{"text":"\\section{Introduction}\n\n\\indent\n\nThe notions and facts used but not described here can be found in \\cite{S}.\n\nLet $S$ be a finite set of points in $\\mathbb{R}^d$, $X \\subseteq S$, and \n$z \\in \\mathbb{R}^d \\setminus S$.\n\\\\[1ex]\nA set $X$ is called a {\\em $z$-containing set} (a {\\em $z$-avoiding set}) if $z$ is in the interior of the convex hull of $X$ \n(respectively, $z$ is not in the interior of the convex hull of $X$). \n\\\\[1ex]\nA $z$-containing set $X$ is {\\em minimal} if $X$ has no proper $z$-containing subset.\n\\\\[1ex]\nA $z$-avoiding set $X$ is {\\em maximal in} $S$ if $S$ has no $z$-avoiding subset containing $X$ properly.\n\\\\[1ex]\nLet ${\\cal C}(S)$ and ${\\cal A}(S)$ denote the sets of minimal \n$z$-containing and maximal $z$-avoiding subsets of $S$, respectively.\n\\\\[1ex]\n\\indent\nIn 2002 E. Boros and V. Gurvich raised the following interesting question.\n\\begin{question}\n\\label{question} Suppose that $S$ is a finite set of points in $\\mathbb{R}^d$,\n $z \\in \\mathbb{R}^d \\setminus S$, and $S$ is a $z$-containing set. Is it true that \n$|{\\cal A}(S)| \\le 2 d~|{\\cal C}(S)|$?\n\\end{question}\n\nQuestions of this type arise naturally in the algorithmic theory of a so-called {\\em efficient} enumeration of different type of geometric or combinatorial objects\n(see, for example, \\cite{Rtcr, IIS}).\n\\\\[1ex]\n\\indent\n For an affine subspace $F$ of $\\mathbb{R}^d$, let $dim(F)$ denote the affine dimension of $F$ \nand $R(X)$ denote the minimal affine subspace in $\\mathbb{R}^d$ containing $X$.\n\\\\[1ex]\n\\indent\nA finite set $S$ of points in $\\mathbb{R}^d$ is said to be {\\em in a general position} if for every $X \\subseteq S$, \n\\\\[0.3ex]\n$|X| - 1 \\le d$ $\\Rightarrow$ $dim (R(X)) = |X| - 1$.\n\\\\[1ex]\n\\indent\nWe say that {\\em $z$ is in a general position with respect to $S$} if\n$dim(R(X)) < d$ $\\Rightarrow$ $z \\not \\in R(X)$ for every $X \\subseteq S$.\n\\\\[1ex]\n\\indent\nOne of our main results is the following theorem.\n\\begin{theorem} \n\\label{!}\nLet $S$ be a finite set of points in the \n$d$-dimensional space $\\mathbb{R}^d$, $z \\in \\mathbb{R}^d \\setminus S$.\nSuppose that $z$ is in a general position with respect to $S$.\nThen $|{\\cal A}(S)| \\le d~|{\\cal C}(S)| + 1$.\n\\end{theorem}\n\nThis theorem was announced in \\cite{CKK} and its proof was presented at the RUTCOR seminar directed by E. Boros and V. Gurvich in June 2002.\n\\\\[1ex]\n\\indent\nLater L. Khachian gave a construction providing for every \n$d \\ge 4$ a counterexample $(S,z)$ in $\\mathbb{R}^d$ to the inequality in Question \\ref{question} such that $S$ is not in a general position (see \\cite{Rtcr}).\nFrom Theorem \\ref{!} it follows that in all these counterexamples $z$ is not in a general position with respect to $S$.\n\\\\[1ex]\n\\indent\nIn \\cite{CKK} it is shown that if $S$ is a finite set of points in the plane \n$\\mathbb{R}^2$, $z \\in \\mathbb{R}^2 \\setminus S$, and $S$ is a $z$-containing set, then \n$|{\\cal A}(S)| \\le 3|{\\cal C}(S)| +1$, and so \nthe inequality in Question \\ref{question} is true for the plane.\n\\\\[1ex]\n\\indent\nIn this paper we also give some strengthenings of Theorem \\ref{!}.\n\n\n\n\n\n\n\n\\section{Some notions, notation, and auxiliary facts}\n\\label{notions}\n\n\\indent\n\nGiven a convex polytope $P$ in $\\mathbb{R}^d$, a face $F$ of a polytope $P$ is called \na {\\em facet} of $P$ if $dim (R(F)) = d - 1$.\nObviously, $R(F)$ is a hyperplane; we call $R(F)$ \na {\\em facet hyperplane of $P$}.\nLet, as above, $S$ be a finite set of points in $\\mathbb{R}^d$, $X \\subseteq S$, and \n$z \\in \\mathbb{R}^d \\setminus S$. Let $s \\in S$.\n\\\\[1ex]\n\\indent\nWe will use the following notation:\n\n\\begin{itemize}\n\\item $conv(X)$ is the convex hull of $X$,\n\n\\item if $P$ is a convex polytope, then $V(P)$ is the set of vertices of $P$ and $v(P) = |V(P)|$, \n\n\\item as above, ${\\cal C}(S)$ is the set of minimal \n$z$-containing subsets of $S$; \nalso ${\\cal C}_s(S)$ is the set of members of ${\\cal C}(S)$ containing $s$,\n\n\\item as above, ${\\cal A}(S)$ is the set of maximal \n$z$-avoiding subsets of $S$,\n\n\\item ${\\cal A}^s(S)$ is the set of subsets $A$ in $S$ such that $A$ is maximal $z$ avoiding in $S$ and \n$A \\setminus s$ is maximal $z$-avoiding in \n$S \\setminus s$, and so $A \\in {\\cal A}(S)$ and \n$A \\setminus s \\in {\\cal A}(S \\setminus s)$,\n\n\\item ${\\cal A}_s(S) = {\\cal A}(S) \\setminus {\\cal A}^s(S)\n = \\{X \\in {\\cal A}(S): s \\in X~and~X \\setminus s \\not \\in {\\cal A}(S \\setminus s\\}$,\n\n\n\\item ${\\cal S}mpl(S)$ is the set of simplexes $C$ such that\n$z$ is an interior point of $C$ and $V(C) \\subseteq S$ \nand\n${\\cal S}mpl_s(S)$ is the set of simplexes $C$ in \n${\\cal S}mpl(S)$ such that $s \\in V(C)$, \n\n\\item ${\\cal C}onv(S) = \\{conv(X): X \\in {\\cal C}(S)\\}$\nand ${\\cal C}onv_s(S)$ is the set of members of \n${\\cal C}onv(S)$ containing $s$, \n\n\\item ${\\cal H}(S)$ is the set of hyperplanes $H$ such that $H$ is a facet hyperplane of a simplex in ${\\cal S}mpl(S)$ and ${\\cal H}_s(S)$ is the set of hyperplanes in ${\\cal H}(S)$ containing $s$, and\n\n\\item ${\\cal F}(S)$ is the set of subsets $T$ of $S$ such that $|T| = d$ and $conv(T)$ is a face of a simplex in ${\\cal S}mpl(S)$ and ${\\cal F}_s(S)$ is the set of subsets of $S$ in ${\\cal F}(S)$ containing $s$.\n\n\\end{itemize}\n\nObviously, \nwe have the following.\n\\begin{lemma}\n\\label{observation}\nLet $S$ be a finite set of points in $\\mathbb{R}^d$.\nThen\n\\\\[1ex]\n$(a1)$ $|{\\cal C}onv(S)| = |{\\cal C}(S)|$,\n\\\\[1ex]\n$(a2)$\n $|{\\cal H}(S)| \\le |{\\cal F}(S)| \\le (d+1)|{\\cal S}mpl(S)|$, and \n $|{\\cal C}onv_s(S)| = |{\\cal C}_s(S)|$,\n\\\\[1ex]\n$(a3)$\n$|{\\cal H}_s(S)| \\le |{\\cal F}_s(S)| \\le d~|{\\cal S}mpl_s(S)|$, \nand\n\\\\[1ex]\n$(a4)$\n$|{\\cal A}^s(S)| = |{\\cal A}(S\\setminus s)|$, and so \n$|{\\cal A}(S)| - |{\\cal A}(S\\setminus s)| = |{\\cal A}_s(S)|$.\\end{lemma}\n\nWe recall that {\\em $z$ is in a general position with respect to $S$} if\n$dim(R(X)) < d$ $\\Rightarrow$ $z \\not \\in R(X)$ for every $X \\subseteq S$.\n\\\\[1ex]\n\\indent\nWe will use the following well known and intuitively obvious fact.\n\\begin{lemma}\n\\label{SimplexInPcontainingZero}\nLet $P$ be a convex polytope in $\\mathbb{R}^d$ and \n$z$ a point in the interior of $P$. Suppose that\n$z$ is in general position with respect to $V(P)$.\nThen there exists $X \\subseteq V(P)$ such that\n$conv(X)$ is a simplex of dimension $d$ and $z$ is in the interior of $conv(X)$.\n\\end{lemma}\n\n\\noindent{\\bf Proof} \\ \nWe prove our claim by induction on the dimension $d$.\nThe claim is obviously true for $d = 1$.\nWe assume that the claim is true for $d = n-1$ and will prove that the claim is also true for \n$d = n$, where $n \\ge 2$. Thus, $P$ is a convex polytope in $\\mathbb{R}^n$.\nSince $z$ is in a general position with respect to $V(P)$, clearly $z \\not \\in V(P)$. Let $p \\in V(P)$ and $L$ the line containing $p$ and $z$. Then there exists the point $t$ in $L \\cap P$ such that the closed interval\n$pLt$ contains $z$ as an interior point and $t$ is not an interior point of $P$.\nIn a plain language, $pL$ is the ray going from point $p$ through point $z$ and $t$ is the first point of the ray which is not the interior point of $P$. Then $t$ belongs to a face $F$ of $P$ which is a convex polytope of dimension at most $n - 1$. Since $z$ is in a general position with respect to $V(P)$, the dimension of $F$ is $n-1$ and $t$ is in a general position with respect to $V(F)$. By the induction hypothesis, there exists \n$T \\subseteq V(F)$ such that $conv(T)$ is a simplex of dimension $n - 1$ and $t$ is in the interior of \n$conv(T)$. Put $X = T\\cup p$. Then $X \\subseteq V(P)$, \n$conv(X)$ of dimension $n$ is a simplex and $z$ is in the interior of $conv(X)$.\n\\hfill $\\Box$\n\\\\[1.5ex]\n\\indent\nLemma \\ref{SimplexInPcontainingZero} also follows from the Caratheodory Theorem (see \\cite{S}) and Lemma \\ref{z-in-conv(X)} below.\n\\\\[1.5ex]\n\\indent\nGiven $X, Y \\subset \\mathbb{R}^d$ and a hyperplane $H$, we say that $H$ {\\em separates $X$ and $Y$} (or {\\em separates $X$ from $Y$}) if\n$X \\setminus H$ and $Y \\setminus H$ belong to different \nhalf-spaces of $\\mathbb{R}^d \\setminus H$.\n\\\\[1.5ex]\n\\indent\nIt is easy to see the following.\n\\begin{lemma}\n\\label{(A,z)-separating-hyperplane}\nLet $A$ and $A'$ be $z$-avoiding subsets of $S$. Then\n\\\\[1ex]\n$(a1)$ there exists a hyperplane separating $A$ from $z$,\n\\\\[1ex]\n$(a2)$\nif there exists a hyperplane separating both $A$ and $A'$ from $z$, then $A \\cup A'$ is also a $z$-avoiding subsets of $S$, and therefore \n\\\\[1ex]\n$(a3)$ if $A$ and $A'$ are maximal $z$-avoiding subsets of $S$ and there exists a hyperplane separating both $A$ and $A'$ from $z$, then $A= A'$.\n\n\\end{lemma}\n\nWe also need the following simple facts.\n\\begin{lemma}\n\\label{z-in-conv(X)}\nLet $z$ be in a general position with respect to \n$S$ and $X \\subseteq S$.\nThen the following are equivalent:\n\\\\[0.7ex]\n$(a1)$ $z \\in conv(X)$ and\n\\\\[0.7ex]\n$(a2)$ $z$ is in the interior of $conv(X)$.\n\\end{lemma}\n\n\\noindent{\\bf Proof} \\ \nObviously, $(a2)$ implies $(a1)$.\nSuppose, on the contrary, $(a1)$ does not imply $(a2)$.\nThen $z$ belongs to $conv(X')$ for some \n$X' \\subseteq X$ with $dim (R(X')) < d$. \nTherefore $z$ is not in a general position with \nrespect to $S$, a contradiction.\n\\hfill $\\Box$\n\n\\begin{lemma}\n\\label{Min-z-containingSet}\nLet $z$ be in a general position with respect to \n$S$. If $C$ is a minimal $z$-containing subset of $S$,\nthen $conv(C)$ is a simplex, and so\n${\\cal S}mpl(S) = {\\cal C}onv(S)$. \n\\end{lemma}\n\n\\noindent{\\bf Proof} \\ (uses Lemmas \n\\ref{SimplexInPcontainingZero} and \\ref{z-in-conv(X)}).\nSince $C$ is $z$-containing, there exists $X \\subseteq C$ such that $z \\in conv(X)$ and $conv(X)$ is a simplex.\nBy Lemma \\ref{z-in-conv(X)}, $z$ is in the interior of $conv(X)$.\nSince $C$ is minimal $z$-containing, clearly \n$C = X$.\nNow by Lemma \n\\ref{SimplexInPcontainingZero}, $conv(C)$ is a simplex.\n\\hfill $\\Box$\n\n\\begin{lemma}\n\\label{sz-ray}\nLet $z$ be in a general position with respect to $S$. Suppose that $X \\subset S$ and $s \\in S \\setminus X$.\nLet $L$ be the line in $\\mathbb{R}^d$ containing $s$ and $z$.\nIf $dim (R(X)) \\le d-2$, then $L \\cap R(X) = \\emptyset $.\n\\end{lemma}\n\n\\noindent{\\bf Proof} \\ Suppose, on the contrary, $L \\cap R(X)\\ne \\emptyset $.\nThen $z \\in R(X \\cup s)$ and $dim (R(X \\cup s)) < d$. \nThen $z$ is not in a general position with respect to $S$, \na contradiction.\n\\hfill $\\Box$\n\n\\begin{lemma}\n\\label{sLz-A,A'}\nLet $S$ be a $z$-containing set and $z$ in a general position with respect to $S$.\nLet $A$ be a maximal $z$-avoiding set in $S$ and\n$s \\in S \\setminus A$.\nThen $A$ has a subset $T$ such that\n\\\\[1ex]\n$(a1)$\n$|T| = d$, \n\\\\[1ex]\n$(a2)$\n$T$ belongs to a facet of $conv(A)$, \n\\\\[1ex]\n$(a3)$ \n$conv(T \\cup s)$ is a simplex of dimension $d$ containing $z$ as an interior point, and \n\\\\[1ex]\n$(a4)$\n$R(T)$ is a facet hyperplane of $A$ separating\n$A$ from $z$.\n\\end{lemma}\n\n\n\\noindent{\\bf Proof} \\ (uses Lemmas \\ref{SimplexInPcontainingZero} and \\ref{sz-ray}).\nLet $L$ be the line in $\\mathbb{R}^d$ containing $s$ and $z$. \nSince $s \\not \\in A$ and $A$ is a maximal $z$-avoiding subset in $S$, clearly $z$ is in the interior of \n$conv(A \\cup s)$.\n\nLet $I = L \\cap conv(A \\cup s)$. Since $conv(A \\cup s)$ is a convex set, clearly $I$ is a line segment $sLr$, where $r$ \nis not an interior point of $conv(A \\cup s)$.\nWe claim that $r \\in conv(A)$. Indeed, if not, then $r$ belongs to a facet $R$ of $conv(A \\cup s)$ containing $s$.\nSince $sLr$ is a convex set and $s, r \\in R$, we have:\n$z \\in sLr \\subseteq R$, and so $z \\in R$.\nIt follows that $z$ is not an interior point of $conv(A \\cup s)$, a contradiction.\nThus, $conv(A) \\cap L\\ne \\emptyset $.\n\nSince $s \\not \\in A$, \nthere exists a unique point $t$ in $L$ such that \nthe closed interval $sLt$ in $L$ with the end-points $s$ and $t$ has the properties: $z$ is in the interior of $sLt$ \n and $conv(A) \\cap sLt = t$. \n In a plane language, $sL$ is the ray going from point $s$ through point $z$ and $t$ is the first point of the ray belonging to $conv(A)$. \nBy Lemma \\ref{sz-ray}, $t$ belongs to the interior of a \nfacet $F$ of $conv(A)$.\nHence, by Lemma \n\\ref{SimplexInPcontainingZero}, $F$ has a set $T$ of \n$d$ vertices such that $T \\subset A$ and \n$conv (T)$ is a simplex of dimension $d - 1$ containing \n$t$ as an interior point.\nThen $conv (T \\cup s)$ is a simplex of dimension $d$ containing \n$z$ as an interior point, and so $t$ is an interior point of $conv(T)$. \n\\hfill $\\Box$\n\n\n\n\n\n\n\n\n\\section{Main results}\n\n\\indent\n\nFirst we will prove a weaker version of our main result\nto demonstrate the key idea concerning the relation between the minimal $z$ containing and maximal\n$z$-avoiding subsets of $S$.\n\\begin{theorem}\n\\label{w}\nSuppose that $S$ is a $z$-containing set and $z$ is in a general position with respect \n\\\\[0.5ex]\nto $S$.\nThen $|{\\cal A}(S)| \\le (d +1) |{\\cal C}(S)|$.\n\\end{theorem}\n\n\\noindent{\\bf Proof} \\ (uses Lemmas\n\\ref{(A,z)-separating-hyperplane}$(a4)$, \n\\ref{Min-z-containingSet}, and\n \\ref{sLz-A,A'}).\nLet $A$ be a maximal $z$-avoiding subset of $S$.\nSince $S$ is a $z$-containing set, there exists $s \\in S \\setminus A$.\nSince $A$ is a maximal $z$-avoiding subset of $S$,\n$A \\cup s$ is not a $z$-avoiding subset of $S$. Hence \n$A \\cup s$ is a $z$-containing set, and so $z$ is in the interior of $A \\cup s$.\nBy Lemma \\ref{sLz-A,A'}, \n$A$ has a subset $T$ such that\n$|T| = d$, $T$ belongs to a face of $conv(A)$, \nand \n$conv(T \\cup s)$ is a simplex containing $z$ as an interior point.\n\\\\[0.7ex]\n\\indent\nLet, as above, ${\\cal H}(S)$ denote the set of hyperplanes $H$ such that $H$ is a facet hyperplane of a simplex in ${\\cal S}mpl(S)$ and ${\\cal F}(S)$ denote the set of subsets $T$ of $S$ such that $|T| = d$ and $conv(T)$ is a facet of a simplex in ${\\cal S}mpl(S)$.\nObviously, $|{\\cal H}(S)| \\le |{\\cal F}(S)| = (d+1)|{\\cal S}mpl(S)|$.\nBy Lemmas \\ref{(A,z)-separating-hyperplane}$(a4)$ and \\ref{sLz-A,A'}, $|{\\cal A}(S)| \\le |{\\cal H}(S)|$.\nClearly,\n$|{\\cal C}onv(S)| = |{\\cal C}(S)| $ and, by Lemma \n\\ref{Min-z-containingSet},\n${\\cal S}mpl(S) = {\\cal C}onv(S)$. \nThus, $|{\\cal A}(S)| \\le (d+1) |{\\cal C}(S)| $.\n\\hfill $\\Box$\n\\\\[1.5ex]\n\\indent\nOne of the referees informed us that Theorem \\ref{w}\nwas formulated in terms of minimal infeasible subsystems and proved in a different way in \\cite{IIS}.\n\\\\[1.5ex]\n\\indent\nA hyperplane $H$ in ${\\cal H}(S)$ is said to be \n{\\em essential}\nif $H$ is a facet hyperplane of a maximal $z$-avoiding subset $A$ in $S$ separating $A$ from $z$, and\n{\\em non-essential}, otherwise.\nLet ${\\cal H}^e(S)$ and ${\\cal H}^e_s(S)$ denote the sets of essential hyperplanes in ${\\cal H}(S)$ and ${\\cal H}_s(S)$, respectively.\n\\begin{lemma}\n\\label{lemma!}\nLet $S$ be a finite set of points in $\\mathbb{R}^d$, $z \\in \\mathbb{R}^d \\setminus S$, and $s \\in S$.\nSuppose that $S$ is a $z$-containing set and $z$ is in a general position with respect to $S$.\nThen \n\\\\[1ex]\n$|{\\cal A}(S)| - |{\\cal A}(S\\setminus s)| = \n|{\\cal A}_s(S)| \\le |{\\cal H}^e_s(S)| \\le |{\\cal H}_s(S)| \\le |{\\cal F}_s(S)|\\le \n d~|{\\cal S}mpl_s(S)| =d~|{\\cal C}_s(S)|$.\n\\end{lemma}\n\n\\noindent{\\bf Proof} \\ (uses Lemmas \n\\ref{(A,z)-separating-hyperplane}$(a4)$,\n\\ref{Min-z-containingSet},\n and \\ref{sLz-A,A'}).\nWe prove that $|{\\cal A}_s(S)| \\le |{\\cal H}^e_s(S)|$.\nLet $A \\in {\\cal A}_s(S)$. Then $s \\in A $ and \n$A' = A \\setminus s$ is a $z$-avoiding but not maximal \n$z$-avoiding set in $S' = S - s$. Therefore there exists \n$s' \\in S' \\setminus A'$ such that $A' \\cup s'$ is also a \n$z$-avoiding set. \nObviously, $A' \\cup \\{s,s'\\}$ is a $z$-containing set in $S$.\nBy Lemma \\ref{sLz-A,A'}, \n$A$ has a subset $T$ such that\n$|T| = d$, $T$ belongs to a face of $conv(A)$, \nand \n$conv(T \\cup s')$ is a simplex containing $z$ as an interior point. Since $A' \\cup s'$ is a $z$-avoiding set, clearly\n$s \\in T$.\nNow by Lemmas \\ref{(A,z)-separating-hyperplane}$(a4)$ and \\ref{sLz-A,A'}, $|{\\cal A}_s(S)| \\le |{\\cal H}^e_s(S)|$.\nBy Lemma \\ref{Min-z-containingSet},\n${\\cal S}mpl_s(S) = {\\cal C}onv_s(S)$, and clearly,\n$|{\\cal C}onv_s(S)| = |{\\cal C}_s(S)| $.\nAll the other inequalities in our claim are obvious.\n\\hfill $\\Box$.\n\\\\[1.5ex]\n\\indent\nNow we are ready to prove the following strengthening of \nTheorem \\ref{w} which is also an extension of \nTheorem \\ref{!}.\n\\begin{theorem} \n\\label{m}\nLet $S$ be a finite set of points in the \n$d$-dimensional space $\\mathbb{R}^d$.\nSuppose that $z$ is in a general position with respect to $S$ {\\em (and so $z \\in \\mathbb{R}^d \\setminus S$)}.\nThen \n\\\\[1ex]\n$(a1)$ if either \n$S$ is $z$-avoiding or $S$ is $z$-containing and $|S| = d+1$ {\\em (and so $conv(S)$ \nis a $d$-dimensional simplex)}, then \n$|{\\cal A}(S)| = d~|{\\cal C}(S)| + 1$, \n\\\\[1ex]\n$(a2)$ $|{\\cal A}(S)| \\le d~|{\\cal C}(S)| + 1$, \n\\\\[1ex]\n$(a3)$ if $S$ is $z$-containing and $|S| = d + 2$, then\n$|{\\cal A}(S)| = d~|{\\cal C}(S)| - d +1$, and\n\\\\[1ex]\n$(a4)$ if $S$ is $z$-containing and $|S| \\ge d + 3$,\nthen $|{\\cal A}(S)| \\le d~|{\\cal C}(S)| - d$.\n\\end{theorem} \n\n\\noindent{\\bf Proof} \\ (uses Lemmas \n\\ref{Min-z-containingSet} \nand \n\\ref{lemma!}).\n\\\\[1.5ex]\n\\indent\n{\\bf (p1)} First we prove $(a1)$. If $S$ is $z$-avoiding, then \n$|{\\cal A}(S)| = 1$ and $|{\\cal C}(S)| = 0$, and so \n$|{\\cal A}(S)| = d~|{\\cal C}(S)| + 1$.\nIf $S$ is $z$-containing and $|S| = d+1$, then\n$conv(S)$ is a $d$-dimensional simplex, $z$ is in the interior of $conv(S)$, $|{\\cal A}(S)| = d+1$, and $|{\\cal C}(S)| = 1$, and therefore \n$|{\\cal A}(S)| = d~|{\\cal C}(S)| + 1$.\n\\\\[1.5ex]\n\\indent\n{\\bf (p2)}\nWe prove $(a2)$.\nOur claim is obviously true if $S$ is a $z$-avoiding set.\nTherefore we assume that $S$ is a $z$-containing set.\nWe prove our claim by induction on $|S|$.\nBy Lemma \\ref{Min-z-containingSet}, $|S| \\ge d+1$.\nIf $|S| = d+1$, then $conv(S)$ is a simplex, and our claim is obviously true. Thus, we assume that our claim is true for every $z$-containing set $S$ with $|S| = k \\ge d+1$ and will prove that the claim is also true if $|S| = k + 1$.\nSince $S$ is $z$-containing, by Lemma \\ref{Min-z-containingSet}, there exists $X \\subseteq S$ such that $conv(X)$ is a simplex and $z$ is the interior point of \n$conv(X)$. Since $|X| = d+1 < |S|$, there exists \n$s \\in S \\setminus X$. Obviously, $S' = S \\setminus s$ is a $z$-containing set. Since $k = |S'| < |S| = k+1$, by the induction hypothesis, our claim is true for $S' = S \\setminus s$, i.e. $|{\\cal A}(S')| \\le d~|{\\cal C}(S')| + 1$.\nBy Lemma \\ref{lemma!}, \n$|{\\cal A}(S)| - |{\\cal A}(S')| = \n|{\\cal A}_s(S)| \\le d~|{\\cal C}_s(S)| $.\nNow since $|{\\cal C}(S)| = |{\\cal C}(S')| + |{\\cal C}_s(S)|$, \nour inductive step follows.\n\\\\[1.5ex]\n\\indent\n{\\bf (p3)} We prove $(a3)$. \nSince $S$ is $z$-containing, there exists $S' \\subset S$ such that $\\Delta = conv(S')$ is a simplex and $z$ is in the interior of $conv(S')$. We can assume that\n$conv(S')$ is a minimal (by inclusion) simplex such that $S' \\subset S$ and $z$ is in the interior of $conv(S')$.\n Then the interior of $\\Delta $ does not contain points from $S$. \nClearly, there is $s \\in S$ such that $S' = S \\setminus s$. Let $L$ be the line containing $s$ and $z$ and \n$sLt$ be the closed interval in $L$ such that $z$ is in the interior of $sLt$ and $t$ belongs to a face of \n$\\Delta $. Let $sLt'$ be the maximal closed interval in $L$ that has no interior point of $\\Delta $.\nObviously, there exist faces $F$ and $F'$ of $\\Delta $ containing $t$ and $t'$, respectively. \nIn plane language, $t'$ and $t$ are the first and the last common points of the ray $sL$ with $\\Delta $.\nSince $z$ is in a general position with respect to $S$, clearly $t$ and $t'$ are the interior points of $F$ and $F'$, respectively, \nand $dim (R(F)) = d-1$, \nand so $v(F) = d$.\nLet $s' = S' \\setminus V(F)$. Then \n$\\Delta ' = conv(F \\cup s) = conv (S \\setminus s')$ is a $z$-containing simplex, and so \n$V(F) \\cup s$ is a minimal $z$-containing subset of $S$.\nBy the above definition, \n $\\Delta = conv (S \\setminus s) = conv (F \\cup s')$ is another $z$-containing simplex, and so \n $V(F) \\cup s' = S \\setminus s$ is another minimal $z$-containing subset of $S$.\n \nWe claim that $S \\setminus x$ is a $z$-avoiding subset of $S$ for every $x \\in V(F)$. \nTo prove this claim we consider two cases.\nSuppose first that $x \\in V(F) \\setminus V(F')$. Then \n$S \\setminus x = V(F') \\cup s$ and by the definition of $F'$, $z$ is not in the interior of $conv(V(F') \\cup s)$. Therefore $S \\setminus x$ is $z$-avoiding.\nNow suppose that $x \\in V(F) \\cap V(F')$. Then \n$S \\setminus x = V(F'') \\cup s$, where $F''$ is a face of $\\Delta $ distinct from $F$ and $F'$. Obviously, $L \\cap F'' = \\emptyset $. Therefore $z$ is not an interior point of $ Conv(V(F'') \\cup s)$, and so again $S \\setminus x$ is $z$-avoiding.\n \nObviously, if $S \\setminus x$ is $z$-avoiding, then $S \\setminus x$ is also maximal $z$-avoiding. Therefore \n$S \\setminus x$ is a maximal $z$-avoiding subset of $S$ for every $x \\in V(F)$.\nAlso $V(F)$ is a $z$-avoiding subset of $S$ and since both $V(F) \\cup s$ and $V(F) \\cup s'$ are $z$-containing, clearly $V(F)$ is a maximal $z$-avoiding subset of $S$.\nThus, \n${\\cal A}(S) = \\{S \\setminus x: x \\in V(F)\\} \\cup \\{V(F)\\}$.\nAlso both $\\Delta = conv(F \\cup s')$ and \n$\\Delta ' = conv(F \\cup s)$ are $z$-containing simplexes, and so ${\\cal C}(S) = \\{S \\setminus s, S \\setminus s'\\}$.\nTherefore, \n $d +1= |{\\cal A}(S)| = d~|{\\cal C}(S)| - d +1 = d+1$.\n It follows that $(a3)$ holds.\n \\\\[1.5ex]\n\\indent\n{\\bf (p4)} The proof of $(a4)$ is similar to that in ${\\bf (p3)}$ because it can be checked that the inequality holds if\n$|S| = d +3$. Claim $(a4)$ is also a particular case of Theorem \\ref{mm} below.\n\\hfill $\\Box$\n\\\\[2ex]\n\\indent\nOur next goal is to prove a strengthening of Theorem \\ref{m} that takes into consideration the size of $S$.\n\\begin{lemma}\n\\label{good-vertex-in-conv(S)}\nLet $S$ be a finite set of points in $\\mathbb{R}^d$, \nand $z$ in a general position with respect to $S$\n{\\em (and so $z \\in \\mathbb{R}^d \\setminus S$)}. Let $P = conv (S)$ and $V = V(P)$.\nSuppose that $|S| \\ge d+2$ and $S$ is a $z$-containing set.\nThen there exist a simplex $\\Delta $ and \n$u \\in S$\nsuch that\n\\\\[0.7ex]\n$(a1)$ $V(\\Delta ) \\subseteq V$,\n\\\\[0.7ex]\n$(a2)$ $z$ is an interior point of $\\Delta $,\n\\\\[0.7ex]\n$(a3)$ $\\Delta $ and $P$ have a common facet hyperplane $H$, and \n\\\\[0.7ex]\n$(a4)$ $u \\in H$ and \n$S \\setminus u$ is a $z$-containing set.\n\\end{lemma}\n\\noindent{\\bf Proof} \\ (uses Lemma \\ref{SimplexInPcontainingZero}).\nWe will consider two cases: $P$ is a simplex or \n$P$ is not a simplex.\n\\\\[1.5ex]\n\\indent\n${\\bf (p1)}$ \nSuppose that $P$ is a simplex.\nLet $\\Delta = P$.\nSince $|S| \\ge d+2$, there exists \n$v \\in S \\setminus V(P)$.\n\nSuppose that $v$ is in the interior of $P$. \nThen obviously, there exists a facet $F$ of $P$ such that $z$ is in the interior of $conv(F \\cup s)$. \nLet $u$ be a (unique) vertex of $V(P)$ not belonging to \n$F$. Then $(\\Delta , u)$ satisfies $(a1)$ - $(a4)$, where $H$ is the hyperplane containing $F$.\n\nNow suppose that $v$ is not in the interior of \n$P$. Then there exists a facet $T$ of $P$ containing $v$. Put $u = v$. Then again $(\\Delta , u)$ satisfies $(a1)$ - $(a4)$, where $H$ is the hyperplane containing $T$.\n\\\\[1.5ex]\n\\indent\n${\\bf (p2)}$ Finally, suppose that $P$ is not a simplex.\nBy Lemmas \\ref{SimplexInPcontainingZero}, there exists a simplex $\\Delta _z$ such that \n$V(\\Delta _z) \\subseteq V$ and $z$ is in the interior of \n$\\Delta _z$.\nSince $P$ is not a simplex, $\\Delta _z \\ne P$.\nTherefore $\\Delta _z$ has a facet $F_z$ which is not a facet of $P$. Let $v$ be a unique vertex of $\\Delta _z$ such that $v \\not \\in F_z$.\nLet $L$ be the line containing $v$ and $z$. Let $vLz'$ be a maximal segment in $L$ such that $z \\in vLz'$ and $vLz' \\subset P$. Obviously, such segment exists (and is unique) and $z'$ belongs to a face $F$ of $P$.\nMoreover, since $z$ is in a general position with respect to $S$, clearly $F$ is a facet of $P$ and point $z'$ is in a general position with respect to $S' = V(F)$ (and so $S'$ is a $z'$-containing set) in $\\mathbb{R}^{d-1}$.\nBy Lemma \\ref{SimplexInPcontainingZero}, $F$ contains an $(d-1)$-dimensional simplex \n$\\Delta '$ such that $z'$ is in the interior of $\\Delta '$.\nThen $\\Delta = conv(\\Delta ' \\cup v)$ is a $d$-dimensional simplex satisfying $(a1)$, $(a2)$, and $(a3)$ with \n$H = R(F) = R(\\Delta ')$.\nSince $F$ is a facet of $P$, clearly $V(F) \\subset V(P) \\subseteq S$ and\n$V(F) \\setminus V(F_z) \\ne \\emptyset $.\nLet $u$ be an arbitrary point of $V(F) \\setminus V(F_z)$. Obviously, $u \\ne v$, for otherwise, \n$z \\in vLz' \\subseteq F$, and so $z \\in F \\subset H$. This is impossible because $z$ is in the interior of $P$. \nThus, $\\Delta \\subset conv(S \\setminus u)$, and so \n$S \\setminus u$ is $z$-containing. \n\\hfill $\\Box$\n\\\\[2ex]\n\\indent\nNow we are ready to prove the following strengthening of Theorem \\ref{m}. \n\\begin{theorem}\n\\label{mm}\nLet $S$ be a finite set of points in the \n$d$-dimensional space $\\mathbb{R}^d$, $z \\in \\mathbb{R}^d \\setminus S$.\nSuppose that $S$ is $z$-containing, $z$ is in a general position with respect to $S$, and \n$|S| \\ge d + 2$.\nThen \n$|{\\cal A}(S)| \\le d~|{\\cal C}(S)| - |S| + 3$.\n\\end{theorem}\n\n\\noindent{\\bf Proof} \\ (uses Lemmas \\ref{lemma!}, \n\\ref{good-vertex-in-conv(S)}, and Theorem \\ref{m}).\nWe prove our claim by induction on $|S|$.\nIf $|S| = d + 2$, then by Theorem \\ref{m} $(a3)$, the claim is true. We assume that our claim is true for $|S| = k \\ge d+2$ and prove that it is also true if $|S| = k +1$.\nBy Lemma \\ref{lemma!},\n\\\\[1ex]\n\\indent\n $|{\\cal A}(S)| - |{\\cal A}(S \\setminus s)| = |{\\cal A}_s(S)| \\le \n |{\\cal H}^e_s(S)| \\le |{\\cal H}_s(S)| \\le \n d~|{\\cal S}mpl_s(S)| = d~|{\\cal C}_s(S)|$\n \\\\[1ex]\n for every $s \\in S$.\n By Lemma \\ref{good-vertex-in-conv(S)}, \n there exist a point $u$ in $S$ and a simplex $\\Delta $ in ${\\cal S}impl_u(S)$ such that \n $S \\setminus u$ is $z$-containing and \n$R(T) = R(F)$ for some facets $T$ and $F$ of $\\Delta $ and $conv(S)$, respectively. Then obviously, $R(T)$ is a non-essential hyperplane in ${\\cal H}_u(S)$, and therefore \n\\\\[0.5ex] \n$|{\\cal H}^e_u(S)| \\le |{\\cal H}_u(S)| - 1$. \nTherefore \n\\\\[1ex]\n\\indent\n$|{\\cal A}(S)| - |{\\cal A}(S \\setminus u)| = |{\\cal A}_u(S)| \\le \n |{\\cal H}^e_u(S)| < |{\\cal H}_u(S)| \\le d~|{\\cal C}_u(S)|$.\n\\\\[1ex]\nBy the induction hypothesis, we have:\n$|{\\cal A}(S \\setminus u)| \\le \nd~|{\\cal C}(S\\setminus u)| - |S \\setminus u| + 3$.\n\\\\[1ex]\nThus, our inductive step follows from the last two inequalities.\n\\hfill $\\Box$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\nThe property of waves to travel over large distances and long time without changing their shape is an important feature used in current technologies, such as communication devices and other electromagnetic products. The governing equations for electromagnetic problems are the Maxwell's equations and it is to these we seek approximate solutions in this paper. To make the numerical approximation to the solution accurate either low order methods on fine meshes, which can be computationally costly, or high-order methods on coarser meshes can be used. The latter approach is usually preferable for large scale problems.