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\def\Lya{Ly$\alpha\ $}
\def\Lyb{Ly$\beta\ $}
\def\Lyg{Ly$\gamma\ $}
\def\Lyd{Ly$\delta\ $}
\def\Lye{Ly$\epsilon\ $}
\def\LCDM{$\Lambda$CDM\ }
\def\HI{\hbox{H~$\rm \scriptstyle I\ $}}
\def\HII{\hbox{H~$\rm \scriptstyle II\ $}}
\def\DI{\hbox{D~$\rm \scriptstyle I\ $}}
\def\HeI{\hbox{He~$\rm \scriptstyle I\ $}}
\def\HeII{\hbox{He~$\rm \scriptstyle II\ $}}
\def\HeIII{\hbox{He~$\rm \scriptstyle III\ $}}
\def\CIV{\hbox{C~$\rm \scriptstyle IV\ $}}
\def\NHI{N_{\rm HI}}
\def\NHeII{N_{\rm HeII}}
\def\cm2{\,{\rm cm$^{-2}$}\,}
\def\kms{\,{\rm km\,s$^{-1}$}\,}
\def\skm{\,({\rm km\,s$^{-1}$})$^{-1}$\,}
\def\kmsmpc{\,{\rm km\,s$^{-1}$\,Mpc$^{-1}$}\,}
\def\hmpc{\,h^{-1}{\rm \,Mpc}\,}
\def\mpch{\,h{\rm \,Mpc}^{-1}\,}
\def\hkpc{\,h^{-1}{\rm \,kpc}\,}
\def\ev{\,{\rm eV\ }}
\def\kel{\,{\rm K\ }}
\def\intunits{\,{\rm ergs\,s^{-1}\,cm^{-2}\,Hz^{-1}\,sr^{-1}}}
\def\ltsima{$\; \buildrel < \over \sim \;$}
\def\lsim{\lower.5ex\hbox{\ltsima}}
\def\gtsima{$\; \buildrel > \over \sim \;$}
\def\gsim{\lower.5ex\hbox{\gtsima}}
\def\etal{{ et~al.~}}
\def\aj{AJ}
\def\apj{ApJ}
\def\apjs{ApJS}
\def\mnras{MNRAS}

\journal{Preprint-00}

\title{Lyman-Alpha Forest Power Spectrum}

\author[A. Meiksin and M. White]{Avery Meiksin${}^{1}$, Martin White${}^{2}$ \\
${}^1$Institute for Astronomy, University of Edinburgh,
Blackford Hill, Edinburgh\ EH9\ 3HJ, UK \\
${}^2$Departments of Astronomy and Physics, University of California,
Berkeley, CA 94720, USA}

%\pubyear{2002}

\maketitle

\begin{abstract}
We look at the effects on the flux power spectrum of fluctuations in the
UV background due to a finite number of sources.
\end{abstract}

\begin{keywords}
methods:\ numerical -- intergalactic medium -- quasars:\ absorption lines
\end{keywords}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction} \label{sec:introduction}

Due in large part to insights gained from hydrodynamical simulations of
structure formation in the universe
(Cen \etal 1994; Zhang, Anninos \& Norman 1995; Hernquist \etal 1996;
Zhang \etal 1997; Bond \& Wadsley 1997; Theuns, Leonard \& Efstathiou 1998a)
the \Lya forest, as measured in high redshift Quasi-Stellar Object (QSO)
spectra, has emerged as one of our premier probes of the universe at
high redshift.
It is now believed that photoionization equilibrium for the gas results in a
tight density--temperature relation for the absorbing material with the
neutral hydrogen density proportional to the baryon density squared
(REFS).
Since pressure filtering is subdominant to thermal broadening (REF),
the neutral hydrogen density closely traces the total matter density
(squared) on the scales relevant to the forest ($0.1-10\,h^{-1}$Mpc).
The structure in QSO absorption thus traces, in a calculable way, slight
fluctuations in the matter density of the universe back along the line of
sight to the QSO, with most of the \Lya forest arising from overdensities
of a few times the mean density.

While much work has been done on the physics of the \Lya forest and the
cosmology which can be extracted, the role of fluctuations in the UV
background on the absorption properties of the \Lya forest have been
largely unexplored
(see Zuo 1992; Fardal \& Shull 1993; Croft et al. 1999, 2002;
 Gnedin \& Hamilton 2002).
In this paper, we investigate the effect of the fluctuations on the 2-pt
flux distribution (the power spectrum) of the \Lya forest.
This extends our earlier work on the 1-pt distribution function reported
in Meiksin \& White (2002; hereafter Paper I).

