\section{Additional preliminaries, lemmas and proofs}
\subsection{Signatures of Knowledge}
\label{sec:sok}
A $\SoK$ for an efficiently decidable binary relation $\REL$ is defined as a tuple of PPT algorithms $\SoK = (\signsetup,  \sign, \allowbreak \verify,  \simsetup, \simsign)$:

\begin{description}
    \item[$\signsetup(1^\secpar, \REL) \rightarrow  \param$:]
	The setup algorithm takes a security parameter $\secpar$ and a binary relation $\REL$
	and returns public parameters $ \param$.  The input $ \param$ is implicit to all subsequent algorithms.

    \item[$\sign(\mesage, \inp, \wit)  \rightarrow \signature$:]
The signing algorithm takes as input a message $ \msg \in \{0,1\}^{*}$,
a statement $\inp$, and a witness $\wit$.
	Outputs a signature $\signature$.

    \item[$\verify(\mesage, \inp, \signature) \rightarrow 1/ 0$:]
The verification algorithm takes as input
 a message $\mesage$,  a statement $\inp$
 together with a signature $\signature$,
	outputs $1$ if the the signature is valid, $0$ otherwise.

    \item[$\simsetup(\REL) \rightarrow (\param, \td)$:]
    	A simulated setup algorithm which takes as input a relation $\REL$ and returns public parameters $\param$ and a trapdoor $\td$.

   \item[$\simsign(\td, \mesage, \inp) \rightarrow \signature'$:]
   	A simulated signing  that takes as input  a trapdoor $\td$, a message $\mesage$ and a statement $\inp$ and returns a simulated signature $\signature'$.
\end{description}

A SoK scheme should satisfy correctness, extractability and simulatability:

\begin{description}
\item[Perfect Correctness.] This guarantees that a signer with a valid witness can always produce a signature that
will convince the verifier:
For all $\secpar\in \NN$, for
all efficiently decidable binary relation $\REL$,
for all  $(\inp, \wit) \in \REL$, and for all $ \mesage \in \{0,1\}^{*}$:
   \[
  \condprob{
	  \begin{matrix}
~\verify(\mesage, \inp, \signature) = 1
	 \end{matrix}
}{
	  \begin{matrix}
~(\param, \td) \gets \signsetup(1^\secpar, \REL)\\
~ \signature \gets  \sign(\mesage, \inp, \wit)
 \end{matrix} }  =1.
\]
%
\item[Simulation Extractability:] This guarantees that an adversary is not able to issue a new signature
unless it knows a witness. This should hold even if the adversary gets to see signatures on
arbitrary messages under arbitrary statements. We model this notion in a strong sense, by
letting the adversary see simulated signatures for arbitrary messages and statements, which
potentially includes false statements. Even under this strong attack model, we require that
whenever the adversary outputs a valid signature not queried before, it is possible to extract a
witness for the signature. More formally,  for any PPT adversary $\adv$ there exists a PPT extractor $\ext_\adv$ such that:
   \[
  \condprob{
	  \begin{matrix}
~ \verify(\mesage, \inp, \signature) = 1   \\
\land (\inp, \wit) \notin \REL \\
\land (\mesage, \inp, \signature) \notin Q^{\simsign_\td}
	 \end{matrix}
}{
	  \begin{matrix}
~(\param, \td) \gets \simsetup(1^\secpar, \REL)\\
~ (\mesage, \inp, \signature) \gets \adv^{\simsign_\td} (\param)\\
\wit \gets \ext_\adv({\sf trans}_\adv)
 \end{matrix}
} \leq \negl
\]
where the adversary has access to simulated signatures via the oracle $\simsign_\td(\inp_i, \mesage_i) \coloneq \simsign(\td, \inp_i, \mesage_i)$ and the extractor $\ext_\adv$ takes as input the transcript ${\sf trans}_\adv$ of all queries made by $\adv$ including its randomness.

\vspace{4pt}
%
\item[Perfect Simulatability:] The verifier should not learn anything  about the witness from the signature.  The secrecy of the witness is modelled by the ability to
simulate signatures without the witness. More precisely,  we say the signatures of knowledge
are simulatable if there is a simulator that can create public parameters together with an associated trapdoor that enables producing signatures without a witness that are indistinguishable from real ones.  More formally, for any number of queried $\mesage_i\in \{0,1\}^{*}$ and $(\inp_i, \wit_i) \in \REL$:

$
\Big	\vert	   \condprob{
		\begin{matrix}
		\adv(\param) = 1
		\end{matrix}
		}{
		\begin{matrix}
	\param \gets  \signsetup(1^\secpar, \REL)\\
	~\signature_i \gets \sign(\mesage_i, \inp_i, \wit_i)
		\end{matrix}
		 }
		- $
\[	  \condprob{
		\begin{matrix}
		\adv (\param) = 1
		\end{matrix}
}{
		\begin{matrix}
	~(\param,\td) \gets  \simsetup(1^\secpar, \REL) \\
	~\signature_i \gets   \simsign(\td,\mesage_i, \inp_i)
		\end{matrix}} \Big	\vert
		 \leq \negl.
		\]
  \end{description}

  \subsection{Dlog assumptions}
\label{sec:dlog_assumptions}
\begin{definition}[$(q_1, q_2)\mhyph\dlog$ assumption]
	Let $\adv$ be a $\ppt$ adversary that gets as input
  $\gone{1, \chi, \ldots, \chi^{q_1}}, \gtwo{1, \chi, \ldots, \chi^{q_2}}$, for
  some randomly picked $\chi \in \FF_p$, then
	\[
		\condprob{\chi \gets \adv(\gone{1, \chi, \ldots, \chi^{q_1}}, \gtwo{1, \chi,
        \ldots, \chi^{q_2} })}{\chi \sample \FF_p} \leq \negl.
	\]
\end{definition}

\begin{definition}[$(q_1, q_2)\mhyph\ldlog$ assumption]
  Let $\adv$ be a $\ppt$ adversary that gets as input
  $\gone{\chi^{-q_1}, \ldots, 1, \chi, \ldots, \chi^{q_1}}, \gtwo{\chi^{-q_2},
    \ldots, 1, \chi, \ldots, \chi^{q_2}}$, for some randomly picked
  $\chi \in \FF_p$, then
	\[
    \condprob{\chi \gets \adv(\gone{\chi^{-q_1}, \ldots, 1, \chi, \ldots,
        \chi^{q_1}}, \gtwo{\chi^{-q_2}, \ldots, 1, \chi, \ldots, \chi^{q_2}
      })}{\chi \sample \FF_p} \leq \negl.
	\]
\end{definition}

\subsection{Uber assumption}
\label{sec:uber_assumption}
\ourpar{BBG uber assumption.}
Also, to be able to show trapdoor-less simulatability of
$\plonk$ in the standard model, what is required by our reduction, we rely on the
\emph{uber assumption} introduced by Boneh et
al.~\cite{EC:BonBoyGoh05} as presented by Boyen in \cite{PAIRING:Boyen08}.

Let $r, s, t, c \in \NN \setminus \smallset{0}$, Consider vectors of polynomials
$\pR \in \FF_p[X_1, \ldots, X_c]^r$, $\pS \in \FF_p[X_1, \ldots, X_c]^s$ and
$\pT \in \FF_p[X_1, \ldots, X_c]^t$. Write $\pR = \left( \p{r}_1, \ldots,
  \p{r}_r \right)$, $\pS = \left( \p{s}_1, \ldots, \p{s}_s \right)$ and $\pT =
\left( \p{t}_1, \ldots, \p{t}_t \right)$ for polynomials $\p{r}_i, \p{s}_j,
\p{t}_k$.