\n\nSeveral high-order methods in computational electromagnetics have been proposed, such as high-order finite-difference time-domain (FDTD) methods \\cite{Yee1966,Xie2002}, discontinuous Galerkin (DG) methods \\cite{Hesthaven2002,cockburn2001runge} and pseudo-spectral methods \\cite{Fan2002,Galagusz2016,Yang1997}, to name a few. High-order explicit FDTD methods require a restrictive stability condition and wide stencils, which complicate the enforcement of boundary conditions. Unconditionally stable alternating-direction-implicit (ADI) FDTD methods have been developed to circumvent the time step constraints \\cite{Namiki1999,Zheng1999,Tan2008,Chen2010,Liang2013}, however, methods are difficult to generalize to high order and treating complex geometry is not straightforward. \n \nDiscontinuous Galerkin methods achieve high-order convergence rates by approximating the function using local high order polynomials and are an excellent choice for problems where a high quality mesh can be generated. The main drawbacks of DG methods is their restrictive time step at high order of accuracy and the duplication of degrees of freedom on the edges of elements. \t\n\t \nAnother avenue to handle time dependent wave problems is the Hermite-Taylor method, which consists of a Hermite interpolation procedure in space and a Taylor method in time \\cite{Goodrich2005}. The key idea is to evolve, in time, the numerical solution as well as its $m$ first space derivatives to achieve a $(2\\,m+1)$-order accurate method using only $(m+1)^d$ degrees of freedom per element in $d$-dimensions. \n\nAs was shown for linear symmetric hyperbolic problems in \\cite{Goodrich2005}, this method provides a stability condition that only depends on the largest wave-speed, independent of the order. Hence, large time-step sizes can be used for these high-order methods and therefore ease the computational burden for large-scale problems. As the $(m+1)^d$ degrees of freedom in a Hermite method are collocated at a single node the imposition of general boundary conditions can be challenging. Typically, in addition to the physical boundary conditions the method needs to be augmented with a relatively large number of numerical boundary conditions (sometimes called compatibility boundary condition). While this has been successfully done for the wave equation on both Cartesian and curvilinear meshes in \\cite{compat_wave_hermite_AAL_DEAA_WDH}, it has proven difficult to use this technique for first order hyperbolic systems. \n\nA possible solution to this is to use a hybrid DG-Hermite method \\cite{Chen2014} for Maxwell's equations. The method in \\cite{Chen2014} takes advantage of the flexibility of DG solvers to handle complex geometries and boundary conditions by considering two non-overlapping meshes, an unstructured mesh for the DG method and a staircased Cartesian mesh where the Hermite method is used. This approach requires a hybrid structured-unstructured mesh and the use of local time-stepping to maintain large time-step sizes in the Hermite method. In \\cite{OversetHermiteDG} an overset grid method that combines a Hermite method (on Cartesian meshes) and a DG method (on structured curvilinear meshes) for the wave equation is proposed. This method does not require a hybrid non-overlapping mesh and as such it is somewhat more geometrically flexible but again, it is not easy to extend to first order hyperbolic systems. \t\n\nIn this work, we propose an alternative solution for imposing boundary conditions for Maxwell's equations within the framework of Hermite methods. Our new method is based on the correction function method (CFM). The CFM was first proposed in \\cite{Marques2011} to handle Poisson's equation with interface conditions and continuous coefficients in a finite-difference context. Given a numerical solution (for example from a finite difference method) that has been updated near but not on the boundary from the CFM seeks a polynomial approximation to the solution in the vicinity of a boundary or interface using a minimization procedure. A functional that is based on a square measure of the residual of the original PDE problem and that also contains terms from the finite difference solver is minimized over a suitable space of polynomials. Once this polynomial approximation, also called the correction function, is found, the numerical solution can be corrected so that it satisfies the boundary conditions to high order of accuracy. The CFM method has been used for Poisson's equation \\cite{Marques2017,Marques2019}, the wave equation \\cite{Abraham2018} and for electromagnetic problems with both interface and boundary \\cite{LawMarquesNave2020,LawNave2021,LawNave2022}. \n\nIn this paper we introduce a CFM - Hermite-Taylor method. An advantage with using Hermite based methods for the base scheme is that the Hermite stencil remains the same regardless of its order. This is not the case for FDTD methods. Additionally, the Hermite-Taylor method directly provides a space-time polynomial approximating the solution that is required in the CFM functional. In this paper we focus exclusively on the case when the geometry of the problem can be represented on a Cartesian mesh or on a logically Cartesian curvilinear mesh. Already in this setting the Hermite stencil provides a good advantage but we expect that in future work where we treat interfaces and non-grid aligned boundaries the advantage will be even greater. \n\t\nThe paper is organized as follows. We introduce Maxwell's equations with the considered boundary conditions in Section~\\ref{sec:def_problem}. In Section~\\ref{sec:HermiteTaylor}, the 1-D Hermite-Taylor method is described in detail and some remarks are provided for higher dimensional cases. The correction function method is introduced and described in detail in the Hermite-Taylor setting in Section~\\ref{sec:CFM}. Finally, numerical examples in 1-D and 2-D that verify the properties of the Hermite-Taylor correction function method are presented in Section~\\ref{sec:num_examples}. \n\n\\section{Problem Definition} \\label{sec:def_problem}\nIn this work, we seek approximate solutions to Maxwell's equations\n\\begin{equation} \\label{eq:problemMaxwell}\n\\begin{aligned}\n\t\\mu\\,\\partial_t \\mathbold{H} + \\nabla\\times \\mathbold{E} =&\\,\\, 0, \\\\\n\t\\epsilon\\,\\partial_t \\mathbold{E} - \\nabla\\times\\mathbold{H} =&\\,\\, 0,\\\\\n\t\\nabla\\cdot(\\epsilon\\,\\mathbold{E}) =&\\,\\, 0,\\\\\n\t\\nabla\\cdot(\\mu\\,\\mathbold{H})=&\\,\\, 0,\n\\end{aligned}\n\\end{equation}\nin the domain $\\Omega \\subset \\mathbb{R}^d$ with $d=1,2$ and the time interval $I=[t_0,t_f]$.\nHere $\\mathbold{H}$ is the magnetic field, \n\t$\\mathbold{E}$ is the electric field, \n\t$\\mu$ is the magnetic permeability and $\\epsilon$ is the electric permittivity.\nTo complete the system \\eqref{eq:problemMaxwell}, \n\twe consider the initial conditions \n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\mathbold{H}(\\mathbold{x},t_0) =&\\,\\, \\mathbold{H}_0 \\quad \\text{in } \\Omega,\\\\ \n\t\t\t\\mathbold{E}(\\mathbold{x},t_0) =&\\,\\, \\mathbold{E}_0 \\quad \\text{in } \\Omega,\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tand either of the boundary conditions,\n\\begin{equation} \\label{eq:nxE}\n\t\\mathbold{n}\\times\\mathbold{E} = \\mathbold{g}_E \\quad \\text{on } \\Gamma \\times I,\n\\end{equation}\n\tand \n\\begin{equation} \\label{eq:nxH}\n\t\\mathbold{n}\\times\\mathbold{H} = \\mathbold{g}_H \\quad \\text{on } \\Gamma \\times I.\n\\end{equation}\nHere $\\Gamma$ is the boundary of the domain $\\Omega$, $\\mathbold{n}$ is the outward unit normal to $\\Gamma$, and $\\mathbold{g}_E$ and $\\mathbold{g}_H$ are known functions. The case when $\\mathbold{n}\\times\\mathbold{E} = 0$ corresponds to the boundary condition of a perfect electric conductor (PEC). \n\t\n\\section{Hermite-Taylor Method} \\label{sec:HermiteTaylor}\nIn the following, a brief review of the Hermite-Taylor method, introduced by Goodrich et al. \\cite{Goodrich2005}, is provided. For simplicity, we consider the 1-D case and include some comments regarding higher dimensions. \n\t\nThe Hermite method uses a mesh staggered in both space and time as illustrated in Fig.~\\ref{fig:Hermite_method}.\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=3.5in]{Hermite_method.pdf}\n \\caption{Illustration of the Hermite-Taylor procedure to evolve the data from $(x_{i+1},t_n)$ to $(x_{i+1},t_{n+1})$. The Hermite interpolation procedure and the Taylor method are \n denoted respectively by $\\mathcal{I}$ and $\\mathcal{T}$. The primal and dual nodes are respectively \n represented by black squares and blue circles.}\n \\label{fig:Hermite_method}\n\\end{figure}\nConsider the domain $\\Omega = [x_\\ell, x_r]$ and a time interval $I = [t_0,t_f]$. \nWe then define the primal mesh to be \n\t$$x_i = x_\\ell + i\\,\\Delta x, \\quad i=0,\\dots,N_x, \\quad \\Delta x = \\frac{x_r-x_\\ell}{N_x}.$$\nHere $N_x$ is the number of cells on the primal mesh.\nThe dual mesh is then defined as the cell centers of the primal mesh \n\t$$x_{i+1\/2} = x_\\ell + (i+1\/2)\\,\\Delta x, \\quad i=0,\\dots,N_x-1.$$\nThe approximate solution on the primal mesh is centered at times \n\t$$ t_n = t_0 + n\\,\\Delta t, \\quad n=0,\\dots,N_t, \\quad \\Delta t = \\frac{t_f-t_0}{N_t},$$\n\twhile the approximation on the dual mesh is centered at times \n\t$$t_{n+1\/2} = t_0 + (n+1\/2)\\,\\Delta t, \\quad n=0,\\dots,N_t-1.$$\nHere $N_t$ is the number of time steps. \n\nThe Hermite-Taylor method requires three processes:\n\\begin{itemize}\n\t\\item[1.] \\underline{Hermite interpolation:}\n\t\n\t\n\tAssume that the values of the electromagnetic fields and their $m$ first space derivatives \n\t\t(or sufficiently accurate approximation of these) are available on the primal mesh at $t_{n-1}$. \n\tThen, \n\t\tfor each cell in the primal mesh, \n\t\tfor each electromagnetic field,\n\t\twe construct the unique polynomial of degree $2\\,m+1$ coinciding with the electromagnetic field and \n\t\tits $m$ first derivatives at the endpoints of the cell,\n\t\tthat is the Hermite interpolant of the electromagnetic field. \n\tIn Fig.~\\ref{fig:Hermite_method}, \n\t\tthis step is represented by $\\mathcal{I}$.\n\t\n\t\\item[2.] \\underline{Recursion relation:}\n\t\n\tThe recursion relation constructs a space-time polynomial,\n\t\treferred as a Hermite-Taylor polynomial in this work, \n\t\tapproximating each electromagnetic field.\n\tConsidering a cell and a given Hermite interpolant of each electromagnetic field on this cell, \n\t\twe identify the derivatives of the electromagnetic field as scaled coefficients of the polynomial at the cell center.\n\tBy expanding,\n\t\tin time, \n\t\teach scaled coefficient in a Taylor polynomial and enforcing the PDE at the cell center, \n\t\twe obtain a recursion relation for the coefficients of the Hermite-Taylor polynomials. \n\tThis step is represented in Fig.~\\ref{fig:Hermite_method} by either blue dashed circles or black dashed squares.\n\t\n\t\\item[3.] \\underline{Time evolution:}\n\t\n\tFinally, \n\t\twe update the electromagnetic fields and their $m$ first space derivatives at the dual mesh points by simply evaluating the Hermite-Taylor polynomials. \n\tThis step is represented by $\\mathcal{T}$ in Fig.~\\ref{fig:Hermite_method}.\n\t\t\n\\end{itemize}\nLet us now detail each time step of the method.\n\n\\subsection*{Hermite Interpolation} \nAssuming that the $m$ first space derivatives of the electromagnetic fields at the initial time $t_0$ are available on the primal mesh, \n\twe compute the $(2\\,m+1)$ degree Hermite interpolant $p_i^f(x)$ on each cell $[x_i, x_{i+1}]$ satisfying \n\t\\begin{equation*}\n\t\t\\frac{d^\\ell p_i^f(x_i,t_0)}{dx^\\ell} = \\frac{d^\\ell f(x_i,t_0)}{dx^\\ell}, \\quad\n\t\t\\frac{d^\\ell p_i^f(x_{i+1},t_0)}{dx^\\ell} = \\frac{d^\\ell f(x_{i+1},t_0)}{dx^\\ell}, \\quad \\ell=0,\\dots,m.\n\t\\end{equation*}\nHere $f$ is either the magnetic field $H$ or the electric field $E$. \nWe then obtain a polynomial approximating each electromagnetic field on the cell $[x_i,x_{i+1}]$ and centered at the cell center $x_{i+1\/2}$,\n\\begin{equation*}\n\t\\begin{aligned}\n H(x,t)|_{t=t_0} \\approx&\\,\\, p^H_i(x) = \\sum_{\\ell=0}^{2\\,m+1} c^H_\\ell(t)|_{t=t_0}\\,\\bigg(\\frac{x-x_{i+1\/2}}{\\Delta x}\\bigg)^\\ell, \\\\\n E(x,t)|_{t=t_0} \\approx&\\,\\, p^E_i(x) = \\sum_{\\ell=0}^{2\\,m+1} c^E_\\ell(t)|_{t=t_0}\\,\\bigg(\\frac{x-x_{i+1\/2}}{\\Delta x}\\bigg)^\\ell,\n \t\\end{aligned}\n\\end{equation*}\n\twhere $c^H(t)$ and $c^E(t)$ are time-dependent coefficients.\n\n\\subsection*{Recursion Relation} \n\nLet us now compute a Hermite-Taylor polynomial approximating each electromagnetic field. \t\nTo do so, \n\twe expand the coefficients in a Taylor polynomial of degree $q$ centered at $t_0$, \n\twhich leads to \n\\begin{equation} \\label{eq:Hermite_Taylor_polynomials}\n\t\\begin{aligned}\n H(x,t) \\approx&\\,\\, p^H_i(x,t) = \\sum_{\\ell=0}^{2\\,m+1} \\sum_{s=0}^q c^H_{\\ell,s}\\,\\bigg(\\frac{x-x_{i+1\/2}}{\\Delta x}\\bigg)^\\ell\\,\\bigg(\\frac{t-t_{0}}{\\Delta t}\\bigg)^s, \\\\\n E(x,t) \\approx&\\,\\, p^E_i(x,t) = \\sum_{\\ell=0}^{2\\,m+1} \\sum_{s=0}^q c^E_{\\ell,s}\\,\\bigg(\\frac{x-x_{i+1\/2}}{\\Delta x}\\bigg)^\\ell\\,\\bigg(\\frac{t-t_{0}}{\\Delta t}\\bigg)^s.\n \t\\end{aligned}\n\\end{equation}\nHere $c^H_{\\ell,0}$ and $c^E_{\\ell,0}$ are known from the initial data and the interpolation step. \nConsider Maxwell's equations in 1-D with constant coefficients, \n\t\\begin{equation*}\n\t\t\\begin{aligned}\n \t\t\t\\frac{\\partial H}{\\partial t} =&\\,\\, -\\frac{1}{\\mu}\\frac{\\partial E}{\\partial x}, \\\\\n \t\t\t\\frac{\\partial E}{\\partial t} =&\\,\\, -\\frac{1}{\\epsilon}\\frac{\\partial H}{\\partial x}.\n\t\t\\end{aligned}\n\t\\end{equation*}\nFor smooth solutions, \n\twe then have \n\t\\begin{equation} \\label{eq:recursion_Maxwell_1D}\n\t\t\\begin{aligned} \n\t\t\\frac{\\partial^k}{\\partial x^k}\\frac{\\partial^{r+1}H}{\\partial t^{r+1}}=&\\,\\, -\\frac{1}{\\mu}\\frac{\\partial^{r}}{\\partial t^{r}}\\frac{\\partial^{k+1}E}{\\partial x^{k+1}},\\\\\n\t\t\\frac{\\partial^k}{\\partial x^k}\\frac{\\partial^{r+1}E}{\\partial t^{r+1}}=&\\,\\, -\\frac{1}{\\epsilon}\\frac{\\partial^{r}}{\\partial t^{r}}\\frac{\\partial^{k+1}H}{\\partial x^{k+1}}.\n\t\t\\end{aligned}\n\t\\end{equation}\nSubstituting $H$ and $E$ by their Hermite-Taylor approximations $p_i^H(x,t)$ and $p_i^E(x,t)$, \n\tin the system \\eqref{eq:recursion_Maxwell_1D} and evaluating it at $(x_{i+1\/2},t_0)$,\n\twe obtain the following recursion relations for the coefficients\n\\begin{equation*} \n\tc^H_{\\ell,s} = -\\frac{(\\ell+1)\\,\\Delta t}{\\mu\\,s\\,\\Delta x}\\,c^E_{\\ell+1,s-1}, \\quad\n\tc^E_{\\ell,s} = -\\frac{(\\ell+1)\\,\\Delta t}{\\epsilon\\,s\\,\\Delta x}\\,c^H_{\\ell+1,s-1}, \\quad\n\t\\ell = 0,\\dots,2\\,m+1, \\quad s=1,\\dots,q.\n\\end{equation*}\nKnowing $c^H_{\\ell,0}$ and $c^E_{\\ell,0}$, \n\tthese recursion relations allow the computation of the Hermite-Taylor polynomials approximating the electromagnetic fields.\n\n\\subsection*{Time Evolution}\nFinally, \n\twe evolve\tthe electromagnetic fields and their $m$ first space derivatives on the dual mesh nodes,\n\tlocated at $(x_{i+1\/2},t_{1\/2})$ for the cell $[x_i, x_{i+1}]$, \n\tby evaluating \\eqref{eq:Hermite_Taylor_polynomials}\n$$\\frac{\\partial^{\\ell} p^H_i(x_{i+1\/2},t_{1\/2})}{\\partial x^\\ell}, \\quad \\frac{\\partial^{\\ell} p^E_i(x_{i+1\/2},t_{1\/2})}{\\partial x^\\ell}, \\quad \\ell=0,\\dots,m.$$\n\t\nA similar process is repeated to evolve the data from the dual mesh at $t_{1\/2}$ to the primal mesh at $t_1$ and therefore to complete the time step. \nThe overall procedure is repeated until the final time is reached. \nFig.~\\ref{fig:Hermite_method} illustrates the Hermite-Taylor method at a given primal node.\n\t \n\\begin{remark}\nFor linear constant coefficients hyperbolic problems, \n\tthe Taylor expansion in time of the coefficients of the Hermite polynomials is computed exactly for $q$ sufficiently large \\cite{Goodrich2005},\n\tfor example $q=2\\,m+1$ in \\eqref{eq:Hermite_Taylor_polynomials} for the 1-D case.\nIn general, \n\t we set $q=\\nu\\,(2\\,m+1)$ in $\\mathbb{R}^{\\nu}$ to obtain an exact time expansion of the coefficients.\n\\end{remark}\n\t\t\n\\begin{remark}\nIn higher dimensions, \n\tthe primal mesh is defined as the classical Cartesian mesh while the dual nodes are defined \n\tat the cell center. \nHence, \n\tthis differs from the mesh used in FDTD methods. \nAs for the Hermite interpolation procedure, \n\tapproximations are computed using a tensor product of 1-D Hermite polynomials.\nWe refer the interested reader to \\cite{Goodrich2005} for more details \n\ton the Hermite-Taylor setting for higher dimensions. \n\\end{remark}\n\t\nAs mentioned before,\n\ta challenge for the Hermite-Taylor method is to enforce general boundary conditions. \nIndeed, \n\tthis method requires to know all information on the boundary,\n\tincluding the $m$ first space derivatives, \n\twhich are usually not available. \nIn the next section,\n\twe present a way to obtain the needed information using the correction function method.\n\n\\section{Correction Function Method} \\label{sec:CFM}\n\nIn this section, we describe the correction function method that computes approximations to the electromagnetic fields and their $m$ first space derivatives at the nodes located on the boundary of the domain. There are two key ingredients to the CFM: the minimization of functionals describing the electromagnetic fields near the boundary, and careful definition of the space-time domains of the functionals along the boundary. We refer to a space-time domain of a functional as a local patch. Once the minimization procedure is completed, we obtain space-time polynomials, called correction functions, approximating each electromagnetic field in the vicinity of the boundary. The correction functions are used to update the solution at the boundary nodes. In the following, we first describe the method in detail in 1-D and then generalize it in higher dimensions.\n\n \n\n\n\\subsection{The Hermite CFM Method in One Dimension} \\label{sec:cfm_1D}\nOn the mesh in Fig.~\\ref{fig:Hermite_method}, the first step has allowed for the update of the Hermite solution on the dual mesh at time level $t_{n+1\/2}$ and the second step has allowed for the update of the numerical solution on the primal mesh at $t_{n+1}$, \texcept near the boundary. At $(x_0,t_n)$ and $(x_{N_x},t_n)$ for $n=1,\\dots,N_t$ the solution will be updated using the CFM. \n\t\nWe define a node where the numerical solution is updated using the Hermite-Taylor method as a Hermite node and a node where the numerical solution is computed using the CFM we denote as a CF node. In the following, the subscript $i$ refers to the $i^{\\text{th}}$ CF node in the mesh and the superscript $n$ refers to the time level $t_n$. In the 1-D case, $i=0$ and $i=1$ refer respectively to the boundary nodes $x_0$ and $x_{N_x}$. \n\nWe further note that although the functional just to be defined can depend on time, as manifested by the $n$ superscript, (for example to account for a moving geometry) but for all the problems considered here it will not. When there is no time dependence all the small linear system of equations (one at each CF node) resulting from the quadratic optimization problem, will not change in time and can thus be formulated, factored and stored once and for all before the time stepping loop. Consequently the complexity of the Hermite-CFM method will approach that of the Hermite method in the limit $h \\rightarrow 0$. \n\nThe CFM minimizes a functional unique to each CF node composed of three parts \n\\begin{equation} \\label{eq:functional_J}\n\tJ_i^n = \\mathcal{G}_i^n + \\mathcal{B}_i^n + \\mathcal{H}_i^n.\n\\end{equation}\nHere, $\\mathcal{G}_i^n$ weakly enforces the governing equations, $\\mathcal{B}_i^n$ weakly enforces the boundary conditions and $\\mathcal{H}_i^n$ weakly enforces that the correction functions match the Hermite solution near the $i^\\text{th}$ CF node.\n\nThe domains over which the different terms in the functional are computed are not the same.\nThe domain of $\\mathcal{B}_i^n$ should include the part of the boundary in the vicinity of the $i^{\\text{th}}$ CF node to weakly enforce the boundary conditions. The domain of $\\mathcal{H}_i^n$ should be the same as the space-time domain of the Hermite node closest to the $i^{\\text{th}}$ CF node. We then weakly enforce the correction functions to match the Hermite solution in the domain of $\\mathcal{H}_i^n$ while avoiding extrapolation procedures of the Hermite solution. \nFinally, the domain of integration for $\\mathcal{G}^n_i$ should enclose the $i^{\\text{th}}$ CF node, the domain of integration for $\\mathcal{B}_i^n$ and the domain of integration for $\\mathcal{H}_i^n$ to enforce Maxwell's equations over the whole local patch of the functional $J_i^n$.\n\nAs an example for the CF node $x_0$ at time level $t_n$, $\\mathcal{G}_0^n$ contains the residual of the PDE and it is integrated over the rectangular space-time region (the local patch) consisting of the direct product of the space interval $S_0=[x_0,x_{3\/2}]$ with the time interval $I_n=[t_{n-1\/2},t_n]$ as illustrated in Fig.~\\ref{fig:local_patch_1D_x0}. \n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=3.0in]{integration_domain_G0_1D.pdf}\n\t\\caption{Illustration of the domain of integration $S_0\\times I_n$ of $\\mathcal{G}_0^n$. \n\t\t The primal CF and Hermite nodes are respectively represented by green squares and black squares \n\t\t while the dual Hermite nodes are represented by blue circles.\n\t\t The CFM seeks the information located at $(x_0,t_{n})$ which is enclosed by the red circle.\n\t\t The space-time local patch $S_0\\times I_n$ is denoted by a dashed magenta box.}\n\t\\label{fig:local_patch_1D_x0}\n\\end{figure}\nWe then have \n\\begin{equation*}\n\t\\mathcal{G}_0^n(H^n_{h,0},E^n_{h,0}) = \\frac{\\ell_0}{2} \\,\\int\\limits_{I_n}\\!\\int\\limits_{S_0} (\\mu\\,\\partial_t H^n_{h,0}+\\partial_x E^n_{h,0})^2 + (\\epsilon\\,\\partial_t E^n_{h,0} + \\partial_x H^n_{h,0})^2\\,\\mathrm{d}x\\,\\mathrm{d}t,\n\\end{equation*}\nwhere $\\ell_0 = x_{3\/2}-x_0 = 1.5\\,\\Delta x$ is the characteristic length of the space interval $S_0$. Here $H^n_{h,0}$ and $E^n_{h,0}$ are the sought correction functions approximating the electromagnetic fields and are used to update the numerical solution at $(x_0,t_n)$.\n\n\nThe term $\\mathcal{B}_0^n$ contains the residual of the boundary condition at $x_0$ and it is integrated over the time interval $I_n$ as shown in Fig.~\\ref{fig:boundary_1D_x0}. Hence, we have either \n\\begin{equation*}\n\t\\mathcal{B}_0^n(E^n_{h,0}) = \\frac{1}{2} \\, \\int\\limits_{I_n} (E^n_{h,0}(x_0,t)-g_E(t))^2\\,\\mathrm{d}t,\n\\end{equation*}\nfor the boundary condition \\eqref{eq:nxE} or \n\\begin{equation*}\n\t\\mathcal{B}_0^n(H^n_{h,0}) = \\frac{1}{2} \\, \\int\\limits_{I_n} (H^n_{h,0}(x_0,t)-g_H(t))^2\\,\\mathrm{d}t,\n\\end{equation*}\nfor the boundary condition \\eqref{eq:nxH}.\n\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=3.0in]{integration_domain_B0_1D.pdf}\n\t\\caption{Illustration of the domain of integration $I_n$ at $x_0$ of $\\mathcal{B}_0^n$. \n\t\t The primal CF and Hermite nodes are respectively represented by green squares and black squares \n\t\t while the dual Hermite nodes are represented by blue circles.\n\t\t The CFM seeks the information located at $(x_0,t_{n})$ which is enclosed by the red circle.\n\t\t The intersection between the boundary and the local patch,\n\t\t\tthat is the line connecting $(x_0,t_{n-1\/2})$ to $(x_0,t_n)$, \n\t\t\tis denoted by a dashed purple line.}\t\n\t\\label{fig:boundary_1D_x0}\n\\end{figure}\nWe now require the correction functions to weakly match the Hermite solution over the space-time domain of the primal Hermite node $x_1$. This is what connects the two methods and is needed for the minimization problem to be well-posed. The term $\\mathcal{H}_0^n$ contains the Hermite-Taylor polynomials $H^*(x,t)=p_1^H(x,t)$ and $E^*(x,t)=p_1^E(x,t)$, and it is integrated over the rectangular space-time region consisting of the direct product of the space\tinterval $S^{\\mathcal{H}}_0 = [x_{1\/2},x_{3\/2}]$, which is the cell associated with the primal Hermite node $x_1$, \twith the time interval $I_n$. The space-time region $S^{\\mathcal{H}}_0\\times I_n$ is illustrated in Fig.~\\ref{fig:HermiteTaylor_patch_1D_x0}. We then have \n\\begin{equation} \\label{eq:infoHermite}\n\t\\mathcal{H}_0^n(H^n_{h,0},E^n_{h,0}) = \\frac{c_H(\\Delta x)}{2}\\,\\int\\limits_{I_n}\\!\\int\\limits_{S_0^\\mathcal{H}} (H^n_{h,0}-H^*)^2 + (E^n_{h,0}-E^*)^2\\,\\mathrm{d}x\\,\\mathrm{d}t,\n\\end{equation}\nwhere $c_H(\\Delta x)$ is a given penalization function that depends on the mesh size $\\Delta x$ and is such that $a\\leq c_H(\\Delta x)\\leq 1$ for $a>0$. \n\n\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=3.0in]{integration_domain_H0_1D.pdf}\n \\caption{Illustration of the domain of integration $S_0^\\mathcal{H} \\times I_n$ of $\\mathcal{H}_0^n$. \n\t\t The primal CF and Hermite nodes are respectively represented by green squares and black squares \n\t\t while the dual Hermite nodes are represented by blue circles.\n\t\t The CFM seeks the information located at $(x_0,t_{n})$ which is enclosed by the red circle.\n\t\t The domain $S_0^\\mathcal{H} \\times I_n$,\n\t\t\twhere we enforce the correction functions to match the Hermite-Taylor polynomials,\n\t\t\tis denoted by a dashed blue \n\t\t box.}\n\t\\label{fig:HermiteTaylor_patch_1D_x0}\n\\end{figure}\nA similar procedure is used to define the local patch and the functional associated with the second CF node $x_{N_x}$ at the time level $t_n$.\n\n\\subsubsection{The Linear System of Equations that Solves the Optimization Problem}\nAt each CF node we must solve the following problem. \n\t\\begin{equation} \\label{eq:minPblm_1D}\n\t\t\\begin{aligned}\n\t\t\t&\\text{Find } (\\mathbold{H}^n_{h,i},\\mathbold{E}^n_{h,i}) \\in V \\times V \\text{ such that }\\\\\n \t\t\t&\\qquad (\\mathbold{H}^n_{h,i},\\mathbold{E}^n_{h,i}) \\in \\underset{\\mathbold{v},\\mathbold{w}\\in V}{\\arg\\min}\\, J_i^n(\\mathbold{v},\\mathbold{w}).\n\t\t\\end{aligned}\n\t\\end{equation}\nHere $V = P^k\\big(S_i\\times I_n \\big)$ is the space of polynomials of degree $k$. In this work, we use space-time Legendre polynomials. In our one dimensional example $i=0,1$. Note that although $n=1,\\dots,N_t$, since the boundary does not change in time, there is in fact only one optimization problem for each CF node.\n \nWe formally compute the gradient of $J_i^n$ with respect to the coefficients of the polynomial approximations $H^n_{h,i}$ and $E^n_{h,i}$, and use that it vanishes at a minimum to find a solution to the minimization problem \\eqref{eq:minPblm_1D}.\nThis leads to a linear system \n\\begin{equation*}\n\tM_i^n\\, \\mathbold{c}_i^n = \\mathbold{b}_i^n,\n\\end{equation*}\nwhere $\\mathbold{c}_i^n$ contains the coefficients of $H^n_{h,i}$ and $E^n_{h,i}$. \n\nAgain, since the boundary of the domain does not move, we have $M_i = M_i^n$, so the matrices $M_i$, their scaling and LU factorization are found in a pre-computation step. Consequently, the only computations needed at each time step is the computation of the right-hand side $\\mathbold{b}_i^n$, followed by forward and backward substitutions to find $\\mathbold{c}_i^n$.