Lensing by Lyman Limit Systems: Determining the Mass to Gas Ratio

20 Pages·0.24 MB·English

by  Ariyeh Maller

Lensing by Lyman Limit Systems:
Determining the Mass to Gas Ratio
Ariyeh H. Maller, Tsafrir S. Kolatt
Racah Institute for Physics,The Hebrew University, Jerusalem, 91904 ISRAEL
1
0
0
Matthias Bartelmann
2

n Max-Planck-Institut fu¨r Astrophysik, P.O. Box 1317, D-84741 Garching, GERMANY
a
and
J

1
1
George R. Blumenthal

1
UCO/Lick Observatory, Department of Astronomy & Astrophysics, University of California,
v
2 Santa Cruz, CA 95064
8
1
1
0
1
0 ABSTRACT
/
h
We present a new method to determine Υ , the total mass-to-neutral gas ratio in
p HI
- Lyman-limits systems. The method exploits the relation between the neutral hydrogen
o
column density and the magnification of background sources due to the weak gravita-
r
t tional lensingthatthesesystemsinduce. Becauseweak lensingdoesnotprovideadirect
s
a measure of mass, one must use this relation in a statistical sense to solve for the aver-
:
v age Υ and its distribution. We use a detailed mock catalog of quasars (sources) and
HI
i
X Lyman-limit systems (lenses) to demonstrate the applicability of this approach through
r our ability to recover the parameter ΥHI. This mock catalog also allows us to check for
a
systematics in the method and to sketch its limitations. For a universal constant Υ
HI
and a sample of n quasars, we obtain an unbiased estimate of its value with 95%
QSO
confidence limits (independent of its actual value) of Υ ±140 105/n .
HI QSO
p
Subject headings: galaxies: formation—quasars: absorption lines—dark matter —gravitational
lensing
1. Introduction tic disks (Wolfe et al. (1986); see, however,
Rauch et al. (1997)), while lower column den-
The Lyman limit systems of neutral hy-
sity systems are more likely to correlate with
drogen column density N > 2 × 1017cm−2
HI galactic or proto-galactic halos (Bahcall &
are believed to be closely related to galax-
Spitzer 1969; Churchill et al. 1996; Mo &
ies at all redshifts. Among these the rare
Miralda-Escude 1996; Abel & Mo 1998). Ha-
damped Ly−α systems (DLAS) with N >
HI los are the basic building blocks of structure
2×1020cm−2 areusuallyidentifiedwithgalac-
formationwhenitcomestodarkmatter(DM)
1
gravitational collapse and virialization in any between the results from the simulations and
hierarchical scenario. Hence, low column den- the observed column density function added
sity Lyman limit systems (hereafter simply credibility to this picture.
LLS) provide a unique opportunity to study The appeal to N-body simulations can be
the connection between the DM component computationally expensive, and it is unclear
and the gas component at different redshifts whether the higher resolution needed for the
whilethetwoarestillinhydrodynamicalequi- LLS can be achieved today in a large cos-
librium. mological simulation (Gardner et al. 1999).
The relation between the DM component Thesimulationsalsomakeassumptionsonthe
and the gas component in LLS can becharac- physical conditions and the equation of state
terized in terms of the “mass-to-gas” ratio, in of the gas, which also limits their use. It is
analogy to the mass-to-light ratio commonly therefore desirable to construct an indepen-
used for galactic disks. In the large scale dent method by which the mean mass-to-gas
structure picture this ratio represents the ratio and other parameters can be measured.
mean value of the biasing factor, which con- Here we propose to use gravitational lens-
nects the spatial distribution of the baryons ing to determinethe matter content, baryonic
(gas as well as stellar disks) with that of the fraction and ionization state of the LLS. The
dark matter. Generally, the biasing may be absorption features, being a direct probe of
a linear relation between the gas over-density the neutral hydrogen content in LLS, provide
and the DM over-density, or a more elab- complementarydataneededtorevealthecon-
orated functional connection (e.g., Dekel & nection between the matter and HI densities.
