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Measuring the Radiative Histories of QSOs with the Transverse Proximity Effect

Measuring the Radiative Histories of QSOs with the Transverse Proximity Effect
Measuring the Radiative Histories of QSOs with the Transverse Proximity Effect

a r X i v :a s t r o -p h /0405505v 1 25 M a y 2004

Received 2004February 20;Accepted 2004May 21

Preprint typeset using L A T E X style emulateapj

MEASURING THE RADIATIVE HISTORIES OF HIGH-REDSHIFT QSOS WITH THE

TRANSVERSE PROXIMITY EFFECT

Kurt L.Adelberger 1

Carnegie Observatories,813Santa Barbara St.,Pasadena,CA 91101

Received 2004February 20;Accepted 2004May 21

ABSTRACT

Since the photons that stream from QSOs alter the ionization state of the gas they traverse,any changes to a QSO’s luminosity will produce outward-propagating ionization gradients in the surrounding intergalactic gas.This paper shows that at redshift z ~3the gradients will alter the gas’s Lyman-αabsorption opacity enough to produce a detectable signature in the spectra of faint background galaxies.By obtaining noisy (S:N ~4)low-resolution (~7?A )spectra of a several dozen background galaxies in a

R ~20′

?eld surrounding an isotropically radiating 18th magnitude QSO at z =3,it should be possible to detect any order-of-magnitude changes to the QSO’s luminosity over the previous 50–100Myr and to measure the time t Q since the onset of the QSO’s current luminous outburst with an accuracy of

~5Myr for t Q <~

50Myr.Smaller ?elds-of-view are acceptable for shorter QSO lifetimes.The major uncertainty,aside from cosmic variance,will be the shape and orientation of the QSO’s ionization cone.This can be determined from the data if the number of background sources is increased by a factor of a few.The method will then provide a direct test of uni?cation models for AGN.

Subject headings:quasars:general —galaxies:high-redshift —intergalactic medium —black hole

physics

1.INTRODUCTION

The 108-M ⊙black holes that lie at the heart of nearby bulge galaxies are believed to have accreted much of their mass in their youths when they shone brie?y and brightly as QSOs (e.g.,Yu &Tremaine 2002).A great deal could be learned about the physical processes that produce super-massive black holes if we could observe how their bright-nesses varied during this time.For example,a black hole that radiates at a fraction l ≡L/L E of its Edding-ton luminosity and accretes mass with radiative e?ciency

?≡L/˙Mc

2requires t S ~M BH /˙M BH ~4×107l (?/0.1)yrs to change its mass signi?cantly.Showing that QSO out-bursts have a much shorter duration would con?rm the popular belief (e.g.,Haehnelt &Kau?mann 2000;Cava-liere &Vittorini 2000;Wyithe &Loeb 2002;Di Matteo et al.2003)that super-massive black holes build up their masses through numerous accretion episodes.Various the-oretical attempts to explain the observed correlation be-tween a black hole’s mass and its bulge’s velocity disper-sion (e.g.,Silk &Rees 1998;Adams,Gra?,&Richstone 2001;Burkert &Silk 2001;Miralda-Escud′e &Kollmeier 2004)each postulate di?erent physical mechanisms that quench the QSO’s luminosity when it reaches a certain level.To the extent that these mechanisms operate on di?erent time scales,measuring the duration of QSO out-bursts should help us distinguish between them.

The challenge is to ?gure out from a few short nights or decades of observations how a QSO’s brightness changed over the preceding few million years.Investigators have resorted to a number of indirect schemes.By making as-sumptions about the link between QSOs and either local black holes or massive potential wells at the redshift of observation,several authors have deduced that individ-ual QSOs are unlikely to have shone for fewer than 1Myr or more than 100Myr (e.g.,Haehnelt,Natarajan,&Rees 1998;Richstone et al.1998;Martini &Weinberg 2001;Hosokawa 2002).Unfortunately the uncertain underlying

assumptions do not seem to allow a much more precise limit on the typical QSO lifetime.In any case statistical arguments like these will never be able to tell whether a QSO’s total radiative lifetime of (say)107yr occurred in one contiguous chunk or was instead split among numer-ous shorter bursts,though this distinction could be crucial in developing a physical picture of black-hole formation.This paper describes a method that is somewhat less indirect and that can provide a rough indication of how any given QSO’s luminosity varied over the 50–100Myr preceding the time of observation.The simple idea behind the method is illustrated in Figure 1.Although we cannot detect the photons a QSO emitted in the past,we can de-tect their e?ect on its surroundings.As hydrogen-ionizing photons from a QSO propagate outwards,they destroy neutral hydrogen in the intergalactic medium and reduce the number of Lyman-αabsorption lines in the spectra of background objects.The change in the Lyman-αopacity at radius r should therefore provide some indication of the QSO’s luminosity at the earlier time t ~r/c .The rest of the paper works out this simple idea in more detail.§2presents a brief review of the relevant intergalactic physics and justi?es two assumptions that will be needed later.§§3and 4work out the e?ect of changes in the QSO’s luminosity on background galaxies’spectra.The next two sections discuss the signi?cance with which the e?ect can be detected:§5treats the uncertainties from a theoretical point-of-view and concludes that cosmic variance will be the primary problem,while §6discusses the spectroscopic exposure times that will be necessary on a 10m telescope and argues that neither continuum removal nor interstel-lar absorption lines will be a major obstacle.The results from the preceding sections are brought together in one place in §7,which presents a sample analysis of a simu-lated QSO with a known radiative history.Some readers may wish to skip directly to this section.§8discusses the time resolution that can be achieved with this technique.