For a function $f$ and vector $(x_1, \ldots, x_c)$ we write $f(\pR)$ to
denote application of $f$ to each element of $\pR$, i.e.
\(
	f(\pR) = \left( f(\p{r}_1 (x_1, \ldots, x_c), \ldots, f(\p{r}_r
	(x_1, \ldots, x_c) \right).
\)
Similarly for applying $f$ to $\pS$ and $\pT$.

\begin{definition}[Independence of $\pR, \pS, \pT$]
	\label{def:independence}
	Let $\pR, \pS, \pT$ be defined as above. We say that polynomial $\p{f} \in
  \FF_p[X_1, \ldots, X_c]$ is \emph{dependent} on $\pR, \pS, \pT$ if there
  exists $rs + t$ constants $a_{i, j}, b_k$ such that $ \p{f} = \sum_{i = 1}^{r}
  \sum_{j = 1}^{s} a_{i, j} \p{r}_i \p{s}_j + \sum_{k = 1}^{t} b_k \p{t}_k. $ We
  say that $\p{f}$ is \emph{independent} if it is not dependent.
\end{definition}

To show (standard-model) zero knowledge of $\plonk$ we utilize a generalization
of Boneh-Boyen-Goh's \emph{uber assumption} \cite{EC:BonBoyGoh05} stated as
follows (the changed element has been put into a \dbox{dashbox})
\begin{definition}[$(\pR, \pS, \pT, \p{F}, 1)$-uber assumption]
	\label{def:uber_assumption}
	Let $\pR, \pS, \pT$ be defined as above,
    $(x_1, \ldots, x_c, y_1, \ldots, y_{d}) \sample \FF_p^{c + d}$ and let
    $\p{F}$ be a cardinality-$d$ set of pair-wise independent polynomials which are also
    independent of $(\pR, \pS, \pT)$, cf.~\cref{def:independence}.  Then, for
    any $\ppt$ adversary $\adv$
	\begin{multline*}
      \Pr\left[\adv(\gone{\pR(x_1, \ldots x_c)}, \gtwo{\pS(x_1, \ldots, x_c)},
		\gtar{\pT(x_1, \ldots, x_c)}, \dboxed{\gone{\p{F}(x_1, \ldots, x_c)}}) = 1\right] \approx_\secpar \\
      \Pr\left[\adv(\gone{\pR(x_1, \ldots x_c)}, \gtwo{\pS(x_1, \ldots, x_c)},
        \gtar{\pT(x_1, \ldots, x_c)}, \dboxed{\gone{y_1, \ldots, y_{d}}}) =
        1\right].
	\end{multline*}
  \end{definition}

  Compared to the original uber assumptions, there are two major changes. First,
  we require not target group $\GRP_T$ elements to be indistinguishable, but
  elements of $\GRP_1$. Second, Boneh et al.'s assumption works for
  distinguishers who are given only one challenge polynomial $\p{f}$,
  i.e.~$\abs{\p{F}} = 1$.

We show security of our version of the uber assumption using the generic group
model as introduced by Shoup \cite{EC:Shoup97} where all group elements are
represented by random binary strings of length $\secpar$. That is, there are
random encodings $\xi_1, \xi_2, \xi_T$ which are injective functions from
$\ZZ_p^+$ to $\bin^{\secpar}$. We write
$\GRP_i = \smallset{\xi_i(x) \mid x \in \ZZ_p^+}$, for
$i \in \smallset{1, 2, T}$. For the sake of clarity  we denote by $\xi_{i, j}$
the $j$-th encoding in group $\GRP_i$.

Let
$\p{P}_i = \smallset{p_1, \ldots, p_{\tau_i}} \subset \FF_p[X_1, \ldots, X_n]$,
for $i \in \smallset{1, 2, T}, \tau_i, n \in \NN$, be sets of multivariate
polynomials. Denote by $\p{P}_i(x_1, \ldots, x_n)$ a set of evaluations of
polynomials in $\p{P_i}$ at $(x_1, \ldots, x_n)$. Denote by
$L_i = \smallset{(p_j, \xi_{i, j}) \mid j \leq \tau_i}$.

Let $\adv$ be an algorithm that is given encodings $\xi_{i, j_i}$ of polynomials
in $\p{P}_i$ for $i \in \smallset{1, 2, T}, j_i = \tau_i$. There is an oracle $\oracleo$
that allows to perform $\adv$ the following queries:
\begin{description}
\item[Group operations in $\GRP_1, \GRP_2, \GRP_T$:] On input
  $(\xi_{i, j}, \xi_{i, j'}, i, op)$, $j, j' \leq \tau_i$,
  $op \in \smallset{\msg{add}, \msg{sub}}$, $\oracleo$ sets $\tau'_i \gets \tau_i + 1$,
  computes
  $p_{i, \tau'_i} = p_{i, j}(x_1, \ldots, x_n) \pm p_{i, j'}(x_1, \ldots, x_n)$
  respectively to $op$. If there is an element  $p_{i, k} \in L_i$ such
  that $p_{i, k} = p_{\tau'_i}$, then the oracle returns encoding of $p_{i,
    k}$. Otherwise it sets the encoding $\xi_{i, \tau'_i}$ to a new unused
  random string, adds $(p_{i, \tau'_i}, \xi_{i, \tau'_i})$ to $L_i$, and returns
  $\xi_{i, \tau'_i}$.
\item[Bilinear pairing:] On input $(\xi_{1, j}, \xi_{2, j'})$ the oracle sets
  $\tau' \gets \tau_T + 1$ and computes
  $r_{\tau'} \gets p_{i, j}(x_1, \ldots, x_n) \cdot p_{i, j'}(x_1, \ldots,
  x_n)$. If $r_{\tau'} \in L_T$ then return encoding found in the list $L_T$,
  else pick a new unused random string and set $\xi_{T, \tau'}$ to it. Return
  the encoding to the algorithm.
\end{description}

Given that, we are ready to show security of our variant of the Boneh et
al.~uber assumption. The proof goes similarly to the original proof given in
\cite{EC:BonBoyGoh05} with minor differences.

