, Neal Katz
, David H. Weinberg
,
Jordi Miralda-Escudé
Lars Hernquist, Neal Katz, David H. Weinberg,
Jordi Miralda-Escudé
E-mail: lars@helios.ucsc.edu, nsk@astro.washington.edu, dhw@payne.mps.ohio-state.edu, jordi@sns.ias.edu
[1] University of California, Lick Observatory, Santa Cruz, CA 95064
[2] Sloan Fellow, Presidential Faculty Fellow
[3] University of Washington, Department of Astronomy, Seattle, WA 98195
[4] Ohio State University, Department of Astronomy, Columbus, OH 43210
[5] Institute for Advanced Study, Princeton, NJ 08540
cold dark
matter model. Our simulation reproduces many of the observed
properties of the Ly
forest surprisingly well.
The distribution of HI column densities agrees with existing data to
within a factor of 
two over most of the range from
to 
; i.e., from unsaturated
Ly
forest lines to damped Ly
systems. The equivalent width
distribution matches the observed exponential form with a
characteristic width 
Å. The distribution of
b-parameters appears consistent with that derived from QSO spectra.
Most of the low column density absorption arises in large, flattened
structures of moderate or even relatively low overdensity,
so there is no sharp distinction between the Ly
forest and the
``Gunn-Peterson'' absorption produced by the smooth intergalactic medium.
Our results demonstrate that a Ly
forest like that observed develops
naturally in a hierarchical clustering scenario with a photoionizing
background. Comparison between simulations and high-resolution QSO
spectra should open a new regime for testing theories of cosmic
structure formation.
quasars: absorption lines, Galaxies: formation, large-scale structure of Universe
The Ly
absorption systems in QSO spectra provide an excellent
observational probe of the high-redshift universe. At 
,
they easily outnumber all other detectable tracers of cosmic
structure. As a tool for testing cosmological theories, Ly
absorbers offer several advantages: they sample a broad range of
redshifts, they trace baryonic material over a wide range of densities,
temperatures, and ionization states,
they yield good statistical constraints on structure
because of their large numbers, and they may be primitive enough to
retain a more direct memory of their initial conditions than highly
nonlinear objects like galaxies and quasars. Several attempts have been
made to describe the formation of Ly
absorbers as arising within the
general gravitational instability theory of cosmic structure formation
(Arons 1972; Rees 1986; Ikeuchi 1986; Bond, Szalay, & Silk 1988;
McGill 1990).
With cosmological simulations that incorporate gravity and gas
dynamics, one can make detailed predictions of Ly
absorption in
a priori theoretical models (Cen et al. 1994; hereafter CMOR),
while circumventing many of the idealizations required by analytic
calculations.
This paper describes an analysis of the Ly
forest in a numerical
simulation of an 
cold dark matter (CDM) model,
with 
, normalized
to yield a present day rms mass fluctuation 
in
spheres of radius 16 Mpc (close to the value advocated by
White, Efstathiou & Frenk 1993).
Our investigation
is similar in spirit to that of CMOR, but we examine a different
theoretical model (
CDM instead of a low density model with
a cosmological constant), and we use a very different numerical
method, based on smoothed-particle hydrodynamics (SPH; see, e.g.,
Lucy 1977; Gingold & Monaghan 1977; Hernquist & Katz 1989;
Monaghan 1992). For this
application, SPH has the advantage of providing an unusually large
dynamic range in density, enabling us to study both low density,
Lyman-forest systems and dense, radiatively cooled, Lyman-limit and damped
Ly
systems in a single simulation. Resolving high densities
requires short time steps, so the price of this dynamic range is a
limitation in particle number, and hence in mass resolution.
Our results for Lyman-limit and damped Ly
absorption are described
in a companion paper (Katz et al. 1995, hereafter KWHM), so here we
focus on systems with HI column densities below 
.
The simulation volume represents a periodic cube of comoving size 22.222 Mpc
drawn randomly from a CDM universe with 
and baryon density
. The initial
conditions are identical to those employed by Katz, Hernquist, &
Weinberg (1992; hereafter KHW) and Hernquist, Katz, & Weinberg (1995),
but the present simulation differs from our previous efforts
in two important respects. First, we impose a uniform
photoionizing radiation field, with intensity
, where 
is the Lyman-limit
frequency, 
, and
We compute radiative cooling and heating rates assuming optically
thin gas in ionization equilibrium with this radiation field.