Lahav 1999). Biasing can be deterministic,
The absorption features are well known
but in reality it is more likely to be stochas-
and well measured (see Petitjean 1998; Rauch
tic. Itis possibletocalculate such parameters
1998, for recent reviews). Their interpreta-
as the mean biasing in any scheme in which
tion, although subject to uncertainties in fit-
there is a functional relation between the bias
ting the line profile, is rather straightforward
and the overdensity.
and can be expressed in terms of their col-
In order to determine the mass-to-gas ra- umn density N , their equivalent width, and
HI
tio, one has to observe at least two quantities, their velocity parameter. The lensing due to
where one of them is directly related to the the moderate column densities of the LLS has
gas density andtheother probestheDMden- not previously been considered because of the
sity field. This has been in part applied by low signal expected.
Croft et al. (1998) who presented a method
So far only lensing by DLAS (N > 2 ×
HI
by which they recover the DM density fluc- 1020cm−2) has been studied for its effect on
tuations from the Ly−α forest. A detailed
the magnification bias for the background
comparison to N-body simulations provided
source population, as well as on the distri-
them with the “experimental” bias parame-
bution of the observed impact parameters
ter and spared the need to solve for it inde-
with respect to the center of the lens (Smette
pendently. Based on these simulations, Katz
et al. 1997; Bartelmann & Loeb 1996; Smette
etal.(1996) andHernquistetal.(1996) previ-
1995). Lensing by column densities of N <
HI
ously realized that Ly−α absorption systems 1020cm−2 seemed unreasonable to consider
consist of a variety of physical objects: col-
because of the very small effect it has on
lapsing halos, cuspsin phasespace for density
the background sources; a typical quasar is
shells, and galactic disks. The nice agreement
only magnified by about one percent. The-
2
oretically, only the overall effects of lensing In a large sample, the magnification of
on quantities like the luminosity function of point sources can have a strong statistical ef-
quasars have been considered (Hamana et al. fect. Treating all quasars as a set eliminates
2000; Pei 1995a), while observationally a pos- in part the need to know the absolute mag-
sibleexcessinthenumbercountsofmetalline nitudes of the individual sources. The mag-
systemshasbeendetected (vandenBerketal. nification shifts the luminosity function of the
1996). combinedpopulationinacalculable way. The
Other manifestations of weak lensing have lensed population can be “de-magnified” (as-
been detected and used to map the mass dis- suming a mass-to-gas scheme) on an individ-
tribution in clusters andlarger scale structure ualbasis. Thecriterionforasuccessfulcorrec-
(see Bartelmann & Schneider 2000 for a re- tion is the return to the statistical properties
view). The standard lore in weak gravita- of an un-lensed population.
tional lensing is to use the ellipticity as in- Every statistical approach requires a large
duced by the intervening gravitational shear data base in order to maximize the signal
field in order to resolve the projected mass to noise ratio. Fortunately in the upcoming
distribution along the line of sight (Kaiser years such a data base will become available
& Squires 1993). Due to the small level of when the quasar sample from the Sloan Dig-
magnificationandtheunavailableabsolutelu- ital Sky Survey (SDSS) is completed. This
minosity of the sources, the observed flux of sample should serve as an excellent target list
the background sources has never been used by itself or for follow-up observations. Com-
in weak lensing studies. For this reason, bined with high resolution spectra for a se-
point sources, for which ellipticities cannot be lective sub sample from ground- and space-
measured, cannot be used to map interven- based telescopes it will allow coverage of
ing mass concentrations in this fashion. The Ly−α absorption features over a wide range
weak magnification of point sources has usu- of redshifts.
ally been considered only as a nuisance in Here, we propose a recipe for an obser-
extracting information from the background vational project to determine the mass-to-
sources (e.g., in the case of supernovae Ia neutral-hydrogen ratio in LLS,check different
Wambsganss et al. 1997; Porciani & Madau bias schemes, or apply a similar technique to
2000). the mass-to-metal-line ratio in these systems.