\begin{theorem}[Security of the uber assumption]
  \label{thm:uber_assumption}
  Let $\p{P}_i \in \FF_p[X_1, \ldots, X_n]^{m_i}$, for
  $i \in \smallset{1, 2, T}$ be $\tau_i$ tuples of $n$-variate polynomials over
  $\FF_p$ and let $\p{F} \in \FF_p[X_1, \ldots, X_n]^m$. Let
  $\xi_0, \xi_1, \xi_T$, $\GRP_1, \GRP_2, \GRP_T$ be as defined above. If
  polynomials $f \in \p{F}$ are pair-wise independent and are independent of
  $\p{P}_1, \p{P}_2, \p{P}_T$, then for any $\adv$ that makes up to $q$ queries to the
  GGM oracle holds:
  \begin{equation*}
    \begin{split}
     \left|\,
    \Pr\left[
    \adv\left(
      \begin{aligned}
        \xi_1(\p{P}_1(x_1, \ldots, x_n)), \\
        \xi_2(\p{P}_2(x_1, \ldots, x_n)), \\
        \xi_T(\p{P}_T(x_1, \ldots, x_n)), \\
        \xi_{1}(\p{F}_0), \xi_{1}(\p{F}_1)
      \end{aligned}
    \right) = b
    \, \left|\,
      \begin{aligned}
        x_1, \ldots, x_n, y_1, \ldots, y_m \sample \FF_p,\\
        b \sample \bin, \\
        \p{F}_b \gets \p{F}(x_1, \ldots, x_n),\\
        \p{F}_{1 - b} \gets (y_1, \ldots, y_m)
      \end{aligned}
    \right.  \right] - \frac{1}{2} \, \right| \\
     \leq \frac{d(q + m_1 + m_2 + m_T +
      m)^2 }{2p}
    \end{split}
  \end{equation*}
\end{theorem}
\begin{proof}
  Let $\cdv$ be a challenger that plays with $\adv$ in the following
  game. $\cdv$ maintains three lists
  \[
    L_i = \smallset{(p_j, \xi_{i, j}) \mid j \in \range{1}{\tau_i}},
  \]
  for $i \in \smallset{1, 2, T}$. Invariant $\tau$ states that
  $\tau_1 + \tau_2 + \tau_T = \tau + m_1 + m_2 + m$.

  Challenger $\cdv$ answers $\adv$'s oracle queries. However, it does it a bit
  differently that the oracle $\oracleo$ would:
  \begin{description}
  \item[Group operations in $\GRP_1, \GRP_2, \GRP_T$:] On input
    $(\xi_{i, j}, \xi_{i, j'}, i, op)$, $j, j' \leq \tau_i$,
    $op \in \smallset{\msg{add}, \msg{sub}}$, $\cdv$ sets
    $\tau' \gets \tau_i + 1$, computes
    $p_{i, \tau'}(X_1, \ldots, X_n) = p_{i, j}(X_1, \ldots, X_n) \pm p_{i,
      j'}(X_1, \ldots, X_n)$ respectively to $op$. If there is a polynomial
    $p_{i, k}(X_1, \ldots, X_n) \in L_i$ such that
    $p_{i, k}(X_1, \ldots, X_n) = p_{\tau'}(X_1, \ldots, X_n)$, then the
    challenger returns encoding of $p_{i, k}$. Otherwise it sets the encoding
    $\xi_{i, \tau'}$ to a new unused random string, adds
    $(p_{i, \tau'}, \xi_{i, \tau'})$ to $L_i$, and returns $\xi_{i, \tau'}$.
  \item[Bilinear pairing:] On input $(\xi_{1, j}, \xi_{2, j'})$ the challenger
    sets $\tau' \gets \tau_T + 1$ and computes
    $r_{\tau'}(X_1, \ldots, X_n) \gets p_{i, j}(X_1, \ldots, X_n) \cdot p_{i,
      j'}(X_1, \ldots, X_n)$. If $r_{\tau'}(X_1, \ldots, X_n) \in L_T$, $\cdv$
    returns encoding found in the list $L_T$. Else it picks a new unused random
    string and set $\xi_{T, \tau'}$ to it. Finally it returns the encoding to
    the algorithm.
\end{description}

After at most $q$ queries to the oracle, the adversary returns a bit $b'$. At
that point the challenger $\cdv$ chooses randomly $x_1, \ldots, x_n, y_1 \ldots, y_m$,
random bit $b$, and sets $X_i = x_i$, for $i \in \range{1}{n}$, and $Y_i = y_i$,
for $i \in \range{1}{m}$; furthermore, $\p{F}_b \gets \p{F}(x_1, \ldots, x_n)$
and $\p{F}_{1 - b} \gets (y_1, \ldots, y_m)$. Note that $\cdv$ simulates
perfectly unless the chosen values $x_1, \ldots, x_n, y_1, \ldots, y_m$ result
in equalities between polynomial evaluations that are not equalities between the
polynomials. That is, the simulation is perfect unless for some $i, j, j'$ holds
\[
  p_{i, j}(x_1, \ldots, x_n) - p_{i, j'}(x_1, \ldots, x_n) = 0,
  \]
  for $p_{i, j}(X_1, \ldots, X_n) \neq p_{i, j'}(X_1, \ldots, X_n)$.  Denote by
  $\bad$ an event that at least one of the three conditions holds. When $\bad$
  happens, the answer $\cdv$ gives to $\adv$ differs from an answer that a real
  oracle would give. We bound the probability that $\bad$ occurs in two steps.

  First we set $\p{F}_b = \p{F}(X_1, \ldots, X_n)$. Note that symbolic
  substitutions do not introduce any new equalities in $\GRP_1$. That is, if for
  all $j, j'$ holds $p_{1, j} \neq p_{1, j'}$, then $p_{1, j} \neq p_{1, j'}$
  even after setting $\p{F}_b = \p{F}(X_1, \ldots, X_n)$. This follows since all
  polynomials in $\p{F}$ are pairwise independent and $\p{F}$ independent on
  $\p{P}_1, \p{P}_2, \p{P}_T$. Indeed, $p_{1, j} - p_{1, j'}$ is a polynomial of
  the form
  \[
    \sum_{j = 1}^{m_1}a_j p_{1, j} + \sum_{j = 1}^{m} b_j f_j (X_1, \ldots, X_n),
  \]
  for some constants $a_j, b_j$. If the polynomial is non-zero, but setting
  $\p{F}_b = \p{F}(X_1, \ldots, X_n)$ makes this polynomial vanish, then some
  $f_k$ must be dependent on some $\p{P}_1, \p{F} \setminus \smallset{f_k}$.

  Now we set $X_1 \ldots, X_n, \p{F}_{1 - b}$ and bound probability that for
  some $i$ and $j, j'$ holds
  $(p_{i, j}(x_1, \ldots, x_n) - p_{i, j'}(x_1, \ldots, x_n) = 0$ for
  $p_{i, j} \neq p_{i, j'}$. By the construction, the maximum total degree of
  these polynomials is
  $d = \max(d_{\p{P}_1}+ d_{\p{P}_2}, d_{\p{P}_T}, d_{\p{F}})$, where $d_f$ is
  the total degree of some polynomial $f$ and for a set of polynomials
  $F = \smallset{f_1, \ldots, f_k}$, we write
  $d_F = \smallset{d_{f_1}, \ldots, d_{f_k}}$. Thus, for a given $j, j'$ probability that a random assignment to
  $X_1, \ldots, X_n, Y_1, \ldots, Y_n$ is a root of $p_{i, j} - p_{i, j'}$ is,
  by the Schwartz-Zippel lemma, bounded by $\infrac{d}{p}$, which is
  negligible. There is at most $2 \cdot {q + m_0 + m_1 + m  \choose 2}$ such
  pairs $p_{i, j}, p_{i, j'}$ we have that
  \[
    \prob{\bad} \leq  {q + m_0 + m_1 + m  \choose 2} \cdot \frac{2d}{p} \leq (q
    + m_0 + m_1 + m)^2 \frac{d}{p}.
  \]