1

2Measuring the Radiative Histories of QSOs

My main conclusions are reviewed and criticized in §9.I should state at the outset that this is not the only method for constraining the radiative histories of individ-ual QSOs.Readers may judge for themselves the relative merits of the alternatives that are listed (e.g.)in the ex-cellent review by Martini (2003).Nor is the idea behind the method new.Jakobsen et al.(2003)have used it,for example,to estimate an age of t >107yr for the QSO Q03022-0023from the lack of absorption lines in the spec-trum of a neighboring QSO.Schirber,Miralda-Escud′e ,&McDonald (2004)and Croft (2004)applied a similar anal-ysis to a larger sample of QSO pairs and inferred signif-icantly shorter lifetimes.What is new,as far as I know,is the assumption that the background sources will be nu-merous faint galaxies rather than a single bright QSO.This complicates the analysis in a number of ways but yields a considerably more detailed view of the foreground QSO’s radiative

history.

Fig. 1.—Schematic view of the underlying concept.QSOs emit H-ionizing radiation that will destroy some of the HI in the inter-galactic medium out to a maximum radius ct Q ,where t Q is the time since the onset of QSO activity.HI in the intergalactic medium pro-duces absorption lines in the spectra of background objects.The absorption is represented by a broken line between the objects and the observer;breaks in the line mark positions with signi?cant HI absorption.By obtaining spectra of numerous galaxies that lie be-hind a high-redshift QSO,one can (a)map the spatial distribution of intergalactic HI around the QSO,(b)use the shape of the region with lowered HI absorption to estimate the rough geometry of the in-tergalactic volume that has been hit by the QSO’s radiation,and (c)deduce the QSO’s lifetime and the level of anisotropy in its emission.

2.PRELIMINARIES

The low observed level of Lyman-αabsorption from in-tergalactic gas at z ~3implies that the gas must be al-most completely ionized,with perhaps only one neutral H atom per million (Gunn &Peterson 1965).As a result the typical hydrogen ionizing photon will travel far before it is absorbed,~50proper Mpc (?M =0.3,?Λ=0.7,h =0.65;Madau,Haardt,&Rees 1999),and one can safely assume that intergalactic gas is optically thin on length scales signi?cantly smaller.Radiation from distant sources will permeate this optically thin gas at a roughly uniform level J bg ,ionizing residual hydrogen atoms at a rate Γbg γn HI where n HI is the neutral hydrogen density and Γbg

γ≡

13.6eV

dE 4πJ bg (E )σi (E )/E (1)is an integral over energy of the photon number density

times the hydrogen photoionization cross-section σi .For plausible intensities of the background radiation ?eld,e.g.,

J bg (ν)~10?21.3(hν/13.6eV)?1.8erg s ?1cm ?2sterad ?1Hz ?1,

(2)

and for densities near the cosmic mean,photon absorption will be the dominant ionization pathway for hydrogen,and the neutral fraction ηof intergalactic gas will adjust itself until the photoionization rate is equal to the recombination rate αHII n HII n e :

η≡

n HI

Γγ

.

(3)

The recombination coe?cient has a temperature depen-dence that is well ?t by the expression αHII ?2.11×

10?14T ?0.76(1+T 0.76)?1

cm 3s ?1where T 6is the tempera-ture in units of 106K (Cen 1992).

Now consider what happens to intergalactic gas when it is hit by a blast of ionizing radiation from a nearby QSO.The photoionization rate increases by an amount ΓQ γ,given by equation 2with the QSO’s radiation ?eld J Q replacing J bg ,and the neutral fraction ηfalls to its

new equilibrium value ηnew =ηold Γbg γ/(Γbg γ+ΓQ

γ)on the

time scale |n HI /˙n HI |~1/ΓQ γ,or ~104

yr if J Q is equal to the background intensity J bg .After the blast subsides,recombination will raise the neutral density back to its previous level on the same time-scale.