\n\n\\subsubsection{Summary of the Hermite-CFM Method in One Dimension}\nGiven the numerical solution on the primal mesh at $t_{n-1}$, the algorithm of the Hermite-Taylor correction function method to evolve the numerical solution at $t_{n}$ is:\n\\begin{itemize}\n\\item[1.] Update the numerical solution on the dual mesh at $t_{n-1\/2}$ using the Hermite-Taylor method;\n\\item[2.] Update the numerical solution on the primal Hermite node at $t_{n}$ using the Hermite-Taylor method and store the Hermite-Taylor polynomials needed for the CFM;\n\\item[3.] Update the numerical solution at the CF nodes using the CFM by computing the right hand sides $\\mathbold{b}_i^n$ and solve for $\\mathbold{c}_i^n$. This is done independently for each $i$ and can thus be done in parallel without any communication step.\n\\end{itemize}\n\n\t\n\\subsection{The Hermite-CFM Method in Two Dimensions}\nWe only consider piecewise rectangular domains composed of straight lines between primal nodes. For higher dimensions, the spatial domain of a local patch is adapted depending on the geometry of the boundary and where the Hermite solution is available \n\tin the vicinity of its CF node while the time domain $I_n$ remains the same. \nThe spatial domain $S_i$ of a local patch needs to satisfy three constraints:\n\t\\begin{itemize}\n\t\\item[1.] The $i^{\\text{th}}$ CF node must be inside;\n\t\\item[2.] Part of the boundary of the domain close to the $i^{\\text{th}}$ CF node must be contained in it;\n\t\\item[3.] It must contain the cells of the Hermite nodes closest to the $i^{\\text{th}}$ CF node.\n\t\\end{itemize}\nExamples of the spatial domains of local patches in 2-D that satisfy these constraints are shown in Fig.~\\ref{fig:edgePatch}, \n\tFig.~\\ref{fig:cornerPatch} and Fig.~\\ref{fig:reentrant_corner_patch}.\nFor simplicity, \n\twe omit the subscript associated with the CF node in the description of the local patches.\n\t\nLet us first consider a CF node $(x_i,y_0)$ along an edge as depicted in Fig.~\\ref{fig:edgePatch}.\nIn this case,\n\tthe spatial domain of the local patch is $S = [x_{i-1\/2},x_{i+1\/2}]\\times[y_0,y_{3\/2}]$ while its intersection with \n\tthe boundary of the domain, \n\t$S\\cap\\Gamma$, \n\tis the line connecting the points $(x_{i-1\/2},y_0)$ and $(x_{i+1\/2},y_0)$.\nThe spatial domain where we weakly enforce the Hermite solution is $S^{\\mathcal{H}} = [x_{i-1\/2},x_{i+1\/2}]\\times[y_{1\/2},y_{3\/2}]$.\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=3.0in]{integration_domain_J_edge_2D.pdf}\n \\caption{Illustration of a 2-D local patch for a bottom edge CF node. \n \t\t The primal CF and Hermite nodes are respectively represented by green squares and black squares \n\t\t while the dual Hermite nodes are represented by blue circles.\n\t\t The CFM seeks the information located at $(x_i,y_0)$,\n\t\t\twhich is enclosed by the red circle.\n\t\t The spatial domain $S$ of local patches is denoted by a dashed magenta box.\n\t\t The part of the boundary $\\Gamma$ include in the local patch is represented by a dashed purple line.\n\t\t The spatial domain $S^{\\mathcal{H}}$ where we enforce the correction functions to match the Hermite-Taylor polynomials \n\t\t \tis denoted by a dashed blue box.\n\t\t }\n\t\\label{fig:edgePatch}\n\\end{figure}\n\nFor a CF node located at a corner $(x_0,y_0)$ as illustrated in Fig.~\\ref{fig:cornerPatch}, \n\twe have $S = [x_0,x_{3\/2}]\\times[y_0,y_{3\/2}]$ and $S^{\\mathcal{H}} = [x_{1\/2},x_{3\/2}]\\times[y_{1\/2},y_{3\/2}]$.\nThe intersection of $S$ with the boundary is composed of the line connecting $(x_0,y_0)$ to $(x_0,y_{3\/2})$ and that connecting $(x_0,y_0)$ to $(x_{3\/2},y_0)$.\n\n\\begin{figure} \n \t\\centering\n\t\\includegraphics[width=3.0in]{integration_domain_J_corner_2D.pdf}\n\t\\caption{Illustration of a 2-D local patch for a bottom-left corner CF node. \n \t\t The primal CF and Hermite nodes are respectively represented by green squares and black squares \n\t\t while the dual Hermite nodes are represented by blue circles.\n\t\t The CFM seeks the information located at $(x_0,y_0)$,\n\t\t\twhich is enclosed by the red circle.\n\t\t The spatial domain $S$ of local patches is denoted by a dashed magenta box.\n\t\t The part of the boundary $\\Gamma$ include in the local patch is represented by a dashed purple line.\n\t\t The spatial domain $S^{\\mathcal{H}}$ where we enforce the correction functions to match the Hermite-Taylor polynomials \n\t\t \tis denoted by a dashed blue box.\n\t\t }\n\t\t \\label{fig:cornerPatch}\n\\end{figure}\n\nAs a last example, \n\twe consider the situation in Fig.~\\ref{fig:reentrant_corner_patch} where a CF node is located at a reentrant corner $(x_i,y_i)$.\nWe then have $S=[x_{i-1\/2},x_{i+3\/2}]\\times[y_{j-1\/2},y_{j+3\/2}]$. \nThe spatial domain where the Hermite solution is enforced $S^\\mathcal{H}$ is the union of $[x_{i-1\/2},x_{i+3\/2}]\\times[y_{j+1\/2},y_{j+3\/2}]$\n \tand $[x_{i+1\/2},x_{i+3\/2}]\\times[y_{j-1\/2},y_{j+1\/2}]$.\nThe intersection between the spatial domain $S$ of the local patch and the boundary is composed of the line connecting $(x_i,y_{j-1\/2})$ to $(x_i,y_j)$ and \n\tthat connecting $(x_{i-1\/2},y_j)$ to $(x_i,y_j)$.\n\\begin{figure} \n \t\\centering\n\t\\includegraphics[width=3.0in]{integration_domain_J_reentrant_2D.pdf}\n \\caption{Illustration of a 2-D local patch for a reentrant corner CF node. \n \t\t The primal CF and Hermite nodes are respectively represented by green squares and black squares \n\t\t while the dual Hermite nodes are represented by blue circles.\n\t\t The CFM seeks the information located at $(x_i,y_i)$,\n\t\t\twhich is enclosed by the red circle.\n\t\t The spatial domain $S$ of local patches is denoted by a dashed magenta box.\n\t\t The part of the boundary $\\Gamma$ include in the local patch is represented by a dashed purple line.\n\t\t The spatial domain $S^{\\mathcal{H}}$ where we enforce the correction functions to match the Hermite-Taylor polynomials \n\t\t \tis denoted by a dashed blue box.\n\t\t }\n\t\t \\label{fig:reentrant_corner_patch}\n\\end{figure}\n\nLet us now consider Maxwell's equations in 3-D and seek polynomial approximations of the magnetic field and the electric field \n\tin each local patch, \n\tthat is $\\mathbold{H}^n_{h,i}$ and $\\mathbold{E}^n_{h,i}$ for $i=0,\\dots,N_\\Gamma$ and $n=1,\\dots,N_t$.\nHere $N_\\Gamma$ is the total number of CF nodes.\nThe first part of the functional \\eqref{eq:functional_J} becomes\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\mathcal{G}_i^n(\\mathbold{H}^n_{h,i},\\mathbold{E}^n_{h,i}) =&\\,\\, \\frac{\\ell_i}{2} \\,\\int\\limits_{I_n}\\!\\int\\limits_{S_i} (\\mu\\,\\partial_t \\mathbold{H}^n_{h,i} + \n\t\t\t\\nabla\\times\\mathbold{E}^n_{h,i})\\cdot(\\mu\\,\\partial_t \\mathbold{H}^n_{h,i} + \\nabla\\times\\mathbold{E}^n_{h,i}) \\\\\n\t\t\t+&\\,\\, ( \\epsilon\\,\\partial_t \\mathbold{E}^n_{h,i} - \\nabla\\times\\mathbold{H}^n_{h,i})\\cdot( \\epsilon\\,\\partial_t \\mathbold{E}^n_{h,i} - \n\t\t\t\\nabla\\times\\mathbold{H}^n_{h,i})\\\\\n\t\t\t+&\\,\\, (\\nabla\\cdot (\\mu\\, \\mathbold{H}^n_{h,i}))^2 + (\\nabla\\cdot (\\epsilon\\, \\mathbold{E}^n_{h,i}))^2\\,\\mathrm{d}\\mathbold{x}\\,\\mathrm{d}t,\n\t\t\\end{aligned}\n\t\\end{equation*}\nwhere $\\ell_i = \\beta\\,h$ is the characteristic length of the spatial domain $S_i$ that depends on the mesh size $h$ and $\\beta >0$.\nThe second part of the functional $J_i^n$ that weakly enforces the boundary conditions is either\n\t\\begin{equation*}\n\t\t\\mathcal{B}_i^n(\\mathbold{E}^n_{h,i}) = \\frac{1}{2} \\,\\int\\limits_{I_n}\\!\\int\\limits_{\\Gamma\\cap S_i} (\\mathbold{n}\\times\\mathbold{E}^n_{h,i} - \\mathbold{g}_E)\\cdot(\\mathbold{n}\\times\\mathbold{E}^n_{h,i} - \\mathbold{g}_E)\\, \\mathrm{d}s\\,\\mathrm{d}t,\n\t\\end{equation*}\n\tfor the boundary condition \\eqref{eq:nxE} or \n\t\\begin{equation*}\n\t\t\\mathcal{B}_i^n(\\mathbold{H}^n_{h,i}) = \\frac{1}{2} \\, \\int\\limits_{I_n}\\!\\int\\limits_{\\Gamma\\cap S_i} (\\mathbold{n}\\times\\mathbold{H}^n_{h,i} - \\mathbold{g}_H) \\cdot (\\mathbold{n}\\times\\mathbold{H}^n_{h,i} - \\mathbold{g}_H)\\, \\mathrm{d}s\\,\\mathrm{d}t,\n\t\\end{equation*}\n\tfor the boundary condition \\eqref{eq:nxH}.\nThe final part of $J_i^n$ that weakly enforces the correction functions to match the Hermite solution is given by\n\t\\begin{equation} \n\t\\begin{aligned}\n\t\t\\mathcal{H}_i^n(\\mathbold{H}^n_{h,i},\\mathbold{E}^n_{h,i}) =&\\,\\, c_H(h) \\int\\limits_{I_n}\\!\\int\\limits_{S^\\mathcal{H}_i} (\\mathbold{H}^n_{h,i}-\\mathbold{H}^*)\\cdot(\\mathbold{H}^n_{h,i}-\\mathbold{H}^*) \\\\ \n\t\t+&\\,\\, (\\mathbold{E}^n_{h,i}-\\mathbold{E}^*)\\cdot(\\mathbold{E}^n_{h,i}-\\mathbold{E}^*)\\,\\mathrm{d}\\mathbold{x}\\,\\mathrm{d}t.\n\t\\end{aligned}\n\t\\end{equation}\n \nWe then have the following problem statement: \n\t\\begin{equation} \\label{eq:minPblm}\n\t\t\\begin{aligned}\n\t\t\t&\\text{Find } (\\mathbold{H}^n_{h,i},\\mathbold{E}^n_{h,i}) \\in V \\times V \\text{ such that }\\\\\n \t\t\t&\\qquad (\\mathbold{H}^n_{h,i},\\mathbold{E}^n_{h,i}) \\in \\underset{\\mathbold{v},\\mathbold{w}\\in V}{\\arg\\min}\\, J_i^n(\\mathbold{v},\\mathbold{w}),\n\t\t\\end{aligned}\n\t\\end{equation}\n\tfor $i=0,\\dots,N_\\Gamma$ and $n=1,\\dots,N_t$.\nHere \n\\begin{equation*}\n\tV = \\big\\{ \\mathbold{v} \\in \\big[P^k(S_i\\times I_n )\\big]^3 \\big\\}.\n\\end{equation*}\n\nAs in 1-D, \n\twe use that the gradient of the functional $J_i^n$ with respect to the coefficients of the polynomial approximations $\\mathbold{H}_{h,i}^n$ and $\\mathbold{E}_{h,i}^n$ \n\tvanishes at a minimum to obtain a linear system of equations to solve.\nThe dimension of the minimization problems is independent of the mesh size and the time step size, \n\tand is $3\\,(k+1)^{3}$ in 2-D and $6\\,(k+1)^{4}$ in 3-D. \nHowever, \n\tthe number of minimization problems $(N_\\Gamma+1)\\,N_t$ increases as the mesh size and the time step size diminish.\nOnce the minimization problem is solved on a local patch, \n\tthe electromagnetic fields and their $m$ first space derivatives are estimated at its CF node using \n\t$\\mathbold{H}^n_{h,i}$ and $\\mathbold{E}^n_{h,i}$.\n\n\t\n\\begin{remark}\nThe terms in $\\mathcal{G}_i^n$ enforcing the residual of Maxwell's equations \\eqref{eq:problemMaxwell} \n\t\tare scaled by $\\ell_i$ to guarantee that all the terms in $\\mathcal{G}_i^n$ and $\\mathcal{B}_i^n$ behave in a \n\t\tsimilar way as the mesh size diminishes \\cite{Marques2011}. \nLet us assume that the correction functions are polynomials of degree $k$ that leads to an accuracy \n\tof $\\mathcal{O}(\\ell_i^{k+1})$ and that $k=2\\,m$.\nUsing $\\mathbold{H}_i^n = \\mathbold{H} + \\mathcal{O}(\\ell_i^{k+1})$ and $\\mathbold{E}_i^n = \\mathbold{E} + \\mathcal{O}(\\ell_i^{k+1})$ in the functional $J_i^n$, \n\twe have that the terms in $\\mathcal{G}_i^n$ and $\\mathcal{B}_i^n$ behave \n\tas $\\mathcal{O}(\\ell_i^{2\\,k+5})$ \n\twhile the term in $\\mathcal{H}_i^n$ scales as $\\mathcal{O}(\\ell_i^{2\\,k+6})$. \nHence, \n\tthe functional $J_i^n$ is dominated by the boundary conditions and Maxwell's equations as $\\ell_i$ diminishes.\n\\end{remark}\n\n\\begin{remark}\nThe number of matrices to construct can be further reduced depending on the geometry of the domain and the physical properties of the material $\\mu$ and $\\epsilon$.\nAs an example, \n\tlet us consider a 2-D geometry discretized with a Cartesian mesh with $\\Delta x = \\Delta y$.\nWe also assume the boundary $\\Gamma$ of the domain \n\tto coincide only with primal nodes.\nFor problems with constant coefficients on a rectangular domain, \n\tthe number of matrices is reduced to eight because the spatial domain $S_i$ of local patches on an edge translates along it.\nIf reentrant corners are also considered, \n\tthere is a maximum of twelve matrices to compute. \n\\end{remark}\n\n\\begin{remark} \\label{rem:propertyHTCFM}\n\tAssuming that the correction functions are polynomials of degree $k$ that lead to an accuracy of $\\mathcal{O}(\\ell_i^{k+1})$, \n\t\twe then have $k\\geq2\\,m$ to preserve the accuracy of a $(2\\,m+1)$-order Hermite-Taylor method.\n\tAs was remarked for FDTD methods in \\cite{LawNave2021}, \n\t\tthe CFM impacts the stability of the original method because of the Hermite-Taylor polynomials $\\mathbold{H}^*$ and $\\mathbold{E}^*$.\n\tSince a rigorous proof of the stability of the proposed method is out of reach for the moment, \n\t\twe investigate numerically its stability properties in Section~\\ref{sec:num_examples}. \n\\end{remark}\n\n\n\n\\section{Numerical Examples} \\label{sec:num_examples}\nIn this section, we numerically investigate the stability of the proposed method and perform convergence studies in 1-D and 2-D. \n\n\\subsection{Examples in One Dimension}\n\nLet us seek approximate solutions to Maxwell's equations \n\\begin{equation*} \\label{eq:1DMaxwellEq}\n\\begin{aligned}\n \\mu\\,\\partial_t H + \\partial_x E =&\\,\\, 0, \\\\\n \\epsilon\\,\\partial_t E + \\partial_x H =&\\,\\, 0,\n\\end{aligned}\n\\end{equation*}\n \tin the domain $\\Omega = [x_\\ell, x_r]$ and the time interval $I=[t_0,t_f]$.\nThe initial conditions are $H(x,t_0) = a(x)$ and $E(x,t_0) = b(x)$, \n\tand we focus on the boundary conditions $E(x_\\ell,t) = g_\\ell(t)$ and $E(x_r,t) = g_r(t)$.\n Here $a(x)$,\n $b(x)$, \n $g_\\ell(t)$ and $g_r(t)$ are known functions.\n\nIn this subsection, \n\twe use the Hermite-Taylor correction function method with $1\\leq m \\leq 5$. \nWe set the degree of the correction functions to be $2\\,m$.\nThe CFM should not therefore impact the convergence rate of the Hermite-Taylor method.\n\n\\subsubsection{Stability}\nLet us first investigate the stability of the Hermite-Taylor correction function method. \nWe consider $\\Omega = [0,1]$, \n\tand set $\\mu=1$ and $\\epsilon =1$.\nThe stability condition of the Hermite-Taylor method depends only on the largest wave speed and is given here by \n\t$\\Delta t < h$,\n\twhere $h$ is the mesh size. \nAs mentioned in Remark~\\ref{rem:propertyHTCFM}, \n\tthe stability of the Hermite-Taylor method is impacted by the CFM because we use Hermite-Taylor polynomials $\\mathbold{H}^*$ and $\\mathbold{E}^*$ \n\tin the minimization problem \\eqref{eq:minPblm}.\nAlthough we do not have a rigorous proof of the stability of the Hermite-Taylor correction function method, \n\twe provide numerical evidences of it by investigating the eigenvalues of the global matrix \n\tassociated with the method. \n\t\nSince Maxwell's equations is a linear system of PDEs and assuming $g_\\ell = g_r = 0$,\n\tthe proposed numerical method can be written as \n\\begin{equation*}\n\t\\mathbold{W}_p^{n+1} = A\\,\\mathbold{W}_p^n,\n\\end{equation*}\n\twhere $A$ is a square matrix of dimension $2\\,(N_x+1)\\,(m+1)$ and \n\t$\\mathbold{W}_p^n$ is a vector containing all the degrees of freedom on the primal mesh at time $t_n$.\nA stable method should have all the eigenvalues of $A$ inside the unit circle of the complex plane. \nIn the following, \n\twe compute numerically the eigenvalues of $A$ and consider that the scheme is stable if the spectral radius $\\rho(A)$ of the matrix $A$ \n\tis at most one with an error of $\\mathcal{O}(10^{-10})$.\n \\begin{figure} \n\\begin{adjustbox}{max width=1.0\\textwidth,center}\n \\centering\n\t\\includegraphics[width=3.0in]{pblm_1D_err_specRad_vs_h.pdf}\n\t\\hspace*{-0.1in}\\includegraphics[width=3.0in]{pblm_1D_err_specRad_vs_CFL_N80.pdf}\n\t\\hspace*{-0.1in}\\includegraphics[width=3.0in]{pblm_1D_err_specRad_vs_cH_N80.pdf}\n\\end{adjustbox}\n \\caption{Absolute difference between one and the spectral radius of the matrix $A$ as a function of the mesh size, the CFL constant and the penalization parameter $c_H$ for various values of $m$. For the left plot, the CFL constant is set to $0.9$ and $c_H = 1$. For the middle plot, the mesh size is $h=\\tfrac{1}{80}$ and $c_H=1$. For the right plot, the CFL constant is set to $0.9$ and $h=\\tfrac{1}{80}$. The circled marker indicates that $1-\\rho(A)>0$ and therefore the eigenvalues of $A$ are inside the unit circle.}\n \\label{fig:spect_1D}\n\\end{figure}\n\nThe left plot of Fig.~\\ref{fig:spect_1D} illustrates the absolute difference between one and the spectral radius of the matrix $A$,\n\tdenoted $\\rho(A)$,\n\tas a function of the mesh size for a CFL constant of $0.9$, \n\t$c_H=1$ and various values of $m$.\nFor $m\\leq3$, \n\twe observe that the method is stable for a sufficiently small mesh size. \nIn other words,\n\tthe eigenvalues of $A$ are moving inside the unit circle as the mesh is refined.\nThis is expected since the terms in $\\mathcal{H}_i^n$ impacting the stability\n\tscale as $\\mathcal{O}(\\ell_i^{2\\,k+6})$ while the other terms in $J_i^n$ scale as $\\mathcal{O}(\\ell_i^{2\\,k+5})$.\nFor $m\\in\\{4,5\\}$, \n\twe do not observe a clear improvement as the mesh size diminishes for the considered CFL constant.\nThis motivates us to diminish the CFL constant and the value of $c_H$ in order to improve the stability of the Hermite-Taylor correction function method.\n\t\nThe middle plot of Fig.~\\ref{fig:spect_1D} illustrates the absolute difference between one and $\\rho(A)$ \n\tas a function of the CFL constant for $h=\\tfrac{1}{80}$, \n\t$c_H=1$ and various values of $m$.\nFor $m\\leq2$,\n\twe clearly have a stable method as the CFL constant diminishes. \nHowever, \n\tfor $m\\geq3$, \n\tthere is a lower bound on the CFL constant for which values under it lead to a unstable method.\n\t\nThe right plot of Fig.~\\ref{fig:spect_1D} illustrates the absolute difference between one and the spectral radius of the matrix $A$ \n\tas a function of $c_H$ for a CFL constant of $0.9$, \n\t$h=\\tfrac{1}{80}$ and various values of $m$.\nFor all $m$,\n\ta smaller value of the penalization coefficient $c_H$ helps to obtain a stable method. \n\t\nTo give further evidences of that, \n\tFig.~\\ref{fig:spect_1D_CFL_m_3} illustrates the absolute difference between one and $\\rho(A)$ as a function of the CFL constant for $m=3$,\n\t$h\\in\\big\\{\\tfrac{1}{20},\\tfrac{1}{100},\\tfrac{1}{250},\\tfrac{1}{500},\\tfrac{1}{750},\\tfrac{1}{1000}\\big\\}$ and \n\t$c_H \\in \\big\\{1,\\tfrac{1}{10},\\tfrac{1}{100}\\big\\}$.\nA smaller penalization coefficient $c_H$ does not improve the stability of the proposed method \n\tfor coarser meshes.\nIn these cases, \n\twe therefore need to lower the CFL constant.\n \\begin{figure} \n\\begin{adjustbox}{max width=1.00\\textwidth,center}\n \\centering\n\t\\includegraphics[width=3.0in]{pblm_1D_err_specRad_vs_CFL_m_3_cH_1.pdf}\n\t\\hspace*{-0.1in}\\includegraphics[width=3.0in]{pblm_1D_err_specRad_vs_CFL_m_3_cH_1div10.pdf}\n\t\\hspace*{-0.1in}\\includegraphics[width=3.0in]{pblm_1D_err_specRad_vs_CFL_m_3_cH_1div100.pdf}\n\\end{adjustbox}\n \\caption{Absolute difference between one and the spectral radius of the matrix $A$ as a function of the CFL constant for $m=3$, \n \tand various mesh sizes and values of $c_H$. The circled marker indicates that $1-\\rho(A)>0$ and therefore the eigenvalues of $A$ are inside the unit circle.}\n \\label{fig:spect_1D_CFL_m_3}\n\\end{figure}\nBased on these results, \n\tthe stability of the Hermite-Taylor correction function method improves by reducing the CFL constant and the value of \n\tthe penalization coefficient $c_H$.\t\nMoreover,\n\tfor $m\\leq3$,\n\tthe stability of this method improves as the mesh size diminishes, \n\twhich suggests that larger CFL constants could be used for finer meshes.\n\n\\subsubsection{Condition Number of CFM Matrices}\nLet us now investigate the impact of $h$, \n\t$c_H$ and the CFL constant on the condition number of the matrices $M_i$ coming from the minimization procedure used in the CFM.\nFig.~\\ref{fig:cond_1D} illustrates the maximum condition number of these matrices as a function of the mesh size, \n\tthe CFL constant and the penalization parameter $c_H$ for various values of $m$.\nWe observe that the condition number increases as the mesh size diminishes and, \n\tmore precisely, \n\tscales as $\\tfrac{1}{h}$ for all different settings. \nThe condition number also increases as the CFL constant and $c_H$ diminish. \nHence,\n\tan arbitrary small value of $c_H$ cannot be taken to avoid poorly conditioned matrices coming from the CFM.\n \\begin{figure} \n\\begin{adjustbox}{max width=1.0\\textwidth,center}\n \\centering\n\t\\includegraphics[width=3.0in]{pblm_1D_cond_vs_h.pdf}\n\t\\hspace*{-0.1in}\\includegraphics[width=3.0in]{pblm_1D_cond_vs_CFL.pdf}\n\t\\hspace*{-0.1in}\\includegraphics[width=3.0in]{pblm_1D_cond_vs_cH.pdf}\n\\end{adjustbox}\n \\caption{Maximum condition number of the matrices coming from the CFM as a function of the mesh size, the CFL constant and the penalization parameter $c_H$ for various values of $m$. For the left plot, the CFL constant is set to $0.9$ and $c_H = 1$. For the middle plot, the mesh size is $h=\\tfrac{1}{80}$ and $c_H=1$. For the right plot, the CFL constant is set to $0.9$ and $h=\\tfrac{1}{80}$.}\n \\label{fig:cond_1D}\n\\end{figure}\nTo preserve the advantage of the Hermite-Taylor method to take large time-step sizes for large $m$ and \n\tlimit the impact on the condition number of CFM matrices, \n\twe propose to use \n\\begin{equation} \\label{eq:penalization_coeff}\n\tc_H(h) = \\left\\{\n\t\\begin{aligned}\n\t&1 &\\text{if} \\quad h < h^*, \\\\\n\t&a &\\text{otherwise}, \n\t\\end{aligned}\n\t\\right.\n\\end{equation}\n\twhere $0 2\\,{\\mathrm M}_{\\odot}$), differentially\nrotating neutron star~\\cite{shibata:06bns, giacomazzo:11,\n hotokezaka:11, bauswein:12}. The stability of this postmerger\nneutron star (PMNS) against gravitational collapse depends on its mass\nand on the details of the nuclear EOS. Less\nmassive systems whose component masses add up to less than the maximum\nmass that can be supported by the EOS in combination with uniform\nrotation (the supramassive limit, e.g., \\cite{cook:94c,kaplan:14}) ,\nresult in long-lived stable PMNSs. For more massive systems,\nstrong differential rotation temporarily supports the remnant and it\neventually undergoes gravitational collapse due to redistribution of\nangular momentum via viscous processes and radiation of GWs. The\nrole of thermal pressure support is secondary, because at the\ndensities involved, the temperature-insensitive pressure due to the\nnuclear force dominates the EOS~\\cite{kaplan:14}. Sufficiently\nhigh-mass systems that cannot be supported even by extreme\ndifferential rotation will result in prompt collapse to a black hole\n(BH) upon merger or very shortly after merger, emitting a\nhigh-frequency ring-down \\gw{} signal at $\\sim$\\,6\\,--\\,7\\,kHz (e.g.,\n\\cite{shibata:06bns}).\n\nTransient non-axisymmetric deformations in the surviving postmerger\nremnant lead to a short duration ($\\sim$\\,10\\,--\\,100\\,ms) burst\nof \\gw{s} with rich high frequency content, dominated by emission from\n$f$-mode oscillations at $\\sim$\\,2\\,--\\,4\\,kHz and generally\nlower-frequency sub-dominant peaks from nonlinear couplings between\ncertain oscillation modes~\\cite{stergioulas:11}. The general\nmorphology of the \\gw{} signal thus emitted resembles an\namplitude-modulated damped sinusoid, the phase and amplitude evolution\nof which are not yet well modeled by numerical simulations of neutron\nstar coalescence and postmerger evolution. However, the spectral\nproperties of this signal carry a particularly distinct signature of\nthe EOS.\n\n\\begin{table}\n \\centering\n\\begin{tabular}{l l l l l l}\n\\toprule\nEoS & $M_{\\mathrm{max}}$ & $R_{\\mathrm{max}}$ & $R_{1.35}$ & $\\rho_{\\mathrm{c}}\/\\rho_0$ & $f_{\\mathrm{peak}}$\\\\\n & $[{\\mathrm M}_{\\odot}]$ & [km] & [km] & & [Hz] \\\\ \\hline\nAPR~\\cite{1998PhRvC..58.1804A} (approx) & 2.19 & 9.90 & 11.33 & 10.4 & 3405 \\\\\nDD2~\\cite{2010PhRvC..81a5803T,2010NuPhA.837..210H} (full) & 2.42 & 11.90 & 13.21 & 7.2 & 2589 \\\\\nShen~\\cite{1998NuPhA.637..435S} (full) & 2.22 & 13.12 & 14.56 & 6.7 & 2263 \\\\\nNL3~\\cite{1997PhRvC..55..540L,2010NuPhA.837..210H} (full) & 2.79 & 13.43 & 14.75 & 5.6 & 2157 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[EoS comparisons]{The nuclear EOS employed in this study.\n References are provided in the first column. EOS indicated by\n ``approx'' refer to models which rely on an approximate treatment of\n thermal effects, whereas ``full'' marks EOS which provide the full\n temperature dependence. $M_{\\mathrm{max}}$, $R_{\\mathrm{max}}$, and\n $\\rho_{\\mathrm{c}}$ are the gravitational mass, circumferential\n radius, and central energy density of the maximum-mass\n Tolman-Oppenheimer-Volkoff configurations. We list\n $\\rho_{\\mathrm{c}}$ in units of the nuclear saturation density\n $\\rho_0=2.7\\times 10^{14}~\\mathrm{g\\,cm}^{-3}$. $R_{1.35}$ is the\n circumferential radius of a $1.35\\,M_\\odot$ NS. The final column\n $f_{\\mathrm{peak}}$ gives the dominant postmerger oscillation\n frequency.}\n \\label{tab:eos}\n\\end{table}\n\nA number of studies~\\cite{bauswein:12,hotokezaka:13,bauswein:14} have\nidentified and confirmed a correlation between the dominant postmerger\noscillation frequency (i.e., half the peak \\gw{} emission frequency) and the\nradius of a fiducial cold, non-rotating neutron star. For example,\nin~\\cite{bauswein:12}, the authors perform a Fisher matrix analysis and find\nthat it may be possible to use aLIGO observations of postmerger signals to\nmeasure the dominant postmerger oscillation frequency to an accuracy of\n$\\sim$\\,40\\,Hz and thus determine the radius of a fiducial\n$1.6$\\,M$_\\odot$ NS to an accuracy of 100\\,--\\,200\\,m with an expected\ndetection horizon of approximately 15\\,Mpc\\footnote{Assuming an optimal SNR\ndetection threshold of 5, justified below.}, corresponding to a detection rate\nof just under 1 per century, assuming the ``realistic'' rate in~\\cite{ratesdoc}.\nFurthermore, a systematic Monte-Carlo analysis using a variety of postmerger\nwaveforms corresponding to different component masses and EOS and data from the\ninitial-LIGO\/Virgo instruments, recolored to the nominal advanced detector\ndesign sensitivities was presented in~\\cite{clark:14}. There, the results\nof~\\cite{bauswein:12} are essentially confirmed, albeit at a reduced expected\ndetection range of $\\sim\\,5$\\,Mpc, where this is now the angle-averaged range\nsince a network of detectors was used in that analysis and so the notion of\nhorizon distance is not well-defined. This corresponds to an expected detection\nrate of $\\sim\\,0.5$ events per century. The reduction in sensitivity arises\nsince a generic burst search was used in the absence of accurate templates.\nFisher analysis, by contrast, assumes that an optimal filtering strategy is\nfeasible.\n\n\\begin{table}\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{cccccccccccccc}\n\\toprule\n& \\multicolumn{3}{c}{aLIGO} & \\multicolumn{3}{c}{Blue LIGO Voyager{} Baseline} & \\multicolumn{3}{c}{Red LIGO Voyager{} Baseline} & \\multicolumn{3}{c}{Green LIGO Voyager{} Baseline} & \\multicolumn{1}{c}{Frequency Recovery} \\\\\n\\cmidrule(lr{0.2em}){2-4} \\cmidrule(lr{0.2em}){5-7} \\cmidrule(lr{0.2em}){8-10}\\cmidrule(lr{0.2em}){11-13}\\cmidrule(lr{0.2em}){14-14}\n& $\\langle \\rho \\rangle$ & $D_\\text{H}$ & $\\dot{\\mathcal{N}}$ & $\\langle \\rho \\rangle$ & $D_\\text{H}$ & $\\dot{\\mathcal{N}}$ & $\\langle \\rho \\rangle$ & $D_\\text{H}$ & $\\dot{\\mathcal{N}}$ & $\\langle \\rho \\rangle$ & $D_\\text{H}$ & $\\dot{\\mathcal{N}}$ & $\\tilde{\\delta f}\\pm\\mathrm{IQR}$\\\\\nEoS & @ 10 Mpc & [Mpc] & [yr$^{-1}$] & @ 10 Mpc & [Mpc] & [yr$^{-1}$] & @ 10 Mpc & [Mpc] & [yr$^{-1}$] & @ 10 Mpc & [Mpc] & [yr$^{-1}$] & [Hz] \\\\\n\\cmidrule(lr{0.2em}){1-1} \\cmidrule(lr{0.2em}){2-4} \\cmidrule(lr{0.2em}){5-7} \\cmidrule(lr{0.2em}){8-10} \\cmidrule(lr{0.2em}){11-13}\\cmidrule(lr{0.2em}){14-14}\nAPR & $1.49$ & $8.3$ & $3.0\\times10^{-3}$ & $5.24$ & $39.63$ & $0.09$ & 4.71 & 33.20 & 0.05 & 9.00 & 52.41 & 0.31 & $10\\pm54$ \\\\\nDD2 & $2.55$ & $14.1$ & $6.6\\times10^{-3}$ & $9.03$ & $53.26$ & $0.23$ & 8.07 & 52.10 & 0.22 & 14.76 & 94.42 & 0.74 & $-2\\pm14$ \\\\\nShen & $2.64$ & $13.8$ & $6.4\\times10^{-3}$ & $9.37$ & $70.19$ & $0.43$ & 8.34 & 54.08 & 0.24 & 14.73 & 94.81 & 0.74 & $8\\pm22$ \\\\\nNL3 & $3.13$ & $15.7$ & $7.7\\times10^{-3}$ & $11.15$ & $73.84$ & $0.49$ & 9.91 & 67.04 & 0.37 & 17.26 & 115.03 & 0.74 & $5\\pm10$ \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption[BNS postmerger detection parameters v. Interferometer]\n {Detectability of the post-merger oscillation in binary neutron star\n mergers. Angle-averaged\n SNR $\\langle\\rho\\rangle$, horizon distances\n $D_\\text{H}$, and expected annual detection rate $\\dot{\\mathcal{N}}$\n (at SNR = 5). Local merger rates are based on ``realistic'' rates~\\cite{ratesdoc},\n but are uncertain within three orders of magnitude.