rm
sr}^{-1}. \label{eq:JGRB}
\end{equation}
This neutrino spectrum extends to $\sim10^{16}$~eV, and is suppressed at higher energy due to energy loss of pions and muons (Waxman \& Bahcall 1997, 1999, Rachen \& P. M\'esz\'aros 1998; for the contribution of Kaon decay at high energy see Asano \& Nagataki 2006). Eq.~(\ref{eq:JGRB}) implies a detection rate of $\sim20$ neutrino-induced muon events per year (over $4\pi$~sr) in a cubic-km detector (Waxman \& Bahcall 1997, Alvarez-Mu\~{n}iz et al. 2000, Guetta et al. 2004).

Since GRB neutrino events are correlated both in time and in direction with gamma-rays, their detection is practically background free. The main background is due to atmospheric neutrinos, which produce neutrino-induced muons, travelling in a direction lying within a cone of opening angle $\Delta\theta$ around some direction, at a rate
\begin{equation}\label{eq:atmo}
    J_{\nu\rightarrow\mu}^A\simeq4\times10^{-3}\left(\frac{\Delta\theta}{0.5^o}\right)^2
\left(\frac{E}{100\rm TeV}\right)^{-\beta}\,{\rm km^{-2}yr^{-1}},
\end{equation}
with $\beta=1.7$ for $E<100$~TeV and $\beta=2.5$ for $E>100$~TeV. At high energies, the neutrino induced muon propagates at nearly the same direction as the incoming neutrino, and km-scale neutrino telescopes will be able to determine the incoming neutrino direction to better than $\sim0.5^o$. For a known source direction, therefore, the neutrino search is practically background free.

\subsection{TeV neutrinos}
\label{sec:TeV}

The 100~TeV neutrinos discussed in the previous section are produced at the same region where GRB $\gamma$-rays are produced. Their production is a generic prediction of the fireball model. It is a direct consequence of the assumptions that energy is carried from the underlying engine as kinetic energy of protons and that $\gamma$-rays are produced by synchrotron emission of shock accelerated particles. Neutrinos may be produced also in other stages of fireball evolution, at energies different than 100~TeV. The production of these neutrinos is dependent on additional model assumptions. We discuss below some examples related to the GRB progenitor. For a discussion of neutrino emission during the afterglow phase see, e.g., Dai \& Lu (2001). For a more detailed discussion see M\'esz\'aros (2002) and references therein.

The most widely discussed progenitor scenarios for long-duration GRBs involve core collapse of massive stars. In these "collapsar" models, a relativistic jet breaks through the stellar envelope to produce a GRB. For extended or slowly rotating stars, the jet may be unable to break through the envelope. Both penetrating (GRB producing) and "choked" jets can produce a burst of $\sim5$~TeV neutrinos by interaction of accelerated protons with jet photons, while the jet propagates in the envelope (M\'esz\'aros \& E. Waxman 2001, Razzaque et al. 2004). The estimated event rates may exceed $\sim10^2$ events per yr in a km-scale detector, depending on the ratio of non-visible to visible fireballs. A clear detection of non-visible GRBs with neutrinos may be difficult due to the low energy resolution for muon-neutrino events, unless the associated supernova photons are detected. In the two-step "supranova" model, interaction of the GRB blast wave with the supernova shell can lead to detectable neutrino emission, either through nuclear collisions with the dense supernova shell or through interaction with the intense supernova and backscattered radiation field (Dermer \& Atoyan 2003, Guetta \& Granot 2003, Razzaque et al. 2003).

\subsection{GRB neutrinos and SWIFT}
\label{sec:swift}

Recent observations with SWIFT revealed new and unexpected GRB characteristics, which may require modifications and extensions of GRB models. Such extensions and modifications may have implications for the predicted neutrino signal. The detection of GRB neutrinos may therefore allow one to discriminate between different models suggested for explaining these unexpected GRB characteristics.

Possibly the most intriguing new characteristic is the existence of strong X-ray flares following the prompt $\gamma$-ray emission with delays of hundreds of seconds to days, sometimes carrying energy comparable to that of the prompt emission (Burrows 2006). These flares may be explained as due to delayed internal shocks within the fireball wind, due to a prolonged activity of the "central engine" (e.g. Fan \& Wei 2005). If this interpretation is correct, these delayed flares should be accompanied by emission of PeV neutrinos, which may be detectable by km-scale neutrino telescopes (Murase \& Nagataki 2006).