The redshift history in equation (1) is consistent with observational
constraints, but these have large uncertainties. The intensity 
is lower than most estimates; we shall comment on this point in § 3.
The second important difference from the KHW simulation is
a factor of eight improvement in mass resolution.
We use 
SPH particles and 
dark matter particles,
lowering the masses of individual particles to
and 
, respectively.
The softening length for gravitational forces is 20 kpc in
comoving coordinates (13 kpc equivalent Plummer softening).
The gas resolution varies from 
kpc in the highest density regions
to 
kpc in the lowest density regions.
Because of the larger number of particles, we have only
evolved this new simulation to z=2, which is adequate for many
aspects of absorption line studies.
A detailed description of the simulation code, TreeSPH , and the treatment of
cooling and ionization appears in Katz, Weinberg & Hernquist (1995).
Other results from this simulation are discussed by KWHM and by
Weinberg, Hernquist & Katz (1995).
Figure 1:
Examples of artificial spectra at z=2. Solid lines show
transmission against velocity along four random lines of sight. At
this redshift, the physical size of the periodic simulation box is
7.41 Mpc, corresponding to a Hubble flow of 1924.5 km/s. Dashed and
dotted lines show spectra along lines of sight displaced arbitrarily
from that of the primary spectrum by physical separations of
100 kpc and 300 kpc, respectively.
At each output, the simulation provides positions, velocities, densities, and temperatures of the baryonic fluid elements represented by SPH particles. It is straightforward to compute the neutral hydrogen absorption that would be produced in the light of a background QSO along an arbitrary line of sight through the simulation volume. First, we calculate the neutral fraction associated with each gas particle. We then spread each particle over its 3-dimensional SPH smoothing volume and take a line integral through the smoothed distribution to determine the neutral hydrogen mass, neutral mass-weighted velocity, and neutral mass-weighted temperature along the line of sight. Finally, we use these 1-dimensional profiles to calculate the optical depth as a function of frequency (see, e.g., the description in CMOR).
Figure 1 shows artificial spectra along four randomly
chosen lines of sight at redshift z=2. The physical size of the
periodic simulation box is 7.41 Mpc, corresponding to a Hubble flow of
1924.5 km/s. In each panel, the solid line
shows the transmission 
, where 
is the Ly
optical
depth. The translation from velocity v to wavelength 
is
simply 
Å, where z=2.
In each panel of Figure 1, the dashed line shows a
spectrum along a line of sight 100 kpc away from the ``primary''
spectrum represented by the solid line. The dotted line shows a
spectrum at 300 kpc separation (200 kpc from the dashed spectrum).
Many absorption features appear in both of the first two spectra, and
there are significant matches even for lines of sight separated by 300
kpc, though these are
often accompanied by substantial changes in the features'
depth or shape.
(For the typical absorption features in Figure 1,
the SPH smoothing
length of the gas particles is 
.)
As shown in Figure 1 of KWHM, much of the low
column density absorption arises in long, filamentary structures,
explaining the correlation between separated lines of sight (see also
Figure 1 of CMOR). Qualitatively, it appears that our results can
account for the large coherence scale found in absorption studies of
QSO pairs (e.g. Bechtold et al. 1994; Dinshaw et al. 1994, 1995),
though quantitative tests (e.g. Charlton, Churchill & Linder 1995)
are needed to assess the agreement or lack of agreement with recent
observations.
The standard technique for identifying and measuring properties of QSO
absorption lines involves fitting a spectrum with a superposition of
Voigt profiles. Such a procedure is difficult to automate, and,
moreover, our simulations imply that the underlying assumption that
individual features are well represented as Voigt profiles is not
valid in detail. We have therefore adopted a simpler
prescription for measuring line properties.
Moving along
a transmission spectrum (T vs. 