In contrast to the potentially large swings in the gas’s neutral fraction,any changes to its temperature should be imperceptibly slight.Although the gas will warm as photo-electrons collide with other particles and distribute their kinetic energy,the change in its total thermal energy will be negligible:photo-electrons are necessarily as rare as neutral atoms (~1ppm)and their typical kinetic energy at ejection

?HI =(ΓQ γ)

?1

∞E th

dE 4πJ Q (E )σi (E )(E ?E th )/E (4)=E th /(2+s )?3.6eV for

J Q (E )∝E ?s ,

s =1.8,σi (E )∝E ?3,E th =13.6eV

is not very di?erent from the energy per particle in the ~20000K undisturbed gas.The gas received its energy of about an eV per particle when it was almost completely ionized at earlier times,and its temperature will hardly be a?ected by giving a few eV to the particle per million that remained neutral.2This shows that the temperature of the gas will not rise appreciably while its ionization balance adjusts to the increased ionizing background.Afterwards the gas will be heated by photoionizations at the same rate as before—the increase in photoionization rate per HI atom will be exactly compensated by a decrease in the density of HI atoms—and so the equilibrium temperature will be the same in parts of the IGM that are and are not illuminated by the QSO’s radiation.

2I

am neglecting the possibility that the gas around the QSO may be heated by photo-ionization of atoms other than hydrogen.In order to cause a signi?cant temperature change,the product of the photo-ionized atom’s number density n X and typical ejection energy ?X must be comparable to n H kT .HeII will satisfy this con-dition before it is reionized;it is abundant and its photo-electrons’ejection energies are large due both to its high ionization potential (see equation 5)and to its optical thickness.The latter means that essentially all photons more energetic than the ionization threshold will be absorbed,not merely the lower energy photons whose ioniza-tion cross-section is greatest (Abel &Haehnelt 1999).By redshift z ~3the reionization of HeII should be nearly complete,and my ne-glect of this heating source is justi?ed.It would not be if the redshift were much greater.

Adelberger

3

Taken together,the results reviewed in this section jus-tify two assumptions that I will adopt for the remainder of the paper.(a)If a QSO’s luminosity towards solid an-gle ?at time t is L (?,t ),then the intensity of the QSO’s radiation ?eld at position r ,?at time t +|r |/c will be proportional to L (?,t )/r 2.In other words,intergalactic gas at larger distances will not be signi?cantly shielded from the QSO’s radiation by intergalactic gas at smaller distances.(b)The intergalactic temperature will not be a?ected by changes in the intensity of ionizing radiation.This implies ?rst that a blast of radiation from a QSO will alter the ionization balance of intergalactic gas but not its spatial or velocity structure,and second that any changes in neutral fraction will be precisely proportional to the change in the ionizing radiation density (equation 3with constant αHII )averaged over the last ~104yr.

3.MEAN TRANSMISSIVITY VERSUS J ν

We will be able to measure changes in a QSO’s ioniz-ing history with this method only if we can detect spatial changes in the surrounding density of neutral hydrogen.Intergalactic HI density is normally measured by ?tting Voigt pro?les to the numerous Lyman-αabsorption lines in the spectra of background QSOs.This approach is not fea-sible when the background sources are faint galaxies,since individual Lyman-αforest absorption lines are hopelessly blended and confused in their noisy low-resolution spec-tra.Instead we can only hope to measure ˉf

≡ e ?τLy α ,the mean transmissivity along a spectral segment whose length of a few ?A is many times larger than the typical absorption-line spacing but comparable to the instrumen-tal resolution.Although a reliance on ˉf

is forced on us,ˉf

has two advantages over n HI as a probe of the ionizing background radiation.First,it receives signi?cant con-tributions from the parts of the IGM with middling HI optical depths τLy α~1whose response to changes in the ionizing radiation are easiest to measure.In contrast ˉn HI is dominated by systems with large optical depths,and since e ?τ?e ?2τ?0for large τ,big changes in the col-umn densities of optically thick systems can be hard to detect.Second,the value of ˉn HI along any particular line-of-sight is strongly a?ected by whether the line happens to pierce an especially dense system.This adds to ˉn HI large random ?uctuations that may obscure underlying changes

in the radiation ?eld.ˉf

weights more evenly across sys-tems of di?erent column densities and is consequently less a?ected by chance ?uctuations in the density of matter,a point that we will elaborate upon in §5below.

Let us work out,then,how ˉf

responds to changes in the ionizing background.As argued in §2,τat ?xed to-tal hydrogen density is inversely proportional to the ioniz-ing radiation intensity J ,and so if the radiation intensity changes from J to bJ ,with b an arbitrary constant,the new mean transmissivity will be

ˉf = ∞0

dτP (τ)e ?τ/b (5)

where P (τ)is the Lyman-αoptical-depth distribution that existed before the change.

To estimate P (τ),I converted published Lyman-αVoigt pro?le lists for 5QSOs 3near z =3into lists of the ob-3HS1946+7658

from Kirkman &Tytler (1997;z =2.994)and

served Lyman-αoptical depth τhttps://www.sodocs.net/doc/219766541.html,oving distance,

applied a QSO-dependent scaling to every τto make each QSO’s Lyman-αforest have the same mean transmissiv-ity ˉf

=0.67appropriate to z =3.00(McDonald et al.2001),and ?nally constructed a histogram of the resulting τvalues.