Until now, a key aspect of thephenomenon In §2 we define our notation and present
of weak lensing by LLS has been neglected, the formalism of the technique. In order to
the fact that the same systems that cause describe the main features of the proposed
the magnification are also responsible for the methodin§3, weintroduceatoy modelwhich
troughs in the continuum emission of the avoids many complications and which allows
background source. Both are proportional to us to outline the ingredients of the technique.
projected surface densities along the line of The next section (4) is devoted to the gener-
sight. The two are therefore correlated. It is ation of a realistic mock quasar and Lyman-
this correlation, and the fact that it holds for limit system catalog which enables various es-
eachindividualsystemthatwewishtoexploit timates of the method’s biases and errors, as
in order to extract statistically the relation well as the functional dependence of the ob-
between the total mass of these systems and tained signal-to-noise ratio on the input pa-
their HI content. In addition, the absorption rameters. In the same section, we use Monte-
system gives the redshift of the lens. Carlorealizations ofthatcatalog anddescribe
3
our procedure for measuring the mass-to-gas and f is a geometry factor that relates the
g
ratio. The method is implemented, and its possible different geometries of the baryons
advantages and limitations are discussed. We and the dark matter (e.g., gas disk in a spher-
concludethissectionbyadditionalchecksthat ical dark matter halo, etc.). The fraction of
verify that the measured effect indeed comes neutral hydrogen is defined by
from gravitational lensing. In section 5 we
ρ N
HI HI
sketch the preferred procedure for actual ob- YHI = = , (5)
ρ N
H H
servations, consider extensions of themethod,
and estimate additional effects like dust. Sec- whereinthelastequalityweexplicitlyassume
tion6summarizesthemethod’sstrengthsand that the volume density fraction is identical
weaknesses. to the surface density fraction. The geometry
factor, f is of order unity if LLS can be ade-
g
2. The Theory quately described by a roughly spherical halo
with the DM and baryons well mixed.
Itis common ingravitational lensing toex-
We can define a projected mass-to-gas ra-
pressthemagnificationintermsoftwodimen-
tio,
sionless quantities; κ, the convergence and γ,
Σ
the shear. The magnification µ is then given Υ ≡ , (6)
HI
m N
by HI HI
whichis determinedbythesedifferentfactors.
1
µ = , (1) Observationssuggest valuesof f ∼ 0.1 (Kirk-
(1−κ)2−γ2 b
man et al. 2000), f ≃ 0.75, and Y in the
H HI
(Narayan&Bartelmann1999;Schneideretal. range of 10−2−10−3 for z = 0−5 (Prochaska
1992). In the weak lensing limit, where κ ∼ 1999; Haardt & Madau 1996). Hence from
γ ≪ 1 this equation becomes Eqs. (4) and (6) we expect ΥHI to be in the
range of 102 −104.
µ = 1+2κ. (2)
In a similar manner it is possible to define
a mass-to-ionic-column-density ratio that re-
The convergence, κ, is the projected surface
lates the magnification to a metal line column
density, Σ, along the line of sight divided by
density. Thisintroducesanadditionalparam-
the critical surface density,
eter, the metallicity of the gas, or, more ac-
c2 D D curately, the specific abundance of a certain
l ls
Σ = D ; D = , (3)
cr 4πG eff eff D metal in a specific ionization state. Through-
s
out this paper, except in §5, we will focus
where D are the angular diameter dis-
l,s,ls specifically on HI absorption. However, ev-
tances between observer, lens and source.
erything we will present can also be applied
For absorption systems the column density to any observed metal line.
in neutral hydrogen, N , along the line of
HI With the above definitions we can now ex-
sight is related to Σ via
press the magnification of a quasar by an ab-
f sorption system along its path as
H
N = Y f f Σ, (4)
HI HI b g
mH ΥHINHImH
µ = 1+2 . (7)
Σ
where f the fraction of baryons that are in cr
H
hydrogen, m is the hydrogen mass, f is the Ifwecouldmeasurethemagnificationofasin-
H b
fraction of total mass made up by baryons, gle quasar, we could determine Υ for every
HI
4
observed absorption system. However since range 18 < logN < 19.72 is taken to be
HI
there is no way to know a quasar’s un-lensed
luminosity,wemustexploitthecorrelationbe- dn= 0.096N−181.5(1+z)1.55dN18dz (8)
tween magnification and the observable N
HI where N = N /(1018cm−2). The choice of
to determine Υ statistically. 1 18 HI
HI
these parameters will be justified in section
For a statistical sample of quasars, mag-
4.1. Throughout this paper we adopt the cos-
nification has two effects. It causes the ob- mological parameters H = 65kms−1Mpc−1,
0
served solid angle to correspond to a smaller
Ω = 0.3 and Ω = 0.7.