\section{Neutrino physics and astrophysics}
\label{sec:nu_phys}

Detection of high energy neutrinos from GRBs will provide information on fundamental neutrino properties (Waxman \& Bahcall 1997).

Detection of neutrinos from GRBs could be used to test the simultaneity of neutrino and photon arrival to an accuracy of $\sim1$~s. It is important to emphasize here that since the background level of
neutrino telescopes is very low, see eq.~(\ref{eq:atmo}), the detection of a single neutrino from the direction of a GRB on a time sale of months after the burst would imply an association of the neutrino with the burst and will therefore establish a time of flight delay measurement. Such a measurement will allow one to test for violations of Lorentz invariance (as expected due to quantum gravity effects) (Waxman \& Bahcall 1997, Amelino-Camelia et al. 1998, Colman \& Glashow 1999, Jacob \& Piran 2006), and to test the weak equivalence principle, according to which photons and neutrinos should suffer the same time delay as they pass through a gravitational potential. With $1{\rm\ s}$ accuracy, a burst at $1{\rm\ Gpc}$ would reveal a fractional difference in (photon and neutrino) speed of $10^{-17}$, and a fractional difference in gravitational time delay of order $10^{-6}$ (considering the Galactic potential alone). Previous applications of these ideas to supernova 1987A (see Bahcall 1989 for review), yielded much weaker upper limits: of order $10^{-8}$ and $10^{-2}$ respectively.

High energy neutrinos are expected to be produced in GRBs by the decay of charged pions, which lead to the production of neutrinos with flavor ratio $\Phi_{\nu_e}:\Phi_{\nu_\mu}:\Phi_{\nu_\tau}=1:2:0$ (here $\Phi_{\nu_l}$ stands for the combined flux of $\nu_l$ and $\bar\nu_l$). Neutrino oscillations then lead to an observed flux ratio on Earth of
$\Phi_{\nu_e}:\Phi_{\nu_\mu}:\Phi_{\nu_\tau}=1:1:1$ (Learned \& Pakvasa 1995). Up-going $\tau$'s, rather than $\mu$'s, would be a distinctive signature of such oscillations. It has furthermore been been pointed out that flavor measurements of astrophysical neutrinos may help determining the mixing parameters and mass hierarchy (Winter 2006), and may possibly enable one to probe new physics (Learned \& Pakvasa 1995,Athar et al. 2000).

\begin{figure}[htbp]
\epsfxsize=14cm
\centerline{\epsfbox{fig2.eps}}
\caption{Flavor and anti-particle content of the flux of astrophysical neutrinos produced by pion decay, for pion energy spectrum (at production) $dn/dE_\pi\propto E_\pi^{-2}$ and electromagnetic pion energy loss rate $d E_\pi/dt\propto E_\pi^2$, as expected for internal shocks in GRB fireballs. Left: The ratio between $\Phi_{\nu_{\mu}(\nu_\tau)}$ and $\Phi_{\nu_e}$ (solid line), with 90\% CL lines of $\nu_\mu$ (dashed) and $\nu_\tau$ (dotted) fluxes (here, $\Phi_{\nu_l}$ stands for the combined flux of $\nu_l$ and $\bar\nu_l$). Right: The ratio of $\bar\nu_e$ to total $\nu$ flux on Earth, solid (dashed) line for neutrinos produced by  $p\gamma$ ($pp$) interactions. $E_{0,\mu}$ is the muon energy for which the muon life time is comparable to its electromagnetic energy loss time. $E_{0,\mu}\approx10^3$~TeV for internal shocks in GRB fireballs.}
\label{fig:fNu}
\end{figure}
It is interesting to point out here that flavor ratio measurements may provide interesting constraints not only on fundamental neutrino properties, but also on the astrophysical models. Although a $1:1:1$ flavor ratio appears to be a robust prediction of models where neutrinos are produced by pion decay, energy dependence of the flavor ratio is a generic feature of models of high energy astrophysical neutrino sources (Kashti \& Waxman 2005). Pions are typically produced in environments where they may suffer significant energy losses prior to decay, due to interaction with radiation and magnetic fields (Waxman \& Bahcall 1997, Rachen \& M\'esz\'aros 1998). Since the pion life time is shorter than the muon's, at sufficiently high energy the probability for pion decay prior to significant energy loss is higher than the corresponding probability for muon decay. This leads to suppression at high energy of the relative contribution of muon decay to the neutrino flux. The flavor ratio is modified to $0:1:0$ at the source, implying $1:1.8:1.8$ ratio on Earth (Kashti \& Waxman 2005). The energy at which the transition from $0:1:0$ to $1:1.8:1.8$ ratio takes place depends on the energy density (of magnetic fields and radiation) at the source. Detection of the flavor ratio transition will therefore provide constraints on the properties of the source. Figure~\ref{fig:fNu} presents the expected energy dependence of flavor and anti-particle content for internal shocks in GRB fireballs. Since the transition is expected at $\sim100$~TeV, it may be detectable by km-scale neutrino telescopes.