  As noted, if $\bad$ does not occur then the simulation is perfect. Also the
  bit $b$ has been chosen independently on the $\adv$'s view, thus $\condprob{b
    = b'}{\neg \bad} = \infrac{1}{2}$. Hence,
  \[
    \begin{aligned}
      \prob{b = b'} & \leq \condprob{b = b'}{\neg \bad}(1 - \prob{\bad}) + \prob{\bad} =
      \frac{1}{2} + \frac{\prob{\bad}}{2} \\
      \prob{b = b'} & \geq \condprob{b = b'}{\neq \bad}(1 - \prob{\bad}) =
      \frac{1}{2} - \frac{\prob{\bad}}{2}.
    \end{aligned}
  \]
  Finally,
  \[
    \abs{\Pr[b = b'] - \frac{1}{2}} \leq \prob{\bad}/2 \leq (q
    + m_0 + m_1 + m)^2 \frac{d}{2p}
  \]
  as required.
\end{proof}

\subsection{Special simulation-extractability of sigma protocols and forking lemma}
\label{sec:forking_lemma}
\begin{theorem}[Special simulation extractability of the Fiat--Shamir transform
  \cite{INDOCRYPT:FKMV12}]
	Let $\sigmaprot = (\prover, \verifier, \simulator)$ be a non-trivial sigma
  protocol with unique responses for a language $\LANG \in \npol$. In the random
  oracle model, the NIZK proof system $\sigmaprot_\fs = (\prover_\fs,
  \verifier_\fs, \simulator_{\fs})$ resulting by applying the Fiat--Shamir
  transform to $\sigmaprot$ is special simulation extractable with extraction error
  $\eta = q/h$ for the simulator $\simulator$. Here, $q$ is the number of random
  oracle queries and $h$ is the number of elements in the range of $\ro$.
\end{theorem}

The theorem relies on the following \emph{general forking lemma} \cite{JC:PoiSte00}.

\begin{lemma}[General forking lemma, cf.~\cite{INDOCRYPT:FKMV12,CCS:BelNev06}]
	\label{lem:forking_lemma}
	Fix $q \in \ZZ$ and a set $H$ of size $h > 2$. Let $\zdv$ be a $\ppt$
  algorithm that on input $y, h_1, \ldots, h_q$ returns $(i, s)$, where $i
  \in\range{0}{q}$ and $s$ is called a \emph{side output}. Denote by $\ig$ a
  randomised instance generator. We denote by $\accProb$ the probability
	\[
		\condprob{i > 0}{y \gets \ig; h_1, \ldots, h_q \sample H; (i, s) \gets
		\zdv(y, h_1, \ldots, h_q)}\,.
	\]
	Let $\forking_\zdv(y)$ denote the algorithm described in
  \cref{fig:forking_lemma}, then the probability $\frkProb$ defined as $
  \frkProb := \condprob{b = 1}{y \gets \ig; (b, s, s') \gets \forking_{\zdv}(y)}
  $ holds
	\[
		\frkProb \geq \accProb \brak{\frac{\accProb}{q} - \frac{1}{h}}\,.
	\]
	%
	\begin{figure}
		\centering
		\fbox{
		\procedure{$\forking_\zdv (y)$}
		{
			\rho \sample \RND{\zdv}\\
			h_1, \ldots, h_q \sample H\\
			(i, s) \gets \zdv(y, h_1, \ldots, h_q; \rho)\\
			\pcif i = 0\ \pcreturn (0, \bot, \bot)\\
			h'_{i}, \ldots, h'_{q} \sample H\\
			(i', s') \gets \zdv(y, h_1, \ldots, h_{i - 1}, h'_{i}, \ldots,  h'_{q};
			\rho)\\
			\pcif (i = i') \land (h_{i} \neq h'_{i})\ \pcreturn (1, s, s')\\
			\pcind \pcelse \pcreturn (0, \bot, \bot)
		}}
		\caption{Forking algorithm $\forking_\zdv$}
		\label{fig:forking_lemma}
\end{figure}
\end{lemma}

\subsection{Proof of the generalized forking lemma (\cref{lem:generalised_forking_lemma})}
\label{sec:forking_proof}
\begin{proof}
First denote by $\accProb(y)$ and $\frkProb(y)$ the following probabilities
\begin{align*}
\accProb(y) & =  \condprob{i \neq 0}{h_1, \ldots, h_q \sample H;\ (i, s)
\gets \zdv(y, h_1, \ldots, h_q)}\,.\\
	\frkProb(y) & = \condprob{b = 1}{(b, \vec{s}) \gets
\genforking_{\zdv}^{m}(y, h_1, \ldots, h_q)}\,.
\end{align*}

We start by claiming that for all $y$
\begin{equation}\label{eq:frkProb_y}
	\frkProb(y) \geq
	\frac{\accProb(y)^m}{q^{m - 1}} - \accProb(y) \cdot \left(1 -
  \frac{h!}{(h - m)! \cdot h^{m}}\right)
	\end{equation}
Then with the expectation taken over $y \sample \ig$, we have
\begin{align}
	\frkProb & = \expected{\frkProb(y)} \geq
	\expected{\frac{\accProb(y)^m}{q^{m - 1}} -  \accProb(y) \cdot \left(1 -
  \frac{h!}{(h - m)! \cdot h^{m}}\right)} \label{eq:use_eq1}\\
	& \geq \frac{\expected{\accProb(y)}^m}{q^{m - 1}} -
	\expected{\accProb(y)} \cdot \left(1 - \frac{h!}{(h - m)! \cdot
  h^{m}}\right) \label{eq:by_lemma_jensen}\\
	& = \frac{\accProb^m}{q^{m - 1}} -  \accProb \cdot \left(1 -
  \frac{h!}{(h - m)! \cdot h^{m}}\right)\label{eq:by_accProb}\,.
\end{align}
Where \cref{eq:use_eq1} comes from \cref{eq:frkProb_y};
\cref{eq:by_lemma_jensen} comes from linearity of expected value and \cref{lem:jensen}; and
\cref{eq:by_accProb} holds by the fact that $\expected{\accProb(y)} =
\accProb$.

We now show \cref{eq:frkProb_y}.
Denote by $J = \range{1}{m}^2 \setminus \smallset{(j, j)}_{j \in \range{1}{m}}$.
For any input $y$, with probabilities taken over the coin tosses of
$\genforking_{\zdv}^{m}$ we have
\begin{align*}
	\frkProb (y) & = \prob{i_j = i_{j'} \land i_j \geq 1 \land
h_{i_j}^{j} \neq h_{i_{j'}}^{j'} \text{ for } (j, j') \in J}	\\
	& \geq \prob{i_j = i_{j'} \land i_j \geq 1 \text{ for } (j, j') \in J} %\\
   - \prob{i_j \geq 1 \land h_{i_j}^{j} = h_{i_{j'}}^{j'} \text{ for some } (j, j') \in J}\\
	& = \prob{i_j = i_{j'} \land i_j \geq 1 \text{ for } (j, j') \in J} -
	\prob{i_j \geq 1} \cdot
  \left(1 - \frac{h!}{(h - m)! \cdot h^{m}}\right) \\
	& = \prob{i_j = i_{j'} \land
	i_j \geq 1 \text{ for } (j, j') \in J} - \accProb(y) \cdot \left(1 -
\frac{h!}{(h - m)! \cdot h^{m}}\right)\,.
\end{align*}

Probability that for some $(j, j') \in J$ and $i_j = i_{j'}$ holds
$h_{i_j}^{j} \neq h_{i_{j'}}^{j'}$ equals
\[
  \frac{h \cdot (h - 1)
\cdot \ldots \cdot (h - m - 1)}{h^m} = \frac{h!}{(h - m)! \cdot h^m}.
\]
That is, it equals the number
of all $m$-element strings where each element is different divided by
the number of all $m$-element strings, where elements are taken from a
set of size $h$.