The top panel of ?gure 2shows the dependence of the mean transmissivity on the radiation intensity,calculated numerically from equation 5using a spline ?t to this histogram as an approximation to P (τ).Also shown is ?ˉf

≡(d ˉf/d ln J Q ),with J Q ≡J ν?J bg the QSO’s con-tribution to the radiation ?eld.This provides some in-dication of how strongly ˉf

responds to large changes in J Q .The bottom middle panel presents the J ν–ˉf

relation-ship in a slightly di?erent way,as the mean transmissivity as a function of distance r from an isotropically radiating z =3.0QSO with AB magnitude m 912=18or m 912=20at rest-frame 912?A that has been radiating forever at a constant rate.This panel assumes that the background radiation ?eld J bg (equation 2)would have been spatially uniform in the absence of the QSO,which implies that the actual radiation ?eld is

J (r )= 1+

r

eq

4π(1+z )

1/2

,

(7)

f ν≡10?0.4(m 912+48.60)is the ?ux from the QSO received on earth at wavelength (1+z )×912?A and d L (z )is the QSO’s luminosity distance.

The top panel implies (for example)that if the mean transmissivity in a given region can be estimated with an uncertainty of σf ~0.04,then a signi?cant change to the QSO’s luminosity (?ln J Q ~1)will be marginally detectable if the region’s distance to the QSO implies an

expected mean transmissivity 0.72<~

ˉf <

~0.93.The bot-tom panel shows that the required distance is 12<~r <~

2proper Mpc for a QSO with m 912=18.Adopting the crude approximation t delay ~?2r/c leads to the prelim-inary guess that the method should provide reasonable constraints on the QSO’s luminosity for time delays of

?10>~t >

~?80Myr.A more careful treatment is deferred until §7.

4.TIME-DELAY SURF ACE

The actual situation is slightly more complicated than Figure 1suggests,since light from the background sources does not pass the QSO instantaneously.Instead it encoun-ters material that lies behind the QSO before it encounters material that lies in front,and as a result the observed in-tergalactic absorption from material behind the QSO will be sensitive to the QSO’s luminosity at an earlier time.If we de?ne t =0as the time when the QSO emitted the light that is just now reaching earth,and if we place the QSO at the origin of a polar coordinate system where R measures proper displacements along the plane of the sky and z measures proper displacements in the redshift di-rection,then the light from a background galaxy passes through the intergalactic gas that lies a proper distance z

Q0636+680,Q0956+122,Q0302-003,and Q0014+813from Hu et al.(1995;z =3.180,3.288,3.294,3.366)

4Measuring the Radiative Histories of

QSOs

Fig.2.—Top panel:the solid line shows dependence of intergalac-

tic Lyman-αtransmissivityˉf≡ e?τLyα at z=3on the intensity

Jνof the ionizing radiation?eld.The transmissivityˉf?0.67is

appropriate to the actual background ionizing-radiation?eld J bg.

Increases in the radiation intensity destroy neutral hydrogen and

increase the transmissivity.The curve shown was calculated with

equation5from Voigt-pro?le?ts to the Lyman-αforest at z~3

(see text).The dashed line shows?ˉf≡(dˉf/dJν)(Jν?J bg),which

is roughly the change inˉf that would result if the perturbation to

the radiation?eld(i.e.,the di?erence between Jνand J bg)were dou-

bled.Bottom panel:the expected intergalactic transmissivity as a

function of distance from a constantly shining QSO with apparent

AB magnitude m912=18(solid line)or m912=20(dotted line)at

z=3.The QSO’s radiation adds to the background radiation J bg,

which we are taking to be J bg=5×10?22erg s?1cm?2Hz?1sr?1

at912?A,and increases the transmissivity in its vicinity.

behind the QSO at the time t=?z/c,and the ionization

balance of the gas at this time is sensitive to the QSO’s

luminosity at the earlier time

t I(R,z)≡ ?z?(R2+z2)1/2 /c.(8)

Figure3shows the parabolic contours of this function in

astronomically useful units.The?gure implies,for ex-

ample,that the ionization balance of the intergalactic gas

that lies5Mpc behind the QSO and2Mpc to the left will

re?ect the QSO’s luminosity at time t=?34Myr,while

the material that lies directly in front of the QSO will have

an ionization balance that re?ects the QSO’s luminosity at

t=0(i.e.,its observed luminosity).

Now the?eld of view of a large optical imagers is around

40′,or~19.9h?165proper Mpc at z=3(?M=0.3,

?Λ=0.7),and the largest optical multi-object spectro-

graph(Dressler,Sutin,&Bigelow2003)on an8m-class

telescope is not much smaller.This makes it relatively

easy to obtain spectra of high-redshift galaxies throughout

an r~20′region centered on a QSO at z=3,probing

the ionization balance of the IGM in the region of this

plot with?10

background galaxies with redshift z>~3.0and

magnitude

Fig. 3.—The time-delay surface.A QSO at the origin

emits hydrogen-ionizing radiation into the surrounding intergalactic

medium.Concentric circles show the distance that has been trav-

eled traveled by the photons emitted by the QSO at the earlier times

t=?10,-20,-30,-40,-50Myr.Photons from background sources at

z→∞traverse this region from top to bottom as they travel to earth

at z→?∞.At each point along the way they pass photons that

were emitted by the QSO at a di?erent time in its past.The shaded

contours show the emission time of the QSO photons that were il-

luminating a given intergalactic region when the observed photons

from a background source passed by.The strength of the Lyman-α

absorption lines from this region in the spectrum of the background

source will depend on the QSO’s luminosity at the indicated time.