\section{Discussion}
\label{sec:discussion}

Several large volume high energy neutrino telescopes are currently operating and under construction (see \S~\ref{sec:nu_tel}): The optical Cerenkov detector AMANDA is operating at the south pole with an effective detector mass of 0.1~Gton for neutrinos in the 1~TeV--1~PeV energy range, and being expanded to a 1~Gton detector (IceCube); 0.1~Gton (for neutrinos in the 1~TeV--1~PeV energy range) optical Cerenkov detectors (ANTARES, NESTOR) are being constructed in the Mediterranean sea, and a 1~Gton Mediterranean detector is at an R\&D phase (km3NeT); Still larger detectors are being built for higher energy neutrinos, $\gg1$~PeV, utilizing the coherent radio Cerenkov emission of neutrino induced showers in the ice (ANITA, RICE). These detectors will reach in the coming few years a sensitivity which may allow the detection of extra-Galactic neutrino signals (see \S~\ref{sec:WB} and \S~\ref{sec:nu_tel}).

The detection of extra-Galactic neutrinos will allow us to answer some of the most interesting and important questions of high energy astrophysics (see \S~\ref{sec:WB}): It may allow us to identify the sources of UHECRs, to identify the UHECR primaries, and to probe the physics of particle acceleration to ultra-high energy.

GRBs are likely sources of UHECRs and high energy neutrinos (see \S~\ref{sec:GRBnu}). The production of 100~TeV neutrinos in the region where GRB $\gamma$-rays are produced is a generic prediction of the fireball model (see \S~4a). It is a direct consequence of the {\it assumptions} that energy is carried from the underlying engine, most likely a (few) solar mass black hole, as kinetic energy of protons and that $\gamma$-rays are produced by synchrotron emission of shock accelerated particles. The detection of the predicted neutrino signal will therefore provide strong support for the validity of underlying model assumptions, which is difficult to obtain using photon observations due to the high optical depth in the vicinity of the GRB "engine." For example, an alternative model where the energy is carried away from the source as pure Poynting flux and where $\gamma$-rays are produced by magnetic field dissipation (e.g. Lyutikov et al. 2003) is unlikely to be ruled out by photon observations, but will be strongly disfavored by the detection of 100~TeV neutrinos. The predicted neutrino intensity implies a detection of $\sim20$ muon induced neutrino events per yr in a km-scale neutrino detector. Since these events should be correlated in direction with GRB $\gamma$-rays, the search for GRB neutrinos is essentially background free (see \S~4a).

Recent observations with SWIFT revealed new and unexpected GRB characteristics, which may require modifications and extensions of GRB models. Such extensions and modifications may have implications for the predicted neutrino signal and the detection of GRB neutrinos may allow one to discriminate between different models. In particular, if the delayed X-ray flares are due to delayed internal shocks within the fireball wind, due to a prolonged activity of the "central engine," emission of PeV neutrinos should accompany the flares and may be detectable by Gton neutrino telescopes (see \S~4c). Neutrino observations may also allow one to probe GRB progenitors, as lower energy, $\sim1$~TeV, neutrinos are expected to be produced if GRBs are associated with the collapse of massive stars (see \S~4b).