M Λ
solid angle in the source plane, and therefore
Using these distribution functions, we con-
the density of quasars with fluxes intrinsically
struct a sample of quasars and the associ-
above a fluxlimit decreases. However, magni-
ated absorption line systems along their lines
fication also increases the flux received from
of sight. The magnification value for each
each quasar and therefore more quasars are
quasar is calculated according to the systems
detected above the flux limit. The net effect
in its foreground, and the apparent luminos-
depends on the slope of the luminosity func-
ity of the quasar is correspondingly corrected
tion of quasars. An increase in the number
by applying Eq. 7. The dilution of quasars
of quasars is detected if the slope is steeper
by solid-angle magnification is taken into ac-
than −2 and a decrease otherwise. In the sit-
count by randomly removing quasars. This is
uation we will be considering, magnification
performed for values of Υ of 0, 1000, and
will tend to increase the number of quasars HI
2000 (we loosely denote the un-lensed case by
brought into a sample.
Υ = 0).
HI
In the next section, we illustrate the
A flux cutoff at 45 µJy (B brighter than
essence of the effect studied here with a toy
20m) is then applied to the magnified sample.
model. Most of its simplifications will be
Quasars which initially were below this flux
droppedin later sections, to which readers fa-
limit are magnified by their LLS, exceed the
miliar with gravitational lensing might want
flux limit, and introduce more high column
to skip right away.
density systems into the sample.
3. Simple Model Figure 1 shows the quasar LF in this toy
model. The upper lines depict the LF of all
Inthissection weintroduceasimplemodel quasars in the sample with the three Υ val-
HI
that demonstrates the basic concepts of the ues. Since the fraction of significantly mag-
method for a statistical recovery of ΥHI with- nified quasars is negligible, the magnification
out the complications that will be introduced (different Υ values) has no detectable effect
HI
in the following, more realistic case. on the total LF, in accord with the conclu-
Let us assume all quasars are at z = 3 and sions of Pei (1995a). The lower three lines
their luminosity function (LF) is given by a depict the LF as derived from a subset of
power law of slope −3.0. The number of ab- quasars with at least one absorption system
sorption systems as a function of column den- with logN > 19.0. For this subset it is ev-
HI
sity and redshift with column densities in the ident that the LF is shifted to the right with
increasing Υ , i.e. increasing magnification.
HI
1Strictly speaking we determine the cross-section
weighted average of ΥHI. 2Throughout this paper logarithmic column densities
are assumed to be measured in cm−2.
5
log (S/µJy)
10
1.8 2.0 2.2 2.4
s ϒ =2000
O0 ϒHI=1000
-1010 ϒϒϒHHHIII===21 00 00 000 1000 QS100 ϒHHII= 0
) r
-13 LO • ) peHI500
-c N
p 0
G 1
g
Φ ( (lo
d
1 n/
1 d
-0
1
18.0 18.5 19.0 19.5
log N
13.00 13.10 13.20 13.30 13.40 13.50 13.60 10 HI
log (L/L )
10 O •
Fig. 2.— Thedistributionofcolumn densities
Fig. 1.— The observed luminosity function of observed absorption systems for values of
of our toy model for all quasars and for ΥHI of 0, 1000, and 2000. The number of ab-
quasars with at least one absorption system sorption systems observed per 1000 quasars is
with logN > 19.0. Each is plotted for three plotted versus the log of the column density.
HI
valuesofΥHI(0,1000,2000). Theupperthree For higher values of ΥHI more high column
lines lie on top of each other as weak lensing density systems are brought into the sample
has a negligible effect on the total quasar lu- becauseofthemagnification biasontheback-
minosity function, as was previously found. ground sources.
However the magnification by gravitational
lensing is discernible in the subset of quasars
at the distribution in column density of the
that contain high column density absorption
absorption systems. Figure 2 shows this dis-
systems.
tribution for different values of Υ . The ef-
HI
fect of increasing Υ is immediately obvi-
HI
In principle, the relative heights of the dif- ous; more high column density systems are
ferent lines in Fig. 1 are proportional to Υ broughtabovethefluxlimitduetomagnifica-
HI
and could be used to find its value. Unfortu- tion bias. The distribution also deviates from
nately, inpracticethereisonlyonerealization the original power law as the effect becomes
of the Universe so only one of the lines (for progressively more pronounced with increas-
each NHI cutoff) in Fig. 1 is observed. In or- ing NHI.