\section{Numerical simulations}

To investigate the effects of UV background fluctuations on the power spectrum
of the \Lya forest, we have performed simulations of 3 different cosmological
models within the $\Lambda$CDM family.  In each case we used pure dark matter
simulations, mimicking the temperature of the gas using a polytropic equation
of state and assuming gas and dark matter have the same spatial distribution.
This has been shown
(Gnedin \& Hui 1998; Petitjean, M\"ucket \& Kates 1995;
 Croft \etal 1998; Meiksin \& White 2001)
to produce results comparable to the full hydrodynamical simulations at
the level of 10 per cent.  The agreement is expected to be even better
on the large scales of interest to this work.
The parameters for the simulations can be found in Table~\ref{tab:models}
and a description of the parallel PM code used can be found in Paper I.
In each case we used $512^3$ particles and a $1024^3$ force mesh, in a
cubic box with (comoving) side length $25\,h^{-1}$Mpc.  Convergence tests,
described in the Appendix, suggest that this is adequate for our purposes.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
Model & $\Omega_M$ & $\Omega_\Lambda$   & $\Omega_b h^2$ & $h$   & $n$ &
$\sigma_8$ \\
\hline 
\hline
 1  & 0.30   & 0.70   & 0.02 & 0.70  & 1.05 & 0.970 \\ \hline
 2  & 0.35   & 0.65   & 0.02 & 0.70  & 0.95 & 0.885 \\ \hline
 3  & 0.40   & 0.60   & 0.02 & 0.55  & 1.10 & 0.902 \\ \hline
\end{tabular}
\end{center}
\caption{Parameters for the cosmological models.
$\Omega_M$ is the total mass density parameter, $\Omega_\Lambda$ the 
cosmological constant density parameter, $\Omega_b$ the baryonic mass fraction,
$h=H_0/100$\kmsmpc, where $H_0$ is the Hubble constant at $z=0$,
$n$ the slope of the primordial density perturbation power spectrum,
$\sigma_8$ the fluctuation normalization at $z=0$ in a sphere of
radius $8h^{-1}$ Mpc.}
\label{tab:models}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
$z$ & $\tau_{\rm eff}$ & $r_0$ & $N_0$ & $N_{\rm box}$ \\ \hline\hline
8.5 &   6.76   &  ?    & ?    & ?   \\
6.0 &   2.34   &  11.6 & 0.5  & 1.2 \\
5.5 &   1.82   &  23.9 & 5.1  & 1.4 \\
5.0 &   1.38   &  49.1 & 49   & 1.5 \\
4.0 &   0.73   &  179  & 3190 & 2.1 \\
\end{tabular}
\end{center}
\caption{Some of the output times for the simulations.
For each redshift the mean opacity, $\tau_{\rm eff}$, the attenuation length,
$r_0$ in comoving $h^{-1}$Mpc, and the mean number of sources per attenuation
volume, $N_0$, is listed for Model 1.  The last column, $N_{\rm box}$, shows
the number of sources in the simulation volume of side $25\,h^{-1}$Mpc
(comoving).}
\label{tab:aout}
\end{table}

Given a set of final particle positions and velocities, we compute the spectra
as follows.  First the density and density-weighted line-of-sight velocity
are computed on a grid (using CIC interpolation) and Gaussian smoothed
using FFT techniques in order to adequately sample the velocity field.
This forms the fundamental data set.
A regular grid of $32\times 32$ sightlines is drawn through the box,
parallel to the box sides.  Along each sightline we integrate (in real space)
to find $\tau(u)$ at a given velocity $u$.  Specifically we define
\begin{equation}
  \tau(u) = \int dx\, A(x) \left[ {\rho(x)\over\bar{\rho}} \right]^2
            T(x)^{-0.7} b^{-1} e^{-(u-u_0)^2/b^2}
\label{eqn:taudef}
\end{equation}
where $u_0=xaHL_{\rm box}+v_{\rm los}$ and $b=\sqrt{2k_B T/ m_{\rm H}}$ is the
Doppler parameter, where $m_{\rm H}$ is the mass of a hydrogen atom.
The flux at velocity $u$ is $\exp[-\tau]$. The integration variable $x$
indicates the distance along the box in terms of the expansion velocity across
the box.
In evaluating Eq.~(\ref{eqn:taudef}) we assume a power-law relation between
the gas density and temperature
\begin{equation}
  T \equiv T_0 \left( {\rho\over\bar{\rho}} \right)^{\gamma-1}
\end{equation}
In practice, $T_0$ and $\gamma$ will depend on the reionization
history of the IGM, but we shall fix $T_0=2\times 10^4$K and $\gamma=1.5$
throughout as described in Paper I.