The discussion of GRB neutrino emission demonstrates that in addition to identifying the sources of UHECRs, high-energy neutrino telescopes can provide a unique probe of the physics of these sources. In addition, detection of high energy neutrinos from GRB sources will provide information on fundamental neutrino properties (see \S~\ref{sec:nu_phys}). Detection of GRB neutrinos could be used to test the simultaneity of neutrino and photon arrival to an accuracy of $\sim1$~s. This will allow one to test for violations of Lorentz invariance and to test the weak equivalence principle, according to which photons and neutrinos should suffer the same time delay as they pass through a gravitational potential. With $1{\rm\ s}$ accuracy, a burst at $1{\rm\ Gpc}$ would reveal a fractional difference in (photon and neutrino) speed of $10^{-17}$, and a fractional difference in gravitational time delay of order $10^{-6}$ (considering the Galactic potential alone). Previous applications of these ideas to supernova 1987A, yielded much weaker upper limits: of order $10^{-8}$ and $10^{-2}$ respectively.

The decay of charged pions in GRB fireballs is expected to produce electron and muon neutrinos only. Neutrino oscillations would then lead to an observed flux ratio on Earth of $\Phi_{\nu_e}:\Phi_{\nu_\mu}:\Phi_{\nu_\tau}=1:1:1$. Up-going $\tau$'s, rather than $\mu$'s, would be a  distinctive signature of such oscillations. Flavor measurements of astrophysical neutrinos may furthermore help determining the mixing parameters and mass hierarchy, and may possibly enable one to probe new physics.

Finally, it is interesting to point out that flavor ratio measurements may provide interesting constraints not only on fundamental neutrino properties, but also on the astrophysical models. Pion and muon energy losses are expected to modify the flavor ratio observed on Earth to $1:1.8:1.8$ at high energies (see \S~\ref{sec:nu_phys}). The energy at which the transition from $0:1:0$ to $1:1.8:1.8$ ratio takes place depends on the energy density (of magnetic fields and radiation) at the source and therefore provides constraints on the properties of the source. For GRB sources the transition is expected at $\sim100$~TeV, and may be detectable by km-scale neutrino telescopes.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
\label{intro}

The fate of hadrons under extreme conditions is one of the outstanding questions in the theory of strong interactions, quantum chromodynamics (QCD).
As the temperature increases, the hadron gas --- with confined quarks and broken chiral symmetry --- transitions into a quark--gluon plasma (QGP), with deconfined light degrees of freedom and chiral symmetry restored.  In the vicinity of the crossover transition, thermal modifications of hadron properties are expected as a precursor to chiral symmetry restoration and deconfinement.  Hadrons built out of heavier (charm and beauty) quarks may survive in the QGP, and these states may serve as probes of the hot medium created in heavy-ion collisions.

Hadron properties are encoded in the spectral functions $\rho_\Gamma(\omega)$ (where $\Gamma$ denotes the quantum numbers), which are related to the euclidean correlators $G(\tau)$ computed on the lattice by an integral transform,
\begin{equation}
G(\tau;T) = \frac{1}{2\pi}\int_0^\infty\rho(\omega;T)K(\tau;T)d\omega\,,
\label{eq:spectral}
\end{equation}
where $K(\tau;T)$ is a known kernel that depends on the nature of the correlator.  Determining $\rho(\omega)$ from $G(\tau)$ is an ill-posed problem, and no single method has to date been shown to yield reliable results in all circumstances.  Where possible, avoiding this inversion problem by extracting information on thermal modifications directly from the correlator may be preferable.  A useful tool in this context is the \emph{reconstructed correlator}, which is obtained by integrating eq.~\eqref{eq:spectral} with $\rho(\omega;T)$ at some reference temperature $T=T_r$, which is usually taken to be the lowest available temperature.  In many cases this may be constructed directly from the correlator $G(\tau;T_r)$ \cite{Ding:2012sp} without having to determine $\rho(\omega;T_r)$.

The FASTSUM collaboration has employed anisotropic lattice QCD to study properties of charmonium and open-charm hadrons \cite{Aarts:2007pk,Kelly:2018hsi}, beautonium \cite{Aarts:2010ek,Aarts:2011sm,Aarts:2013kaa,Aarts:2014cda} and light and strange baryons \cite{Aarts:2015mma,Aarts:2017rrl,Aarts:2018glk}, as well as the electrical conductivity of QCD matter \cite{Amato:2013naa,Aarts:2014nba} and properties of the chiral transition \cite{Aarts:2014nba,Aarts:2020vyb}.  In these proceedings we will present recent results for mesons and baryons containing light, strange, charm and beauty quarks using our most recent gauge ensembles, and report on the status of our next generation ensembles.