Averaging together the absorption lines from large regions with the

same time delay(e.g.,the region marked A)can provide a constraint

on the QSO’s luminosity at that https://www.sodocs.net/doc/219766541.html,paring many such regions

allows one to chart historical?uctuations in the QSO’s luminosity.

A particularly promising strategy is to estimate the luminosity at

the time corresponding to region A(~?90Myr)by comparing the

absorption lines in A to the absorption lines in a similar region(B)

on the opposite side of the QSO that is illuminated by QSO photons

emitted at t~0.Most systematics will cancel out in this binary

comparison.See text.

R≤25.5will be found in a?eld of this size with the

“UV drop-out”technique(e.g.,Steidel et al.2003),the

Lyman-αabsorption lines in these galaxies’spectra can

in principle provide an enormously detailed view of the

IGM’s ionization balance near the QSO.

Assume,then,that we have measured the intergalactic

Lyman-αtransmissivity f≡exp(?τLyα)throughout the

region of the?gure with?10

ways to estimate the evolution of the QSO’s ionizing lumi-

nosity suggest themselves immediately.One is to divide

the region behind the QSO(i.e.,the region with z>0

in?gure3)into a number of bins whose edges align with

contours of the time delay surface,then see how the mean

transmissivityˉf obs in each compares toˉf exp,the expected

transmissivity if L(t)were constant.Region A in?gure3

is one such bin.The problem is estimatingˉf exp.This

suggests a slight variation:compareˉf obs in each bin to

ˉf

con

,the mean transmissivity in a bin of the same shape

Adelberger5

located on the opposite side of the QSO.See(e.g.)regions A and B in the?gure.Since A is illuminated by photons emitted at t~?80Myr and B is illuminated by pho-tons emitted at?10Myr<~t<0,the di?erence in their mean transmissivities will be sensitive to the any di?er-ences in the QSO’s luminosity at those two times.This approach is also not ideal,since the mean transmissivity in the control region(B)is an unnecessarily noisy indica-tor of the QSO’s luminosity at time?10Myr<~t<0;only a small fraction of the volume illuminated by the QSO’s luminosity at?10Myr<~t<0falls in region B.The best approach is probably to?nd a maximum-likelihood?t of the data to an appropriate surface.Since the results of unbinned maximum-likelihood?ts are harder to present in a simple graphical way,however,I will continue with a paired-bin analysis for the remainder of this paper but will make one assumption that should cause the estimated un-certainties to more closely resemble those of a maximum-likelihood?t:I will assume that the uncertainties in mean transmissivities of the control bins are negligibly small. In other words,I will assume that the uncertainty in the mean transmissivity in region B of?gure3is small com-pared to the uncertainty for region A.The justi?cation is that random?uctuations in B can be removed to a large extent by analyzing the large volume illuminated by the QSO’s luminosity at?10Myr<~t<0.4

The fact that so large a region is illuminated by light emitted at t~0is crucial to the success of this approach, since it allows historical changes in the ionizing luminosity to be measured by comparing regions that are distributed symmetrically about the QSO.Various systematics(e.g., the r?2decrease in?ux from the QSO,bipolar beaming, errors in the continuum?ts to the background galaxies’spectra,peculiar velocities,gradients in the matter den-sity,and so on)should also be symmetric about the QSO, at least in cosmic average,and so they will cancel out in a front-to-back comparison of the HI absorption.No sys-tematic that I can think of will have a shape that resembles the time delay surface in much detail.

The closest candidate might be the gradual expansion of the universe,which causes the intergalactic material be-hind the QSO to be slightly denser than the material in front,changing the recombination time and producing a slight systematic gradient in the intergalactic HI density along the line of sight.This gradient could mimic a bright-ening QSO.The change in expansion scale factor as light traverses a region of proper length40Mpc at z=3.0is ~4%(?M=0.3,?Λ=0.7,h=0.65),and the cor-responding change in mean transmissivity of the IGM is ?ˉf~0.035(e.g.,McDonald et al.2000).?ˉf~0.035 is comparable to the smallest radiation-related change we might hope to measure(see below),so this e?ect cannot be 4It may help to illustrate this point with a concrete example. Here is a crude recipe for reducing the noise in the control bins:?t a low-order polynomial to a plot of the mean transmissivity in each control bin versus the bin’s distance to the QSO,then use the value of this function in each bin as the control transmissivity,rather than the measured transmissivity itself.Since we know a priori that the mean transmissivity in the control bins ought to be a smoothly and monotonically declining function of distance to the QSO,deviations from this behavior must be noise.They can be largely removed with the polynomial?t.Maximum-likelihood?tting of a surface to the unbinned data is a more sophisticated implementation of the same idea.ignored.One way to shrink it to insigni?cance is to divide each measured transmissivity by ˉf (z),the global mean transmissivity at its redshift.I will assume below that every transmissivity f has been rescaled in a way(e.g., divided by ˉf (z)and then multiplied by ˉf (3)=0.67) that makes this e?ect negligible.