In terms of the baryon density parameter $\Omega_b$, hydrogen baryonic mass
fraction $X$, the metagalactic photoionization rate $\Gamma_{-12}$, in units
of $10^{-12}\, {\rm s^{-1}}$, and the number $f_e$ of electrons per
hydrogen nucleus, the function $A(x)$ is given by:
\begin{equation}
  A=18.0f_e\left(\frac{X}{0.76}\right)\left(\frac{\Omega_bh^2}{0.02}\right)
  \Gamma_{-12}^{-1}L_{\rm box}(1+z)^5,
\label{eq:Adef}
\end{equation}
where $\Gamma_{-12}$ is evaluated at position $x$, $L_{\rm box}$ is in Mpc
and $b$ in eq.~(\ref{eqn:taudef}) is in \kms.
For both constant and spatially varying $A(x)$ we scale the final $\tau$ so
that the mean opacity follows (REF)
\begin{equation}
  \tau_{\rm eff} \equiv -\ln\left\langle e^{-\tau}\right\rangle
                   = 0.0028\ (1+z)^{3.46}
  \qquad .
\end{equation}
Values for $\tau_{\rm eff}$ at each of the outputs of the simulation can
be found in Table \ref{tab:aout}.

\section{UV background from sources}

We need to assign luminosities to halos to compute $A(x)$.  It is most
natural to take the $N_{\rm box}$ most massive sources in the box, in
decreasing order of mass, and assign them luminosities.  This can be done
in a number of ways.

First we could equate the cumulative mass distribution to the cumulative
luminosity distribution to assign luminosities.

Second, we could take the most massive halos in the simulation and measure
their velocity dispersions then convert these to black hole masses using
(Tremaine et al.~2002)
\begin{equation}
  M_{BH} = 10^8\,M_\odot
    \left( {\sigma\over 200{\rm km}\,{\rm s}^{-1}} \right)^4
\end{equation}
The Eddington luminosity scales as the mass of the QSO, so we could take
$L\propto\sigma^4$ also.

Finally, we could notice that $N_{\rm box}$ is a very small number.  So we
could just pick flux ratios between the first few most massive sources and
set the overall scaling to get $\tau_{\rm eff}$ right.

\begin{figure}
\begin{center}
\leavevmode \epsfxsize=3.3in \epsfbox{f1.eps}
\end{center}
\caption{A simulated spectrum from a `typical' sightline at z=5 from Model 1.
The upper panel shows the spectrum with a uniform ionizing background
($A(x)\equiv $const, dashed) and where the UV background comes from a single
source situated at the center of the box (dotted).
The lower panel shows $0.1\times\rho/\bar{\rho}$ (dotted) and $A(x)/\bar{A}$
(dashed) for the single source case.}
\label{fig:spectrum}
\end{figure}

\section{History}

{\bf
Avery, you know the history and context here better than I do.  We
probably need to have some discussion somewhere in the paper about
what has gone before and where we differ/do better.  I've put the
following in as a placeholder for now.
}

Zuo (1992) and Fardal \& Shull (1993) did randomly distributed clouds and
$J$ fluctuations.
Croft et al.~(1999) looked at whether UVBG fluctuations and uniform IGM
could give spectra like what is observed (no) and how big an effect
putting fluctuating $J$ into observed spectra was (their Fig.~12).
Croft et al.~(2002) took halos in the simulation, set $L\propto$SFR and
added a uniform component to look at effect on spectra and $P(k)$ at
$z\sim 3$ (their Figs.~11 and 12).
Finally Gnedin \& Hamilton (2002) discuss fluctuating UVBG using hydro
simulations with radiative transfer.
They show UVBG fluctuations are a small effect on the power spectrum,
but cause an underestimate of the photo-ionization rate by ${\cal O}(20\%)$.


\bigskip
\section*{acknowledgments}

M.~White was supported by the NSF and NASA.
Parts of this work were done on the IBM-SP2 at the National Energy Research
Scientific Computing Center.

\appendix

\section{Convergence Tests} \label{sec:tests}

Even including only gravitational physics, simulations of the \Lya forest
are restricted by two size constraints.
High spatial resolution is required to resolve the structure of the
absorbers, while a large box size is required both to capture the
large-scale power which will affect the velocity widths of the absorption
lines as well as to obtain a fair sample of the universe.
Although a small box may be adequate to resolve the line structure, if the
box is too small the fluctuations corresponding to the scale of the box will
become nonlinear, in which case the results are no longer representative of
the cosmological model simulated.
These numerical effects have been investigated in different contexts by a
number of authors (see references in McDonald 2001 for example).