\section{Simulation details}
\label{sec:lat-params}

\begin{table}[ht]
\centering
\begin{tabular}{c|crrcrrr}\hline
Gen & $N_f$& $\xi$  & $a_s$ (fm) & $a_\tau^{-1}$ (GeV) & $m_\pi$ (MeV) & $N_s$
& $L_s$ (fm)  \\ \hline
1 \cite{Aarts:2007pk} & 2   & 6.0 &  0.162 &   7.35 & 490 & 12 & 1.94 \\
2 \cite{Aarts:2014nba} & 2+1 & 3.45 &  0.121 &   5.63 & 390 & 24 & 2.95 \\
 & & & & & & 32 & 3.94 \\
2L \cite{Aarts:2020vyb} & 2+1 & 3.45 &  0.112 &   6.08 & 240 & 32 & 3.58 \\\hline
2P & 2+1 & 3.45 & *0.100 &  *6.80 & 140 & 48 & 4.80 \\
3 & 2+1 & 7.0 & *0.120 & *11.66 & *390 & 32 & 3.94 \\
\hline
\end{tabular}
\caption{FASTSUM ensemble parameters: number of flavours $N_f$, spatial lattice spacing $a_s$, inverse temporal lattice spacing $a_\tau^{-1}$, anisotropy $\xi=a_s/a_\tau$, pion mass $m_\pi$, number of lattice sites $L_s$ in the spatial directions and spatial extent $L_s$.  The Gen2P and Gen3 ensembles are in preparation (see section~\ref{sec:tuning}) and the starred numbers are target or expected values.}
\label{tab:params}
\end{table}

Our simulations are carried out using anisotropic lattices with an
${\cal O}(a^2)$ improved gauge action and an ${\cal O}(a)$ improved Wilson
fermion action with stout smearing.  The Gen2 and Gen2L ensembles were produced following the parameter tuning and ensembles generated by the Hadron Spectrum Collaboration \cite{Edwards:2008ja,Lin:2008pr}.  Details about all our ensembles
used to date as well as our planned new ensembles (Gen2P and Gen3) are
given in table~\ref{tab:params}.  The temperature is given by $T=(a_\tau N_\tau)^{-1}$ and is varied by changing the number of sites $N_\tau$ in the temporal direction.  For more details about the Gen2 and Gen2L ensembles, see Ref.~\cite{Aarts:2020vyb} and references therein. Most of the results presented in
this update are from our most recent ensemble, Gen2L.

\section{Results}
\label{sec:results}

\subsection{Light hadrons}
\label{sec:light}

We have previously \cite{Aarts:2015mma,Aarts:2017rrl,Aarts:2018glk}
investigated light and strange baryon
correlators below and above the chiral crossover.  We have found that
in all channels studied, the positive-parity baryon masses remain
constant to within the accuracy of our calculations below $T_c$, while
the negative-parity baryon masses decrease with increasing temperature
and become degenerate with the positive-parity masses at $T_c$.  The
parity doubling in the correlators can be encoded in the $R$-parameter
defined by
\begin{align}
R &= \frac{\sum\limits_{n=n_0}^{N_\tau/2-1}R(\tau_n)/\sigma^2(\tau_n)}
{\sum\limits_{n=n_0}^{N_\tau/2-1}1/\sigma^2(\tau_n)}\qquad\text{with}\qquad&
R(\tau) &= \frac{G(\tau)-G(N_\tau-\tau)}{G(\tau)+G(N_\tau-\tau)}\,\,
\label{eq:baryon-R}
\end{align}
where $\sigma(\tau)$ is the uncertainty in the correlator $G(\tau)$ and we choose $n_0\sim4$ to reduce contamination from lattice artefacts and excited states.
We have found that $R$ behaves qualitatively like an order parameter
for the chiral transition, and that the inflection point in $R$ is
consistent between different baryon (spin and flavour) channels and
with $T_c$ determined from the renormalised chiral condensate and
chiral susceptibility.  These studies were carried out using standard
exponential fits with extended (smeared) sources to enhance the
overlap with the ground state, but the results have also been
corroborated by extracting spectral functions with the maximum entropy
method (MEM).