It remains to show that $\prob{i_j = i_{j'} \land i_j \geq 1 \text{ for } (j,
  j') \in J} \geq \infrac{\accProb(y)^m}{q^{m - 1}}$. Let $\RND{\zdv}$ denote
the set from which $\zdv$ picks its coins at random. For each $\iota \in
\range{1}{q}$ let $X_\iota \colon \RND{\zdv} \times H^{\iota - 1} \to [0, 1]$ be
defined by setting $X_\iota(\rho, h_1, \ldots, h_{\iota - 1})$ to
\[
  \condprob{i = \iota}{h_\iota, \ldots, h_q \sample H; (i, s) \gets \zdv(y, h_1,
    \ldots, h_q; \rho)}
\]
for all $\rho \in \RND{\zdv}$ and $h_1, \ldots, h_{\iota - 1} \in H$. Consider
$X_\iota$ be a random variable over the uniform distribution on its domain. Then
\begin{align*}
	& \prob{i_j = i_{j'} \land i_j \geq 1 \text{ for } (j, j') \in J}
	 = \sum_{\iota = 1}^{q} \prob{i_1 = \iota \land \ldots \land i_m = \iota} \\
	& = \sum_{\iota = 1}^{q} \prob{i_1 = \iota} \cdot \condprob{i_2 = \iota}{i_1 = \iota} \cdot \ldots \cdot \condprob{i_m = \iota}{i_1 = \ldots = i_{m - 1} = \iota} \\
	& = \sum_{\iota = 1}^{q} \sum_{\rho, h_1, \ldots, h_{\iota - 1}} X_{\iota}
   (\rho, h_1, \ldots, h_{\iota - 1})^{m} \cdot \frac{1}{\abs{\RND{\zdv}} \cdot \abs{H}^{\iota - 1}}
   = \sum_{\iota = 1}^{q} \expected{X_\iota^m} \,.
\end{align*}
Importantly, $\sum_{\iota = 1}^q \expected{X_{\iota}} = \accProb(y)$.

By \cref{lem:jensen} we get
\[
	\sum_{\iota = 1}^{q} \expected{X_\iota^m} \geq \sum_{\iota = 1}^{q} \expected{X_\iota}^m\,.
\]
Note that for e.g.~$X_i = 1$, $i \in \range{1}{q}$ the inequality becomes equality, that is, it is tight.

We now use the H\"older inequality, cf.~\cref{lem:holder}, for $x_i = \expected{X_i}$, $y_i = 1$, $p = m$, and $q = m/(m - 1)$ obtaining
\begin{gather}
	\left(\sum_{i = 1}^{q} \expected{X_i}\right)^{m}  \leq \left(\sum_{i = 1}^{q} \expected{X_i}^m\right) \cdot q^{m - 1}\\
	\frac{1}{q^{m - 1}} \cdot \accProb(y)^{m} \leq \sum_{i = 1}^{q} \expected{X_i}^m\,.
\end{gather}
Finally, we get
\[
	\frkProb(y) \geq \frac{\accProb(y)^m}{q^{m - 1}} -
	 \accProb(y) \cdot \left(1 - \frac{h!}{(h - m)! \cdot h^m}\right)\,.
\]
\qed
\end{proof}
\begin{lemma}\label{lem:jensen}
	Let $\RND{\zdv}$ denote the set from which $\zdv$ picks its coins at random.
	For each $\iota \in \range{1}{q}$ let $X_\iota \colon \RND{\zdv} \times
	H^{\iota - 1} \to [0, 1]$ be defined by setting $X_\iota(\rho, h_1, \ldots,
h_{\iota - 1})$ to
\[
  \condprob{i = \iota}{h_\iota, \ldots, h_q \sample H; (i, s) \gets \zdv(y, h_1,
    \ldots, h_q; \rho)}
	\]
	for all $\rho \in \RND{\zdv}$ and $h_1, \ldots, h_{\iota - 1} \in H$. Consider
  $X_\iota$ as a random variable over the uniform distribution on its domain.
  Then $\expected{X_\iota^m} \geq \expected{X_\iota}^m$.
\end{lemma}
\begin{proof}
	First we recall the Jensen inequality \cite{W:Weissten20}, if for some random
  variable $X$ holds $\abs{\expected{X}} \leq \infty$ and $f$ is a Borel convex
  function then
	\[
		f(\expected{X}) \leq \expected{f(X)}\,.
	\]
	Finally, we note that $\abs{\expected{X}} \leq \infty$ and taking to the
  $m$-th power is a Borel convex function on $[0, 1]$ interval. \qed
\end{proof}

\begin{lemma}[H\"older's inequality. Simplified.]\label{lem:holder}
	Let $x_i, y_i$, for $i \in \range{1}{q}$, and $p, q$ be real numbers such that
  $1/p + 1/q = 1$. Then
	\begin{equation}
    \label{eq:tightness}
		\sum_{i = 1}^{q} x_i y_i \leq \left(\sum_{i = 1}^{q}
      x_i^p\right)^{\frac{1}{p}} \cdot \left(\sum_{i = 1}^{q}
      y_i^p\right)^{\frac{1}{q}}\,.
	\end{equation}
\end{lemma}

\begin{remark}[Tightness of the H\"older inequality]
	In is important to note that Inequality (\ref{eq:tightness}) is tight. More
  precisely, for $\expected{X_i} = x$, $i \in \range{1}{q}$ we have
	\begin{gather*}
		\sum_{i = 1}^q x = \left(\sum_{i = 1}^{q} x^m\right)^\frac{1}{m} \cdot \left(\sum_{i = 1}^{q} 1^{\frac{m}{m - 1}}\right)^{\frac{m - 1}{m}} \\
		qx = \left(qx^m\right)^\frac{1}{m} \cdot q^{\frac{m - 1}{m}} \\
		(qx)^m = qx^m \cdot q^{m - 1} \\
		(qx)^m = (qx)^m\,.
	\end{gather*}
\end{remark}

\begin{lemma}
  \label{lem:root_prob}
  Let $\p{f}(X)$ be a random degree-$d$ polynomial over $\FF_p[X]$. Then the
  probability that $\p{f}(X)$ has roots in $\FF_p$ is at least $\infrac{1}{d!}$.
\end{lemma}
\begin{proof}
  First observe that there is $p^{d}$ canonical polynomials in $\FF_p[X]$.  Each
  of the polynomials may have up to $d$ roots. Consider polynomials which are
  reducible to polynomials of degree $1$, i.e.~polynomials that have all $d$
  roots. The roots can be picked in $\bar{C}^{p}_{d}$ ways, where
  $\bar{C}^{n}_{k}$ is the number of $k$-elements combinations with repetitions
  from $n$-element set. That is,
  \[
    \bar{C}^n_k = \binom{n + k - 1}{k}\,.
  \]
  Thus, the probability that a randomly picked polynomial has all $d$ roots is
  \begin{multline*}
    p^{-d} \cdot \bar{C}^p_d = p^{-d} \cdot \binom{p + d - 1}{d} =
    p^{-d} \cdot \frac{(p + d - 1)!}{(p + d - 1 - d)! \cdot d!} = \\
    p^{-d} \cdot \frac{(p + d - 1) \cdot \ldots \cdot p \cdot (p - 1)!}{(p - 1)!
      \cdot d!} = p^{-d} \cdot \frac{(p + d - 1)\cdot
      \ldots \cdot p}{d!} \\
    \geq p^{-d} \cdot {\frac{p^d}{d!}} = \frac{1}{d!}
  \end{multline*}
  \qed
\end{proof}