To investigate these effects of numerical resolution further we chose a
relatively large box, $30\,h^{-1}$Mpc (comoving) on a side, and simulated
the concordance cosmology using $128^3$, $256^3$ or $512^3$ particles
(corresponding to $256^3$, $512^3$ and $1024^3$ mesh cells).
At the high redshifts of interest to us the fundamental mode of the box is
safely in the linear regime, suggesting that the simulation represents a fair
sample of the universe.
To isolate the effects of numerical resolution from sample variance, we
sampled the initial conditions starting at low-$k$ and moving `outwards'
to higher $k$ so that the initial phases and amplitudes of the common modes
were identical from simulation to simulation.

\begin{figure}
\begin{center}
\leavevmode \epsfxsize=3.3in \epsfbox{fA.eps}
\end{center}
\caption{The (real space) 1D matter power spectrum in the concordance
cosmology at $z=5$ from our resolution sequence with $128^3$, $256^3$
and $512^3$ particles and force meshes of $256^3$, $512^3$ and $1024^3$.
The solid line shows the predicted 1D spectrum, using the
Peacock \& Dodds (1996) fitting formulae for the 3D matter power spectrum,
Eq.~\protect\ref{eqn:pk1d}, and a Gaussian filtering of 3 mesh cells (FWHM).}
\label{fig:dk_seq}
\end{figure}

\begin{figure}
\begin{center}
\leavevmode \epsfxsize=3.3in \epsfbox{fB.eps}
\end{center}
\caption{The 1D flux power spectrum in the concordance cosmology at $z=5$ from
our resolution sequence with $128^3$, $256^3$ and $512^3$ particles and force
meshes of $256^3$, $512^3$ and $1024^3$.  The box size in each run is
$30\,h^{-1}$Mpc.  These results indicate that our highest resolution runs are
converged at better than the 10\% level (in power).}
\label{fig:pk_seq}
\end{figure}

The 3D matter power spectrum (not shown) exhibits the expected behaviour,
with the higher resolution simulations having progressively more small-scale
power.  At $z=5$ the fundamental mode is safely in the linear regime, with
$\Delta^2(k_{\rm fund})\simeq 0.04$, and the mesh scale is non-linear
with $\Delta^2(k_{\rm mesh})>10$.  The fitting formula of
Peacock \& Dodds (1996) underestimates the non-linear power by
${\cal O}(50)$ per cent.
Fig.~\ref{fig:dk_seq} shows the 1D matter power spectrum, in real space.
This is an integral of the 3D spectrum over wavenumbers greater than the
1D wavenumber under consideration:
\begin{equation}
  {kP(k)\over\pi} = k \int_{k}^\infty {dy\over y^2}\ \Delta^2_{3D}(y)
  \qquad .
\label{eqn:pk1d}
\end{equation}
We find that the numerical results are relatively well fit by the
Peacock \& Dodds (1996) formalism, if we filter the 3D power spectrum,
$\Delta^2_{3D}(k)$, with a Gaussian filter of FWHM $3.2$ mesh cells
(plotted as the solid line for the highest resolution run)
to approximate the finite force resolution of the PM code.
Note that missing high-$k$ power in the lower resolution simulations biases
the 1D power spectrum low over the whole $k$-range plotted since much of
the long-wavelength power comes from aliasing of 3D short wavelength power.
Finally we show in Fig.~\ref{fig:pk_seq} the power spectra of the flux
produced by these 3 runs.
{}From these results we infer that, for the relatively high redshifts of
interest, a $30\,h^{-1}$Mpc box with $512^3$ particles and a $1024^3$ force
mesh produces spectra converged to better than the $10$ per cent level.
Similar conclusions were found by McDonald (2001), though our convergence is
not quite at the level quoted therein.
The convergence becomes fractionally better if we increase $T_0$ or decrease
$\gamma$.  It becomes only slightly worse if we lower $T_0$ to $10^4$K.
A further sequence reducing the box size (not shown here) suggests that we can
safely reduce the volume simulated to a box of side $25\,h^{-1}$Mpc if we
are interested in the \Lya forest at high redshift.  This makes the mesh scale
slightly smaller, capturing the features in the forest slightly better, thus
we have used $25\,h^{-1}$Mpc thoughout.

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