\begin{figure}[t]
\centering
\includegraphics[width=\colw]{V_AX_uu.png}
\includegraphics[width=\colw]{mPS_uu.png}
\caption{Light meson masses as function of temperature, from analysis
  of smeared correlators. Left: vector
  and axial-vector mesons, $\rho$ and $a_1$.  Right: pion.
  The vertical band indicates the chiral crossover.  The solid point for the $\rho$ at $T\approx0$ represent the mass determined by the Hadron Spectrum collaboration on the same ensemble in a coupled-channel calculation using distillation and a variational analysis \cite{Wilson:2015dqa,Wilson:2019wfr}, while that for $a_1$ is the Particle Data Group value for the mass.}
\label{fig:light}
\end{figure}

We have recently applied the methods used for studying light baryons
to light, strange and charmed mesons as well as charmed baryons.
Specifically, we have performed exponential fits to correlators of
extended operators with optimised overlaps with the ground state, to
determine hadron masses below and in the vicinity of $T_c$.
Fig.~\ref{fig:light} shows the result of this analysis for light
mesons in the vector ($\rho$), axial-vector [$a_1(1260)$] and pseudoscalar
(pion) channels.  The vector and axial-vector are chiral partners and
should be degenerate if chiral symmetry is restored.  As we can see in
fig.~\ref{fig:light} (left) this near-degeneracy\footnote{Since the light quark mass is nonzero we do not expect exact degeneracy.} becomes manifest at $T_c$
with the mass of the axial-vector meson dropping substantially to
become approximately degenerate with the $\rho$ meson.  It is interesting to note that the
$\rho$ meson also appears to exhibit a small negative mass shift below
$T_c$.  A similar effect is seen for the kaon.

In the right panel of fig.~\ref{fig:light} we show the pion mass as a function of temperature.  We see that unlike the $\rho$, the pion mass increases with temperature.  This can be understood as a precursor to chiral symmetry restoration, as the pion is the pseudo-Goldstone boson of the broken chiral symmetry and as a result of this is much lighter than the other hadrons below $T_c$.  This mass difference will disappear once chiral symmetry is restored.  At this point, the pion (pseudoscalar) and scalar meson (isovector $a_0$) should become degenerate; however, it is numerically difficult to reliably extract masses of scalar mesons and we have as yet not been able to observe this degeneracy.

\subsection{Charmed hadrons}
\label{sec:charm}
\begin{figure}[h]
\centering
\includegraphics[width=\colw]{Dmesons_gen2L.png}
\includegraphics[width=\colw]{R_12_gvar_curves.pdf}
\caption{Left: open charm masses from analysis of smeared correlators \cite{Aarts:2022krz}.
  Right: the parity doubling parameter $R$ for spin-$\frac{1}{2}$
  charmed baryons \cite{Bignell:InPrep}. The vertical band indicates the chiral crossover.}
\label{fig:opencharm}
\end{figure}

Temporal correlators and spectral functions of open-charm mesons were studied for the first time by us using the Gen2 ensemble \cite{Kelly:2018hsi,Quinn:2019uwq}.  Recently, this has been updated with a detailed study \cite{Aarts:2022krz} of mass modifications below and around $T_c$ using our Gen2L ensemble.  In the left panel of fig.~\ref{fig:opencharm} we show the resulting masses of the ground state open-charm mesons in the pseudoscalar ($D,D_s$) and vector ($D^*,D_s^*$) channels.  We find no change for $T\lesssim120\,$MeV, but a decrease of about 40\,MeV in all channels at $T\sim150\,$MeV.  We find somewhat larger modifications in the vector than the pseudoscalar channel, and an indication that the strange $D_s$ meson is less affected than the others.  This may have implications for the interpretation of experimental results suggesting an increased $D_s/D$ ratio in $A-A$ vs $p-p$ collisions.

We have also investigated charmed baryons \cite{Bignell:InPrep}.  The right panel of fig.~\ref{fig:opencharm} shows the parity-doubling $R$ parameter \eqref{eq:baryon-R} for light, singly charmed and doubly charmed spin-$\half$ baryons.  We see that the singly-charmed baryons exhibit the same qualitative behaviour as light baryons, even to the extent that the inflection point is consistent.  For doubly charmed baryons on the other hand we do not find any clear signal of a transition.  Similar results are found for spin-$\frac{3}{2}$ baryons, but in this case also some of the doubly-charmed baryons exhibit features consistent with a transition.  We are in the process of analysing these results in terms of thermal mass modifications.