\n Note that the true SNR recovered by a\n burst search will be factors of a few below the values given\n here (which are based on matched filtering).\n We also estimate the accuracy in the determination of the\n dominant postmerger oscillation frequency, based on a Monte-Carlo\n analysis (see text for details).}\n\\label{tab:pmns_estimates}\n\\end{table}\n\nHere, we estimate the detectability of the postmerger signal from a surviving\nPMNS for four different EOS using three figures of merit: the angle-averaged\nsignal-to-noise ratio, described in \\ref{s:snr_estimates}; the horizon\ndistance, the distance at which an optimally-oriented face-on source yields an\noptimal matched-filter SNR of some fiducial value and the expected annual\ndetection rate based on the ``realistic'' rates from~\\cite{ratesdoc}\\footnote{As\nwith the estimates for core-collapse supernovae, these FOMs assume that the\nwaveform is sufficiently well-modeled to permit a matched filtering detection\nstrategy.}. Our interest here is in the detectability of the merger\/postmerger\nsignal arising from surviving PMNS. We therefore choose to window the waveforms\nprior to computing their optimal and angle-averaged SNR such that the\ntime-domain amplitude of the waveform is zero at times prior to the merger\n(taken as the point of maximum amplitude). This helps to ensure that there is\nessentially no contribution to the SNR (and hence, our detectability estimates)\nfrom the pre-merger inspiral signal. Finally, the SNR is computed for\nfrequencies of 1\\,kHz and above.\n\nSince the postmerger signal is likely to only be observable for relatively\nnearby, rare events, it is reasonable to assume that its time is known to\nextremely high accuracy from the time of coalescence measured from the much\nhigher SNR inspiral precursor. We therefore adopt a relatively low nominal SNR\nthreshold of $5$ in computing the horizon distance and detection rate. The EOS\nused for these estimates are a subset of those used in the more extensive study\nof~\\cite{clark:14} and range from the rather soft APR (high frequency \\gw{}\nsignal) to the somewhat softer NL3 (lower frequency \\gw{} signal).\nTable~\\ref{tab:pmns_estimates} reports the FOMs described above for each\nwaveform, for both the aLIGO noise curve and that of the red, blue, and green\nLIGO Voyager{} baseline designs.\n\nWhile the mere detection of the postmerger signal will itself have\nsignificant consequences for our understanding of the neutron star EOS\nby excluding those EOS unable to support a long-lived postmerger\nobject, one of the most useful, and simple, measurements that will be\npossible is the identification of the dominant postmerger oscillation\nfrequency. It is, therefore, informative to also estimate the\naccuracy with which the postmerger frequency may be determined.\nTo that end, we have performed a Monte-Carlo study wherein a\npopulation of postmerger waveforms with a uniform distribution in SNR\nhave been injected into Gaussian noise with the noise spectral density\nof the blue LIGO Voyager{} baseline design. The signals are recovered using a Bayesian\nnested sampling algorithm with a simple sine-Gaussian template and the peak\nfrequency of the postmerger signal is recovered from the maximum-likelihood\nestimate (ML) of the sine-Gaussian template. This is a reasonable approximation\nto a realistic burst search; the sine-Gaussian yields a 70\\% match, or better,\nwith the postmerger waveforms and allows for accurate recovery of the peak\nfrequency. We quantify the accuracy of the frequency estimation via the error\n$\\delta f = f_{\\mathrm{True}} - f_{\\mathrm{ML}}$.\nTable~\\ref{tab:pmns_estimates} reports the median value and the interquartile\nrange\\footnote{the difference between the 25$^{\\mathrm{th}}$ and 75$^{\\mathrm{th}}$\npercentiles, representing a robust measure of the spread in the frequency error.}\nof the frequency error for each waveform, for all injections recovered with a\nsine-Gaussian SNR $>6$. This higher threshold is chosen to reflect the fact\nthat this burst-like analysis will recover a factor of order unity less of the\nSNR that an optimal matched-filter strategy would. Since this frequency\nestimate is essentially only SNR-dependent (i.e., the differences in the shapes\nof the aLIGO and LIGO Voyager{} baseline design noise curves at such high frequencies do\nnot affect this analysis), the frequency error is common to both noise curves.\n\nIn conclusion, we find that the blue LIGO Voyager{} baseline design will provide a\nrealistic chance of the detection, and characterization, of the \\gw{} signal\nassociated with postmerger oscillations following binary NS coalescence. A\nsearch for these signals with data from LIGO Voyager{} would still require some level of\noptimism and a relatively nearby (i.e., 10\\,--\\,50\\,Mpc) event, but, given the\nuncertainties in the expected rate of binary NS coalescence in the local\nUniverse~\\cite{ratesdoc}, the uncertainties\nin the numerical modeling of the postmerger signal and the science possible with\nthe detection and accurate measurement of the dominant postmerger oscillation\nfrequency ($\\delta f \\sim 10$\\,Hz), this constitutes an important high-frequency\nsource for the next generation \\gw{} observatories.\n\n\n\n\n\n\n\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\label{table:pmns_jacobian}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{>{\\tiny} l >{\\tiny}c >{\\tiny}c >{\\tiny}c >{\\tiny}c >{\\tiny}c >{\\tiny} c >{\\tiny} c >{\\tiny} c}\n\\toprule\n\\textbf{Waveform} &\\textbf{NN} & \\textbf{Sei} & \\textbf{SUS} & \\textbf{SPOT\/CTN} & \\textbf{SQZ} & \\textbf{POW} & \\textbf{FCL} & \\textbf{MASS}\\\\\n\n\\toprule\nAPR & 0 & 0 & $4.50\\times10^{-7}$ & $4.89\\times 10^{-3}$ & 0.81 & 0.45 & $-2.73\\times 10^{-5}$ & $-2.79\\times 10^{-5}$ \\\\\nDD2 & 0 & 0 & $1.62\\times10^{-7}$ & $7.89\\times 10^{-3}$ & 0.81 & 0.45 & $-4.14\\times 10^{-5}$ & $-4.26 \\times 10^{-5}$ \\\\\nShen & 0 & 0 & $1.46\\times10^{-7}$ & 0.01 & 0.81 & 0.45 & $-5.18\\times 10^{-5}$ & $-5.31\\times 10^{-5}$ \\\\\nNL3 & 0 & 0 & $1.83\\times10^{-8}$ & 0.01 & 0.81 & 0.45 & $-5.76\\times 10^{-5}$ & $-5.93\\times 10^{-5}$ \\\\\n\\hline\n\\bottomrule\n\\end{tabular}}\n\\caption[Jacobian of Science Return for various postmerger waveforms]\n{Jacobian of science goals as a function of interferometer upgrade technology.\nThe baseline FOM is the ideal optimally-oriented single-detector\nmatched filtering SNR for various postmerger waveforms (cf. \\refsection{sec:pmns}).\nThe true SNR recovered by a burst search will be factors of order unity\nbelow the values given for aLIGO and the baseline LIGO Voyager{} design.}\n\\label{tab:changeSNRpmns}\n\\end{table}\n\n\n\\subsubsection{Dense Matter Equation of State from the Tidal Deformation of Neutron Stars}\n\\label{sec:DensMatterEoS}\nThe inspiral and merger of BH\/NS or NS\/NS binaries can\nprovide a wealth of information about the NS Equation of State (EOS).\nThis may come about through observing the phase evolution of a NS\/NS or BH\/NS in\nthe late inspiral to merger, the pulsations of a hypermassive NS (or\nnewly-born stable NS), which may form during a NS\/NS merger, and the frequency of tidal disruption of the NS in a BH\/NS inspiral; see \\cite{GW170817:EOS} for inference on the EOS from the gravitational-wave signal of the NS\/NS merger GW170817. \n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{Figures\/NSNS-detector-plot.pdf}\n \\caption[The Amplitudes of BNS waveforms for different EoS]\n {The amplitudes of NS\/NS waveforms,\ncompared to noise spectra of various GW detectors, showing the frequency range\nwhere various EoS effects may be seen. Hybrid waveforms for merging binary\nneutron stars with two different EoS from~\\cite{2009PhRvD..79l4033R} are shown.\nEoS HB produces a neutron-star radius of 11.6\\,km and EoS 2H a radius of\n15.2\\,km.\nBoth waveforms are from 1.35\\,--\\,1.35\\,M$_\\odot$\nbinaries at an effective distance of 100\\,Mpc. Differences in waveform\namplitudes due to strong tidal interactions are seen before the merger\n(up to $\\sim\\,1000$\\,Hz for EoS 2H, and up to $\\sim\\,2000$\\,Hz for EoS HB.) The\neffects of the EoS on the phase evolution are not visible in the amplitudes\nbelow $\\sim\\,500$\\,Hz but would be measurable in Advanced LIGO for extreme\nEoS like 2H~\\cite{hinderer:10}. Post-merger oscillations of a hypermassive\nneutron star formed by the merger produce peaks in the spectrum near $2000$\\,Hz\nfor EoS 2H, and near $3500$\\,Hz for EoS HB.}\n \\label{fig:NS_EoS_Strains}\n\\end{figure}\n\n\nFor most of the NS\/NS or BH\/NS inspiral, NSs are well approximated as\npoint particles; detections and rate estimates can be made with templates that\nignore finite size effects. However, as the size of the orbit becomes comparable\nto the size of the neutron stars, the EoS will begin to modify the phase\nevolution as tidal effects deform the neutron stars. These modifications in the high-frequency portions of the waveforms are shown in Figure~\\ref{fig:NS_EoS_Strains} for two representative equations of state.\n\nThe leading order effect of the EoS on the GWs is characterized\nby a parameter that relates the size of the quadrupole deformation induced in\nthe star to the strength of the external tidal field. A dimensionless form of\nthis parameter is\n\\begin{equation}\n\\Lambda = \\frac{2}{3} k_2 \\left(\\frac{R}{M}\\right)^5,\n\\end{equation}\nwhich depends on the Love number $k_2$, the radius of the star $R$, and the\nmass $M$ of the star. Both $k_2$ and $R$ are determined by the EoS. Tidal effects\ncontribute to the waveform formally at 5th and higher post-Newtonian (PN) orders\n\\cite{hinderer:08,vines:11}.\nThe PN model will break down at high frequency, as the stars\ninteract more strongly, and eventually as the stars collide at a frequency which\ndepends on the EoS. However, the magnitude of the waveform effects predicted in\nPN models is approximately equal to the magnitude of effects seen in both EOB\ncalculations\\,\\cite{Damour:2012yf} and estimates from hybrid waveforms using\nnumerical simulations~\\cite{read:11c}. In this paper, we therefore\nuse the leading order tidal contribution in TaylorF2 post-Newtonian models to\nestimate the measurability of EoS effects. In practice, a more accurate waveform\nmodel will be required to measure this parameter without significant systematic\nerrors~\\cite{2014PhRvL.112j1101F, 2014PhRvD..89j3012W, GW170817:EOS}.\n\nThe figure of merit for neutron star tidal deformability is related to how well we can differentiate between different values of $\\Lambda$ (and hence different equations of state) for various noise curves. We estimate how well we can differentiate between two waveforms using the distinguishability\ncriteria $\\delta h$, as discussed in \\cite{read:11c}. We find $\\delta h$ for a given system at 100\\,Mpc, and then\ncompute 100\\,Mpc $\\times \\delta h$ to find the effective distance at which $\\delta h = 1$ (the minimum distance for distinguishability).\n\nWe consider a $1.4 - 1.4$\\,${\\mathrm M}_{\\odot}$ NS-NS binary, and find how well we can\nrecover a given $\\Lambda$ by computing overlaps of Taylor T4 waveforms.\nFor our figure of merit, we consider the maximum effective\ndistance at which the SLY~\\cite{Haensel:2002cia} and \nAP4~\\cite{Lattimer:2012nd} equations of state can be distinguished, where $\\Lambda_\\mathrm{SLY} = 323$\nand $\\Lambda_\\mathrm{AP4} = 270$ for the given binary system. The accuracy in the recovery of the tidal deformability parameter $\\Lambda$ is plotted in Figure~\\ref{fig:tidal_deformability}. For NS\/BH binaries,\nwe consider the maximum effective distance where these two equations of state would be \ndistinguishable in a $1.4 - 10.0$\\,M$_\\odot$ binary. The Jacobian quantities\nusing this FOM are then computed by the same prescription as in the rest of this\nstudy.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.7\\columnwidth]{Figures\/tidal_deformability.pdf}\n \\caption[Tidal deformability figure of merit]\n {Tidal deformability figure of merit. $\\Lambda$ is the parameter\n characterizing the tidal deformability of various equations of state.\n We show the maximum effective distance at which a given difference in \n $\\Lambda$ is distinguishable. This figure considers a NS-NS, $1.4 - 1.4$\\,M$_\\odot$ binary, with waveforms\n generated using Taylor T4, with tidal terms parametrized by $\\Lambda$. Note that to distinguish between the SLY and AP4 equations of state,\n for example, one would need to find $\\Delta \\Lambda \\approx 50$.}\n \\label{fig:tidal_deformability}\n\\end{figure}\n\n\\iffalse\nWe also consider the effect of mass. If we, for the same value of $\\Lambda$,\ncompute the effective distances for distinguishing NS\/NS waveforms with\nvarious masses from a $1.4\/1.4$\\,${\\mathrm M}_{\\odot}$ test NS-NS binary, we find that,\nfor example, for the SLY equation of state, a 1.35\/1.4 mass binary will have\nan effective distance of $19.8 \\times 10^3$\\,Mpc for the LIGO Voyager{} noise curve,\na factor $\\sim4$ greater than for the aLIGO noise curve. Similarly, for the\nAP4 equation of state, we find an effective distance of $19.7 \\times 10^3$\\,Mpc,\nalso a factor of $\\sim4$ better than the aLIGO noise curve.\n\\fi\n\n\\section{Interferometer Parameters}\n\\label{s:IFOparams}\n\n\n\\input{aLigoAnhinga.tex}\n\\input{RedWarbler1.tex}\n\\input{BlueBird5.tex}\n\n\\begin{table}[h]\n\\resizebox{\\textwidth}{!}{%\n \\centering\n \\begin{tabular}{lccc}\n \\toprule\n\n Parameter & Advanced LIGO & Red & Blue \\\\\n\n \\midrule\n\n Laser wavelength & \\GwincVal{aLIGO.Laser.Wavelength_nm}\\,nm &\n \\GwincVal{Red.Laser.Wavelength_nm}\\,nm &\n \\GwincVal{Blue.Laser.Wavelength_nm}\\,nm \\\\\n\n Laser power & \\GwincVal{aLIGO.Laser.Power}\\,W &\n \\GwincVal{Red.Laser.Power}\\,W & \\GwincVal{Blue.Laser.Power}\\,W \\\\\n\n Mirror substrate & \\GwincVal{aLIGO.Substance} &\n \\GwincVal{Red.Substance} & \\GwincVal{Blue.Substance} \\\\\n\n Mirror radius & \\GwincVal{aLIGO.Materials.MassRadius_cm}\\,cm &\n \\GwincVal{Red.Materials.MassRadius_cm}\\,cm &\n \\GwincVal{Blue.Materials.MassRadius_cm}\\,cm \\\\\n\n Mirror thickness & \\GwincVal{aLIGO.Materials.MassThickness_cm}\\,cm\n & \\GwincVal{Red.Materials.MassThickness_cm}\\,cm &\n \\GwincVal{Blue.Materials.MassThickness_cm}\\,cm \\\\\n\n Beam radius on ITM\/ETM &\n \\GwincVal{aLIGO.Optics.ITM.BeamRadius_cm}\/\\GwincVal{aLIGO.Optics.ETM.BeamRadius_cm}\\,cm\n &\n \\GwincVal{Red.Optics.ITM.BeamRadius_cm}\/\\GwincVal{Red.Optics.ETM.BeamRadius_cm}\\,cm~\\tablefootnote{The\n Red design projects a factor of 3.2 improvement in all coating\n and substrate noises. A factor of 1.6 is to be achieved through\n increasing the beam size, and a further factor of 2 through\n coating technology improvements. For the\n noise budget, this has been modeled as an increase in the beam\n size by the full factor of 3.2.} &\n \\GwincVal{Blue.Optics.ITM.BeamRadius_cm}\/\\GwincVal{Blue.Optics.ETM.BeamRadius_cm}\\,cm\n \\\\\n\n Mass per stage &\n \\GwincVal{aLIGO.Suspension.Stage(4).Mass}\/\\GwincVal{aLIGO.Suspension.Stage(3).Mass}\/\\GwincVal{aLIGO.Suspension.Stage(2).Mass}\/\\GwincVal{aLIGO.Suspension.Stage(1).Mass}\\,kg\n &\n \\GwincVal{Red.Suspension.Stage(4).Mass}\/\\GwincVal{Red.Suspension.Stage(3).Mass}\/\\GwincVal{Red.Suspension.Stage(2).Mass}\/\\GwincVal{Red.Suspension.Stage(1).Mass}\\,kg\n &\n \\GwincVal{Blue.Suspension.Stage(4).Mass}\/\\GwincVal{Blue.Suspension.Stage(3).Mass}\/\\GwincVal{Blue.Suspension.Stage(2).Mass}\/\\GwincVal{Blue.Suspension.Stage(1).Mass}\\,kg\n \\\\\n\n Final stage temperature & \\GwincVal{aLIGO.Suspension.Temp}\\,K &\n \\GwincVal{Red.Suspension.Temp}\\,K &\n \\GwincVal{Blue.Materials.Substrate.Temp}\\,K \\\\\n\n Final stage construction & \\GwincVal{aLIGO.Substance}\n \\GwincVal{aLIGO.Suspension.FiberType_str} &\n \\GwincVal{Red.Substance}\n \\GwincVal{Red.Suspension.FiberType_str}~\\tablefootnote{The noise\n of the optimized suspension fiber in the Red design was roughly\n fitted in GWINC, using the specific heat of fused silica as the\n fit parameter.} & \\GwincVal{Blue.Substance}\n \\GwincVal{Blue.Suspension.FiberType_str} \\\\\n\n Final stage length &\n \\GwincVal{aLIGO.Suspension.Stage(1).Length}\\,m &\n \\GwincVal{Red.Suspension.Stage(1).Length}\\,m &\n \\GwincVal{Blue.Suspension.Stage(1).Length}\\,m \\\\\n\n Newtonian noise suppression & N\/A & \\GwincVal{Red.Seismic.Omicron}\n & \\GwincVal{Blue.Seismic.Omicron} \\\\\n\n Squeeze factor & N\/A &\n $\\GwincVal{Red.Squeezer.AmplitudedB}$\\,dB~\\tablefootnote{Squeezer\n parameters have been altered to provide an equal ground for\n comparison between the two designs. T1200005 originally assumed\n 20~dB of squeezing.} &\n $\\GwincVal{Blue.Squeezer.AmplitudedB}$\\,dB \\\\\n\n Squeeze injection loss & N\/A &\n \\GwincVal{Red.Squeezer.InjectionLoss} &\n \\GwincVal{Blue.Squeezer.InjectionLoss} \\\\\n\n Squeeze filter cavity length & N\/A &\n \\GwincVal{Red.Squeezer.FilterCavity.L}\\,m &\n \\GwincVal{Blue.Squeezer.FilterCavity.L}\\,m \\\\\n\n Squeeze filter cavity loss & N\/A &\n \\GwincVal{Red.Squeezer.FilterCavity.Lrt_ppm}\\,ppm-rt &\n \\GwincVal{Blue.Squeezer.FilterCavity.Lrt_ppm}\\,ppm-rt \\\\\n\n \\bottomrule\n \\end{tabular}}\n \\captionof{table}{Parameters varied by the Red and Blue designs,\n relative to Advanced LIGO.}\n \\label{tab:compare}\n\\end{table}\n\n\n\\section{Overview of the Astrophysical Sources}\n\\section{Astrophysical Metrics}\n\\label{s:sources}\nThe main aim of this paper is to introduce a number of metrics to quantify the ability of the LIGO Voyager detectors to perform various astrophysical measurements, and study the variation of these figures of merit with respect to changes in different design parameters of the detector. Here we provide a brief overview of the astrophysical science that can be potentially performed by these detectors and to discuss figures of metrics related to these astrophysical measurements.\n\n\\input{cbcIntro}\n\n\\input{CompactBinaries}\n\n\\input{StellarCollapseIntro}\n\\input{StellarCollapse}\n\n\\input{NeutronStarsIntro}\n\\input{NeutronStars}\n\n\n\n\n\\input{cosmo}\n\n\n\nWhenever a new vista onto the cosmos has been exposed in the past, it\nhas revolutionized our understanding of the Universe and its denizens.\nWe anticipate a similarly dramatic upheaval as the gravitational wave\nUniverse reveals itself.\nA detector with broadband sensitivity is best suited for exploring the full range of serendipitous discoveries.\n\n\n\n\n\\section*{Conclusions}\n\\label{s:conclusion}\nWe have shown that a number of significant quantitative improvements can be achieved relative to a\nwide array of known astrophysical targets by upgrading the LIGO interferometers within the existing facilities.\nPrecision tests of extreme spacetime curvatures can be made with\nthese improved instruments, perhaps even shedding light on what really happens at the black hole horizons.\nIn order to aid with making design tradeoffs for the LIGO Voyager\ndetector, we have numerically computed derivatives for these targets,\nindicating how much scientific value there is in incremental improvements in the interferometers.\nIt is clear from the Jacobian tables that there are significant astrophysical gains to be made for modest investments in the reduction of technical noise in the audio band (40\\,--\\,8000\\,Hz).\nTo make improvements for the low frequency (10\\,--\\,40\\,Hz) science targets (e.g. GW memory or mergers of higher mass black holes) would require order-of-magnitude improvements in the seismic isolation, suspension thermal noise, and Newtonian gravity noise. \n\nThis work should serve as a guide in making these detector design choices as well a starting point for more exhaustive evaluation of other science targets.\n\n\n\\section*{Acknowledgements}\nRXA, PA and IM performed part of this work at the Aspen Center for Physics, which\nis supported by National Science Foundation grant PHY-1066293.\nJAC acknowledges support under NSF PHYS-1505824 and PHYS-1505524.\nJSR acknowledges support from NSF PHYS-1307545 and the Research Corporation for Science Advancement.\nRXA, PA and YC acknowledge support from the Indo-US Centre for the\nExploration of Extreme Gravity funded by the Indo-US Science and\nTechnology Forum (IUSSTF\/JC-029\/2016).\nIn addition, P.~A.'s research was supported by a Ramanujan Fellowship from the Science and Engineering Research Board (SERB), India, and by the Max Planck Society through a Max Planck Partner Group at ICTS.\n\n\n\\clearpage\n\n\\begin{appendices}\n\n\\input{appendix1}\n\\clearpage\n\\input{paramtable}\n\n\\end{appendices}\n\n\n\\bibliographystyle{iopart-num}\n\n\\subsection{Astrophysics with Stellar Collapse}\n\\label{s:supernova}\n\nThe core collapse of massive stars has long been considered an interesting \nsource of GWs~\\cite{mtw}. While the intricacies of the core-collapse supernova (CCSN) explosion mechanism \nare not well understood, state-of-the-art 3D simulations (see, e.g.,\n\\cite{roberts:2016,kuroda:2016,takiwaki:2016,radice:2016,abdikamalov:2015,lentz:2015,melson:2015-strangeq,bmueller:2015,moesta:2015,couch:2014,takiwaki:2014,murphy:2013,hanke:2013,couch:2013,dolence:2013,ott:2013,emueller:2012,takiwaki:2012,hanke:2012,burrows:2012,kotake:2011,scheidegger:2010}) suggest \nthat rapid rotation, turbulent convection, and instabilities of the stalled \naccretion shock play important roles in re-energising the shock \nand aiding stellar explosion. GW emission from the initial core collapse and \nsubsequent explosion is strongly influenced by the physical processes driving \nthe explosion. For this reason, GW observations can be used to directly probe \nthe CCSN central engine and gain insight into the explosion\nmechanism~\\cite{logue:12,powell:2016}. \n\nThe angular momentum and degree of differential rotation of the precollapse stellar \ncore are thought to strongly influence the dynamics of the initial collapse, the subsequent \nexplosion, and the compact remnant formed (see, \ne.g.~\\cite{macfadyen:1999,woosley:2006,yoon:2006,georgy:2009}). Observations of the evolving pulsar population suggest a broad distribution of \nmoderately rotating NSs at birth with initial spin periods around \n10\\,--\\,100\\,ms~\\cite{faucher-giguere:2006,popov:2010,gullon:2014}. Wave-driven \nangular momentum transport in massive stars during the late stages of shell \nburning may strongly impact the pre-collapse rotation rate, predicting \na distribution of initial NS periods consistent with\nobservations~\\cite{fuller:2015}. Binary interactions are also expected to have a\nmarked effect on the rotation of massive stars~\\cite{demink:2013,Zaldarriaga:2017,Qin:2019}.\n\nFor stellar cores with pre-collapse periods exceeding a few tens of seconds, \ndelayed revival of the stalled shock is thought to be driven by\nthe neutrino mechanism. In this scenario, some small fraction of the binding\nenergy released in neutrinos is absorbed in a layer between the stalled shock\nand proto-NS. Increased pressure behind the shock from neutrino heating and \nmulti-dimensional hydrodynamic instabilities drive it outwards and aid explosion. \n\nState-of-the-art 3D simulations suggest that turbulent\nconvection and the standing accretion-shock instability (SASI) are expected to \ndominate the explosion dynamics~\\cite{andresen:etal:2017,roberts:2016,radice:2016,abdikamalov:2015,lentz:2015,melson:2015-strangeq,bmueller:2015,yakunin:2015-gw}. \nExtensive research on the GW signature from slowly rotating core collapse has been done in 2D \n(see, e.g.,\\cite{yakunin:2015-gw,bmueller:2013}, for recent studies) and 3D \n(see, e.g.,~\\cite{yakunin:2017,kuroda:2016,andresen:etal:2017,ott:2013,emueller:2012} \nfor recent studies).\n\nProto-neutron star (PNS) oscillation modes can source appreciable GW emission from \n$\\sim(100-200)\\,\\mathrm{ms}$ after core bounce, following a short quiescent\nperiod, for up to $\\sim1\\,\\mathrm{s}$. The GW frequency naturally follows the \ndominant PNS surface g-mode frequency, increasing linearly with time from \n$\\sim(100-200)\\,\\mathrm{Hz}$ to over $1\\,\\mathrm{kHz}$ as the PNS \nevolves~\\cite{murphy:2009,marek:2009,bmueller:2013,yakunin:2015-gw,yakunin:2017}.\nStrong fluid downflows associated with the SASI can modify the accretion rate \nat the PNS, inducing quadrupolar oscillations at $\\sim(100-200)\\,\\mathrm{Hz}$ at later\ntimes~\\cite{murphy:2009,marek:2009,kuroda:2016}.\n\nGW memory, a non-oscillatory contribution to the GW amplitude at leading\nquadrupole order (see, e.g.~\\cite{braginskii:87,favata:10}), may be \ncreated by anisotropic neutrino emission\n\\cite{epstein:78,mueller:04,ott:09,kotake:11,marek:09b,yakunin:10,\n muellere:12,mueller:13gw,yakunin:2015-gw,yakunin:2017} and aspherical \nexplosive outflows (e.g., in jets or if the shock acceleration is not spherically symmetric)\nof matter and magnetic stresses\n\\cite{obergaulinger:06a,obergaulinger:06b,takiwaki:11,ott:09}. For\nanisotropic neutrino emission, the GW memory effect causes emission at less than\n10\\,Hz~\\cite{yakunin:10,marek:09b,ott:09,yakunin:2015-gw,yakunin:2017}.\n\nRapidly rotating stars, which are expected to make up\n$\\sim$\\,1\\,--\\,10\\% of the massive star population\\,\\cite{ott:06spin,woosley:06,heger:05},\ncould explode via a bipolar magnetohydrodynamic explosion leading to \nlarge explosion asymmetries and relativistic\noutflows~\\cite{burrows:07b,takiwaki:09,scheidegger:2010,kuroda:2014,moesta:14b,moesta:2015,masada:2015,takiwaki:2016,richers:2017}.\nThe inner cores in rotating progenitor stars become centrifugally deformed in the late\nstages of collapse. This results in a large quadrupole moment, which\nchanges rapidly at core bounce, leading to a strong and pronounced\npeak in the GW signal that is followed by ring-down oscillations of\nthe PNS~\\cite{dimmelmeier:08,richers:2017,abdikamalov:14,kuroda:2014}.\n\nPNSs with strongly differentially rotating profiles are often \nsubject to a rotational shear instability that drives the development\nof nonaxisymmetric dynamics in the PNS \ncore~\\cite{scheidegger:10b,ott:07prl,ott:09,fu:11,muhlberger:14}. Also known as \nthe co-rotational (or low $T\/|W|$) instability, typical GW energy emitted can be \nas high as $10^{-7}M_\\odot c^2$. \n\nA number of energetic SN explosions have been seen in coincidence with\nnearby long gamma-ray bursts (GRBs), providing an\nobservationally robust connection between long GRBs and stellar\ncollapse \\cite{hjorth:11,modjaz:11}. The central engine in a long GRB is \nthought to be either a nascent black hole surrounded by a fallback accretion \ndisk (a collapsar~\\cite{woosley:93,macfadyen:01,wb:06}) or millisecond \nproto-magnetar~\\cite{wheeler:02,metzger:11}. In systems with accretion disks \nfrom fallback material, various instabilities may develop and lead to GW \nemission (e.g., \\cite{piro:07,korobkin:11,kiuchi:11pp}). Classical dynamical \ninstabilities are unlikely to occur in regular core collapse events, but may be \nrelevant in extreme cases that lead to long GRBs and\/or black hole formation\n\\cite{fryer:02,ott:09,ott:11a,oconnor:11,corsi:09,piro:11}. \n\nBH formation as a consequence of fallback accretion onto the PNS\nis thought to be the formation channel for most stellar\nBHs~\\cite{oconnor:11,ugliano:12}. The timescale on which this occurs is\ndependent on the accretion rate (directly influenced by\nthe properties of the progenitor star), the angular momentum of the PNS, and the\nnuclear matter equation of state (EOS). In most systems, this happens\n$\\sim0.5-3\\,\\mathrm{s}$ after core bounce, and is characterised by a short GW\nburst with a broad spectrum peaking at\n$2.5$--$3.5\\,\\mathrm{kHz}$~\\cite{cerda-duran:2013,oconnor:11,ott:2011}.