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author = {{Wette}, K.},
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author = {{Prix}, R.},
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\begin{document}
\title{MCMC follow-up methods for continuous gravitational wave candidates including
glitching and transient waveforms}
\author{G. Ashton}
\email[E-mail: ]{gregory.ashton@ligo.org}
\affiliation{Max Planck Institut f{\"u}r Gravitationsphysik
(Albert Einstein Institut) and Leibniz Universit\"at Hannover,
30161 Hannover, Germany}
\author{R. Prix}
\affiliation{Max Planck Institut f{\"u}r Gravitationsphysik
(Albert Einstein Institut) and Leibniz Universit\"at Hannover,
30161 Hannover, Germany}
\date{\today}
\begin{abstract}
We detail methods to follow-up potential CW signals (as identified by
wide-parameter space semi-coherent searches) leverging MCMC optimisation of the
$\mathcal{F}$-statistic. First, we demonstrate the advantages of such an
optimisation whilst increasing the coherence time, namely the ability to
efficiently sample an evolving distrubution and consider multiple modes.
Subsequently, we illustrate estimation of parameters and the Bayes factor which
can be used to understand the signficance of the candidate. Finally, we explain
how the methods can be simply generalised to allow the waveform model to be
transient or undergo glitches.
\end{abstract}
\pacs{04.80.Nn, 97.60.Jd, 04.30.Db}
\input{git_tag.tex}
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\maketitle
\section{Introduction}
A possible target for the advanced gravitational wave detector network of LIGO
and Virgo are long-lived periodic sources called continuous waves (CWs).
Rapidly rotating nonaxisymmetric neutron stars are potentially capable of
producing detectable CWs which last much longer than typical observation spans.
There exists three well known sources of the nonaxisymmetry: `mountains',
precession, and r-mode oscillations; each of which make a prediction for the
scaling between $\nu$, the NS spin frequency and $f$, the gravitational wave
frequency. In any case, observing neutron stars through their gravitational
wave emmission would provide a unique astrophysical insight and has hence
motivated numerous searches.
As shown by \citet{jks1998}, the gravitational wave signal from a
nonaxisymmetric source produces a strain in the detector $h(t, \A, \blambda)$;
where $\A{=}\{h_0, \cos\iota, \psi, \phi_0\}$ is a vector of the four
\emph{amplitude-parameters} (expressed in `physical coordinates') and
$\blambda$ is a vector of the \emph{Doppler-parameters} consisting of the
sky-location, frequency $f$, and any spindown terms required by the search.
CW searches typically use a fully-coherent matched-filtering methods whereby a
template (the signal model at some specific set of parameters) is convolved
against the data resulting in a detection statistic. Since the signal
parameters are unknown, it is usual to perform this matched filtering over a
grid of points. Three search categories can be identified: \emph{targeted}
searches for a signal from a known electromagnetic pulsar where the Doppler
parameters are considered `known'; \emph{directed} searches in which the
location is known, but not the frequency and spin-down (i.e. searching for the
neutron star in a supernova remnant which does not have a known pulsar); and
\emph{all-sky} searches where none of the parameters are considered known.
Searching over more parameters amounts to an increase in the dimension of the
search space. Since the density of grid points required to resolve a signal
scales inversely with total coherence time $\Tcoh$ (the span of data used in
a fully-coherent matched filter), wide-parameter searches (such as the all-sky)
with many search dimensions over long durations of data are computationally
demanding.
At a fixed computing cost, it has been shown (see for example \citep{brady1998,
prix2012}) that a semi-coherent search is more sensitive for unknown signals
than a fully-coherent search. While the exact details of how the search works
depends on the implementation, semi-coherent search work by splitting the total
observation span $\Tspan$ into $\Nseg$ segments (each lasting for $\Tcoh$) and
in each segment computes the fully-coherent detection statistic; the
semi-coherent detection statistic is then computed by some combination of all
segments summed at the same point in parameter space. Fundamentally, this gain
in sensitivity is because the width of a peak in the detection statistic due to
a signal is inversely propotional to the cohrence time: shorter coherence times
make the peak wider and hence the a lower density of templates. This idea was
first proposed by \citet{brady2000} along with the first implementation, the
`Stack-slide' search. Since then, several modifications such as the
`Hough-transform' \citep{krishnan2004, astone2014}, and the `Powerflux' method
(first described in \citet{allyskyS42008}) have been proposed, implemented and
applied to gravitational wave searches.
Wide parameter space searches produce a list of candidates with an associated
detection statistic which passes some threshold. In order to verify these
candidates, they are subjected to a \emph{followed-up}: a process of increasing
the coherence time, eventually aiming to calculate a fully-coherent detection
statistic over the maximal span of data. In essense, the semi-coherent search
is powerful as it spreads the significance of a candidate over a wider area of
parameter space, so a follow-up attempts to reverse this process and recover
the maximum significance and tightly constrain the candidate parameters. The
original hierarchical follow-up of \citet{brady2000} proposed a two-stage method
(an initial semi-coherent stage followed directly by a fully-soherent search.
However, it was shown in a numerical study by \citet{cutler2005} that allowing
an arbitrary number of semi-coherent stages before the final fully-coherent
stage can significantly improve the efficiency: ultimately they concluded that
three semi-coherent stages provide the best trade-off between sensitivity and
computational cost.
The first implementation of a two-stage follow-up was given by
\citet{shaltev2013} and used the Mesh Adaptive Direct Search algorithm for
optimisation. At the time of publication, this method was limited to two stages