\begin{figure}[h]
\centering
\includegraphics[width=\colw]{charmonium_Gen2L.png}
\includegraphics[width=\colw]{Jpsi_BR_Gen2L_jack.png}
\caption{Left:
  $J/\psi$ and $\eta_c$ masses from analysis of smeared correlators.
  The vertical band indicates the chiral crossover.  As the analysis assumes narrow bound states, the results for $T>T_c$ cannot be taken as evidence that these states survive in the quark--gluon plasma. Right:
  $J/\psi$ spectral functions from the BR method.}
\label{fig:charmonium}
\end{figure}

In fig.~\ref{fig:charmonium} we show preliminary results for charmonium from our Gen2L ensemble using two different analysis methods.  The left panel shows masses determined with the same approach as already detailed for light and open-charm hadrons.  We see indications of a very slight negative mass shift just below $T_c$.  Whether the $J/\psi$ and $\eta_c$ survive in the QGP and, if so, to what temperature, remains unresolved, and the results shown for $T>T_c$ should not be taken as evidence of survival as they have been obtained assuming a narrow bound state.

The right panel of fig.~\ref{fig:charmonium} shows $J/\psi$ spectral functions determined using the BR method \cite{Burnier:2013nla}.  These results seem to suggest a substantial positive mass shift and broadening already in the hadronic phase.  However, much of this effect has been found \cite{Kelly:2018hsi} to be an artefact of the limited temporal extent at higher temperature, as spectral functions obtained from reconstructed correlators \cite{Ding:2012sp} exhibit the same features.  A comparison with such spectral functions, as was done in Ref.~\cite{Kelly:2018hsi}, will therefore be required to determine the extent of thermal modifications.

\subsection{Beauty hadrons}
\label{sec:beauty}

\begin{figure*}[h]
\centering
\includegraphics[width=\colw,clip]{Upsilon_Gen2L.png}
\includegraphics[width=\colw,clip]{Upsilon_width_fit.png}
\caption{Left: $\Upsilon$ spectral functions from the BR method.  The dashed curves denote the spectral functions obtained from truncated correlators at the lowest available temperature, see text for details.  The uncertainty bands on these spectral functions (not shown) are of the same magnitude as those from the $T>0$ correlators.  Right: $\Upsilon$ width from fits using a Gaussian spectral function Ansatz \cite{Spriggs:2021dsb}.  The top panel shows how the fitted width depends on the final point $\tau_2$ of the fit range for each temperature, while the lower panel shows the width as a function of $T$ using the same fit range for all temperatures.}
\label{fig:beauty}
\end{figure*}

Figure~\ref{fig:beauty} shows selected results for medium modifications of $\Upsilon$ mesons using nonrelativistic QCD (NRQCD).  These are part of an ongoing investigation using a wide range of different methods.  Some results from this investigation have been presented in Ref.~\cite{Spriggs:2021dsb}.  The left hand plot shows spectral functions obtained using the BR method for selected temperatures, compared to the spectral functions $\rho_{\text{rec}}(\omega)$ obtained from correlators at the lowest available temperature, $T=47$, denoted by dashed lines.  Since the NRQCD kernel, $K(\tau;T)=e^{-\omega\tau}$, is temperature-independent, the reconstructed correlator may be obtained by simply discarding points with $\tau>1/T$ from the correlator at the reference temperature.  We find that the spectral functions agree perfectly with those from the reconstructed (truncated) correlators for $T\lesssim T_c$.  For $T>T_c$ the two diverge, with $\rho(\omega)$ having a peak at lower $\omega$ than $\rho_{\text{rec}}(\omega)$ as well as a slightly larger width.  This suggests a small, negative mass shift.  A quantitative analysis is in progress.

The right hand panel of fig.~\ref{fig:beauty} shows the $\Upsilon$ width determined from fits to the correlators using a Gaussian Ansatz for the spectral function \cite{Spriggs:2021dsb}.  Both the mass and the width are found to depend strongly on the fit window used, so a comparison of results using the same fit window at different temperatures is mandatory to disentangle real physical effects from this artefact.  Such a comparison is shown in the upper panel.  It was found \cite{Spriggs:2021dsb} that an extrapolation to $\tau_2=\infty$ at the lowest temperature gave a width consistent with zero.  The bottom panel shows the width as function of temperature using the same fit window for all $T$; we see no change below $T_c$ and an increase of up to 60 MeV at $T=300\,$MeV.