\n\nRapid rotation is also expected to be present in massive accreting\nwhite dwarfs and in the cores of white dwarf merger remnants (e.g.,\n\\cite{yoon:05b}). Such massive degenerate cores are expected to\ncollapse to neutron stars rather than explode as thermonuclear\nsupernovae. Such ``accretion-induced collapse'' (AIC) events are\nexpected to give off a strong burst of gravitational waves~\\cite{ott:09, abdikamalov:10}.\n\n\\subsubsection{Detection Prospects.}\nThe galactic rate of core collapse events is low and estimates vary\nfrom $\\sim$1 in 40 years to 1 per century~\\cite{vdb:91,mannucci:05,keane:08};\nincluding the Large and Small\nMagellanic Clouds and the Andromeda galaxy (M31) roughly doubles the\nrate~\\cite{vdb:91}. A significant increase in the event rate occurs\nonly outside of the Local Group of Galaxies with the M81 group of\ngalaxies at $\\sim$3\\,--\\,5\\,Mpc~\\cite{ando:05, kistler:11}. There,\n$\\sim$\\,0.5 core-collapse supernovae are discovered per year, suggesting\na rate of around 1 event per year, when assuming that $\\sim$50\\% of\nthe events remain undiscovered due to obscuration or weak\/absent EM\nemission. The integrated event rate out to 10\\,Mpc is\nlikely around $\\gtrsim$2 events per year~\\cite{ando:05, kistler:11}.\n\nWhile the search for CCSNe with the first generation of ground-based GW \ndetectors yielded no observations, constraints on energy emitted in GWs by \nCCSNe were made for the first time~\\cite{abbott:etal:2016:1gSNsearch}. With \nthe second-generation instruments, we don't expect to see the typical \nCCSN beyond a few kpc for slowly rotating progenitors, while rapidly rotating \nprogenitors might yield GW emission observable throughout the galaxy and \nMagellanic clouds~\\cite{gossan:etal:2016,hayama:etal:2015,nakamura:etal:2016}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Probing Core-Collapse Supernova Physics.}\n\\label{s:collapse_scipot}\nThe physics that may be learned from a detection of \nGWs from stellar collapse goes far beyond constraining GW emission\nand explosion mechanisms. CCSNe and related\nphenomena are cosmic laboratories for high-energy-density\ngravitational, plasma, nuclear, and particle physics. In particular,\nit may be possible to extract information on the nuclear EOS directly\nfrom GW observations~\\cite{richers:2017,abdikamalov:2014,roever:09,murphy:2009,marek:09b,kuroda:2016}.\nElectromagnetic (EM) observations can tell us little directly about the distribution of rotation rates of\npre-collapse iron cores, such that existing constraints come primarily from \nobservations of the resultant young compact objects. For rapidly rotating\npre-collapse cores, GW observations can be used to directly infer the \nangular momentum distribution~\\cite{abdikamalov:14,engels:2014,edwards:2014}, aided by multimessenger observations with neutrinos~\\cite{yokozawa:2015}.\nCoincident GW and neutrino observations will be of extreme importance\nif the next Galactic core collapse event leads to black hole formation\n(without electromagnetic display). MeV-energy neutrinos from any Galactic or \nnearby extragalatic core collapse event will be observed by current and future neutrino\ndetectors (e.g., \\cite{halzen:09,ikeda:07,icecube:11sn,scholberg:11,hyperkamiokande:11}). \n\nIn the absence of a more specific FOM covering GWs from both slowly and \nrapidly rotating core collapse, we consider the signal SNR as the astrophysical\nFOM. \nLIGO Voyager{}, in the baseline design considered in this report, yields SNRs\nfor core-collapse supernova waveforms (see Table~\\ref{tab:changeSNR} and\n\\refsection{s:supernova}) that are a factor of 4\\,--\\,5 greater than\naLIGO, which means that robust core-collapse supernova model selection\nmay be possible out to distances of $\\sim$\\,8\\,--\\,10\\,kpc,\nproviding coverage virtually throughout the Galaxy. There will \nlikely be at most one core collapse event in the Milky Way in the\nlifetime of LIGO, so extending our reach throughout the Galaxy is\ncrucial.\n\nAccording to the current understanding of core-collapse supernova\ntheory~\\cite{janka:07}, the most likely and most robust\nGW emission mechanism is turbulent neutrino-driven convection in the\ncontext of the ``neutrino mechanism''~\\cite{ott:09,mueller:13gw}. This\nleads to broadband GW emission with most power at\n100\\,--\\,1000\\,Hz. Table~\\ref{tab:changeSNR} shows that most\nimprovement above the baseline LIGO Voyager{} design will come from reducing the\nshot noise either through more squeezing or more laser power.\nNote that the vast majority of stellar collapse events lead to standard-energy\ntype-II supernovae and are unlikely to be strong GW\nemitters. Even LIGO Voyager{} will not be able to observe such events to\ndistances greater than $\\sim 100\\,\\mathrm{kpc}$, which covers the\nMilky Way, and the Magellanic Clouds.\n\nA number of theoretical models (\\refsection{s:supernova}) predict strong\nGW emission connected with hyper-energetic\ncore-collapse supernovae and\/or long-duration GRBs. A potential\ncandidate emission process relies on long lasting non-axisymmetric bar-like\ndeformations of an extremely rapidly spinning PNS (or\n``protomagnetar'') due to a rotational instability\n\\cite{ott:09,ott:10dcc,piro:11,fryer:02}. The GW\nemission is expected to be narrow-band and at high frequency\n($\\sim$\\,400\\,--\\,2000\\,Hz) and Table~\\ref{tab:changeSNR} lists\nresults for a range of potential waveforms, generated using the ad-hoc\nbar model of \\cite{ott:10dcc}. Even the weakest signal considered here\nmay be detectable by aLIGO at a distance of a few hundred\n$\\mathrm{kpc}$. LIGO Voyager{} in its baseline configuration could be able to\ndetect this signal out to 5\\,--\\,10\\,Mpc. At this distance,\n$\\gtrsim$$2$ core collapse events occur per year. It would thus be\npossible to put strong constraint on the presence of such strong\nemission models.\n\n\n\\subsubsection{Gravitational-Wave Memory in Stellar Collapse in the Milky Way}\nGW memory may be left behind by most stellar collapse events, even\nthose that do not result in an explosion. The typical growth timescale\nof the memory is of order $\\gtrsim$$0.1\\,\\mathrm{s}$, which makes it\nthe only known low-frequency GW emission process in stellar\ncollapse. Detecting the GW memory from a galactic\nevent with aLIGO may be a difficult task even if the full projected\nlow-frequency sensitivity is reached, but the baseline LIGO Voyager{} design\nwould allow detection. Searches for GW memory would most\nbenefit from improvements of the coating thermal noise or the arm\ncavity spot size (see Table~\\ref{tab:changeSNR}).\n\n\n\n\\begin{table}[t]\n\\resizebox{\\textwidth}{!}{%\n\\centering\n\\begin{tabular}{lcccccccccc}\n\\textbf{Waveform Type} & \\textbf{aLIGO SNR} & \\textbf{LIGO Voyager{} SNR} & \\textbf{NN} & \\textbf{SEI} & \\textbf{SUS} & \\textbf{SPOT\/CTN} & \\textbf{SQZ} & \\textbf{POW} & \\textbf{FCL} & \\textbf{MASS}\\\\\n& \\textbf{@ 10 kpc} & \\textbf{baseline @ 10 kpc} \\\\\n\\bottomrule\n\\textbf{$\\nu$ GW Memory}\\\\\nB12-WH07-$\\nu$ (2D) \\cite{yakunin:2015-gw} & 19.70 & 75.91 & $4.22\\times10^{-3}$ & \n$3.93\\times10^{-4}$ & $1.00\\times10^{-2}$ & $3.14\\times10^{-1}$ & \n$-2.18\\times10^{-2}$ & $3.40\\times10^{-1}$ & $2.20\\times10^{-1}$ &\n$-1.86\\times10^{-1}$\\\\\nB20-WH07-$\\nu$ (2D) \\cite{yakunin:2015-gw} & 16.54 & 63.73 & $4.18\\times10^{-3}$ &\n$3.89\\times10^{-4}$ & $1.00e-02\\times10^{-2}$ & $3.14\\times10^{-1}$ & \n$-2.19\\times10^{-2}$ & $3.40\\times10^{-1}$ & $2.20\\times10^{-1}$ & \n$-1.86\\times10^{-1}$\\\\\n\\hline\n\\textbf{Rapidly Rot. Core Collapse}\\\\\nR1E1CA$_\\mathrm{L}$ (3D) \\cite{scheidegger:2010} & 1.46 & 5.81 & \n$1.37\\times10^{-6}$ & $5.67\\times10^{-8}$ &\n$5.70\\times10^{-6}$ & $1.29\\times10^{-1}$ & $6.22\\times10^{-1}$ &\n$3.39\\times10^{-1}$ & $-1.62\\times10^{-3}$ & $-6.65\\times10^{-3}$\\\\\nR3E1AC$_\\mathrm{L}$ (3D) \\cite{scheidegger:2010}& 74.38 & 287.87 & \n$2.43\\times10^{-7}$ & $1.14\\times10^{-8}$ &\n$1.09\\times10^{-6}$ & $7.49\\times10^{-2}$ & $7.03\\times10^{-1}$ &\n$3.81\\times10^{-1}$ & $-8.73\\times10^{-4}$ & $-1.84\\times10^{-3}$\\\\\nR4E1FC$_\\mathrm{L}$ (3D) \\cite{scheidegger:2010}& 77.12 & 287.53 & \n$4.42\\times10^{-7}$ & $1.40\\times10^{-8}$ &\n$2.22\\times10^{-6}$ & $5.99\\times10^{-2}$ & $7.38\\times10^{-1}$ &\n$3.99\\times10^{-1}$ & $-7.69\\times10^{-4}$ & $-2.65\\times10^{-3}$ \\\\\nR3 (3D) \\cite{kuroda:2014} & \n30.48 & 123.72 & $1.98\\times10^{-5}$ & $4.95\\times10^{-7}$ &\n$8.17\\times10^{-5}$ & $2.21\\times10^{-1}$ & $5.04\\times10^{-1}$ &\n$2.82\\times10^{-1}$ & $-4.10\\times10^{-3}$ & $-4.28\\times10^{-2}$\\\\\n\\hline\n\\textbf{Convection \\& SASI}\\\\\nL15-3 (3D) \\cite{mueller:12b} & 3.74 & 15.39 & $4.92\\times10^{-4}$ &\n$4.19\\times10^{-5}$ & $1.23\\times10^{-3}$ & $3.28\\times10^{-1}$ &\n$3.58\\times10^{-1}$ & $2.14\\times10^{-1}$ & $-8.59\\times10^{-3}$ &\n$-9.46\\times10^{-2}$\\\\\ns27$f_\\mathrm{heat}1.05$ (3D) \\cite{ott:13a} & 3.45 & 13.70 & \n$1.28\\times10^{-7}$ & $4.30\\times10^{-9}$ & $3.37\\times10^{-6}$ &\n$1.63\\times10^{-1}$ & $5.84\\times10^{-1}$ & $3.20\\times10^{-1}$ &\n$-2.31\\times10^{-3}$ & $-1.48\\times10^{-2}$\\\\\ns20s (3D) \\cite{andresen:etal:2017} & 5.58 & 21.95 & $1.85\\times10^{-4}$ &\n$9.08\\times10^{-6}$ & $6.14\\times10^{-4}$ & $1.88\\times10^{-1}$ &\n$5.52\\times10^{-1}$ & $3.09\\times10^{-1}$ & $-5.15\\times10^{-3}$ &\n$-5.31\\times10^{-2}$\\\\\nTM1 (3D) \\cite{kuroda:2016} & 8.43 & 32.14 & $6.94\\times10^{-7}$ & \n$1.33\\times10^{-8}$ & $2.16\\times10^{-5}$ & $1.33\\times10^{-1}$ & \n$6.39\\times10^{-1}$ & $3.51\\times10^{-1}$ & $-2.67\\times10^{-3}$ & \n$-3.31\\times10^{-2}$\\\\\nSFHx (3D) \\cite{kuroda:2016} & 12.48 & 47.75 & $7.20\\times10^{-7}$ &\n$1.76\\times10^{-8}$ & $2.75\\times10^{-5}$ & $1.79\\times10^{-1}$ &\n$5.83\\times10^{-1}$ & $3.24\\times10^{-1}$ & $-3.87\\times10^{-3}$ & \n$-5.21\\times10^{-2}$ \\\\\n\\hline\n\\textbf{BH Formation in Collapsars}\\\\\nu75rot1 (3D) \\cite{ott:2011} & 11.77 & 43.84 & $1.81\\times10^{-7}$ & \n$3.55\\times10^{-9}$ & $1.76\\times10^{-6}$ & $7.33\\times10^{-2}$ &\n$7.18\\times10^{-1}$ & $3.89\\times10^{-1}$ & $-9.13\\times10^{-4}$ & \n$-3.20\\times10^{-3}$\\\\\nu75rot2 (3D) \\cite{ott:2011} & 30.91 & 114.84 & $5.70\\times10^{-7}$ & \n$1.32\\times10^{-8}$ & $2.98\\times10^{-6}$ & $7.08\\times10^{-2}$ & \n$7.22\\times10^{-1}$ & $3.91\\times10^{-1}$ & $-8.76\\times10^{-4}$ &\n$-2.82\\times10^{-3}$\\\\\n\\hline\n\\end{tabular}}\n\\caption{Jacobian of Science Return as a function of interferometer \n upgrade technology for various stellar collapse waveform families \n introduced in \\refsection{s:supernova}. The baseline FOM is the \n angle-averaged SNR $<\\rho>$.}\n\\label{tab:changeSNR}\n\\end{table}\n\n\n\n\n\n\n\n\\subsubsection{Testing General Relativity.}\n\\label{sec:TestGRProbGWs}\n\\label{sec:TestGRSourcePhysics}\n\nGWs from compact-binary mergers will provide a unique probe of\nstrong-field dynamics~\\cite{Will:2001, Yunes:2011}.\nLIGO's first observations of GWs from binary black holes have already allowed made it possible \nperform the first tests of general relativity (GR) in the highly relativistic \nstrong-field regime~\\cite{GW150914:GR}. LIGO Voyager instruments will allow us to significantly \nimprove the precision of such tests. \n\n\n\n\\paragraph{Speed of propagation of GWs from joint GW-EM observations:}\n\nAccording to GR, GWs travel at the speed of light, $c$. In other\ntheories, the speed $v_g$ of propagation of GWs could be\ndifferent~\\cite{lrr-2006-3}. Coincident observation of electromagnetic\n(EM) and GW signals from astrophysical sources such as GRBs or\ncore-collapse supernovae make it possible to measure the time-delay\n$\\Delta t_a$ between the EM and GW signals, and thus to constrain the\nspeed of GWs. For the case of a source located at a distance $D$,\n\\begin{equation}\n 1-\\frac{v_g}{c} \\simeq \\frac{c \\Delta t}{D}; ~~~~ \\Delta t\n = \\Delta t_a - \\left[(1+z) \\Delta t_s \\right]\n\\label{eq:GWspeedbounds}\n\\end{equation}\nwhere $\\Delta t_s$ is the time-delay between the GW and EM emissions\n\\emph{at the source} and $z$ the cosmological red shift.\n\nThe most promising astrophysical sources for this test are short-hard\nGRBs \\cite{GW170817:GRB}.\nIt can be seen from Eq.(\\ref{eq:GWspeedbounds}) that the sensitivity of this test is\nproportional to the distance to the source, and the best bound is\nprovided by sources located at the horizon distance of the detector.\nThe precise bound that we can place on $v_g$ depends on the the time \ndelay $\\Delta t_s$ at the source, which is currently uncertain.\nHence we use the horizon distance at which a double neutron star inspiral\n$(m_1 = m_2 = 1.4 M_\\odot)$ can be detected with an optimal SNR of 8\nas a figure of merit for this measurement.\n\n\\paragraph{Mass of the graviton from joint GW-EM observations:}\nOne particular scenario in which the speed of GWs $v_g$ could differ from\n$c$ is in the case of graviton having a non-zero rest mass. This is\ncharacterized by the dispersion relation ${v_g^2}\/{c^2} = 1 -\nm_g^2\\,c^4\/{E_g^2}$, where $m_g$ is the rest mass and $E_g \\equiv 2 \\pi \\hbar\nf_{\\mathrm{GW}}$ the rest energy of the graviton with frequency\n$f_{\\mathrm{GW}}$, $\\hbar$ being the reduced Planck constant~\\cite{Will98}.\nFrom this dispersion relation and using Eq.(\\ref{eq:GWspeedbounds}), a lower\nbound on the Compton wavelength $\\lambda_g = 2\\pi\\hbar\/m_g c$ (or, an upper\nbound on the mass $m_g$ of the graviton) can be inferred from joint GW-EM\nmeasurements:\n\n\\begin{equation}\n\\lambda_g \\gtrsim \\left[\\frac{Dc}{2\\Delta t f_\\mathrm{GW}^2}\\right]^{1\/2}\n\\end{equation}\nIt can be seen that the best bound is provided by distant sources.\nHere we also use the horizon distance to a double neutron star inspiral\n$(m_1 = m_2 = 1.4 M_\\odot)$ as a figure of merit for this measurement.\n\n\n\\paragraph{Mass of the graviton from GW observations of CBCs:}\nCBC observations also make it possible to estimate the mass of the graviton even\nin the absence of an EM counterpart. In the case of CBCs, the GW\nfrequency sweeps from lower to higher frequencies. If the graviton is\nmassive, different frequency components travel with different speeds,\ncausing a distortion in the observed waveform~\\cite{Will98}. In\nparticular, the observed GW phase $\\Psi(f)$ in the frequency domain\nwill be deviated from the phase $\\Psi_\\mathrm{GR}(f)$ predicted by GR:\n\\begin{equation}\n\\Psi(f) = \\Psi_\\mathrm{GR}(f) - \\frac{\\pi D}{\\lambda_g^2 (1+z)} \\, f^{-1},\n\\end{equation}\nwhere $\\lambda_g \\equiv h\/m_gc$ is the Compton wavelength of the\ngraviton. LIGO's first observation of a binary black hole system has provided\none of the best lower bound on \n$\\lambda_g \\sim 10^{13}~\\mathrm{km}$~\\cite{PhysRevLett.116.221101}. Here we\nuse the expected lower bound on $\\lambda_g$ from the observation of a binary\nblack hole system with parameters similar to the first LIGO event ($m_1 = m_2 = 30 M_\\odot$,\nlocated at a distance of 500 Mpc) as the figure of merit. The bounds reported\nin Table~\\ref{tabl:sources_baseline} are computed using the Fisher \nmatrix formalism, outlined in~\\cite{Keppel:2010qu}.\n\n\n\n\\paragraph{Decay of GWs during propagation:}\nIf GWs decay during propagation (apart from the expected $1\/r$ falloff; e.g. due to dissipation),\ndistant sources would appear to be systematically dimmer. The detection of this requires a\npopulation of coincident GW+EM observations with redshift $z$ estimation\n(say, from the merger binary neutron stars).\nThen we could look for a systematic suppression of GW amplitude for higher-$z$ sources.\nThe sensitivity of this test would be proportional to the distance traveled by the\nGWs. Assuming that the redshift can be accurately estimated for sources located\nat arbitrary distances, the relevant figure of merit for GW detectors is simply\nthe horizon distance. We take the horizon distance (SNR of 8) to non-spinning binary neutron\nstar inspirals with $m_1 = m_2 = 1.4 M_\\odot$ as the figure of merit for this test.\n\n\n\n\n\n\n\\paragraph{Parametrized deviations from post-Newtonian theory:}\nHere we introduce parametrized deviations from GR in the inspiral\nwaveforms computed using the post-Newtonian (PN) approxmation to GR,\nand examine our ability to constrain these deviations from the data.\nConsistency with their GR values is a null-hypothesis test of\nrelativity~\\cite{Arun:2006a,Mishra:2010,Li:2012}.\n\nThe frequency domain phase of the PN waveforms can be written as:\n\\begin{equation}\n\\Psi(f) = 2\\pi f t_0 + \\phi_0 + \\sum_{k=0}^{7} (\\psi_k + \\psi^\\mathrm{L}_k \\ln f ) f^{(k-5)\/3},\n\\end{equation}\nwhere $\\psi_k$ and $\\psi^\\mathrm{L}_k$ are the PN coefficients of the\nphase at order $k\/2$ PN.\nWe introduce deviations in the PN coefficients in the following way:\n\\begin{equation}\n\\psi_k \\rightarrow \\psi_k + \\Delta \\chi_k, ~~~ \\psi_k^\\mathrm{L} \\rightarrow \\psi^\\mathrm{L}_k + \\Delta \\chi^\\mathrm{L}_k,\n\\end{equation}\nwhere $\\Delta \\chi_k$ and $\\Delta \\chi^\\mathrm{L}_k$ are zero in GR. For a \nnon-spinning binary, the waveform $h(f) = A(f) \\exp [i \\ \\Psi(f)]$ depends on \nthe intrinsic binary parameters $m_1, m_2$, the extrinsic parameters $t_0, \\phi_0, D$ \nand the deviation parameters $\\Delta \\chi_k, \\Delta \\chi^\\mathrm{L}_k$. We deform \nthe GR waveform by introducing one deviation parameter at a time and compute \nthe expected constraint on this from the Cram\\'er-Rao bound, from an \narchetypal binary neutron star system with $m_1 = m_2 = 1.4 M_\\odot$ \nlocated at a distance of 500\\,Mpc.\nThe $1\\sigma$ errors in estimating the deviation parameter\n$\\Delta \\chi_7$ at 3.5PN order, as a fraction of the known value\n$\\psi_7$ of the 3.5PN term in GR, are given in\nTable~\\ref{tabl:sources_baseline}.\n\n\n\n\n\\paragraph{Detecting the tails:}\nThe backscattering of gravitational waves emitted by a binary are by the binary's own gravitational field\\cite{Blanchet1993,blanchet2017first} are known as \\emph{tails}.\nIn a post-Newtonian expansion, this effect appears first in the 1.5\\,PN phasing coefficient. The figure of merit will be our estimation error for this coefficient (that is, $\\Delta \\chi_3$ as defined in the previous section) from a double neutron star system ($m_1 = m_2 = 1.4 M_\\odot$) located at 500\\,Mpc.\n\n\n\\paragraph{Detecting the memory effect:}\nThe {\\it memory wave} can be viewed as the gravitational effect of the\nstress-energy carried by previously emitted waves\n(Christodoulou, or nonlinear memory) with a contribution from the\nfinal momentum distribution of the binary\n(linear memory, e.g., due to kick)~\\cite{Favata:2009,favata:10,Pollney:2011,PhysRevD.98.064031}.\nThe ramping up of the memory wave may be a measurable~\\cite{Lasky2016,Yu2018,PhysRevLett.121.071102} test of GR. Let us provide a simple estimate of this effect, by writing\n\\begin{equation}\n \\label{hmem}\n h^{\\rm mem}(t) \\propto \\frac{R^2}{M} \\int_{-\\infty}^t \\dot h^2(t') dt'\n\\end{equation}\nHere $h$ is the leading gravitational waveform, while $h^{\\rm mem}$ is the \nnonlinear memory effect generated by the stress-energy associated with $h$. \nWe note that $h^{\\rm mem}$ is mostly a slowly increasing function of time, \nexcept during the merger process. We can use the SNR for $h^{\\rm mem}$ as the \nFOM for detecting the memory effect. We choose a fiducial BBH of \n$(30+30)\\,{\\mathrm M}_{\\odot}$ at 500\\,Mpc as representative for this FOM.\n\n\\paragraph{Testing the no-hair theorem with inspirals}\nAccording to the no-hair theorem of General Relativity, the spacetime around \na singularity fully enclosed by an event horizon, with no closed timelike \ncurves outside the horizon, and subject to several additional conditions, must \nbe described by the Kerr metric.\nIn particular, the full structure of the \nspacetime can be described by a multipole moment decomposition with only \ntwo free parameters, mass and spin, and all higher-order moments given by \nthe mass and spin.\n\nRyan proved that the full multipole moment structure of the spacetime\ncan in principle be measured through observations of\ngravitational-wave emission~\\cite{Ryan:1995}. \nTherefore, gravitational waves can be used to test deviations from the null \nhypothesis that massive compact objects are black holes by checking for consistency \nbetween higher-order multipole moments and their values as predicted by the Kerr metric. \nThe mass quadrupole moment is the first measurable term beyond the mass and spin, \nentering the post-Newtonian expansion at the second post-Newtonian order in the phase. \nTherefore, its measurement is likely to be the focus of no-hair theorem tests, \nthough other tests (e.g., through consistency of ringdown \nmodes~\\cite[e.g.,][and see below]{Meidam:2014}) are possible.\n\nIntermediate-mass-ratio inspirals (IMRIs) of low-mass compact objects into intermediate \nmass black holes provide a particularly promising tool for constraining the mass \nquadrupole moment and null hypothesis deviations\\,\\cite{Brown:2007,Rodriguez:2012}. \nGiven the uncertainty in existing waveform families in the intermediate-mass-ratio \nregime\\,\\cite[e.g.,][]{SmithIMRI:2013}, the exact precision of the measurement is \nat present compromised by systematic waveform uncertainty. Therefore, we will use the SNR of an intermediate-mass-ratio inspiral of a $1.4$ solar mass neutron star \ninto a $100\\,{\\mathrm M}_{\\odot}$ black hole at a luminosity distance of 1\\,Gpc \n(consistent with an astrophysically plausible rate of such \ninspirals in globular clusters\\,\\cite{Mandel:2008}) as a proxy for the \ndetectability of IMRIs and the measurability of the mass quadrupole moment.\n\n\n\n\\paragraph{Testing the No-Hair theorem with Ringdowns}\nGW signals from the ringdown phase of a BBH coalescence are expected to be dominated \nby a spectrum of quasi-normal modes (QNMs). According to the no-hair theorem, the \nfrequencies $\\omega_{\\ell m}^\\mathrm{GR}$ and damping times $\\tau_{\\ell m}^\\mathrm{GR}$ \nof these QNMs are unique functions of the mass $M_f$ and spin $a_f$ of the final \nKerr black hole~\\cite{Berti:2009}. Thus, in principle, the mass and spin of the \nfinal black hole can be extracted from different QNMs, which have to be consistent \nwith each other. In practice, one can introduce parameters $\\Delta \\omega_{\\ell m}$ \nand $\\Delta \\tau_{\\ell m}$ that describe deviations from the GR prediction of the \nfrequencies and damping times of the QNMs\n\\begin{equation}\n\\omega_{\\ell m} = \\omega_{\\ell m}^\\mathrm{GR}(M_f, a_f) ~ (1 + \\Delta \\omega_{\\ell m}), ~~~\n\\tau_{\\ell m} = \\tau_{\\ell m}^\\mathrm{GR}(M_f, a_f) ~ (1 + \\Delta \\tau_{\\ell m}).\n\\end{equation}\nand constrain those deviations~\\cite{Meidam:2014}.\nWe use the SNR of the ringdown phase of the expected signal from a BBH system with \n$m_1 = m_2 = 30\\,{\\mathrm M}_{\\odot}$, located at a distance of 500 Mpc as a simple FOM for our \nability to constrain these deviation parameters.\n\n\n\n\\paragraph{Unaccounted loss of energy and angular momentum:}\nIf a GW signal from a BBH coalescence is observed with sufficient SNR,\nthe masses and spins of the black holes can be estimated from just the\ninspiral part of the signal.\nUsing these estimates of the initial parameters of the binary,\nthe mass and spin of the final black hole can be uniquely predicted by\nmaking use of numerical relativity simulations.\nIn addition, the mass and spin of the final black hole can be independently \nestimated from the ringdown part of the signal~\\cite{Hughes:2004vw,Ghosh:2016qgn}. \nAny inconsistency between these two estimates will point to an\nunaccounted loss in the energy \/ angular momentum from the system.\n\nThis consistency test requires binaries with the right masses and\nspins such that the inspiral, merger and ringdown parts of the signal\nare all observed with sufficiently large SNRs.\nAs a simple FOM, we use the SNR of a non-spinning binary with \n$m_1 = m_2 = 30 M_\\odot$, located at 500\\,Mpc.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction} \\label{sec:intro}\n\nObservational evidence suggests the existence of supermassive black holes (SMBHs) at the centers of most, if not all, galaxies \\citep[e.g.,][]{kormendy13}. Galaxies interact and merge according to the current hierarchical cosmological model, thus an expected outcome of those events is the formation of SMBH binaries in the galaxy remnants \\citep[e.g.,][]{begelman80, volonteri03}. To become a bound binary and finally coalesce, the SMBHs will go through different processes that extract their energy and angular momentum, making them fall deeper into the gravitational potential \\citep{begelman80}.\n\nCoalescence can be achieved within a Hubble time \\citep[depending of the environment conditions,][]{merritt05, colpi09, mayer18}, but the details of the mechanisms involved are still a topic of active research. While the galaxies are merging, dynamical friction between the individual SMBHs and the background of stars and gas will drag them into the center of the gravitational potential \\citep{governato94}. After a bound binary is formed, dynamical friction becomes inefficient and the shrinkage will be dominated by scattering of individual stars, a process which efficiency depends on the properties of the stellar background \\citep[e.g.,][]{yu02, vasiliev15}. In the gas-rich case, the gas that followed the black holes through the dynamical friction phase could form a circumbinary accretion disc around the binary \\citep[e.g.,][]{escala05} and allow the system to shrink by angular momentum transfer due to a tidal interaction~\\citep[e.g.][]{cuadra09,roedig11,tang17}. When the SMBHs reach a separation of~O(100) Schwarzchild radii (${\\rm R_S}$), the emission of gravitational waves (GW) dominates and shrinks the binary in a short period of time \\citep{peters64, armitage02, chang10}. Due to the different time-scales involved, most of the (accreting) SMBH binaries are expected to be in the accretion-disc dominated phase, which can take of the order of millions of years for the expected binary parameters \\citep{haiman09, tang17}. Candidates for this kind of system have recently been identified as periodic quasars in the Catalina \\citep{graham15} and Palomar Transient Factory (PTF) surveys \\citep{charisi16}.\n\n\\begin{figure}\n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot0.pdf}\n\\caption{Schematic representation of the binary--disc system for $q \\ll 1$.\nThe whole system is surrounded by the circumbinary (outer) disc, while the primary black hole (mass $M_p$) has an individual (inner) disc extending out to the secondary. For simplicity, we neglect the accretion disc around the secondary black hole (mass $M_s$) and assume the BH and gas orbits are circular, co-planar, and pro-grade. The binary separation $a$ and the surface density $\\Sigma(r)$ of the discs are coupled and evolve over time. The external boundary condition of the system is given by a fixed accretion rate $\\dot{M}_{\\rm ext}$.