5.UNCERTAINTIES

Suppose that we have divided the intergalactic volume surrounding the QSO into di?erent spatial bins,each one illuminated by the light emitted by the QSO at a di?erent period t i

5.1.Arithmetic

Consider a single spatial bin,for example the volume marked A in?gure3,that is being photoionized at the unknown rateΓγ,A.Letˉf obs be the mean Lyman-αtrans-missivity we measure along the sight-line segments that pass through A,and letˉf true be the mean transmissivity that would be measured in an arbitrarily large intergalac-tic volume subjected to the same radiation?eld.ˉf obs and ˉf

true

will di?er because(1)our spectra of the background sources are noisy,(2)we cannot measure the transmissiv-ity throughout the volume A but must instead rely on an uncertain guess from the few thinly scattered probes that background sources supply,and(3)A is not necessarily a fair sample of the universe and could have a mean trans-missivity that di?ers fromˉf true for reasons unrelated to the intensity of ionizing radiation,e.g.,if the random?uc-tuations from in?ation gave it an unusually high or low density of matter.

For simplicity I will obtain estimates of the variance from(1)–(3)by approximating the actual spatial bin A as the volume V that lies between two identically shaped parallel surfaces that are separated by distance l z(see?g-ure4).Generalizing these results to the true geometry is conceptually trivial.In the simpli?ed case,the variance due to(1)is simply

σ2noise?

1

S 2(9) if we have spectra of identical signal-to-noise ratio S for each of the N background sources.5The variance due to (2)is related,in a way that is speci?ed below,to the vari-ance of the mean transmissivity among randomly placed sightline segments of length l z,

σ2l= 1k z l z/2 2,(10)

5S should be calculated for wavelength bins whose size corre-sponds to the depth l z of volume V,of course.

6Measuring the Radiative Histories of QSOs and similarly the variance due to(3)is related to the vari-

ance of the mean transmissivity among randomly placed

volumes of shape V,

σ2V= 1k z l z/2 2.(11)

Here P(k)is the power-spectrum of transmissivity?uctu-ations and W S(k x,k y)is the Fourier transform of surface S.Equations10and11are both special cases of Parseval’s relationship

σ2= 1

(2π)3 d3k P(k)|C k(k x,k y)|2 sin(k z l z/2)

N2 lm exp[i k·(r l?r m)](14)

is the powerspectrum of C,r m≡x m?x+y m?y is a vec-tor specifying the x,y position of the m th background source,and?x and?y are the usual unit vectors.Di?erent arrangements of background sources will result in some-what di?erent variances,but the average variance among all sets C of random galaxy positions7is given by equa-tion13with|C k(k)|2replaced by the expectation value |C k(k)|2 .The latter can be calculated by integrating over all random galaxy positions that lie behind the vol-ume V:

|C k(k)|2 ≡

1

N

σ2V+1

N

σ2V+

1

N ˉf true

Adelberger

7

High signal-to-noise QSO spectra can provide a robust empirical estimate of the variance of the mean transmis-sivity along a line segment of length l z .Figure 5shows the value I ?nd,as a function of l z ,for the primary sample of 7QSOs at redshift z ~3discussed in Adelberger et al.

(2003).

Fig. 5.—The r.m.s.variation of the normalized transmissivity

ˉf / ˉf

(z )along skewers of di?erent lengths.Points with error bars are for the 7primary QSOs from Adelberger et al.(2003);the shaded curve is the prediction of equations 17and 10.The observed values at the largest l are unreliable due to uncertainties in the continuum ?tting.

σ2V is more di?cult.It depends on the three-dimensional power-spectrum of the Lyman-αforest,which has not been measured due to a lack of close QSO pairs.A rough esti-mate of σ2V can be obtained by scaling from σ2l ,σ2V ≡βσ2

l ,with βa constant that can be estimated by assuming a shape for the powerspectrum and numerically integrat-ing equations 10and 11.According to McDonald (2003),the transmissivity powerspectra found in numerical sim-ulations the high-redshift intergalactic medium have the form

P (k R ,k z )∝

1+βk 2z /k 2

2P L (k )D (k R ,k z )

(17)where k z and k R specify the wavenumber in polar coordi-nates,k 2≡k 2z +k 2

R ,P L (k )is the linear powerspectrum of cold dark matter,

D (k R ,k z )≡exp k k p

αp ?

k

z

2

dk R dk z k R P (k R ,k z )

sin(k z l z

/2)

k R R

2

(19)

or

σ2V

=

1

k z l z /2

2

×

2R o J 1(k R R o )?2R i J 1(k R R

i

8Measuring the Radiative Histories of QSOs

where the change in amplitude has been crudely removed by scaling each curve according to the relationship

σl∝[1?ˉf(b)]0.43(21) whereˉf(b)is the mean transmissivity after dividing the actual optical depths by b.Once this scaling is removed, theσl curves have nearly the same shape over the range 1<~l<~10h?1comoving Mpc that is most important in our

calculations.