der to find out what is the relative amplitude, Themagnification whichcauses thelinedi-
thelowest line(un-lensed,ΥHI = 0)shouldbe vergence in Fig. 2 is not only a function of
inferredfromtheexpectedfractionofhighcol- Υ but also of the lens’ redshiftthrough D
HI eff
umn density systems, based on lower column in Eq. (3). We demonstrate this dependence
density systems and Eq. (8). Thus the ef- in Figure 3 where the probability of encoun-
fect of magnification is to change the fraction tering an absorption system (here normalized
of observed quasars with high column density to integrate to unity) as given by Eq. (8) is
absorption systems. weighted by the redshift dependent factors of
This effect is more clearly seen if instead Eq. (3) that contribute to the magnification.
of looking at the luminosity function we look TheresultantfunctionisamodificationofEq.
6
0.8 0
0
P 3
waebsighting function
weighted P
y 0.6 abs Os
nsit QS
De 00 00
bability 0.4 z per 101
o d
Pr 0.2 n/ ϒ =2000
d30 ϒϒHHII== 1 0 0 00
HI
0.0
2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 1
z 1+z
Fig. 3.— Magnification weighted probability
of an absorption system as a function of ab- Fig. 4.—Thedistributionofobservedabsorp-
sorption redshift (thick solid line) for quasars tions systems with column densities greater
at z = 3. The unweighted probability of an then 1019cm−2 in redshift for Υ values of 0,
HI
absorption system (Eq. 8, thin dotted line is 1000 and 2000 for quasars at z = 3. Since
weighted by D of Eq. (3) (dashed line). All lensing depends on the redshift of the lens,
eff
three functions are normalized so that they systems that are at redshifts between 0.5 and
integrate to unity. The combination of mag- 2.0 arepreferentially broughtinto the sample.
nification weighting which peaks at interme-
diate redshifts, and a monotonically increas-
approaches the quasar redshift. A deviation
ing probability of encountering an absorption
fromtheredshiftpower-lawdependenceofEq.
system with redshift,results ina ratherbroad
(8) is thus a statistical tool by which the am-
redshift range in which lensing by absorption
plitude of the magnification can be inferred.
systems is important.
Theseeffectsdodependtosomeextentoncos-
mology.
(8) (again normalized to integrate to unity) Thetoymodelwehaveusedherehighlights
with a broad distribution in redshift. This the main effects of treating the magnification
means that lensing by absorption systems is by intervening absorption line systems in a
relevant over a wide range of redshifts. statistical manner. The luminosity function
This redshift dependence distinctly affects of quasars is shifted to higher luminosities,
thecounts ofabsorptionsystems perunitred- a shift that can be observed if a cut by col-
shiftforafixednumberofquasars. Thiseffect umn density along their line of sight is per-
depends on the value of the magnification co- formed. However, it is cumbersome to get
efficient, Υ . Figure 4 shows the number of a reference line for a non-lensed luminosity
HI
absorption systems with logN > 19.0 as a function, whereas it is more straightforward
HI
function of absorption redshift. The increase to detect the change in the column density
in number of high column density absorption function and its redshift distribution.
systems happens preferentially between red-
shifts of 0.5 and 2.0. This is a clear sign that
the increase is due to lensing for which the
magnification becomes very small as the lens
7
4. Recovery method for Υ • Observational errors in N and flux
HI HI
measurements introduce various biases
The results of the toy model as presented
to the method. These biases must be
in the last section portrayed the basic effects
treated to evaluate their systematic ef-
of lensing on the statistical distributions of
fects on the recovery result.
quasarsandtheirabsorptionsystems. Inreal-
ity, however, therearevariousdeviationsfrom For all these reasons one must appeal to
the idyllic picture of the toy model. In order Monte-Carlo realizations of a realistic mock
to assess our ability to use these altered dis- catalog of quasars and absorption systems.
tributions for the recovery of ΥHI we need to The recovery method is demonstrated and
take all of these complications into account. tested using these Monte-Carlo realizations.