}\n\\label{f:scheme}\n\\end{figure}\n\nThe large computational cost of simulating the long-term evolution of these systems makes 3D or even 2D models prohibitive. Therefore, here we approach the problem with an 1D idealized configuration. We modeled the ``inner\" disc around the primary and the ``outer\" circumbinary disc with the standard hydrodynamics equations. We also include a simple prescription to mimic the binary tidal torque \\citep{armitage02, tazzari15}, and allow for some gas to cross from the outer to the inner disc. For simplicity, we do not model the individual disc around the secondary and neglect the change in the mass ratio due to accretion (as discussed and justified in \\S~\\ref{sec:boundary}). A schematic representation of the system is shown in Fig. \\ref{f:scheme}.\n\nIn this scenario many processes occur simultaneously. The tidal interaction removes angular momentum from the binary and adds it to the circumbinary disc, removing some gas from the vicinity. On the other hand, the inner disc transfer its angular momentum to the binary, slowing its migration \\citep[e.g.][]{papaloizou84}. Viscosity in both discs dissipates kinetic energy and drags the gas inwards \\citep{shakura73, frank02}, which added to the dissipation effect of the tidal torque heats up the gas, shaping its electromagnetic radiation \\citep[e.g.][]{lodato09}. \n\nImportant properties of the system depend on $q \\leq 1$, the mass ratio of the binary. \\citet{dorazio16} found that for $q \\lesssim 0.05$ the tidal interaction digs an annular gap at the location of the secondary orbit, that the disc remains axisymmetric, and drives a constant accretion rate. On the other hand, for $q \\gtrsim 0.05$, the gap becomes a central cavity around the binary, which furthermore becomes lopsided when $q \\gtrsim 0.3$. Short-term 3D simulations of this process show however that streams of material get across the cavity and drive accretion episodically \\citep{artymowicz94, cuadra09, roedig11, farris14, farris15, dunhill15, tang17}, although the efficiency of this feeding decreases as the discs become thinner \\citep{ragusa16}. A similar result has been found in 2D and 3D simulations of $q \\ll 1$ binaries, where material can cross the annular gap \\citep{baruteau12, duffell14, duermann15, duermann17}, again depending on the disc thickness \\citep{cerioli16}.\n\nSeveral previous studies have focused on the long-term evolution of the binary--disc systems in the regime where $q \\ll 1$, but they are either analytical, assuming steady state \\citep{liu10, kocsis12b, rafikov16}, or numerical, but with a relatively small initial binary separation ($a \\leq 10^4 ~{\\rm R_S}$) and without inflow through the secondary orbit \\citep{armitage02, lodato09, chang10, tazzari15}. In this work, we overcome these limitations by developing a self-consistent, time-dependent model that follows the long-term coupled evolution of the system starting at $10^5 ~{\\rm R_S}$, including a prescription for material to cross the gap. \n\nThis paper is organized as follows. In \\S\\ref{sec:phys} we present and explain the physical properties of the system we are considering, and present the equations for each process implemented in our numerical simulation. In \\S\\ref{sec:res} we show our main results on the evolution of a binary system, the surface density profile, the residence time, and the expected spectrum of the discs. In \\S\\ref{sec:pw} we compare our results with previous analytical and numerical works, and discuss the similarities and differences between our approach to the problem. Finally, in \\S\\ref{sec:con}, we present a summary of our main results and their implications.\n\nOur most significant conclusion is that, at least for the mass ratios and inflow speeds included in our study, we find no real steady-state on the discs: the surface density distribution chosen as the initial condition influences the state of the discs during their whole subsequent history.\n\n\\section{System physics and model properties} \\label{sec:phys}\n\nIn this section, we first explain the relevant physics in our study, then we discuss the simplifications needed to implement a 1D simulation, and finally give details of our numerical code.\n\nWe model the inner-- and circumbinary discs assuming their scale height $h$ is much smaller than the distance to the centre $r$, so the system behaves as a standard, two-dimensional thin disc. Furthermore, we consider the primary black hole to be at the center of mass of the system, ignore the individual disc around the secondary, and assume that the secondary and both accretion discs follow circular, co-planar and prograde paths around the primary black hole. These assumptions allow us to model the system in one dimension, with all disc properties functions of the radial coordinate $r$ only.\n\n\\subsection{Viscosity, thickness and energetics} \\label{sec:visc}\n\nViscosity acts as an angular momentum transport mechanism, which produces a torque between contiguous rings of the disc. The standard $\\alpha$-disc model \\citep{shakura73} parametrizes the turbulent kinematic viscosity as a function of the sound speed $c_s$ and the thickness of the disc $h$. While $c_s$ depends on the total pressure, in this study we implement an alternative viscosity prescription called a $\\beta$-disc model, which ensures thermal and viscous stability:\n\\begin{equation}\n\\nu = \\alpha c_s h \\beta, \\label{eq:nu}.\n\\end{equation}\nThis allows us to put all the uncertainties in the constant $\\alpha \\leq 1$. Here, $\\beta$ is the ratio between the gas pressure $p_{\\rm gas}$ and total pressure $p_{\\rm tot} = p_{\\rm gas} + p_{\\rm rad}$ (with $p_{\\rm rad}$ the radiation pressure). For simplicity, we consider the pressures to be determined by the mid-plane temperature of the disc $T_c$: $p_{\\rm rad} = 4 \\sigma T_c^4 \/ (3 c)$ and $p_{\\rm gas} = \\rho k T_c \/ (\\mu m_p)$, with $k$ the Boltzmann constant, $\\sigma$ the Stefan-Boltzmann constant, and $\\mu = 0.615$ the mean particle mass in units of the proton mass $m_p$ for a plasma of solar metallicity.\n\nThe sound speed in the disc is given by $c_s^2 = p_{\\rm tot} \/ \\rho$ with $\\rho = \\Sigma \/ (2 h)$ the volumetric density and $\\Sigma$ the surface density of the discs. Assuming hydrostatic equilibrium in the vertical direction, we can express the sound speed as a function of thickness $h$ and angular velocity: $c_s = \\Omega h$, with $\\Omega = (GM(1+q)\/r^3)^{1\/2}$ the angular velocity of the material.\n\nTo close this system of equations, we relate the central temperature $T_c$ to the surface density $\\Sigma$, assuming photons are transported to the disc surface by vertical diffusion: $F = 8 \\sigma T_c^4 \/ (3 \\Sigma \\kappa)$ (where $\\kappa$ is the opacity \\added{due to electron scattering}), and that viscous and tidal heating are dissipated in the form of radiation \\citep{lodato09, kocsis12a},\n\\begin{equation}\nF = D_\\nu + D_\\Lambda = \\frac{9}{8} \\nu \\Sigma \\Omega^2 - \\frac{1}{2} \\Lambda \\Sigma (\\Omega - \\Omega_s). \\label{eq:F}\n\\end{equation}\nThe tidal term $D_\\Lambda$, as explained in \\citet{lodato09}, comes from angular momentum conservation and from the assumption that orbits remain circular. \n\nUsing the above formulation leads to a system of equations, uniquely determining the disc properties at each radius and time:\n\\begin{equation} \n\\begin{aligned}\nT_c &= \\Bigg[\\dfrac{3 \\kappa \\Sigma^2 \\big(9 \\alpha c_s^2 \\Omega \\beta - 4 \\Lambda(\\Omega - \\Omega_s)\\big)}{64 \\sigma}\\Bigg]^{\\frac{1}{4}},\\\\\n\\beta &= \\left[1 + \\frac{8 \\sigma \\mu m_p T_c^3 c_s}{3 c k \\Sigma \\Omega}\\right]^{- 1},\\\\\nc_s &= \\frac{8 \\sigma T_c^4}{3 c \\Omega \\Sigma (1 - \\beta)}.\n\\end{aligned}\\label{eq:solution}\n\\end{equation}\nIt can be shown that there is only one real and positive solution for the central temperature. Solving the equations allows us to obtain the sound speed $c_s$, the thickness $h$ of the disc, and finally the viscosity itself \\citep{fontecilla17}. Having the central temperature at each radius and assuming that the discs emit as a multi-temperature blackbody \\citep{frank02}, we can obtain the spectral energy distribution (SED) and the bolometric luminosity ($L_{\\rm bol}$) of the system as a function of time.\n\n\\subsection{Surface Density evolution}\n\nAt least two mechanisms will make the surface density of the discs evolve over time: viscosity, already explained in the previous subsection, and the tidal torque produced by the changing gravitational potential of the rotating binary. The latter is a 2D effect, and cannot be implemented directly in a 1D simulation. To bypass this, we adopt a commonly used recipe that models its effect on the accretion discs. Following the literature \\citep{armitage02}, \n\\footnote{See \\citet{dong11a, dong11b, petrovich12, rafikov16} for discussion about the tidal prescription.}\nwe define an orbit-averaged torque,\n\\begin{align}\n\\Lambda = \\begin{cases}\n- \\dfrac{f}{2} q^2 \\Omega^2 r^2 \\bigg(\\dfrac{r}{\\Delta}\\bigg)^4, & \\mbox{if } r < a \\\\[12pt]\n\\enskip \\dfrac{f}{2} q^2 \\Omega^2 r^2 \\bigg(\\dfrac{a}{\\Delta}\\bigg)^4, & \\mbox{if } r \\geq a \\\\\n\\end{cases}\\label{eq:smooth}\n\\end{align}\nwhose functional form depends on whether we are in the inner ($r < a$) or circumbinary ($r \\geq a$) discs. Here \\added{$f = 0.01$} is a dimensionless parameter that controls the strength of the torque, $\\Delta = \\max \\{R_h,~h,~|r - a|\\}$ is a smoothing term, and $R_h = a (q \/ 3)^{1 \/ 3}$ is the Hill radius of the secondary black hole. While this model was originally proposed for $q \\ll 1$, it has been widely used for binaries up to $q = 0.3$ \\citep{lodato09, chang10, kocsis12a, kocsis12b, tazzari15}.\n\nWhile viscosity always transfers the angular momentum of the disc outwards, the tidal effect depends on the position. Inside the binary orbit ($r < a$) the tidal torque $\\Lambda$ adds angular momentum to the binary, producing an inward acceleration on the elements of the disc. On the other hand, for the circumbinary disc ($r \\geq a$), this effect will add angular momentum to the gas, preventing it to cross the binary orbit. Here, the balance between the tidal effect and the viscous mechanism will produce a low density region called cavity or gap, which depends on the mass ratio of the binary. \n\nAs pointed out by \\citet{tazzari15}, for $q \\geq 0.1$, eq. (\\ref{eq:smooth}) gives an unphysically large tidal contribution on the discs outside the Lindblad resonances, which will alter the migration velocity of the binary and the surface density of the discs. Following their work we added an exponential cutoff for the tidal torque outside this region,\n\\begin{align}\n\\Lambda = \\begin{cases}\n- \\dfrac{f}{2} q^2 \\Omega^2 r^2 \\bigg(\\dfrac{r}{\\Delta}\\bigg)^4 \\exp\\left[- \\left(\\dfrac{r - r_{\\rm IMLR}}{w_{\\rm IMLR}}\\right)^2\\right] & \\mbox{if } r \\leq r_{\\rm IMLR}, \\\\[18pt]\n\\enskip \\dfrac{f}{2}q^{2}\\Omega^{2}r^{2}\\bigg(\\dfrac{a}{\\Delta}\\bigg)^4 \\exp\\left[-\\left(\\dfrac{r-r_{\\rm OMLR}}{w_{\\rm OMLR}}\\right)^2\\right] & \\mbox{if } r\\geq r_{\\rm OMLR}.\n\\raisetag{38pt}\n\\end{cases}\\label{eq:smooth2}\n\\end{align}\nHere, $r_{\\rm IMLR} = 0.63 a$ and $r_{\\rm OMLR} = 1.59 a$ are the radius of the innermost and outermost Lindblad resonances, while $w_{\\rm IMLR} = 370 h$ and $w_{\\rm OMLR} = 75 h$ are the widths of the Gaussian smoothing. These values were used by \\citet{tazzari15} to reproduce the gap sizes of a $q = 0.11$ binary from \\citet{artymowicz94}.\n\nFollowing \\citet{frank02}, considering the mass continuity and angular momentum conservation equations in the discs, adding the viscous and tidal torques (eq. \\ref{eq:smooth}), the surface density evolution can be written as:\n\\begin{equation}\n\\frac{\\partial \\Sigma}{\\partial t} = - \\frac{1}{r}\\frac{\\partial}{\\partial r}\\left( - 3 r^{1 \/ 2}\\frac{\\partial}{\\partial r}\\left(\\nu \\Sigma r^{1 \/ 2}\\right) + 2 \\frac{\\Sigma \\Lambda}{\\Omega}\\right), \\label{eq:sigma}\n\\end{equation}\nwhere the first term at the right hand side is the effect of viscosity and the second one is produced by the tidal interaction between the binary and the discs. \n\n\\subsection{Binary separation evolution}\n\nThe separation between the SMBHs will evolve over time by two different mechanisms. The first one, mostly relevant at large binary separation, is the back-reaction between the binary and the disc due to the tidal interaction explained above. The second effect, which dominates at smaller separations, is produced by the GW emission. The following equation for the evolution of the binary separation $a$ incorporates both effects:\n\\begin{equation} \n\\frac{da}{dt} = - \\frac{4 \\pi}{a M_s \\Omega_s} \\int_{r_{\\rm in}}^{r_{\\rm out}} \\Lambda \\Sigma r dr - \\frac{8}{5} q (1 + q) c \\left(\\frac{{\\rm R_S}}{a}\\right)^3.\n\\label{eq:dadt}\n\\end{equation}\nHere the first term of the right-hand side is the binary--disc interaction integrated over the whole disc, and the second term is the GW energy loss \\citep{peters64}; $M_s = q M_p$ is the secondary's mass, $\\Omega_s$ is its angular velocity, ${\\rm R_S} = 2 G M_p \/ c^2$ is the Schwarzchild radius of the primary BH, $G$ the gravitational constant, and $c$ the speed of light.\n\nThe GW emission always tends to bring the SMBHs together. On the other hand, the presence of material inside the binary orbit will push the secondary outwards, while the material in the circumbinary disc will push it inwards. Therefore, the time until coalescence will be influenced by how material is distributed in the discs and how much of it can cross the cavity.\n\n\\subsection{Initial and boundary conditions} \\label{sec:boundary}\n\nAfter obtaining the equations for the surface density and binary separation evolution, we need a few more elements in our implementation to fully simulate the system. First, we need to specify the initial and spatial boundary conditions of the discs.\n\nThe initial surface density profile of the two discs was defined using a standard Shakura--Sunyaev accretion disc \\citep{kocsis12a} with an accretion rate $\\dot{M}_{\\rm ini} = 4 \\dot{M}_{\\rm Edd}$ ($\\dot{M}_{\\rm Edd} = 4 \\pi G M_p \/ (\\kappa c \\eta)$ is the Eddington accretion limit, with $\\eta = 0.1$ the radiative efficiency). To include the effect of the secondary BH, we added an artificial cavity removing all the material within a factor of two of its initial position ($0.5 a_0 < r < 2 a_0$, with $a_0 = 10^5 ~{\\rm R_S}$). We define a local disc mass as\n\\begin{equation}\nM_{\\rm loc} = \\int_{a_0 - R_h}^{a_0 + R_h} 2 \\pi r \\Sigma_{\\rm unp} dr, \\label{eq:ldm}\n\\end{equation}\nwhere $\\Sigma_{\\rm unp}$ is the expected surface density of the disc without the secondary. The ratio between the local disc mass and the secondary BH mass $M_s$ determines the type of migration and its timescale \\citep[e.g.][]{haiman09}. For our fiducial case with $\\alpha = 0.1$, $M_p = 10^7 ~M_\\odot$, $q = 0.1$, and $\\dot{M}_{\\rm ini} = 1 ~{\\rm M_\\odot yr^{-1}}$, we obtain $M_{\\rm loc} \/ M_s \\approx 8$, so we would expect to be in the disc-dominated type II migration regime (but this is not necessarily the case, see discussion below).\n\nIn the outer radius of the circumbinary disc we implemented a constant external accretion rate $\\dot{M}_{\\rm ext}$ which feeds the system during the simulation. For consistency, we use the same accretion rate for the initial condition, $\\dot{M}_{\\rm ext} = \\dot{M}_{\\rm ini}$.\n\nFor the inner radius of the individual disc around the primary BH we used a zero torque boundary condition, which allows material to be freely accreted onto the SMBH.\n\nThe outer boundary of the inner disc and the inner boundary of the circumbinary disc are naturally added in the model by the tidal effect implemented with eq. \\ref{eq:smooth2}. This tidal torque will open a cavity around the secondary black hole, pushing the material in the discs away from its orbit.\n\nFor simplicity, we did not evolve the black hole masses in our model. However, we checked a posteriori if this assumption was justified. To do this we assumed (i) the initial disc inside the secondary orbit is accreted by the primary, and, (ii) all the material that crosses the cavity is accreted by one of the BH. At the end of the simulations we obtained, on average, a variation $\\Delta q \/ q \\approx 25 \\%$ and $\\Delta M_p \/ M_p \\approx 34 \\%$.\n\nMany previous works have shown that the gap opened by the secondary is not an impenetrable wall: gas can cross the binary orbit and enter to the inner disc in the form of dense streams of material. This inflow is crucial because it feeds the inner disc, contributing to the hot part in the SED, determining how much gas will be left just before the coalescence and, indirectly, controlling the migration timescale. Since this process is (at least) a 2D effect, we cannot directly compute it, and we need to use a recipe to include it in our code. Some possibilities could be using a constant accretion \\citep{kocsis12a, kocsis12b}, a fraction of the external accretion \\citep{rafikov16}, or a fraction of the steady-state accretion from a standard single BH Shakura--Sunyaev disc. Using any of these alternative implies the inflow is tied directly to the viscous evolution of the circumbinary disc. However, the flow across the gap or cavity is produced by the tidal (gravitational) torques, and so it should operate on the orbital time of the binary, and not on the viscous time of the gas near the gap\/cavity wall. Recent hydrodynamical simulations confirm this and suggest that this accretion process should be mostly dependent on the dynamical properties of the binary rather than on the local physics of the disc \\citep{farris15, dorazio16}. Motivated by this, in our work we consider a different recipe, which allows us to control the rate at which material crosses the cavity, but ties this rate to the local orbital time:\n\\begin{equation} \n\\dot{M}_{\\rm cross} = \\gamma \\Omega_s r^2 \\Sigma.\n\\label{eq:mdotc}\n\\end{equation}\nHere $\\gamma$ is a free parameter that determines the fraction of the nearby material that crosses the secondary radius per binary orbit. As an example, $\\gamma = 10^{- 4}$ implies that, ignoring viscosity effects, it would take $\\sim 10^4$ orbits to totally deplete the nearby circumbinary disc. To prevent large fluctuations in the accretion rate, eq. (\\ref{eq:mdotc}) is implemented using the average values of the radius ($\\bar{r}$) and the surface density $\\left(\\bar{\\Sigma}\\right)$ in the region of the circumbinary disc where the tidal torque dominates. Based on the 3D simulations of \\citet{cuadra09} the range used was $a < r < 3 a$. From these radii we extract, once per binary orbit, a gas mass given by the product between the accretion rate and the orbital time: $\\dot{M}_{\\rm cross} \\Delta t_{\\rm orb}$. In general, some of this mass will cross the gap (e.g. on horse-shoe orbits for $q\\ll 1$), reach the outer edge of the inner disc, and eventually accrete onto the primary BH. The remaining mass is captured and accreted onto the secondary (in our model, since we do not include the circum-secondary disc, we remove this material from the simulation). We adopt the ratio between these accretion rates from \\citet{farris14}, who computed them explicitly in 2D hydrodynamical simulations over a range of $q$. Specifically, based on their Figure 7, for our simulations with $q = (0.1,~0.3,~0.01)$, we assume that $(6 \\%,~40 \\%,~85 \\%)$ of the mass that leaves the circumbinary disc reach the inner disc. For simplicity, we add this mass at a fixed radius, i.e. as a delta function at $r\/a = (0.4,~0.2,~0.6)$, based on the tidal truncation radii obtained by \\citet{artymowicz94}. Since we do not modify the masses of the BHs, these values are fixed during the whole simulation. \n\n\\added{When we add the gap-crossing material to the inner disc, we endow it with the local Keplerian velocity at the new location.\nAs a result\nour simple inflow implementation does not conserve angular momentum (even if we were to add all the mass to the inner disc).\nWithout following the non-axisymmetric gas flows in 2D or 3D, we do not know where this numerically lost angular momentum would end up. So in order to estimate if this could affect our results, we consider two extreme scenarios: the lost angular momentum could have been (i) added to the secondary BH, or (ii) deposited back in the circumbinary disc. In either case, we found that for $\\gamma \\leq 10^{-5}$ the effects are sub-dominant compared to either (i) the binary's orbital evolution in the current simulation, or (ii) the viscous angular momentum flux at the inner edge of the circumbinary disc.}\n\n\\subsection{Model parameters and numerical implementation}\n\n\\begin{table*}\n \\caption{Simulation properties. From left to right the columns are: a reference simulation number, the logarithm of the primary mass in solar masses, the binary mass ratio ($q$), our parameter to control the inflow through the gap ($\\gamma$), the Shakura--Sunyaev viscosity parameter ($\\alpha$), the initial and external accretion rate in units of the Eddington limit, the grid resolution, the time resolution, and the differences between the current simulation and our fiducial case (simulation $N^{\\rm o} 1$). See text for a more detailed explanation.\n\n }\n\t\\label{tab:params}\n \\begin{tabular*}{0.96\\textwidth}{lcccccccc} \n\t\t\\hline\n \\\\[-2ex]\n\t\t$N^{\\rm o}$ & log$\\left(\\dfrac{M_p}{M_\\odot}\\right)$ & $q$ & $\\gamma$ & $\\alpha$ & $\\dfrac{\\dot{M}}{\\dot{M}_{\\rm Edd}}$ & $n$ & $t^*$ & Differences\\\\\n \\\\[-2ex]\n\t\t\\hline\n \\\\[-2ex]\n $~~1$ & $7$ & $0.1$ & $10^{- 4}$ & $0.1$ & $4$ & $500$ & $10^6$ & - \\\\ \n \\hline\n \\\\[-2ex]\n $~~2 - 7$ & - & - & $10^{- 5},~10^{- 6},~10^{- 7},~10^{- 8},~10^{- 9},~0.0$ & - & - & - & - & $\\gamma$ \\\\ \n $~~8 - 21$ & - & $0.3,~0.01$ & $10^{- 4},~10^{- 5},~10^{- 6},~10^{- 7},~10^{- 8},~10^{- 9},~0.0$ & - & - & - & - & $\\gamma$ and $q$ \\\\ \t \n $22 - 23$& - & - & - & $0.01,~1$ & - & - & - & $\\alpha$ \\\\ \n $24 - 25$ & - & - & - & - & $0.4,~40$ & - & - & $\\dot{M}$ \\\\ \n $26 - 28$ & $6,~8,~9$ & - & - & - & - & - & - & $M_p$ \\\\\n $29$ & - & - & - & - & - & -& - & without inner disc \\\\ \n $30$ & - & - & - & - & - & -& - & without both discs \\\\ \n $31$ & - & - & - & - & - & $10^3$ & $10^5$ & increase resolution \\\\\n $32 - 36$ & - & $0.1$, $0.01$ & $10^{- 5},~0.0$ & - & $0.1$ & - & - & \\citeauthor{kocsis12b} parameters \\\\\n $37 - 38$ & - & $0.01$ & $10^{- 5}$, $0.0$ & $0.3$ & $0.1$ & - & - & \\citeauthor{haiman09} parameters \\\\\n $39$ & $8$ & - & $0.0$ & $0.1$ & $1$ & - & - & \\citeauthor{lodato09} parameters \\\\\n $40$ & - & - & - & - & - & - & - & empty discs, no secondary \\\\\n \\hline \n\t\\end{tabular*}\n\\end{table*}\n\nIn Table \\ref{tab:params}, we list the relevant properties of our simulation runs. The first row refers to our fiducial case. The rest of the table shows how each model deviates from this fiducial case. The first 21 simulations are the core of our work, and are the combinations between $q = \\{0.1,~0.3,~0.01\\}$ and $\\gamma = \\{10^{- 4},~10^{- 5},~10^{- 6},~10^{- 7},~10^{- 8},~10^{- 9},~0.0\\}$. In the next set of simulations, we changed the $\\alpha$ parameter ($22-23$), the external accretion rate ($24-25$) and the primary BH mass ($26-28$). We also started the simulation without either the inner or both discs ($29-30$) to explore the effects of the inflow from the outer boundary and of the initial conditions. We run a high-resolution simulation ($31$) to test the numerical convergence of our results. We have run simulations for the comparisons made in \\S \\ref{sec:pw} with previous work, with parameters following those in \\citet{kocsis12b} ($32-36$), \\citet{haiman09} ($37-38$) and \\citet{lodato09} ($39$). Finally, as a basic test of our code, we ran an additional simulation without a secondary BH, with a constant $\\dot{M}_{\\rm ext}$, and initially empty discs ($40$). We recovered satisfactorily the Shakura--Sunyaev standard solution for the disc for the given accretion rate.\n\nTo convert into code units we define the following quantities: $R_0 = 10^2 ~{\\rm R_S}$, $T_0 = 10^3 ~{\\rm K}$, $\\Sigma_0 = 10^7 ~{\\rm g \/ cm^2}$, $t_0 = t^* 2 \\pi \/ \\Omega(R_0)$ with $t^* = 10^{6}$ a scale factor for the time coordinate, and $m_0 = \\Sigma_0 R_0^2 \\sim 4 \\times 10^2 M_\\odot$ for a primary BH mass of $10^7 M_\\odot$. Then, we divide every physical constant (or variable) used in the simulation with a combination of the previous definitions to obtain dimensionless quantities. Using the dimensionless versions of the equations for the processes shown in the previous section, we compute the state of the discs at each snapshot. To solve the partial differential equation of the surface density evolution we used the backward Euler method, an implicit numerical integration procedure \\citep{hein06}. This method requires that we solve a tridiagonal matrix, which we did using the Thomas algorithm (a special case of Gaussian elimination). We used a front-end code in \\textit{python} to work with the arrays, to set the properties of the simulations, and for visualization. For the heavy calculations, such as the matrix definition and solution, we developed a library in \\textit{C}. We connected the two types of codes using \\textit{ctypes}.\n\nThe surface density of the discs at each snapshot is the key variable in the models: with it we can compute the central temperature $T_c$, the pressure ratio $\\beta$ and the sound speed $c_s$ (Eq. \\ref{eq:solution}). We then obtain the thickness $h$ of the discs and the viscous and tidal interaction (Eq. \\ref{eq:sigma}), which in turn sets how fast the separation between the SMBHs shrinks (Eq. \\ref{eq:dadt}). Once we obtain the updated values of these variables, we evolve the system by one time step, obtaining a new surface density distribution.\n\nWe used a linearly spaced grid for the temporal coordinate and a logarithmic grid for the radial coordinate. We defined a numerical resolution $n$, which translate into a radial step-size $\\Delta r \/ R_0 = r \/ R_0 (1 - 10^{1 \/ n})$ and a temporal step-size $\\Delta t \/ t_0 = 1 \/ n^2$. We have run simulations with different values of $n$, and chose $n = 500$, which we have found to provide a good compromise between precision, convergence and computing time for our study.\n\nSince we cannot directly demonstrate that the logarithmic grid fulfills the conditions to make the truncation error go to zero in the Taylor series \\citep{hein06}, for our fiducial case we also ran a simulation with a linear radial grid, and found no significant differences.\n\nBy definition, throughout the cavity, the surface density should go to zero. This is not always the case in the simulations, due to truncation errors that accumulate over time. To address this numerical noise, we added two control conditions. First, if $\\Sigma \/ \\Sigma_0 \\leq \\Sigma_{\\rm err} = 10^{- 20}$ in some radial cell, we set the surface density of that cell to zero. We expect that, at a given snapshot, when a disc \"ends\" (i.e, the surface density drops to zero at some point), it does not increase again on the same side of the secondary BH orbit. Our second condition is therefore to find the first cell where $\\Sigma \/ \\Sigma_0 = 0$ in each disc, and, for every cell from that point up to the binary separation, we set their surface density to zero. We tested these two control conditions to see if they change the behaviour of the discs and found that, aside from the elimination of some artifacts in the cavity itself, the final results are almost the same with and without them.