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\begin{document}

\section{Post-Newtonian formalism}

\marginpar{Monday\\ 2021-6-21, \\ compiled \\ \today}

We know that the scaling of the perturbation is typically
%
\begin{align}
h \sim \frac{R}{D} \frac{GM}{c^2 R} \qty( \frac{v}{c})^2
\,.
\end{align}

The assumptions for this linearized gravity quadrupole formula are:
\begin{enumerate}
    \item fixed background;
    \item \(v\) is small;
    \item background and source are independent (this is true for non-selfgravitating sources).
\end{enumerate}

However: this formula tells us that the most significant sources of GW are indeed self-gravitating, and not described by it!

So, how do we move forward, and get a formalism for compactness \(GM / c^2R\) approaching \(1/2\), such as BHs or NSs?

A strongly self-gravitating source will also typically have high velocity, because of the virial theorem:
%
\begin{align}
\qty( \frac{v}{c})^2 \sim \frac{2GM}{c^2 R} = \frac{R_s}{R}
\,.
\end{align}

The ``background'' for a strongly self-gravitating source is dependent on the source! It is not really a background.

The idea behind the PN formalism is to iteratively solve the Einstein equations, as an expansion in powers of \(\epsilon = v/c\).

We will still refer to sources with \emph{compact support} \(r<R\), and assume that \(T^{00} \) is the dominant component: \(\abs{T^{00} / T^{0i}} = \order{\epsilon }\) and \(\abs{T^{00 } / T^{ij}} = \order{\epsilon^2}\).

The expansion to order \(\order{\epsilon^{2n}} = \order{ (v/c)^{2n}}\) is called the \(n\)-PN order expansion.

So we can talk about 1PN, 1.5 PN, 2.5PN and so on.

A \emph{historical remark}: the PN formalism started with Einstein; Landau and Lifshitz already included it in their book, also De Sitter and Chandrasekhar worked on it.

Einsten-Infeld-Hoffmann developed the 1PN Lagrangian for the motion of \(N\) particles.
Peter and Hathews (1963).

The formalism was really nailed down systematically by Blanchet and Damour.
Also, Will and Wisemann worked on it in the 90s.

Naîvely, we'd work as such: calculate the motion to \(\epsilon^n\), and then use linear theory to calculate the GWs emitted.
This is wrong: the background is nontrivial, there is a complicated dependence through the Einstein equations.

WE emission subtracts energy to the motion, and GWs at a certain PN order are sources of GW at higher orders through the Isaacson tensor.

The equation \(\dv*{E}{t} = P_{gw}\) is an ansatz: this is used but it not guaranteed to hold at each PN order.

Let us start with the metric at 1PN order:
%
\begin{align}
g_{00} &= - 1 - 2\phi = - 1 + \frac{2 U}{c^2}  \\
g_{0i} &= 0  \\
g_{ij} &= \delta_{ij}
\,,
\end{align}
%
where \(U\) is the solution to the Newtonian Poisson equation:
%
\begin{align}
U &= \frac{G}{c^2} \int \dd[3]{y} \frac{T^{00} (y)}{\abs{x-y}}
\,.
\end{align}

Now, we have seen that different components of \(T_{\mu \nu }\) have different orders of \(c\), so different metric coefficients must be expanded up to different powers, determined by how they appear in the EFE.

If \(g_{00} \) is expanded to order \(\epsilon^n\), then \(g_{0i} \) must be expanded to order \(\epsilon^{n-1}\) because it has a power \(\epsilon \) intrinsically in it, and similarly \(g_{ij} \) must be expanded to the power \(\epsilon^{n-2}\).

\todo[inline]{Understand this better. Look at Maggiore. }

In terms of time reversal symmetry, \(g_{00}\) and \(g_{ij}\) are even while \(g_{0i}\) are odd.

\begin{figure}
\centering
\begin{tabular}{c|ccc}
Component & Newton (0PN) & 1PN & 2PN \\
\hline
\(g_{00} \) & \(-1 + g_{00}^{(2)}\) & \(g_{00}^{(4)}\) & \(g_{00}^{(6)}\) \\
\(g_{0i}\) & 0 & \(g_{0i}^{(3)}\) & \(g_{0i}^{(5)}\) \\
\(g_{ij}\) & \(\delta_{ij}\) & \(g_{ij}^{(2)}\) & \(g_{ij}^{(4)}\)
\end{tabular}
\label{tab:PN-orders}
\caption{}
\end{figure}

This is correct up to when we must include radiation reaction, but it turns out that that comes in at 2.5PN order.

So, in order to write the metric at 1PN order, we can match the EFE at each order taking into account the power counting.

Derivatives have orders \(\partial_{t} = \order{v} \partial_{i}\), while
%
\begin{align}
- \frac{1}{c^2} \partial_{tt} + \triangle = \qty[1 + \order{\epsilon }] \triangle
\,,
\end{align}
%
therefore the retardation effects are ``higher order''.

Therefore, the lowest PN solutions are typically given in terms of \emph{instantaneous potentials} (solution of Poisson equations).

This is expected: the PN expansion reads
%
\begin{align}
F(n) = F( t - \frac{r}{c}) = F(t) - \frac{r}{c} \dot{F} (t) + \frac{r^2}{2 c^2} \ddot{F}( t) + \dots
\,.
\end{align}

In Fourier terms:
%
\begin{align}
\widetilde{F} &\approx \widetilde{F} \qty(1 - \frac{r \omega }{c} + \frac{r^2\omega^2}{c^2} + \dots)  \\
&= \widetilde{F} \qty(1 - \frac{r}{\lambda } - \frac{r^2}{2 \lambda^2})
\,.
\end{align}
%
\todo[inline]{the lambdas are barred --- characteristic length.}

This means that this is also an expansion for nearby fields: the expansion is in small velocities \(\epsilon = v/c \ll 1\), as well as in the \emph{near-zone} \(r / \lambda  \ll 1\).

Therefore, the PN expansion is \textbf{not valid to compute GWs far away from the source}!

Let us define a \textbf{near zone} \(r \ll L\), but \(L \gg R\), such that \(r / \lambda \ll 1\), such that the aforementioned expansion makes sense.

We also have a \textbf{far zone} \(R < r < \infty \), and an \textbf{overlap zone} \(R < r < L\) in which the near and far zones overlap.