Fig.7.—Top panel:σl(l z)(the r.m.s.transmissivity?uctua-tion along a line segment of length l z)recalculated after dividing the observed Lyman-αoptical depths in the QSOs HS1946+7658, Q0636+680,Q0956+122,Q0302-003,and Q0014+813by the num-ber b written next to each curve.Bottom panel:same as top,except the dependence of the amplitude on the mean transmissivityˉf(b) has been crudely removed by dividing each curve by(1?ˉf(b))0.43. The di?erences in shape revealed by this panel are small over the distance range1<~l<~10h?1comoving Mpc that is most important to our analysis.

https://www.sodocs.net/doc/219766541.html,mentary

It is worth considering the relative sizes of the terms in equation16before we move on.Suppose for concreteness that we are interested in measuring changes in the QSO’s luminosity over10Myr time-scales,so we have estimated the mean transmissivity in bins of depth l z~1.5proper Mpc,and suppose further that the radius R of our bins was chosen to?ll the40′?eld-of-view of a mosaicked CCD camera,R~10proper Mpc at z=3for?M=0.3,?Λ=0.7,h=0.65.In comoving units the bin has R= 26h?1Mpc and l z=3.9h?1Mpc,which impliesσl?0.17 (?gure5,forˉf=0.67),σ2l/σ2V?20,and consequently σV?0.04.Inserting these numbers into equation16leads to the two primary conclusions of this section:

(A)The spectra of the background objects do not need to be very good.The second and third terms in equa-tion16contribute equally to the total uncertainty when the signal-to-noise ratio in a spectral segment of length l z~3.9h?1comoving Mpc(~7?A)is S=ˉf/σl~4.Ob-taining better spectra cannot change the total uncertainty by much.Random variations in the intergalactic matter density make the mean HI absorptionˉf1along a single sightline segment a poor indicator of the ionizing back-ground Jνand consequently there is no need to measure ˉf

1

with exquisite precision.

(B)The ultimate limit on our uncertainty is set byσV and it is a limit that we will reach very quickly.For the example considered here,the?rst and second terms in equation16have the same size for N=20.Increasing N further can reduce the second and third terms arbi-trarily but not the total uncertainty.In practice one will want to obtain samples many times larger than N=20 to help address the possibility that the QSO’s radiation is beamed.As a resultσV will probably be the only sig-ni?cant contributor toσf in realistic cases.The bottom panel of?gure2shows thatσV is large enough to prevent us from detecting all but the coarsest changes in a QSO’s luminosity.We can do nothing about this.Only a small region of the universe is bathed in the light that the QSO emitted at one period in its history;the mean density in this region will stray from the global mean to the extent that in?ation requires;the intensity of the QSO’s ionizing radiation cannot be measured if it in?uences the region’s mean HI absorption by less.

Although the numbers quoted above depend on our ar-bitrary choice for the bin size,neitherσl norσV is a strong enough function of R or l for the qualitative conclusions to change signi?cantly as the bin size varies across its useful range.The change of variance with mean transmissivity (equation21)might seem more important.If the QSO were bright enough to drive the mean transmissivity in a bin toˉf~0.9,for example,rather than the valueˉf~0.67 assumed in the preceding three paragraphs,σl would de-crease to0.10,σV would decrease to0.02,and one would ?nd that spectra of slightly higher S:N were desirable.In practice,however,these high transmissivities are unlikely to be reached anywhere except very near the QSO,and here the decrease inσl andσV is largely o?set by the small bin sizes that are required at small radii.This can be seen in the worked example of§7,below.

One quali?cation should be added to my claim(A)that it is a waste of time to obtain high S:N spectra.That is necessarily true only if one is committed to using the mean transmissivityˉf as a probe of the radiation density Jν.At su?ciently high signal-to-noise ratios,however,other op-tions are available.Consider an arbitrary function g of the line-of-sight HI density that has varianceσ2g due to random ?uctuations when the brightness b≡Jν/J bg of the ioniz-ing radiation?eld is kept constant.The random?uctua-tions in g will prevent us from detecting relative changes in b of order?ln b~|d ln b/d ln g|σg/g.If g is the mean transmissivityˉf on a3.9h?1comoving Mpc line segment, which is the only possibility we have treated so far,then d ln b/d ln g~5(near g=0.67)andσg/g~0.26,so the minimum detectable?uctuation in b will have?ln b~1.3. At high enough S:N it would be possible abandonˉf and adopt(say)the total HI column-density N along the same line segment as the probe of b.In this case,with g=N, we have d ln b/d ln g??1andσg/g~6,whereσg/g is the value that obtains among the5QSOs with z~3