\section{Tuning for new ensembles}
\label{sec:tuning}

\begin{figure}[t]
\centering
\includegraphics[width=\colw,clip]{Gen3_xig.png}
\includegraphics[width=\colw,clip]{Gen3_xiq.png}
\caption{Anisotropy tuning for Gen3, showing how the output
  anisotropies $\xi_{g,q}$ depend on the input gauge anisotropy
  $\gamma_g$ (left) and fermion anisotropy $\gamma_q$ (right).}
\label{fig:tuning}
\end{figure}

While analysing data from our Gen2L ensembles, we are preparing for next-generation ensembles which will control the systematic effects of the finite lattice spacing and heavier-than-physical light quark mass.  Specifically, our Gen2P ensemble will bring us to the physical pion mass with a lattice spacing and anisotropy approximately equal to those of Gen2 and Gen2L.  This involves tuning the bare quark mass to reproduce the physical pion mass; this has been done on an $L\approx3\,$fm volume and will be finalised on an $L\approx5\,$fm volume.

The Gen3 ensemble will double the anisotropy and hence halve the temporal lattice spacing while keeping the spatial lattice spacing and pion mass approximately equal to the Gen2 values, $a_s\approx0.12\,$fm and $m_\pi\approx400\,$MeV.  Obtaining the values of the input parameters, the bare gauge and fermion anisotropies $\gamma_g, \gamma_q$, the gauge coupling $\beta$ and the bare quark mass $m_0$, to produce these outputs is a 4-dimensional simultaneous tuning process.  Fig.~\ref{fig:tuning} shows examples of how the physical gauge anisotropy $\xi_g$, determined from the Symanzik flow \cite{Borsanyi:2012zr,Borsanyi:2012zs}, and the fermion anisotropy $\xi_q$, determined from the pion dispersion relation, depend on $\gamma_g$ and $\gamma_q$.  To date we have determined the parameters for the 3-flavour symmetric point with the target value $m_{PS}/m_V=0.545$ \cite{Kelly:PhD} and are in the process of tuning the strange and light quark mass parameters separately.

\section{Summary}

We have presented results for medium modifications, in particular mass shifts, of mesons and baryons containing light, strange, charm and beauty quarks, using anisotropic lattice QCD with 2+1 flavours of clover-improved Wilson quark and $m_\pi\approx240\,$MeV.  We find that the chiral partners $\rho$ and $a_1$ become degenerate at the chiral crossover temperature, with the masses of both decreasing below $T_c$, while the pion mass increases.  For mesons containing charm quarks we find a small negative mass shift below $T_c$.  Singly-charmed baryons exhibit a parity-doubling behaviour similar to that observed for light baryons, while doubly-charmed baryons show a much reduced sensitivity to the chiral crossover.  We report on progress in determining the temperature-dependent mass and width of beautonium states in nonrelativistic QCD.
Finally, we report on the progress of determining the parameters of our next-generation ensembles, which will reach the physical pion mass (Gen2P) and reduce the temporal lattice spacing (Gen3).  These ensembles are expected to come on stream in the coming year.

\section*{Acknowledgments}

This work is supported by the UKRI Science and Technology Facili-
ties Council (STFC) Consolidated Grant No. ST/T000813/1 and by STRONG-2020 ``The strong interaction at the frontier of knowledge: fundamental research
and applications'' which received funding from the European Union's Horizon
2020 research and innovation programme under grant agreement No 824093.
S. K. is supported by the National Research Foundation of Korea under Grant No. NRF-2021R1A2C1092701 funded by the Korean government (MEST).
M. N. A. acknowledges support from The Royal Society Newton International Fellowship. R. H. D. and E. M. acknowledge support from a Maynooth University SPUR scholarship.  R. H. D. is supported by a Maynooth University John and Pat Hume scholarship.

We are grateful to DiRAC, PRACE, ICHEC and Supercomputing Wales for the use of their computing resources and to the Swansea
Academy for Advanced Computing for support. This work was performed using
the PRACE Marconi-KNL resources hosted by CINECA, Italy and the DiRAC
Extreme Scaling service and Blue Gene Q Shared Petaflop system at the University of Edinburgh operated by the Edinburgh Parallel Computing Centre. The
DiRAC equipment is part of the UK's National e-Infrastructure and was funded
by UK's BIS National e-infrastructure capital grant ST/K000411/1, STFC capital grants ST/H008845/1 and ST/R00238X/1, and STFC DiRAC Operations grants ST/K005804/1, ST/K005790/1 and ST/R001006/1.