Back to the 1PN metric: we impose Harmonic gauge, so that \(\partial_{\mu } \qty(\sqrt{-g} g^{\mu \nu }) = 0\), and we start from the 0PN equation \(\triangle U = - (8 \pi G/c^{4}) T^{00}_{(0)}\).
Then, to 1PN order we get:
%
\begin{align}
\triangle g_{ij}^{(2)} &= - \frac{8\pi G}{c^{4}} \delta_{ij} T^{00}_{(0)}  \\
\triangle g_{0i}^{(3)} &= \frac{16 \pi G}{c^{4}} T^{0i}_{(1)}  \\
\triangle g_{00}^{(4)} &= \dots
\,,
\end{align}
%
so, assuming that \(T^{\mu \nu }\) is known, these can be solved in terms of Green's functions.
After all the calculations one finds
%
\begin{align}
\square_\eta V &= - \frac{4 \pi G}{c^{4}} \underbrace{\qty(T^{00} + T^{ii})}_{\mathclap{\text{active gravitational mass density}}}  \\
\square_\eta V^{i} &= \dots
\,,
\end{align}
%
where we have  the 1PN potential \(V\) which defines the metric:
%
\begin{align}
g_{00} &= -1 + \frac{2U}{c^2} + \frac{2 V^2}{c^{4}}  \\
g_{0i} &= - \frac{4 V_{i}}{c^3}  \\
g_{ij} &= \delta_{ij} \qty(1 + \frac{2 V}{c^2})
\,.
\end{align}

The Einstein-Infeld-Hoffmann Lagrangian is the \(N\)-particle one for this metric: the corresponding stress-energy tensor is
%
\begin{align}
T^{\mu \nu } = \frac{1}{\sqrt{-g}} \sum _a \dv{\tau_a}{t} m_a \dv{x_a^{\mu }}{t} \dv{x^{\nu }_a}{t} \delta (x - x_a)
\,.
\end{align}

The action reads
%
\begin{align}
S = - mc^2 \int \dd{t} \sqrt{- g_{00} -2 g_{0i} \frac{v^{i}}{c} - g_{ij} \frac{v^{i} v^{j}}{c^2}}
\,.
\end{align}

This can be written as \(L= L _{\text{Newt}} + L_{1PN}\).

If one takes \(N = 1\) they get the two-body problem, the EOM for a relativistic binary at 1PN order.
This solution can be mapped to a Newtonian problem and solved: this has been done by Damour and Deruelle in 86, which is important since it was applied to the Hulse-Taylor pulsar: the equation they found was
%
\begin{align}
\expval{\dot{\omega}} = \num{2.11353} \qty(\frac{m_1 + m_2 }{M_{\odot}})^{2/3} \SI{}{deg / yr}
\,.
\end{align}

The orbital decay has been observed very accurately, which can be a way to determine the binary mass.

What are the difficulties? We get Poisson equations in the form
%
\begin{align}
\triangle g_{\mu \nu }^{(n)} = \text{matter source} + \text{metric source } g_{\mu \nu }^{(n-1)}
\,.
\end{align}

The second term does not have compact support! The Poisson integrals used so far are not good solutions.
We need different boundary conditions.

For example, consider \(\triangle u = \rho = \const\). The Poisson integral is not a solution since \(u \neq 0\) at infinity: it diverges.

However, a solution exists: \(u(r) = - (1/6) \rho r^2\). We need other formal solutions to the Poisson equation.

How is this solved? We do it by inverting the Poisson equation with boundary conditions by a suitable procedure which involves regularization of the divergences and analytic continuation.
We will not explore this in detail, but it is good to know that the problem exists.

Another difficulty is the following:
the PN expansion of the metric potential is valid in the near zone but it blows up for large \(r \to \infty \).
It is in the form
%
\begin{align}
F (\epsilon , r) = \sum _{n} c_n (r) \epsilon^n
\,.
\end{align}

This is an expansion with two scales, and the series is \emph{not} uniformly convergent. This is analogous to perturbation theory in quantum mechanics: asymptotic series.

The residuals eventually blow up, but there is an optimum \(n\) for the expansion.
The solution is to perform a post-Minkowskian expansion in the far zone:
%
\begin{align}
F = \sum _{n} G^{n} F_n
\,,
\end{align}
%
and match the PN expansion in the near zone.

Also, there is the problem of \textbf{backreaction}, GW influence on the motion.
This breaks the power-counting.

When does it enter the game? We know that
%
\begin{align}
E = K + V \approx - \frac{V}{2} + V = \frac{V}{2} = - K = - \frac{M}{2} v^2
\,,
\end{align}
%
therefore \(\dot{E} \sim - M v \dot{v}\).

Further, we know that
%
\begin{align}
P_{gw} = \dot{E}_{gw} \approx \frac{GM^2}{c^{5}} \frac{v^{6}}{r}
\,,
\end{align}
%
therefore
%
\begin{align}
\dot{v} \approx \frac{GM}{r^2} \qty( \frac{v}{c} )^{5}
\,,
\end{align}
%
which gives the correct order of 5, meaning 2.5PN.

What are the \textbf{Relaxed EFE}?
We define
%
\begin{align}
\mathfrak{h}^{\mu \nu  } = \sqrt{-g } g^{\mu \nu } -\eta^{\mu \nu }
\,,
\end{align}
%
a  new field in the full theory.
We then rewrite the EFE in terms of it, using the gauge \(\partial_{\mu } \mathfrak{h}^{\mu \nu} = 0\) (equivalent to harmonic gauge).

The EFE then read
%
\begin{align}
\square \mathfrak{h}^{\mu \nu } &= \frac{16 \pi G}{c^{4}} \tau^{\mu \nu }  \\
&= \frac{16 \pi G}{c^{4}} \qty[-g T^{\mu \nu } + \tau^{\mu \nu }_{u} + \tau^{\mu  \nu }[ \mathfrak{h}^{\alpha \beta }]]
\,,
\end{align}
%
where \(\tau_u \) is the Landau-Liftschitz pseudotensor.
These are formally similar to the linearized equations, and in the weak-field limit \(\mathfrak{h}^{\mu \nu } = - \overline{h}^{\mu \nu }\).

All the nonlinearity is still there though, this makes no approximations: they are just made to look that way.

This \(\tau \) satisfies \(\partial_{\mu } \tau^{\mu \nu } = 0\) (notice the flat derivative).
Also, these are called ``relaxed'' because a solution of the relaxed equations alone does not imply the conservation of \(\tau^{\mu \nu }\): one needs to add the gauge.

The formal solution of these REFE is
%
\begin{align}
\mathfrak{h}^{\mu \nu } = - \frac{4G}{c^2} \int \dd[3]{y}
\frac{\tau^{\mu \nu }(t - \abs{x-y} / c, y)}{\abs{x-y}}
\,,
\end{align}
%
but we do not truly know \(\tau \); however we can solve this iteratively.

We start by finding a solution for the REFE in the far region with the multipolar formula:
%
\begin{align}
\mathfrak{h}^{\mu \nu } = \sum _{n} \mathfrak{h}_{(n)}^{\mu \nu } G^{n}
\,,
\end{align}
%
there for \(r > R\) we have the unspecified source multipole moments \(\mathfrak{h}^{\mu \nu }_{(n)}\).

Then we find a solution for the REFE in the near zone:
%
\begin{align}
\mathfrak{h}^{\mu \nu } = \sum _{n } \frac{1}{c^{n}} \mathfrak{h}^{\mu \nu }_{(n)}
\,.
\end{align}

Here, for \(r < L\), we have \(^{(n)} \mathfrak{h}^{\mu \nu }\), multipoles of \(\tau^{\mu \nu }\).
We match the two in the overlapping zone to determine \(\tau\).

Finally, we compute the TT solution far away, with a gauge transformation from harmonic to radiative coordinates: from \((I_L, J_L, W_L, \dots)\) to \((U_L,V_L)\), the radiative multipoles of \(h_{ij}^{TT}\).