) in the direction of
increasing wavelength, we identify an absorption feature when the
transmission drops below some threshold 
and later rises back
above it. We determine the equivalent width W by integrating
between the down-crossing wavelength 
and the
up-crossing wavelength 
. We define the b-parameter to be
that of a Voigt profile that would have the same equivalent width W
and interval width 
between the two
threshold-crossing points (notice that the equivalent width here
includes only the contribution between 
and 
, and
not the tails outside this interval, and it is therefore not the
same as what is usually meant by the equivalent width of a line fitted
by a Voigt profile). We separately determine the column density from the
integrated optical depth between 
and 
, using the
relation 
, where 
is the oscillator strength
(Gunn & Peterson 1965).
Throughout this paper we adopt a transmission
threshold 
. At z=2, where line-blending is not severe, our
results are insensitive to the adopted threshold.
The above procedure, developed by one of us (JM), has also been applied to the simulations of Miralda-Escudé et al. (1995). We expect it to yield line properties similar to what would be obtained by Voigt-profile fitting, except at the lowest column densities, where any technique becomes sensitive to the assumptions used to separate blended features; at the same time, our column densities are physical quantities, as explained above, and may differ from the values obtained from line fits if the velocity distribution is non-gaussian. Our approach enables us to make a rough quantitative comparison between the simulation and existing absorption-line data. More precise comparisons with observational data and refinements of this procedure will be presented in future papers.
Figure 2:
Distribution of neutral hydrogen column densities.
The solid line shows the simulation results at z=2.
Points with error bars are taken from Petitjean et al. (1993);
we multiply their values by 1+z=3 to convert from
number of lines per ``absorption distance'' interval 
to
number of lines per redshift interval Delta z.
Figure 2 displays the column-density distribution
, the number of clouds per unit
redshift per linear interval of HI column density. Below
, we obtain 
by applying the procedure
described above to 1200 random lines of sight. Above this column
density we determine 
by creating a map of the HI column density
projected through the simulation cube and measuring the
fractional area above each column density (KWHM). Since our box is
only 7.4 Mpc deep and high column density lines are rare, the
absorption along any line of sight that has
in
the projected map is always dominated by a single absorber. Above
, it is necessary to include the effects of
self-shielding when computing neutral hydrogen fractions. Our
procedure for doing so, and a discussion of the physical nature of the
high column density systems, appear in KWHM. Observational data and
error bars in Figure 2 are taken from Table 2 of
Petitjean et al. (1993).
CDM is an a priori theory ``designed'' to explain galaxy
formation and large-scale structure in a universe with small microwave
background anisotropies. We did not adjust any parameters to obtain
the match in Figure 2.
There is a significant discrepancy with the Petitjean et al. (1993)
data for column densities near 
, which could reflect
either a failure of standard CDM or the presence in the real universe
of an additional population of Lyman-limit systems that are not
resolved by our simulation. However, the level of agreement
over eight orders of magnitude in 
is still remarkable.
The results in Figure 2 depend on our choices for the
mean baryon density, 
, and the intensity of the
ionizing background, 
, which is lower than most
estimates of the expected background from QSOs (e.g. Meiksin & Madau
1993). Changing these two quantities would change the neutral column
densities (and therefore shift the distribution in Figure 2
horizontally) proportionally to 
(neglecting any small
effects from changes in the gas temperature and from the self-gravity
of the baryons). Thus, if 
, we would obtain a similar
result to that in Figure 2 at low column densities for the more
reasonable intensity 
, which also agrees
with observations of the proximity effect (Bechtold 1994 and references
therein). If future observations confirm Songaila
et al. 's (1994) high value for the primordial deuterium abundance, then
the implied low baryon density coupled with the minimal 
from
observed QSOs may make it difficult for many cosmological models
(certainly this
one) to reproduce the observed level of neutral hydrogen absorption in
QSO spectra, unless many of the absorbers originate not from
gravitational collapse as modeled here but from
some other physical mechanism, e.g. pressure-confined clouds
in a shock-heated intergalactic medium
(Sargent et al. 1980; Ikeuchi & Ostriker 1986).