Only a detailed imitation of a realistic sit-
uation will allow us to draw conclusions re- 4.1. The Mock Catalog
garding the success or failure of the recovery
We generate a sample of quasars according
method.
totheirobservedfluxandredshiftdistribution
The main complications that should be in-
dn /dSdz. Although this is the quantity
QSO
cluded in a more realistic picture are;
which is directly observed, it is not reported
in the literature. Instead it is converted to a
• The observed luminosity function of
luminosity function by binning under an as-
quasars resembles a double power law,
sumed cosmology. We thus start with a fit to
ratherthanasingleone,andthusanad-
the luminosity function and recover the ob-
ditional scale (the “knee”) is added to
served flux function. We start with the dou-
theproblem(Boyleetal.1988;Hartwick
blepower-law fittotheluminosity functionas
& Schade 1990; Warren et al. 1994;
given by Pei (1995b).
Boyle et al. 2001).
dnQSO Φ∗/Lz
• Theobservedquasarsspanalargerange Φ(L,z) = = (9)
of redshifts. This together with the dLdV (L/Lz)β1 +(L/Lz)β2
redshift dependence of the LF “knee” where Lz = L∗exp[−(z−z∗)2/(2σ2)] de-
results in a complicated observed flux
scribes the redshift evolution of the lumi-
function. nosity function, Φ∗ = 234(comoving)Gpc−3,
• An observed sample is flux limited, but β1 = 1.83, β2 = 3.70, L∗ = 2.63 × 1013L⊙,
quasarsmaybemagnifiedabovetheflux z∗ = 2.77 and σ = 0.91. We convert this to
the observed flux function by
limit dependingon their absorption sys-
tems and the value of ΥHI. dnQSO dnQSOdLdV
= 4π . (10)
dSdz dLdV dS dz
• Frequently, more than one absorption
system intersects the line of sight to a For the probability of encountering an ab-
quasar. Theseevents,especiallyforhigh sorption system P(N ,z) we assume this
HI
column densities, contribute a substan- function is separable in N and z as observa-
HI
tial signal to the shifts of the distri- tions seem to imply. Storrie-Lombardi et al.
bution functions. Multiple magnifica- (1994) found that the number density of LLS
tion events mustthus beincluded in the increases as (1+z)1.55. Many authors (Tytler
analysis. 1987; Press & Rybicki 1993; Hu et al. 1995;
Lu et al. 1996; Kirkman & Tytler 1997; Kim
8
et al. 1997) showed that the distribution in Armed with these relations we turn to gen-
column density is well fit by N −1.5 over ten erate a mock catalog of quasars and their
HI
decades. Thuswetaketheprobability density associated absorption systems. For the rare
to be high (though still weak) magnification events
which lift quasars above the flux limit of the
dP = 0.096N−1.5(1+z)1.55dN dz (11)
18 18 survey, there is a need for a large sample
of quasars. We have generated a sample of
with N = N /(1018cm−2) as in section
18 HI 8×106 quasars and assigned them fluxes and
3. We only consider absorption systems with
redshifts according to the flux/redshift distri-
column densities between 1018cm−2 and 5×
bution, dn /dSdz (Eqs. 9, 10).
1019cm−2. At the lower limit the magnifica- QSO
In generating this catalog we have in mind
tioninthemostfavorableconfiguration(max-
the specifications of the SDSS and apply a
imal D and high Υ )is only 0.01 sowe can
eff HI
flux limit of 45µJy to approximate the SDSS
safely neglect lower column density systems
limit of 19m.7 in the g′ band. The target for
when considering lensing.
thephotometricerrorsof theSloansurvey are
The upper limit is set because we would
2%. Basedonwhatiscommonlyquotedinthe
like all the systems we consider to contain gas
literature we estimate that the errors in mea-
in a similar ionization state (Y ), i.e, similar
HI
suring N to be around 15%. This catalog is
Υ . This upper limit must be set lower than HI
HI
used in the next section to exercise and test
the column density of damped Ly−α (N >
HI
our proposed method for the Υ recovery.
2×1020cm−2) that consist of mostly neutral HI
gas (Prochaska & Wolfe 1996). A kinematic
4.2. The Recovery Procedure
analysis of damped Ly−α by Maller et al.