The good thing is that this can all be done with a finite amount of terms, however in general one does not know how far the expansion can be pushed.

\todo[inline]{What does the overlap region look like?}

\end{document}


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\title{Stellar Structure and Evolution Numerical Project}
\author{Samuel Factor}
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\begin{document}
\maketitle
%\section{}
%\subsection{}

Code for this project can be found at \\
\url{https://github.com/sfxfactor/StellarNumericalProj}.

\section{ZAMS and Starting Model}
Figure \ref{fig:fig1} is a reproduction of Figure 1 of \citet{BKL}. As I used the assumptions in the paper, and not real models, the curves are strait and not curved as in \citet{BKL}. I used two prescriptions for the effective temperature to see if it would bring the curves into better agreement with the curves in \citet{BKL}. One prescription was from Equation 6 of \citet{BKL} and the other was from the Eddington luminosity,
\begin{equation}\label{eq:tedd}
T_{eff}=\left(\frac{L_{edd}}{4\pi R^2 \sigma_{SB}}\right)^{1/4}.
\end{equation}

\begin{figure}
\begin{center}
    \includegraphics[width=0.6\textwidth]{fig1.pdf}
    \caption{HR Diagram for high mass ($M=100-1000~M_\odot$ in increments of $100~M_\odot$) Pop III (left) and Pop I (right) stars. Different curves correspond to temperatures given by Equation 6 of \citet{BKL} and the Eddington luminosity, Equation \ref{eq:tedd}. }
    \label{fig:fig1}
\end{center}
\end{figure}

Using my code from Problem Set 2, I generated an $n=3$ polytropic model of a $100M_\odot$ Pop III star. The pressure and density structure is shown in Figures \ref{fig:pm} and \ref{fig:dm} respectively. To calculate a temperature I assumed $\beta=0.58$. This value comes from Equation 19.56 in KW$^2$
\begin{equation}\label{eq:beta}
\frac{1-\beta}{\mu^4\beta^4}=3.02\times10^-3\left(\frac{M}{M_\odot}\right)^2.
\end{equation}
The temperature is then
\begin{equation}\label{eq:temp}
T=\left((1-\beta)P\frac{3}{a}\right)^{1/4}
\end{equation}
This produces the temperature structure shown in Figure \ref{fig:tm}.

\begin{figure}
\begin{center}
    \includegraphics[width=0.6\textwidth]{pressure3p0.pdf}
    \caption{Pressure as a function of enclosed mass for an $n=3$ polytropic model of a $100~M_\odot$ Pop III star.}
    \label{fig:pm}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
    \includegraphics[width=0.6\textwidth]{density3p0.pdf}
    \caption{Density as a function of enclosed mass for an $n=3$ polytropic model of a $100~M_\odot$ Pop III star.}
    \label{fig:dm}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
    \includegraphics[width=0.6\textwidth]{temperature3p0.pdf}
    \caption{Temperature as a function of enclosed mass for an $n=3$ polytropic model of a $100~M_\odot$ Pop III star.}
    \label{fig:tm}
\end{center}
\end{figure}

\section{Evolving off the MS}
I worked together brielfy with Brianna and Ben on this part.

I tried very hard to get the \texttt{STELLAR} code to work with a $100~M_\odot$ Pop III (metal free) star though was unsuccessful. You can see my commit history on GitHub to look at the specific changes I tried, but I will outline the important ones below. To see the specific changes I made, go to the link at the begining of this text and click on ``commits" in the top left. You can then click on each commit individually and GitHub will highlight the changes made. If you would rather I attach to this document a copy of the code with changes highlighted send me an email and I can do that.

First, the initial conditions and polytropic index (in \texttt{polytr.inp}, \texttt{pmsstar.start}, \texttt{and pmsstar.inp}) and polytrope constants (in \texttt{polytr.F}) must be changed to match the $n=3$ $100~M_\odot$ metal free star we wish to model. I then changed the pressure prescription in \texttt{invstate.F} to fix $\beta$ at the value mentioned above (see Equation \ref{eq:beta}. Second I replaced all calls to \texttt{opacity} in \texttt{gi.F} with Thompson scattering opacity $0.2(1+X)$. In \texttt{atmos.F} I did the same thing but added H$^-$ opacity, $(1-X-Y)\rho^{1/2}T^9$. This was of course added in inverse according to $1/\kappa=1/\kappa_T+1/\kappa_{H^-}$. Finally I changed the luminosity prescription in \texttt{polytr.F} to Equation 3 of \citet{BKL}.

When looking at the output of the failed stellar run I noticed that the energy generation predicted by the code from the CNO cycle was much too small ($\sim 10^7$ erg/s when the luminosity is $\sim10^{40}$). If this could be brought up the model may have converged. I probably could have changed this part of the code to simply use Equation 3 of \citet{BKL} but that may have effected the chemical evoultion.

If the 100 $M_\odot$ Pop III star would have evolved past the MS, the model would not have been able to model the pulsations caused by pair instability. This causes the rapid burning of oxygen and silicon, stopping the collapse with an enormous explosion. For stars $\sim 140-260 M_\odot$ \citep{pop3}this completely disrupts the star leaving behind no remnent. For masses less than this but greater than $\sim 100 M_\odot$ the pulsations throw off large amounts of mass while the core collapses into a black hole. This is what would probably happen to the star we are modeling.

Since I could not get the $100~M+\odot$ Pop III star to work, I modeled a few solar metalicity stars and tried to push their mass up as high as I could. Shown in Figure \ref{fig:HR} are evolution tracks for 1, 3, 4, 5, and 10 $M_\odot$ solar metalicity stars. You will notice that the 5 and 10 $M_\odot$ models fail when they reach the MS. The \texttt{STELLAR} code seems to be very sensitive to the initial $R$ and $T_{eff}$ of the polytrope model. For example, the initial $R$ and $T_{eff}$ of the 3 and 4 $M_\odot$ (which evolved past the MS) were almost identical. If I changed the 4 $M_\odot$ initial conditions a small amount, the star would evolve to the MS and the model would fail, much like the 5 and 10 $M_\odot$ models.

\begin{figure}
\begin{center}
    \includegraphics[width=\textwidth]{HR.pdf}
    \caption{Evolutionary tracks for solar metalicity stars. The two highest mass models fail when they reach the MS, while the lower mass models evolve off the MS. }
    \label{fig:HR}
\end{center}
\end{figure}

Figures \ref{fig:pse}, \ref{fig:dse}, and \ref{fig:tse} show the initial and final pressure, density, and temperature structures for the 4 $M_\odot$ model, the highest mass model that evolved off the MS.

\begin{figure}
\begin{center}
    \includegraphics[width=0.6\textwidth]{pres.pdf}
    \caption{Pressure as a function of enclosed mass for a $4~M_\odot$ Pop I star at the beginning and end of its life.}
    \label{fig:pse}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
    \includegraphics[width=0.6\textwidth]{dens.pdf}
    \caption{Density as a function of enclosed mass for a $4~M_\odot$ Pop I star at the beginning and end of its life.}
    \label{fig:dse}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
    \includegraphics[width=0.6\textwidth]{temp.pdf}
    \caption{Temperature as a function of enclosed mass for a $4~M_\odot$ Pop I star at the beginning and end of its life.}
    \label{fig:tse}
\end{center}
\end{figure}

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