The simulated 
flattens below 
. The Petitjean et al. (1993) table does not extend below
this column density, but recent analyses of high-resolution spectra
from the Keck telescope suggest that 
continues to rise as a
power law down to 
or even
lower (e.g. Songaila, Hu & Cowie 1995; Hu et al. 1995).
Our identification procedure misses
low column density lines, either because they are blended with more
prominent absorption systems or, when they are isolated, because they
fail to lower the transmission below our threshold 
. An
assessment of this discrepancy will therefore require an analysis of
simulations and data using similar procedures. If real, the discrepancy
might reflect a flaw in standard CDM, a failure of our simulation
to resolve the weakest absorption systems,
or the presence of absorption lines caused by other phenomena.
Figure:
Distribution of equivalent widths of the absorption lines.
Solid histogram shows the simulation result at z=2.
Dotted line shows an exponential distribution with
characteristic width 
Å.
Figure 4:
Distribution of b-parameters.
Solid histogram shows the simulation result at z=2.
Dotted and dashed lines show the distributions that
Press & Rybicki (1993) obtained by fitting two different
functional forms to observed column density and equivalent width
distributions.
The distribution of equivalent widths of our lines, shown in
Figure 3, is almost perfectly described by an exponential
law, 
, with characteristic width 
Å, the same distribution as that derived by Murdoch et al. (1986)
for observed lines with equivalent widths greater than 0.32 Å. For
equivalent widths smaller than 0.2 Å, Murdoch et al. find a weak
overabundance of lines relative to this exponential distribution. We
do not find such an overabundance in the simulation; this discrepancy
is related to
the paucity of very low column density lines in our model,
as discussed above.
As emphasized by Press & Rybicki (1993, hereafter PR), the
equivalent width distribution is not controlled by the column density
distribution alone, because at fixed column density the value of the
b-parameter determines the location of the absorption line on the
curve of growth. The agreement in Figures 2 and
3 thus implies that our simulated lines must have a
reasonable distribution of b-parameters. PR proposed two different
functional forms for the b-parameter profile and determined the
free parameters of their distributions by fitting the observed 
and 
. Figure 4 compares the histogram of
b-parameters from our simulation analysis to the PR results. The
three sets of data agree fairly well with one another, and also with
the distribution obtained by Carswell (1989) from
direct line-fitting.
High-resolution, high signal-to-noise spectra from the Keck telescope should improve estimates of the b-parameter distribution in the near future. However, this property of the lines may be particularly sensitive to the procedure used to analyze the spectra (witness the continuing controversy over the reality of lines with b<20 km/s), so it will be especially important to analyze simulated and real spectra in the same manner. The b-parameters convey information about the physical structure of the absorbers (mainly the gas temperature and the velocity dispersion), so they may have considerable power to distinguish theoretical models.
The agreement between the simulated and observed line populations
suggests that our calculation provides a realistic general picture
for the origin of the Ly
forest, even if the cosmological scenario and
numerical realization are not correct in all their details.
It appears that the
Ly
forest can develop naturally in a hierarchical theory of structure
formation with a photoionizing UV background.
The high column density lines (
)
arise from radiatively cooled gas associated with forming galaxies (KWHM),
in collapsed, high density regions.
Low column density absorption (
)
is produced by systems characterized by an assortment of
scales and in various stages of
gravitational infall and collapse. As a result, the
low column density absorbers are physically diverse: they include
filaments of warm gas, caustics in frequency
space produced by converging velocity flows (McGill 1990),
high density halos of hot, collisionally ionized gas,
layers of cool gas sandwiched between shocks (CMOR),
and modest local undulations in undistinguished regions of the
intergalactic medium. Temperatures of the absorbing gas range from
below 
to above 
.
The ``typical'' low column density
absorbers --- to the extent that we can identify such
a class --- are flattened structures of rather low overdensity
(
), and have b-parameters that are often
set by peculiar motions or Hubble flow rather than thermal broadening.
Gravitational confinement (Rees 1986; Ikeuchi 1986),
pressure confinement by hot gas in halos (Bahcall & Spitzer 1969),
and ram-pressure confinement by infalling gas (CMOR)
all play significant roles, but because of their low overdensities,
most of the absorbers are far from dynamical or thermal equilibrium.