(2001) suggests that cold gas disks at z ≃ 3 Our goal is to present an unbiased recov-
extenddowntoNHI ∼ 4×1019cm−2,andlocal ery method for ΥHI, assumed at first to be
observations of the average surface densities a universal constant. Relaxation of this as-
of gas disks yield N & (few)×1019 (Zwaan sumption will be discussed in §4.3. The re-
HI
et al. 1997). Taking inclination effects into covery is judged by the confidence levels that
account this implies that absorption systems can be put on the value of ΥHI as function of
with column densities below 5×1019cm−2 are its value and the number of quasars above a
dominated by hot gas. We therefore take fluxlimitthatareneededinordertogetthese
this value as a conservative upper limit. This levels. Amagnification signal is manifested in
choice of an upper limit also safely keeps us the change of thedistribution functions of the
away from strong lensing events that would column density: the relative number density
not be described correctly by Eq. (7). of different column density systems and their
redshift distribution. For P(N ,z) there are
Our results for Υ turn out to be quite HI
HI
three independentvariables: the amplitude of
insensitive to a change in the assumed cos-
the function, the slope of the separable N
mology, the quasar flux function, or the less HI
partofthefunction,andtheslopeofthesepa-
constrained column density function. Their
rablez partofit. Notethattheamplitudecan
primary effect is a slight modification in the
only be used once. We assume that the true
number of quasars required to reach a result
underlying slopes can be read off the low col-
with the same level of confidence. In light
umn density systems that produce only neg-
of the current uncertainty of over an order of
ligible magnification, and then extrapolated
magnitude in the value of Υ , we will ignore
HI
to higher column densities. Justifications for
these possible modifications.
9
250 each absorption system along its line of sight.
s
O ThusNT hasastrongercorrelationwithmag-
S HI
Q nification than the column densities of indi-
0
0 200 vidual absorption systems. Another advan-
0
er 1 tage of the NTHI quantity is that its measure-
p mentdoesnotrequireassumptionsoncosmol-
)
TNHI 150 ogy. Figure5showsP(NTHI)forthreevaluesof
0
g1 ΥHI. There is a dramatic dependence on ΥHI
d(lo ϒϒHI==12000000 inthedistributionP(NTHI)forhighertotalcol-
n/ 100 ϒHI= 0 umn densities. This should be compared to
d HI
Fig. 2 where the effect of lensing was much
18.6 18.8 19.0 19.2 19.4 19.6 less evident.
log NT
10 HI
Fig. 5.— The distribution of the total pro- 4.2.2. Rejection of the null hypothesis
jected HI column density, NT , for values of
HI We first examine the rejection levels of the
Υ of 0, 1000 and 2000. Without lensing
HI null hypothesis that the sample under consid-
the distribution would be that of the Υ = 0
HI eration is not magnified at all. Under this
case. Because of the magnification bias, sys-
assumption any deviation from the expected
tems with high values of NT are magnified
HI P(N ,z) is purely because of Poisson errors
HI
over the flux limit. The error bars are the
due to a finite sample and observational er-
Poisson errors from a Monte-Carlo realization
rors. As discussed above the relevant dis-
with n = 105.
QSO tribution function to maximize the lensing
detection is P(NT ) rather than P(N ,z).
HI HI
this assumption and ways to check its validity The equivalent projection along the z axis,
will be presented in §4.4. P (z) = dN P(N ,z) can in principle
NHI HI HI
also be usedR, as we demonstrated in Fig. 2.
4.2.1. The total column density along the Let us first phrase the statistics we use in the
line of sight conventional χ2 terms and then modify it to
Let us now introduce a new variable, NT , get a statistic which is more appropriate for
HI
the problem under consideration.
the total column density of HI along the line
of sight. For the j-th quasar in the sample, ForasampleofnQSOquasars,letnmj odel(NTHI)
be the number of expected absorber in bin j
NTHIj = NHIj,i, (12) spanning the range [NTHI,NTHI +∆NTHI]. This
Xi number depends, of course, on the redshift
where the sum is over all systems along the distribution of the quasars in the sample.
line of sight. Note that this is not weighted Let n˜model(z) be the number of expected ab-
i
by D . The probability distribution of this sorbers in bin i that spans the redshift range
eff
variable, P(NT ), folds in the redshift/flux [z,z +∆z]. If the two observed values of the
HI
distribution of the quasars, the column den- same quantities are nobs(NT ) and n˜obs(z) we
j HI i
sity/redshift distribution of the Lyman-limit can construct the two χ2 functions
systems, and in addition the Poisson process
2
of multiple systems along the line of sight. (nmodel(NT )−nobs(NT )
j HI j HI
χ2 = h i
The total magnification experienced by a 1 ǫ2 +ǫ2
quasar is the product of the magnification by Xj poisson,j error,j
10