\n\n\\section{Simulation results} \\label{sec:res}\n\nIn this section, we present the main results of our work, including the surface density evolution, the migration timescale, the residual mass left in the inner disc, and the electromagnetic emission of the system at different times.\n\n\\subsection{Evolution of the disc surface density ($q = 0.1$)}\n\n\\begin{figure*} \n\\centerline{\\includegraphics[width = \\textwidth]{.\/figures\/plot1.pdf}}\n\\caption{Results for a SMBHB of mass ratio $q = 0.1$ with $\\gamma= 10^{- 4}$ (left, at: $13$ Myr, $4.7$ Myr, $830$ Kyr, $440$ yr and $0.7$ yr before merger) and $\\gamma = 10^{- 5}$ (right, at: $9.6$ Myr, $2.5$ Myr, $340$ Kyr, $440$ yr and $0.7$ yr before merger). The black dashed vertical lines mark the position of the binary at a given separation and the gray regions represent the analytical size of the cavity ($0.5 a$ to $2 a$). The top panel shows the surface density $\\Sigma$, the middle panel the midplane temperature $T_c$, and the bottom panel the surface brightness $2 \\pi r^2 F$. While the $\\gamma = 10^{- 4}$ case approaches a steady-state solution without any pile-up, in the case with $\\gamma = 10^{- 5}$ the inner disc becomes depleted and then refills, while the circumbinary disc suffers a modest pile-up.}\n\\label{f:data}\n\\end{figure*}\n\nIn this subsection, we show the main results for the runs with the fixed fiducial mass ratio $q = 0.1$, but for different inflow rates controlled by the parameter $\\gamma$. Figure \\ref{f:data} shows the radial profiles of different relevant quantities for some of these models. The left panels correspond to the case with $\\gamma = 10^{- 4}$, while the right panels correspond to $\\gamma = 10^{- 5}$. Lines of different colours show the state of the system at different times, labeled by the corresponding binary separation.\n\nThe top row of Figure \\ref{f:data} shows the evolution of the surface density $\\Sigma$. The initial condition can be recognized by the sharp cutoff at the cavity edges and the high surface density in the inner disc (set by a constant accretion rate, \\S~\\ref{sec:boundary}). Once the binary separation has shrank to $a = 10^4 \\ {\\rm R_S}$, the slope of the surface density in the inner boundary of the circumbinary disc has decreased, since the migration timescale of the binary is shorter than the inflow timescale of the material ($t_{\\rm flow} = r \/ |v_r|$) at $r \\geq 4 \\times 10^4 \\ {\\rm R_S}$, so it cannot refill the inner region of the circumbinary disc nor accumulate there.\n\nAt $a = 10^3 \\ {\\rm R_S}$, the decrease in the migration velocity allows the outer disc to recover its initial shape. This slow down in the shrinkage velocity is likely produced by the change in the available mass in the interacting region given by the Lindblad resonances (proportional to the binary separation $a$; \\citealt{tazzari15}). On the other hand, the inner disc has not changed much, except that its outer boundary is depleted by the effect of the tidal torque.\n\nAround $a = 10^2 \\ {\\rm R_S}$ we expect the so-called ``first decoupling'' to happen: at this point the inflow timescale of the circumbinary disc becomes longer than the binary migration timescale due to GW emission. The material of the outer disc will be no longer pushed away by the tidal torque and will evolve only due to viscosity. In this simulation, the inner disc still maintains the same surface density than the previous snapshots, showing that, for this $\\gamma$, we find a configuration approaching a steady state.\n\nDuring the GW emission phase the secondary will migrate inward faster than the viscous time of the inner disc, pushing the gas inward and squeezing it. The final curve shows the moment when the binary separation is $a = 20 \\ {\\rm R_S}$, where a so-called ``second decoupling'' should occur due to the thickening of the disc, which invalidates the thin-disc assumptions \\citep{fontecilla17}. The figures shows that the gradient of the surface density in the inner region of the circumbinary disc becomes shallower, as it is not affected by the tidal barrier. The inner disc, on the other hand, becomes coupled with the secondary black hole at some point between the snapshots, and is squeezed, increasing its surface density by more than a factor of two.\n\nIn our simulations with this value of $\\gamma$, we find that the slope of the circumbinary disc never becomes steeper than the single-BH case \\citep[a process commonly known as pile-up, e.g.,][]{kocsis12a}. We attribute this to the fact that this $\\gamma$ is large enough to constantly make the removed mass a considerable fraction of the available mass in the interaction region at the moment of extraction, preventing a pile-up. \n\nThe right panel in the same row shows a simulation with the same mass ratio, but with $\\gamma = 10^{- 5}$. The reduction in the rate at which material crosses the cavity produces a significant change in the discs: the inner disc becomes slightly depleted at some point after $a = 10^4 \\ {\\rm R_S}$, and the circumbinary disc can pile up gas, increasing its surface density and speeding up the migration. The inner disc refills again before the binary reaches the separation of $a = 10^2 \\ {\\rm R_S}$, showing that the depletion and pile-up do not produce a large difference in the surface density distribution of the inner disc at the end of the simulation, the residual mass (see Fig. \\ref{f:myt} below) or the hot part of the SED.\n\nThe differences between $\\gamma = 10^{- 4}$ and $\\gamma = 10^{- 5}$ can be understood by defining an inflow timescale across the gap, $t_{\\rm cross} \\equiv 2 \\pi r^2 \\Sigma \/ \\gamma \\Omega_s \\bar{r}^2 \\bar{\\Sigma}$. For $\\gamma = 10^{- 4}$, we find that, in the interaction region in the circumbinary disc, $t_{\\rm cross}$ is smaller than the background inflow timescale $t_{\\rm flow}$ of the material in the circumbinary disk, and no pile-up can happen, whereas for $\\gamma = 10^{- 5}$, $t_{\\rm cross}$ becomes slightly longer than $t_{\\rm flow}$, suggesting that a pile-up must occur.\n\nThe exact value of $\\gamma$ for which our inflow model results in steady-state accretion can be estimated by setting $t_{\\rm cross} = t_{\\rm vis} = 2 r^2 \/ (3 \\nu)$. Considering, for simplicity, an $\\alpha$ prescription for the viscosity, \n\\begin{equation}\n\\gamma = 3 \\pi \\alpha \\left(\\frac{h}{r}\\right)^2 \\frac{\\Omega}{\\Omega_s} \\label{eq:gammass},\n\\end{equation}\nfor $\\alpha = 0.1$, $h \/ r = 10^{- 2}$ and $r = 2 a$, we obtain $\\gamma \\sim 3 \\times 10^{- 5}$. We obtain a similar value when we calculate this $\\gamma$ directly from our fiducial simulation. At $r\\sim 2a$, $\\beta \\ll 1$ and eq. \\ref{eq:gammass} explains why the surface density profile presents no pile-up and looks similar to a steady-state solution.\n\n\\subsection{Evolution of the disc temperature and spectrum ($q = 0.1$)}\n\nThe temperature and surface brightness for the same binary separations are shown in the second and third row of Fig.~\\ref{f:data}. Their behaviour is similar to the one of the surface density, as expected.\n\\added{We note that in the outer parts of the circumbinary disc, the temperature falls below $10^4$ K, where our simple opacity implementation due to electron scattering will not be sufficient to properly model the hydrogen ionization instability that should occur.}\n\n\\begin{figure}\n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot2.pdf}\n\\caption{Spectral energy distributions (SED) from the inner and outer discs, obtained assuming they are optically thick and emit as a multi-temperature black body. The top panel shows the results for $q = 0.1$ and $\\gamma = 10^{- 4}$ while the bottom panel shows $\\gamma = 10^{- 5}$. The thick curves correspond to combined total SED, with the black curve the case of a single-BH with $\\dot{M} = \\dot{M}_{\\rm ext}$ for reference. The thin dashed curves indicate the SED from the circumbinary disc alone. Depending on the rate at which material crosses the secondary's orbit, the peak of the SED from each disc can dominate at different stages of the binary evolution.}\n\\label{f:sed}\n\\end{figure}\n\nUsing the temperature profiles, we compute the electromagnetic (EM) emission for the two cases in Fig.~\\ref{f:data}. The top panel of Fig.~\\ref{f:sed} shows the SED for $\\gamma = 10^{- 4}$ at the same binary separations as in Fig.~\\ref{f:data}. For comparison, a single-BH SED is shown in black. The formation of a cavity due to the presence of the binary produces a conspicous feature -- a down-ward depression, or ``notch'', in the spectrum \\citep[e.g.,][]{gultekin12}. The (hotter) inner disc dominates the SED to the right of the notch (i.e higher frequencies), while the (colder) circumbinary disc dominates to the left. As the system evolves, the outer disc can reach closer to the primary BH, heating up and showing an enhancement at shorter wavelengths. The notch in the SED caused by the cavity also moves to the right while the inner disc almost preserves its shape, until $a = 10^2 \\ {\\rm R_S}$, where the squeezing starts and the EM emission peaks at higher frequencies \\citep{farris15, tang18}.\n\nThe case with $\\gamma = 10^{- 5}$ is shown in the bottom panel of the same figure. The behaviour for this simulation is similar, but the pile-up of the material at the inner boundary of the circumbinary disc produces an enhanced contribution \\citep{lodato09, kocsis12b}. For this $\\gamma$, the dominance of the inner disc is not permanent: the circumbinary disc surpasses its luminosity after $a = 10^3 \\ {\\rm R_S}$. However, the inner disc dominates again after the first decoupling.\n\nThe frequency at which the notch appears in the SED is also different in the two cases, because removing material from the circumbinary discs also changes the effective position of the cavity. In the case with smaller $\\gamma$, this allows more material to be hotter and closer to the secondary BH.\n\n\\begin{figure}\n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot3.pdf}\n\\caption{Bolometric luminosity ($L_{\\rm bol}$) for the simulations with $q = 0.1$ and each value of $\\gamma$ as a function of time (running from right to left). The vertical dashed lines mark different binary separations for the case with $\\gamma = 10^{- 4}$, and the horizontal dashed line shows the Eddington luminosity for this system. For the cases when material crosses the cavity rapidly ($\\gamma \\geq 10^{- 5}$), there is relatively more gas near the primary close to the merger, and the peak at the second decoupling ($\\sim$1 year prior to merger) is higher than at the first decoupling ($\\sim$100 days prior to merger). On the other hand, when the secondary ``dam'' is more efficient ($\\gamma \\leq 10^{- 6}$), the circumbinary disc pile-up, peaking at the time of the first decoupling, produces the brightest luminosity during the system's evolution.}\n\\label{f:bol}\n\\end{figure}\n\nSumming over all frequencies, we obtain the bolometric luminosity $L_{\\rm bol}$. The results for each of the $q = 0.1$ simulations are shown in Fig.~\\ref{f:bol}, as a function of time before coalescence. The vertical dashed line indicates the binary separation for the case with $\\gamma = 10^{- 4}$, while the horizontal line marks the Eddington luminosity for this system.\n\nRoughly half of the time in the binary evolution is spent going from the initial separation at $a = 10^5 \\ {\\rm R_S}$ to $a = 10^4 \\ {\\rm R_S}$. At the end of the simulation, all systems become brighter compared to their luminosity at $a \\sim 10^4 \\ {\\rm R_S}$, due to the squeezing of the inner disc. Since, as pointed out by \\citet{lodato09}, we include the tidal heating in both discs, the inner region of the circumbinary disc can dominate the bolometric emission showing a peak around the time the first decoupling happens.\n\nThere are some important, visible differences between the models with different $\\gamma$: the simulation with $\\gamma = 10^{- 4}$ shows a more constant luminosity, since the material crosses the cavity sufficiently rapidly to prevent a pile-up in the circumbinary disc. For $\\gamma = 10^{- 5}$, a pile-up is formed, but material still crosses the cavity rapidly enough to make the emission at the squeezing end-phase ($\\sim$1 year prior to merger) brighter than the one at the first decoupling. Finally, for $\\gamma \\leq 10^6$ this does not happen, and the system is brightest during the first decoupling, when the pile-up is most pronounced (seen at $\\sim$100 days prior to merger in Fig.~\\ref{f:bol}).\n\nThe enhancement in the luminosity after its minimum is due to the refiling of the inner disc and\/or the pile up of the circumbinary disc. After the first decoupling the luminosity decreases again, because the outer disc is no longer influenced by the binary's tidal torques and heating. Reaching the end of the simulation, the secondary BH squeezes the inner disc, increasing the overall luminosity of the system, but, for $\\gamma \\leq 10^6$ there is a relatively small amount of gas left near the primary to be heated, and the luminosity peaks around the first decoupling.\n\n\\subsection{Binary inspiral timescales}\n\nWe next compare the main results among the first 21 simulations, to show how the variation in $q$ and $\\gamma$ affects the time the binary spends at each radius and how much material is left in the inner disc when the first decoupling occurs.\n\nFig. \\ref{f:rest} shows the residence time $t_{\\rm r} \\equiv a \/ (da \/ dt)$, as a function of the binary separation $a$. The thin dashed black line, which every simulation approaches at the end, displays the residence time assuming only GWs were present, while the thin dashed gray line shows an analytical estimate of the viscous time in the inner regions of the circumbinary disc, at the same binary separations. The thick vertical dashed line, on the other hand, indicates the position where the local mass of the disc, defined by eq. \\ref{eq:ldm}, becomes smaller than the secondary BH mass.\n\nAt the beginning of the simulations, the effect of the sharp initial condition is clear: the system needs time to create the right cavity shape before it starts to evolve more smoothly. Following this adjustment, we can distinguish three migration regimes: At the right of the vertical dashed line, the binary should be following a disc-dominated type II migration. When the local disc and secondary masses become similar, the system should change to a slower secondary-dominated type II migration \\citep{haiman09}. This is especially visible in the bottom panel, as a flattening of the slope at this transition. Around the same binary separation, the viscous timescale of the circumbinary disc becomes smaller than the residence time, allowing the material to pile up over time. After the binary separation becomes smaller than $a \\sim 500 \\ {\\rm R_S}$, we see another clear steepening in the slope of each curve, showing the beginning of the GW emission regime.\n\nThe top panel shows the simulations with $q = 0.1$ for every value of $\\gamma$. Except for the case with $\\gamma = 10^{- 4}$, all the other cases have a similar behaviour. As we argued above, values higher than $\\gamma = 10^{- 5}$ have accretion timescales shorter that the inflow timescale. \n\n\\begin{table}\n\\centering\n\t\\caption{Disc properties related with the type of migration. From left to right, the initial disc mass of the inner disc, the initial local mass around the secondary and the binary separation when the local mass becomes equal to the secondary mass}\n\t\\label{tab:dprop}\n \\begin{tabular*}{0.7\\columnwidth}{lccc} \n $q$ & $M_{\\rm int} ~(M_s) $ & $M_{\\rm loc} ~(M_s)$ & $a ~(\\rm{R_S})$ \\\\ \n \\\\[-2ex]\n\t\t\\hline\n \\\\[-2ex]\n $0.1$ & $~~3.38$ & $~~8.09$ & $2.25 \\times 10^4$\\\\\n $0.3$ & $~~1.13$ & $~~3.87$ & $3.81 \\times 10^4$\\\\\n $0.01$ & $33.82$ & $37.30$ & $7.40 \\times 10^3$\\\\\n \\hline \n\t\\end{tabular*}\n\\end{table}\n\nThe bigger deviation happens during the secondary-driven evolution, where the contribution of the circumbinary disc becomes less and less important due to the shrinkage of the interaction zone.\n\nWhen the binary has more similar masses ($q = 0.3$, middle panel) all simulations follow the same trend: more time is needed at the beginning to reach a more physical surface density distribution from the initial condition, while at the end the GW emission becomes dominant faster. We acknowledge that this mass ratio is at the limit of our capabilities with a 1D code, since considering the primary BH to be fixed at the origin, or using the tidal torque implementation, are no longer good approximations. \n\\begin{figure}\n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot4.pdf}\n\\caption{Residence time ($t_r \\equiv a \/ (da \/ dt)$) spent by the binary at each distance $a$ from the primary, shown for each of the $\\gamma$ values used in this work. Top panel: $q = 0.1$, middle panel: $q = 0.3$ and bottom panel: $q = 0.01$. The behaviour at $a \\sim 10^5 \\ {\\rm R_S}$ is produced by the redistribution of the material in the boundaries of the cavity from the sharp edges of the initial condition. At the end, when $a \\sim 5 \\times 10^2 \\ {\\rm R_S}$, GW emission starts to dominate the torques from the gas disc (the diagonal line shows a pure GW residence time). The light diagonal line is the viscous timescale of the circumbinary disc while the vertical thick dashed line marks the separation where the local disc mass becomes equal to the secondary's mass.}\n\\label{f:rest}\n\\end{figure}\n\nFinally, the bottom panel shows the case with a smaller secondary BH ($q = 0.01$). For this mass ratio, we do not show results for $\\gamma = 10^{- 4}$, because in that case the material that is partially added to the inner disc (and removed from the circumbinary disc) changes the direction of the binary migration, increasing their separation instead of shrinking it (see Eq. \\ref{eq:dadt}). We suspect this result is an artefact of our initial condition and inflow implementation: the sharp initial gradients cause an inflow much more rapid than expected from the viscous timescale. We dismiss this behaviour as numerical and not physically relevant.\n\nTable \\ref{tab:dprop} summarizes the initial disc mass in the form of the mass in the inner disc ($M_{\\rm int}$) and as the local mass $(M_{\\rm loc})$), along with the separation when the latter becomes equal to the secondary mass. For each mass ratio, we compared the slope of the residence time obtained here to the analytical estimations from \\citet{haiman09}. We found that, except for $q = 0.01$, our slopes are flatter than the one expected for disc-dominated migration. The results suggest that the condition for this type of migration is that the disc mass needs be higher than the secondary mass by at least an order of magnitude.\n\nWe define a ``residual mass\" as the mass in the inner disc at the time when the binary separation reaches $ a = 100 {\\rm R_S}$, which is when we expect the first decoupling and the inflow through the cavity stops (or at least slows down). This is similar to the ``fossil disc mass'' defined by \\citet{chang10} and \\citet{tazzari15}. \n\nThe top panel of Fig. \\ref{f:myt} presents the residual masses. For each $q$, we find that for smaller $\\gamma$, less material is left in the inner disc, as naively expected. However, the difference between the cases is further affected by the position of the gas added in the inner disc \\citep{artymowicz94} and how much gas is deposited into it \\citep{farris14} Also, removing material from the circumbinary disc changes the surface density distribution itself, affecting the available gas in the vicinity of the secondary. This in turn can change the migration velocity and the inflow rate across the cavity.\nOverall, we find that the dependence of the residual mass on $\\gamma$ is much weaker than linear. Furthermore, when we shut down the inflow through the cavity entirely ($\\gamma = 0$), a fraction of the initial mass of the inner disc still survives, as already found in models that did not allow the gas to flow across the gap \\citep{armitage02, chang10, tazzari15}. \n\\begin{figure}\n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot5.pdf}\n\\caption[Residual mass and time to first decoupling for the first 21 simulations]{Properties of the first 21 simulations. Top panel: Residual mass in the inner disc around the first decoupling, when the binary separation is $a = 10^2 \\ {\\rm R_S}$. Bottom panel: Time elapsed between a starting binary separation of $a = 10^4 \\ {\\rm R_S}$ until it reaches $a = 10^2 \\ {\\rm R_S}$. Both panels are as a function of the mass ratio $q$ and the different values of $\\gamma$. While the residual mass changes considerably between simulations, the migration times are similar for each case. Given a fixed mass ratio, as we reduce $\\gamma$, each property tends to converge to a value for the case when there is no inflow.}\n\\label{f:myt}\n\\end{figure}\n\nThe bottom panel of the same figure shows the time it takes for each system to go from $a = 10^4 \\ {\\rm R_S}$ to $a = 10^2 \\ {\\rm R_S}$. We selected $a = 10^4 \\ {\\rm R_S}$ as a starting separation for this analysis because at this point the shape of the initial condition should long ago have disappeared, making it easier to compare with previous works. The total time that the system needs to reach the final separation does not change much either with the mass ratio $q$ or with the mass influx rate $\\gamma$ -- for all our simulations it is close to $3 \\times 10^6$ yr, which happens to be very similar to the values found in \\citet{tang17}, and also of the same order as the viscous timescale. It is reassuring that our simple torque implementation does not give merger timescales that are wildly different from those found in hydrodynamical simulations, despite the different nature of the torques.\n\nFor a given mass ratio, the similarity between the cases with $\\gamma < 10^{- 5}$ can be understood as the result of a self--regulation: if material crosses the gap more rapidly, the surface density in the inner disc increases, which adds angular momentum to the binary, and slows down the migration. On the other hand, a higher surface density implies a higher viscosity, which increases the accretion onto the primary BH and reduces the total mass in the inner disc. Finally, in the circumbinary disc, removing material more rapidly produces an enhanced density gradient, which causes the region to re-fill more rapidly, and increases its tidal effects.\n\nFor a given $\\gamma$, increasing $q$ increases the time needed to reach $a = 10^2 \\ {\\rm R_S}$. This could be produced by a series of factors: the role of the tidal torque, which becomes stronger for higher mass ratio \\citep[but the smoothing becomes relevant, see][]{tazzari15}, the mass of the secondary in the back reaction term in Eq. \\ref{eq:dadt}, or the position of the circumbinary disc inner edge. After considering the first two factors, by examining the tidal term in Eq.~\\ref{eq:dadt}, we see that the velocity of the secondary increases as $q$ increases, which is the opposite of the trend we found in our simulations. We conclude that the reason for this slow-down comes from the position of the inner boundary of the circumbinary disc: while for $q = 0.01$, $r_{\\rm edge} = 1.2 a$, for $q = 0.1$, $r_{\\rm edge} = 1.7 a$, and for $q = 0.3$, $r_{\\rm edge} = 2.5 a$. Adopting for simplicity the surface density profile for a steady state disc with $\\Sigma \\propto r^{-3 \/ 5}$ \\citep{kocsis12a}, and using Eq.~\\ref{eq:dadt} without the GW contribution, we find that the ratio between the migration velocities are $\\dot{a}_{0.01} \/ \\dot{a}_{0.1} = 5.32$ and $\\dot{a}_{0.1} \/ \\dot{a}_{0.3} = 3.84$, which could explain the slower gas-driven inspiral we found for larger $q$.\n\nAs explained above, $\\gamma = 10^{- 4}$ produces an inflow rate similar to a steady-state accretion without the secondary when $q = 0.1$. In this case, the material added to the inner disc doubles the time the binary needs to coalesce, compared with the lower $\\gamma$ cases. For $q = 0.3$ this $\\gamma$ makes a smaller difference, while for $q = 0.01$ it prevents the binary to merge at all, by driving it outwards.\n\nWe stop our simulations when the binary separation becomes $a = 20 \\ {\\rm R_S}$ because around this point we expect the second decoupling to occur. The inner disc should subsequently become thick, with the secondary ploughing through it. For a more detailed discussion of this last stage of binary evolution, see \\citet{fontecilla17}.\n\n\\subsection{Is the system in steady state?} \\label{sec:steady}\n\n\\begin{figure}\n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot6.pdf}\n\\caption[Accretion rate and scale height of the discs when the binary separation is $a = 20 \\ {\\rm R_S}$, as a function of the radius]{Accretion rate (top) and scale height (bottom) of the discs as a function of radius at the binary separation of $a = 20 \\ {\\rm R_S}$. Results are shown for $\\gamma = 10^{-5}$ and for three different mass ratios $q$ as labeled. Dashed curves show the three preceding snapshots at $10^2, 10^3$, and $10^4 R_S$ of the simulation with $q = 0.1$. The radial dependence of the accretion rate is produced by the binary's history of migration, and shows that the disc has not established steady state, even when the binary is close to merger. The scale height in the bottom panel demonstrates that the inner disc becomes thick at this close binary separation, as discussed in \\citet{fontecilla17}.}\n\\label{f:mdot}\n\\end{figure}\n\nWe next address the question of whether the system follows a series of steady state solutions, as often assumed in previous studies \\citep[e.g.][]{liu10, kocsis12b, rafikov16}. In the top panel of Fig. \\ref{f:mdot}, we show the accretion rate through the disc at the last snapshot of our simulations, for every mass ratio $q$ with $\\gamma = 10^{- 5}$. The thick horizontal dashed line indicates the external accretion rate $\\dot{M}_{\\rm ext}= 1 M_\\odot {\\rm yr^{-1}}$. A higher or lower value indicates that a pile-up or a depletion is occurring in that annulus of the disc. Because this snapshot is right before the merger, if the steady state assumption were fully correct, the system should have had enough time to smooth out those variations and establish a constant $\\dot{M}$ independent of radius. The coloured dashed curves show the three preceding snapshots at $10^2, 10^3$, and $10^4~{\\rm R_S}$ of the simulation with $q = 0.1$. As the figure shows, the system is always far away from a steady-state configuration. At the external boundary we have imposed a constant accretion rate as a boundary condition. Going inward, the accetion rate profile can be understood as follows. For $r \\gtrsim 10^5 \\ {\\rm R_S}$ the enhancement in the accretion rate is mostly produced by the viscous torques while the simulation adjusts the boundaries of the accretion discs after the sharp initial condition. In the range $10^4 \\ {\\rm R_S} \\lesssim r \\lesssim 10^5 \\ {\\rm R_S}$ this accumulation disappears. The migration velocity of the binary was fast enough to make the material unable to pile up at the inner boundary. Also, the viscous time of the gas added as the external boundary condition is longer than the merger time of the system, so this gas does not reach the inner region of the circumbinary disc during our simulation. In the range $10^2 \\ {\\rm R_S} \\lesssim r \\lesssim 10^4 \\ {\\rm R_S}$, a steady increment in the accretion rate is produced by the slow-down of the binary in the cases with $q \\geq 0.1$, while in the case with $q = 0.01$ this barely happens. In this range, the tidal wall formed near the inner edge of the circumbinary disc stays longer at each radius, increasing the surface density and the accretion rate outside the range of the tidal torque.