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\begin{document}
\title{(*Out of date*) On the Fluxes and Rates of Fast Radio Bursts}
\author{Casey J. Law (UCB) et al.}
\date{\today}
\maketitle
\selectlanguage{english}
\begin{abstract}
We reanalyze rates implied by previous surveys with FRB detections to define homogeneous flux limits and FRB fluxes. We show that the fluxes of known FRBs are (in?)consistent with a Euclidean distribution, suggesting... If so, the FRB rate is $1.2\times10^4$ sky$^{-1}$ day$^{-1}$ brighter than roughly 1 Jy-ms.
\end{abstract}
\section{Introduction}
The "fast radio burst" (FRB) is a new class of transient found in a variety of single-dish pulsar surveys \citep{2007Sci...318..777L,2011MNRAS.415.3065K,2013Sci...341...53T,2014arXiv1404.2934S}. FRBs are identified by their large dispersion measure (DM), which has been observed as high as 1100 pc cm$^{-3}$, an order of magnitude larger than expected from the Galaxy. The simplest explanation for this large DM is that the bursts are dispersed by the intergalactic medium (IGM), implying that they originate at distances up to z$\sim$1.
If FRBs are cosmological, then they could be used to probe the intergalactic medium and study processes for their formation \citep[e.g., double neutron star mergers][]{2013PASJ...65L..12T}. However, terrestrial phenomena known as perytons have been discovered at the same telescopes finding FRBs \citep{2011ApJ...727...18B}. Perytons are impulsive radio transients with a width of tens of ms and an apparent DM of a few hundred, partially overlapping with characteristics expected of extragalactic radio transients. \citet{2014arXiv1402.4766K} suggest that perytons and FRBs are the same, terrestrial process seen in different optical regimes of the telescopes. It will be critical to distinguish these populations to be certain that FRBs are are astrophyiscal.
One way to understand their nature is to build statistical tests...
Euclidean distribution...
Rate constraint for future observing campaigns...
We are in the midst of a large survey with the Karl G.\ Jansky Very Large Array (VLA) to detect an FRB \citep[results in][]{law2014a}. The nondetection in that survey inspired us to reanalyze published rates in an attempt to make a reliable prediction for FRB rates for any given survey. Here, we present our estimate of the apparent flux distribution of FRBs to determine if they are consistent with an astrophysical population. We then use a Bayesian technique to estimate the FRB rate for a given telescope flux limit.
\section{FRB Flux Distribution}
\label{rate}
Generally, fast radio transient detections are quantified by a local SNR...
\citet{bsb} developed a formalism for estimating the flux limit of a transient survey based on instrument properties, pointing locations, and the distribution of Galactic DM \citep{2002astro.ph..7156C}. They use the radiometer equation to estimate the flux limit for a given pointing direction. Since most transient surveys were conducted over regions with roughly similar properties (sky temperature, DM, scattering), this technique can be used to measure the mean flux limit for a given survey and estimate the apparent flux of each detection.
These estimated flux limits depend on whether the pulse has a local or extragalactic origin, since extragalactic pulses are subject to Galactic dispersion and scattering. All FRB surveys with detections and the VLA survey were made in regions of small Galactic dispersion, so the flux limits are very weakly dependent ($<5\%$ effect) on our assumption of an extragalactic origin. Below, we assume an extragalactic origin for FRBs, consistent with the predominance of detections at high Galactic latitudes \citep{bsb,2014arXiv1405.5945P}.
With an estimate of the noise of each survey, we scale the apparent SNR to calculate the true flux. Then, for an assumed pulse width, we can estimate the range of fluxes a given survey could detect. Figure \ref{cumflux} shows the rate as a function of apparent flux. Since the primary beam introduces a fundamental ambiguity in the true FRB flux, any event could be brighter than we estimate here. On average, the whole population should be subject to the same bias, so the trend should be preserved in the limit of many FRBs.
**Fitting these distributions with a powerlaw gives...**
The FRB flux distribution tends to be flatter than the Euclideans slope of --3/2. One reason may be that the burst of \citet{2007Sci...318..777L} is not a FRB, but instead a peryton \citep{bsb}.
**Excluding Lorimer gives a best-fit powerlaw of...**
Systematic errors make it difficult to draw a strong conclusion on whether FRBs follow a Euclidean or the Lorimer burst is an FRB. This kind of analysis will allow a more robust conclusion as the number of known FRBs grows.
The new flux limit estimates show that published rates are consistent with single population with a roughly Euclidean flux distribution. This self-consistent rate is equal to that of \citet{2013Sci...341...53T}, $1.2\times10^{4}$ sky$^{-1}$ day$^{-1}$, but for a fluence limit of roughly 1.1 Jy-ms. Assuming all seven published FRB detections come from a single population, the 95\% bound on the rate is 0.6--2.3$\times10^4$ sky$^{-1}$ day$^{-1}$.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/frb-cumflux/frb-cumflux}
\caption{{Replace this text with your caption%
}}
\end{center}
\end{figure}
\section{Estimated Spectral Index}
\begin{equation}
P(\alpha0 | \alpha) = P(\alpha|\alpha0) * P(\alpha)
\end{equation}
\begin{equation}
P(\alpha|\alpha0) = \rm{primary beam bias function}
\end{equation}
and $P(\alpha)$ is proportional to detection probability (Gaussian beam shape).
\section{Bayesian Analysis of FRB Rate}
\begin{equation}
P(\theta|n_det) = P(n_{det}|\theta) * P(\theta) / P(A)
\end{equation}
\begin{equation}
\theta = [A, dt]
\end{equation}
Poisson likelihood for euclidean distribution:
\begin{equation}
P(n_det|\theta) = \lambda^{n_{det}} * exp{-\lambda}
\end{equation}
where $\lambda = A * at * \rm{flim}(dt)^{-3/2})$ and at=area*time on hi-lat sky.
Prior on dt is flat for A>0 and dt from 0.5 to 6 ms (intrinsic).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/frb-rateposterior/frb-rateposterior}
\caption{{Replace this text with your caption%
}}
\end{center}
\end{figure}
\section{Conclusions}
\label{conc}
Our reanalysis of the published rates to include a homogeneous definition of flux limit and shows that the published FRB detections at high Galactic latitude are roughly consistent with a Euclidean distribution, which would indicate a population uniformly distributed with distance. This analysis also highlights anomalous brightness of the burst of \citet{2007Sci...318..777L}, perhaps indicating that it is not an FRB. More detections are needed to strengthen these conclusions.
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