Many systems are
still expanding with residual Hubble flow, so their physical densities
and neutral fractions decrease with time.
We have not quantified the
evolution of the 
histogram in this paper because our threshold
algorithm tends to lose low column density lines to blending at higher
redshifts. However, the mean opacity of the simulated forest
climbs steadily with redshift, and the number of lines above a
specified column density increases with z when blending is not severe.
This evolution is driven primarily by the
increase in physical density with z, which raises the neutral
fraction, and hence the opacity, of individual absorbers. The 
distribution shifts to the right, not up.
Traditional searches for the Gunn-Peterson (1965) effect implicitly
assume a uniform intergalactic medium (IGM) punctuated by discrete
clouds; thus, absorption in identified lines is removed before seeking
a continuum depression. Hydrodynamic simulations, on the other hand,
reveal a smoothly fluctuating IGM, with no sharp distinction between
``background'' and ``Ly
clouds''. According to this interpretation,
one might even say that the Ly
forest is the Gunn-Peterson
effect, if by the latter one means absorption by neutral hydrogen in
the diffuse IGM. For a generalized form of the ``Gunn-Peterson
test,'' one can abandon a distinction between lines and background and
examine the full distribution function of HI optical depth, an
approach we will take in a later paper. Such an analysis could reveal
the signature of gas in underdense regions expanding faster than the
Hubble flow (Reisenegger & Miralda-Escudé 1995).
The mean transmission through our simulated IGM seems in reasonable
agreement with observations. Press, Rybicki & Schneider (1993),
analyzing the Schneider, Schmidt & Gunn (1991) QSO spectra,
find a mean Ly
optical depth 
,
implying a mean transmission 
0.85, 0.64, and 0.38
at z=2, 3, and 4, respectively. The simulation yields
0.83, 0.63, and 0.31 at the same redshifts.
The 
discrepancy at z=4 would be eliminated if
we kept the UV background intensity constant between z=4 and z=3
instead of following equation (1).
CMOR and Miralda-Escudé et al. (1995)
find that a flat, low density CDM model (
,
, 
, 
,
COBE-normalized) can also reproduce the essential properties of the
Ly
forest, including the abundance of low column density systems,
with 
and the same background
intensity 
as employed here.
Zhang, Anninos & Norman (1995)
achieve similar success with a COBE-normalized, 
CDM model,
using a higher background intensity.
(For related studies using approximate
treatments of hydrodynamics see Petitjean, Mücket & Kates 1995,
Mücket et al. 1995, and
Bi, Ge, & Fang 1995.) Clearly, much further work is needed to
determine which cosmological models can match the observed Ly
forest
and which cannot. The success of three somewhat different models in
reproducing 
suggests that the column density distribution alone
may be a rather blunt test, serving largely to constrain the parameter
combination 
within a specific theory. However, one
can check to see whether the required evolution of 
is physically
reasonable, and once this parameter combination is fixed, there is not
much freedom to adjust the predicted higher-order properties of the
Ly
forest --- b-parameters, line shapes, clustering, transmission
distributions, Ly
absorption, and so forth. We can, therefore,
expect that QSO absorption lines will assume an important role in
testing theories of cosmic structure formation.
We acknowledge helpful discussions with Jill Bechtold, Renyue Cen, Craig Hogan, Jeremiah Ostriker, Max Pettini, Tom Quinn, Martin Rees, Wal Sargent, David Tytler, and Ray Weymann. This work was supported in part by the Pittsburgh Supercomputing Center, the National Center for Supercomputing Applications (Illinois), the San Diego Supercomputing Center, the Alfred P. Sloan Foundation, NASA Theory Grants NAGW-2422, NAGW-2523, and NAG5-2882, NASA HPCC/ESS Grant NAG 5-2213, NASA grant NAG-51618, and the NSF under Grant ASC 93-18185 and the Presidential Faculty Fellows Program. DHW acknowledges the support of a Keck fellowship at the Institute for Advanced Study during early phases of this work, and JM acknowledges support from the W. M. Keck Foundation.
THE LYMAN-ALPHA FOREST IN THE COLD DARK MATTER MODEL
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