\n\\footnote{To see this, consider a disc at two different times $t_2 > t_1$, where $\\Sigma_{t_2} > \\Sigma_{t_1}$. We assume the discs have a radial dependence of the form $\\Sigma_{t_i} \\propto r^{d_i}$, where $d_2 < d_1 < 0$ and $d_2 = d_1 - |\\Delta d|$ ($\\Delta d \\ll 1$). Since $\\dot{M} \\propto - r \\Sigma v_r$ and $ v_r \\propto - \\Sigma^{- 1} r^{- 1 \/ 2} \\partial (\\nu \\Sigma r^{1 \/ 2}) \/ \\partial r$, if we use the equations outlined in section \\ref{sec:visc}, considering a gas pressure dominated region without tidal dissipation \\citep[i.e. outside $r_\\Lambda$, see][]{rafikov16}, then $\\nu \\propto \\Sigma^{2 \/ 3} r$ and $ \\dot{M} \\propto r^{1 \/ 2} \\partial (\\Sigma^{5 \/ 3} r^{3 \/ 2}) \/ \\partial r$. With these simplifications, the ratio between the accretion rates becomes:\n\\begin{equation}\n\\frac{\\dot{M}_{t_2}}{\\dot{M}_{t_1}} = \\left(\\frac{\\Sigma_{t_2}}{\\Sigma_{t_1}}\\right)^{5 \/ 3} \\left(1 - \\Delta d \\frac{30}{10 d_1 + 9}\\right),\n\\end{equation}\nwhich for $d_1 < - 0.9$ is alway greater than unity, proving our assertion.}\nWhen $r \\sim 80 \\ {\\rm R_S}$ for $q=0.1$ and at $r \\sim 220 {\\rm R_S}$ for $q = 0.3$, the accretion rate peaks and then falls to zero, showing the decoupling between the binary and the circumbinary disc, since viscosity moves the gas slower than the binary migrates. In the case with $q = 0.01$, the circumbinary disc has not fully decoupled yet, and its inner boundary reaches a distance of less than $2 R_h$ from the binary orbit. This is the closest the disc gets to the secondary in our simulations, in difference from the ``overflow'' scenario described in \\citet[][see discussion below]{kocsis12b}. Near the outer boundary of the inner disc, the seconday's torques have squeezed the material inward and enhanced the accretion rate and the thickness of the inner disc. The bottom panel of the same figure shows the scale height: while it is below unity, the tidal effect can push the material outside the binary orbit; but when it becomes of order unity, the thin disc approximation breaks down.\n\n\\subsection{Sensitivity to model parameters}\n\n\\begin{figure} \n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot7.pdf}\n\\caption{Different tests to see how the surface density at the last snapshot ($a = 20 \\ {\\rm R_S}$) changes with system parameters. The top panel shows simulations with different $\\alpha$ viscosity parameters (runs 22 \\& 23; see Table~\\ref{tab:params} for more details). The second panel shows the results when we change the initial (and boundary) accretion rate (runs 24 \\& 25). In the third panel, we change the mass of the primary black hole (runs 26-28) while in the bottom panel, we remove one or both discs from the initial condition (runs 29 \\& 30).} \n\\label{f:per} \n\\end{figure}\n\nIn order to assess the sensitivity of our results to the model parameters, we ran several additional simulations with a fixed mass ratio $q = 0.1$ and $\\gamma = 10^{- 4}$, but changing one or more parameters of our system, as listed in Table \\ref{tab:params}. Fig.~\\ref{f:per} shows the surface density profile in each case, at a binary separation $a = 20 \\ {\\rm R_S}$.\n\nIncreasing the strength of viscosity (controlled by $\\alpha$, top panel) decreases the initial surface density and enhances the material's inflow speed, which in turn reduces the time the binary needs to merge and produces a less depleted circumbinary disc. On the other hand, increasing the initial and external accretion rates (second panel) increases the initial surface density and reduces the time the binary needs to merge. Still, the overall shape of the surface density, both in the inner and in the circumbinary disc, are overall similar. \n\nWhen we increase the primary black hole mass (third panel), the initial surface density, the external accretion rate and the amount of material that crosses the cavity all increase. Since the Schwarzschild radius also depends on the primary mass, and our initial separation is a fixed amount of $\\rm{R_S}$, the overall effect is a longer time to merger.\n\nFinally, in the simulation without the inner disc, but allowing the inflow through the cavity ($\\gamma = 10^{- 4}$, bottom panel), the final surface density is identical to our fiducial case, but it takes less time to merge. If we remove both discs and allow the external accretion rate to fill the system, it takes ten times longer than in the fiducial case to reach $a = 20 \\ {\\rm R_S}$. At this moment, the surface density is still around $2-3$ times smaller than in the fiducial case, reinforcing our conclusion that the system never reaches a true steady state in its evolution.\n\n\\begin{figure} \n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot9.pdf}\n\\caption{Same simulations as in Fig.~\\ref{f:per}, but here we show the ratio between the binary's residence time in each case to the fiducial run. As expected, changing the primary mass changes the GW timescale, and it dominates from $a = 10^2 \\ {\\rm R_S}$ inwards.} \\label{f:per2} \n\\end{figure}\n\nFig. \\ref{f:per2} shows the ratio between the residence time of each test simulation and our fiducial case. Having no initial discs, reducing $\\alpha$ or $\\dot{M}$, have similar consequences in this ratio, increasing the residence time by a factor of around four, up to the point where GW dominates. Similarly, increasing $\\alpha$ or $\\dot{M}$ reduces the residence time by almost an order of magnitude. The simulation with only the circumbinary disc at the beginning becomes indistinguishable from the fiducial case for $a < 10^4 \\ {\\rm R_S}$. Finally, changing $M_p$ results in an overall scaling of the residence time that can be seen at the end of the simulation.\n\n\\section{Comparison with previous works} \\label{sec:pw}\n\nIn this section we contrast our results with previous studies of the long-term evolution of SMBHB systems. First, we compare our work with analytical estimates that assumed steady-state, and then with other numerical simulations where different conditions were implemented.\n\n\\subsection{Previous analytical works}\n\nThe first and most relevant work to compare our results with is \\citet[][hereafter K12]{kocsis12b}, where the disc properties are obtained analytically, using a similar set of equations, except assuming that the accretion discs go through a series of steady-state solutions as the binary evolve. Since in our simulations we dropped the steady-state approximation and let the binary separation shrink over time by both the influence of the discs and GWs, this comparison can give us insight into whether the steady state assumption is justified.\n\nAs we saw in Fig. \\ref{f:rest}, in the range between $10^2 \\ {\\rm R_S} \\lesssim a \\lesssim 10^4 \\ {\\rm R_S}$ the inflow timescale of the circumbinary disc is shorter than the binary's residence time. While this is a necessary condition for steady state, it is not sufficient, as we already discussed in section \\ref{sec:steady} above (see Fig. \\ref{f:mdot} in particular).\n\nThere are two other important differences between K12's model and the present study. First, K12 did not include the GW contribution in their calculations, since when the GWs dominate, the steady-state assumption is clearly invalidated. Second, K12 treats the gas inflow at the secondary differently. In particular, they assume that the gas near the secondary flows inward at the constant {\\em accretion rate} $\\dot{M}_{\\rm{ext}}$, and demand that the inward {\\em radial velocity} of the gas at the inner edge of the circumbinary disk (at $r\\approx 2a$) is $\\approx$ twice the migration speed of the secondary. This envisions a steady-state configuration, in which the gas just outside the secondary's orbit follows the inward-migrating secondary, preserving the approximately 2:1 ratio of the size of the gap and the binary separation. Depending on the values of the system parameters, solutions with this prescription are found with or without an inner disc. In summary, K12's prescription has no free parameters and describes a steady-state, in difference from ours, in which we move mass inward across the gap at a rate parameterized by the new parameter $\\gamma$, and require no steady state. Since K12 excludes GWs, we can only compare our results with their snapshots at the relatively large binary separation of $a = 500 \\ {\\rm R_S}$. To make a direct comparison, we ran four new simulations with the parameters in K12, $\\dot{M}_{\\rm{ext}} \/ \\dot{M}_{\\rm{Edd}} = 0.1$, $q = \\{0.1, 0.01\\}$ and $\\gamma = \\{10^{- 5}, 0\\}$ while the rest of the parameters are the same as our fiducial case. As our test with $0.1 \\dot{M}_f$ demonstrated, when we vary the accretion rate through the disc, the final shape of the surface density is similar, and the only difference is in its overall scaling. At $a = 500 \\ {\\rm R_S}$, K12 found a solution with a completely depleted inner disc (see their Fig.~2), and a strong pile-up in the outer disc. The increase in the circumbinary disc surface density is a factor of $3$ over the initial condition for $q = 0.1$ and $2.1$ for $q = 0.01$, with no material inside the secondary orbit.\n\nThese results most closely corresponds to our case without any inflow across the gap ($\\gamma = 0$), for which we found a more modest surface density enhancement of a factor of $1.5$ for $q = 0.1$, and no enhancement at all for $q = 0.01$. While in the former case we also obtain a gap that extends all the way down to the innermost stable circular orbit (ISCO), in the latter we found an inner disc, albeit with a very low surface density ($250$ times smaller than the initial condition). Finally, the thickness of the circumbinary disc in our case is three times smaller than in their results, while our $\\beta$ parameter is more than five times higher. At this separation K12 found, for $q = 0.01$, a binary radial velocity $da\/dt \\sim 200 \\ \\rm{cm \/ s}$ and a disc velocity (measured at $r = 2a$) two times faster. Instead, in our simulation with the same mass ratio and $\\gamma = 0$, our binary radial velocity is $da\/dt \\sim 60 \\ \\rm{ cm \/s}$ and the disc velocity is four times faster. Those values are consistent with our surface density being half theirs. Since we run a numerical simulation from farther away and let the system reach this point solving the equations self-consistently, we consider our results to be a more precise estimation of the disc properties at this binary separation.\n\nWe note further that a better definition for a ``pile-up\" is not the overall increase in the surface density, but rather the change in the slope of the discs at different times. The surface density of our discs are generally smaller than K12's because our system never reaches a steady-state configuration. If we were to fix the binary separation and let the discs evolve, the disc masses would increase. To illustrate this, we re-normalized our model output to match their surface density at $ r = 5 \\times 10^3 \\ {\\rm R_S}$, and found a slope very similar to K12's at $a = 500 \\ {\\rm R_S}$.\n\nTo further test K12 set-up, we run a simulation with $\\gamma = 10^{- 5}$. Our result resembles the case on the left panels in our Fig. \\ref{f:data}, where no pile up is found and an inner disc is maintained over all the simulation. An smaller $\\gamma$ could reproduce the circumbinary disc pile-up, with a depletion and subsequent refill of the inner disc, depending on the binary separation.\n\nThe fact that K12's surface density is enhanced by a factor or two in comparison with their initial condition (and our model), also partly explains why their residence time is ($\\approx$ four times) shorter than ours.\n\nTo explore the residence times further, we next focus on the results of \\citet[][hereafter H09]{haiman09}. Unlike K12 or the present work, H09 did not incorporate the binary torques directly into the disc structure equations or the resulting migration rate. Instead, they used the density and viscosity profiles of the usual Shakura-Sunyaev discs, modified by the presence of the secondary as detailed in prior semi-analytic studies \\citep{syer95, ivanov99}. They assumed that the secondary's migration is tied directly to the viscous time in the modified disc, and presented a dedicated analytical study of the resulting residence times.\n\nTo make a meaningful comparison, we run a new simulation with a parameter combination included in H09, i.e, $\\alpha = 0.3$, $\\dot{M}_{\\rm{ext}} \/ \\dot{M}_{\\rm{Edd}} = 0.1$, M = $10^7 M_\\odot$ and $q = 0.01$. The main difference is that we solve the system self-consistently, allowing the disc structure to be modified in a different way, depending on the migration rate of the secondary.\n \nThe residence time we obtained is similar to H09 at both ends, i.e. when the GWs dominate ($a \\lesssim 50 \\ {\\rm R_S}$) and when the binary separation is bigger than $a \\gtrsim 2 \\times 10^4 \\ {\\rm R_S}$, where the type II migration should be controlled by the circumbinary disc (with disc mass exceeding the secondary's mass). In-between these separations, our residence times deviate from H09's results for an unsteady $\\beta$ disc (based on \\citealt{ivanov99}) with the largest difference at $400 \\ {\\rm R_S}$, when ours is two times slower. Then, at $a \\sim 200 \\ {\\rm R_S}$, our solution becomes dominated by GW, decreasing until both results become similar again at $a = 50 \\ {\\rm R_S}$.\n\nOne explanation for this difference could be that in our work we made the direct calculation of the tidal torque, both from the inner and the circumbinary disc, whereas the inner disc is neglected in H09. The presence of an inner disc will add angular momentum to the binary and will slow down the migration. Moreover, as the binary shrinks, the interaction region defined by the Lindblad resonances also shrinks. This effect is stronger for the circumbinary disc, making the contribution of the inner disc more significant at later times. The difference can also be partially produced by the tidal torque used. In their fiducial model, H09 follows the self-similar disc evolution in \\citet{syer95}, which uses a tidal torque:\n\\begin{equation}\n\\Lambda' = q^2 \\Omega^2_s a^2 \\bigg(\\dfrac{a}{\\Delta}\\bigg)^4 = \\frac{2 r}{a f} \\Lambda \\sim (2 - 8) \\times 10^2 \\Lambda,\n\\end{equation}\nin the region of tidal interaction. Therefore, the tidal effect is stronger in their work.\n\nIn summary, Table \\ref{tab:restime} shows our residence times for the simulations mimicking K12 and H09, along with their results for a separation of $a = 400 \\ {\\rm R_S}$ and $a = 10^4 \\ {\\rm R_S}$. We see that, while for $a = 10^4 \\ {\\rm R_S}$ the analytical and numerical results are similar, at closer separations the absence of an enhancement in the circumbinary disc surface density and the presence of an inner disc made our residence times longer. \n\\begin{table}\n\\centering\n\t\\caption{Residence times in our simulations, compared to corresponding semi-analytic results from \\citet{kocsis12b} and \\citet{haiman09} with similar parameters ($q = 0.01$ and $\\gamma = 0$).}\n\t\\label{tab:restime}\n \\begin{tabular*}{0.7\\columnwidth}{lcc} \n & $a = 400 \\ {\\rm R_S}$ & $a = 10^4 \\ {\\rm R_S}$ \\\\ \n \\\\[-2ex]\n\t\t\\hline\n \\\\[-2ex]\n \\citeauthor{kocsis12b} & $1.5 \\times 10^5$ yr & $3.6 \\times 10^6$ yr\\\\\n Run 36 & $~~7 \\times 10^5$ yr & $4.6 \\times 10^6$ yr\\\\\n \\citeauthor{haiman09} & $~~2 \\times 10^5$ yr & $1.5 \\times 10^6$ yr\\\\\n Run 38 & $~~4 \\times 10^5$ yr & $2.1 \\times 10^6$ yr\\\\\n \\hline \n\t\\end{tabular*}\n\\end{table}\n\nFinally, we compare our results with the work of \\citet{rafikov16}, which presented analytical estimates of the circumbinary disc structure, assuming steady-state evolution. Note that \\citet{rafikov16} did not include the binary torque directly in the evolutionary equations. The torques were instead added as an additional flux of angular momentum, injected at $r=0$. The solutions should nevertheless be accurate far outside the binary-disc interaction region, but only if steady state is established. Following \\citet{tang17}, we took Rafikov's equations (3), (11) and (20), and write:\n\\begin{equation}\n3 \\pi \\nu \\Sigma = {\\dot{M}(r_b)} - \\left.\\frac{q M_p}{2 (1 + q) \\sqrt{a r}} \\frac{d a}{d t}\\right|_c = A - \\frac{B}{\\sqrt{r}}. \n\\label{eq:rafikov} \n\\end{equation} \nHere, $\\dot{M}(r_b)$ is a constant accretion rate at a radius $r_b \\gg a$ far away from the secondary, and $d a \/ d t |_c$ is the circumbinary disc contribution to the binary migration. Fig. \\ref{f:rafikov} shows the best fit for $A$ and $B$ in each snapshot, at different binary separations, for our simulations with $\\gamma = 10^{- 4}$. The bottom panel shows our numerical results for $\\gamma = 0$, compared to the same analytic fits as in the top panel. \n\nIn the top panel, the analytic profiles follow our results closely, in each case, beyond 2-3 times secondary's orbit. In the bottom panel, the case with no inflow through the cavity shows a clear pile-up over time, causing much stronger deviations extending to much larger radii. As discussed above, $\\gamma = 10^{- 4}$ is the value at which our mass flow rate across the cavity matches the value expected for steady state accretion, while $\\gamma = 0$ turns off this cross-cavity flow. Since the inflow timescale is shorter than the migration time, material piles up in the inner region of the circumbinary disc. Furthermore, in this case, the profiles deviate from the analytic expectations much farther out. Rather then always remaining approximately correct out to 2-3 times the current binary separation, the analytic solutions are now accurate only beyond a few $\\times 10^4 \\ {\\rm R_S}$, even at late times and much smaller binary separations. Because the system falls increasingly out of steady state, this critical radius (beyond which the analytic solution is valid) now remains tied to the initial conditions, rather than to the current binary separation. \n\n\\begin{figure}\n\\includegraphics[width = 0.93\\columnwidth]{.\/figures\/plot8.pdf}\n\\caption{The radial dependence of the accretion rate in our simulations, compared to the semi-analytic results of \\citet{rafikov16} which are expected to be valid far away from the binary. The solid curves show the simulation results for three different binary separations, as labeled, while the dashed curves show analytic fits to the results of \\citet{rafikov16} from Eq. \\ref{eq:rafikov} for $\\gamma = 10^{- 4}$. In the top panel, the analytic profiles follow our results with $\\gamma = 10^{- 4}$ closely beyond 2-3 times secondary's orbit. In the bottom panel, the case with no inflow through the cavity ($\\gamma = 0.0$) shows a clear pile up over time, causing much stronger deviations at small radii.}\n\\label{f:rafikov} \n\\end{figure}\n\n\\subsection{Previous numerical simulations}\nIn \\citet{lodato09}, the authors study a similar setup as our own, with the difference that their initial condition for the accretion discs has a fixed total mass and they do not have any inflow of material from the outer boundary.\n\nTo compare our results to their work and to see how our initial and boundary conditions change the time to merger, we ran a new simulation with their parameters (No. 39 in Table \\ref{tab:params}). The cutoff from \\citet{tazzari15}, also implemented here, prevents the material outside the interaction region to directly influence the migration of the binary. To compute the mass of the interacting disc, we integrate from ISCO out to a radius where the tidal contribution to the total radial velocity of the gas is less than one percent.\n\n\\citet{lodato09} considered an initial binary separation $a_0 = 0.01 {\\rm pc} \\sim 10^3 \\ {\\rm R_S}$ and concluded that, for an initial disc with half the mass of the secondary, the merger happens after $4.8$ Myr. Our interacting disc mass at the same binary separation is around $0.4 M_s$, and in our simulation the system needs $5.8$ Myr to merge, very close to their results. At the moment of the first decoupling, \\citet{lodato09} found that an inner disc survives even for the case with an initial disc mass ten times smaller than the secondary. In our simulation, the inner disc is completely accreted, which we suspect it due to the different initial condition used (since $\\gamma = 0$).\n\nFinally, we compare our residual mass after the first decoupling with that obtained by \\citet{tazzari15}. Compared with our simulation without inflow, their result is half the one we obtained, and even smaller for the cases where $\\gamma \\neq 0$. \n\\footnote{\\citet{chang10} had found a much smaller residual mass, due to the overestimate of the classical formula for the tidal torque (see eqs. \\ref{eq:smooth} and \\ref{eq:smooth2}, and \\citealt{tazzari15}).} The main differences between our works are:(i) we have an inflow of material from the circumbinary disc, controlled by $\\gamma$ and (ii) our external accretion rate is ten times higher than theirs. In the simulation with no inflow, the external accretion rate has no effect in the inner disc since material cannot cross the cavity. If we compare our inner disc mass at $a = 10^4 \\ {\\rm R_S}$ (their initial condition), ours is three times larger, which could explain the final difference.\n\nIn summary, we emphasize that the key novel factor in our work is the implementation of the inflow through the cavity. This ingredient changes the final state of the binary at the first decoupling, as we can see from the previous comparisons. While we do not know exactly which value of $\\gamma$ is the `right' one, or if it changes over time, our results show how important this inflow rate is for setting the residual mass, while we find that it does not greatly influence the residence time of the binary.\n\n\\section{Summary and Conclusions} \\label{sec:con}\n\nIn this work we developed 1D models to study the long-term, coupled evolution of a SMBH binary + gas disc system, solving the equations for the evolution of the surface density and temperature, and the secondary BH's migration in a self-consistent way. The non-axisymmetric distortions in the background discs, the resulting torques, and the rate at which gas can cross the gap created by the secondary, cannot be computed within our model. We instead incorporate these effects by prescriptions motivated by 2D simulations. In particular, the torques are implemented via a fitting formula calibrated to simulations \\citep{armitage02}. We further introduced the parameter $\\gamma$, which specifies that in the absence of viscosity, the material from the circumbinary disc would be transported to the inner disc in $\\sim \\gamma^{- 1}$ binary orbits, due to the secondary BH's gravitational torques and shocks. We implemented and tested different values of this parameter. While a 1D simulation is unable to capture the full range of physical processes at hand, it remains the only practical way to study the coupled long-term evolution of such systems.\n\nThe main results of our study are the residual mass in the inner disc near the end of the merger (when the binary separation is $a = 10^2 \\ {\\rm R_S}$), the time for the BHs to merge, or, in practice, to reach the GW-driven regime (starting from $a = 10^4 \\ {\\rm R_S}$), as well as the time-evolving deviations in the spectrum and luminosity produced by the tidal heating. All of these quantities are functions of the system parameters, including $q$~and~$\\gamma$.\n\nThe residual mass is very sensitive to system parameters, and ranges between $10^{-2}$ to $10^2 M_\\odot$, with a variation by two orders of magnitude at a given mass ratio. In particular, if gas can cross the gap efficiently (large $\\gamma$), the large amount of residual gas is a promising source of fuel for a possible EM counterpart to LISA events, as proposed in earlier work \\citep{chang10, tazzari15}. By comparison, we find the merger time to be much more robust: the total time for the secondary to migrate from $a = 10^4 \\ {\\rm R_S}$ to $a = 10^2 \\ {\\rm R_S}$ is always within a factor~of~$\\lesssim 3$ of $2 \\times 10^6$ yr. These values are reassuringly similar to those found by directly measuring the total torque in 2D simulations \\citet{tang17}, and also of the same order as the viscous timescale. They reinforce the conclusion that a circumbinary disk, fueled at near-Eddington rates, can drive the binary to coalesce well within a Hubble time. \n\nAs expected, when we decrease $\\gamma$ the pile-up in the circumbinary disc becomes more pronounced and the inner disc depletes. This can be seen in the SED: the hot region of the circumbinary disc is enhanced while the inner disc diminishes. The time before merger when Bolometric Luminosity peaks also depends on the amount of material that crosses the cavity. We find that for $\\gamma \\geq 10^{- 5}$ this happens at the ``second decoupling'', while for smaller values, the tidal heat of the circumbinary disc at the ``first decoupling\" dominates. \n\nWe also found that in general, the binary + disc system never reaches a genuine steady state. This is most clearly seen in our analysis of the accretion rate profile (\\S~\\ref{sec:steady}), where the accretion rate remains a function of radius. The entire evolutionary history of the system generally depends on the initial and outer boundary conditions. When we assume the gas crosses the gap inefficiently (e.g. in our runs with $\\gamma \\leq 10^{- 5}$ and $q = 0.1$), we find that the inner disc is depleted and then re-filled (see Fig. \\ref{f:data}), while the bolometric luminosity increases and decreases by an order of magnitude (Fig. \\ref{f:bol}). When we assume the gas can cross the gap more efficiently (i.e. for higher values of $\\gamma$), this depletion does not occur. In the case with $\\gamma = 10^{- 4}$, an approximately constant surface density develops over time, with the magnitude and the slope of the surface density profiles similar between snapshots, but the accretion rate profiles still show that a steady state is not established (\\S~\\ref{sec:steady}). This non-steady-state behaviour implies that the gas distribution near the time of the merger and close to the BHs depends on the boundary conditions at large radii and at early times. This issue must be studied further in 2- and 3D simulations, and eventually incorporated into the initial conditions of more sophisticated (e.g. 3D GRMHD simulations), designed to follow the last stages of the merger.\n\n\\section*{Acknowledgments}\nCF and JC acknowledge financial support from CONICYT-Chile through FONDECYT (1141175) and Basal (PFB0609) grants. CF acknowledges financial support from CONICYT-PFCHA\/Doctorado Nacional (2017-21171063). CF thanks Columbia University for warm hospitality, as well as financial support through Columbia University's President's Global Innovation Fund, during his visit where this work started. ZH gratefully acknowledges financial support was by NASA through ATP grant NNX15AB19G, ADAP grant NNX17AL82G, and Swift grant 16-SWIFT16-0015, and by the NSF through grant 1715661. JC acknowledges the warm hospitality of MPE, where part of this work was conducted, and funding from the MPG through a Partner Group grant.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}