Adelberger9 discussed at the end of§5.1,so the minimum detectable

?uctuation in b has?ln b~6.N is evidently far inferior

toˉf as an estimator of b.A better choice,letting g equal

n,the number of detected Lyman-αforest lines on the

segment,was advocated by Bajtlik,Duncan,&Ostriker

(1988).If the column density distribution has the form

P(N HI)∝N?α

HI ,then d ln b/d ln g?(1?α)?1.Among

the same5QSOs,σn/n?0.5for line segments of length 3.9h?1comoving Mpc,so the minimum detectable?uctu-ation in b has?ln b~1for the observed slopeα~1.5. Surprisingly,n is only marginally better thanˉf as an esti-mator of Jν.The mean transmissivityˉf along a segment of a low S:N galaxy spectrum can provide almost as strong a constraint on the intensity of the QSO’s radiation as a high S:N spectrum analyzed with the standard approach of Bajtlik,Duncan,&Ostriker(1988).It would be inter-esting to extend this analysis fromσl to the more relevant quantityσV.Since the ratioσl/σV depends on the pow-erspectrum,and di?erent estimators g will have di?erent powerspectra,it is not necessarily true that the best esti-mator forσl will be best forσV.This method would be made far more powerful if one could?nding an estimator that signi?cantly reducesσV.

6.FEASIBILITY

The previous section glibly claimed that observational uncertainties can be made smaller than cosmic variance. This section considers the claim in more detail.The op-timal depth for the spatial bins is~7?A,corresponding roughly to10Myr time resolution(see§8),and for the ?elds-of-view considered here the cosmic variance will con-sequently beσV~0.04(see§§5and7).This is the level to which we must reduce the observational uncertaintyσnoise.

6.1.Signal

Reducing the random errors to the desired levelσnoise~0.04is not much of a challenge.Figure8shows spectra with~10?A resolution of three galaxies in the?eld SSA22a (Steidel et al.2003)that were observed for~29000s with the blue LRIS spectrograph(Steidel et al.2004)on the Keck I telescope.The redshifts and AB-magnitudes of the galaxies(z~3.1and G~24.5,respectively)are shown on the plot.The spectra are preliminary reductions that were selected more-or-less at random from the sample of Shapley et al.(2004,in preparation).These spectra have signal-to-noise ratios in the Lyman-αforest,for7?A bins (~1.5proper Mpc),of around5–6.Averaging together ~10spectra of similar quality would reduce the random errors to the desired level.Roughly forty spectra would be required if the exposure time were2hours instead of8.

6.2.Continuum?tting

Systematic errors in the continuum?tting are a source of greater concern.We are interested not in a spectrum’s ?ux itself but rather in its implied Lyman-αtransmissivity, which is the ratio between the observed?ux and the con-tinuum?ux,i.e.,the?ux that would have been observed in the absence of Lyman-αabsorption.The uncertainty in a bin’s mean transmissivity therefore has an additional term that arises from errors in the estimated continuum. The size of these errors depends on the method that is used to estimate the continuum level.Traditional meth-ods are poorly suited to the present case;they exploit

the

Fig.8.—Examples of the Lyman-αforest in the spectra of galax-ies at redshift z~3.The data were taken with LRIS-B(Steidel et al.2004)and were generously provided by A.E.Shapley.Redshifts and AB magnitudes are indicated;the names(“C11”etc.)refer to the SSA22a catalog of Steidel et al.(2003).Asterisks mark the loca-tions of the interstellar absorption lines discussed in?gure10,below. Dashed lines indicated the wavelengths of Lyman-αand Lyman-βat the redshift of the galaxies’interstellar absorption lines.

occasional presence of spectral regions with little Lyman-αabsorption,and these regions are rare and di?cult to recognize in noisy,low-resolution galaxy spectra.Several authors(e.g.,Hui et al.2001)have presented alterna-tives that are more useful to us.Particularly simple is the method of Croft(2004),in which the continuum level c(λ) is estimated by simply smoothing each object’s spectrum and scaling appropriately:c(λ)?f50(λ)/ˉf(z),where f50(λ)is the object’s Lyman-αforest spectrum smoothed by a Gaussian withσ=50?A andˉf(z)is the published mean transmissivity at redshift z=λ/λLyα?1,which has been calculated by other authors from more-sophisticated continuum?ts to the observed spectra of numerous bright QSOs.

Experimentation on the7QSOs with z~3in the sam-ple of Adelberger et al.(2003)shows that transmissivi-ties estimated with Croft’s(2004)approach and the tra-ditional approach are very similar.Excluding the parts of the Lyman-αforest that fall on the QSOs’Lyman-αand Lyman-βemission lines,the correlation coe?cient between the transmissivities estimated with the two approaches is r~0.96–0.98.This implies that the r.m.s.di?erence be-tween them,(1?r2)1/2σl,is~0.03–0.05for7?A bins with σl?0.17(see Figure5).Since galaxies’and QSOs’con-tinua between Lyman-αand Lyman-βare similarly fea-tureless(see,e.g.,?gures8,9,and10),errors in continuum ?tting should introduce a similar uncertainty in galaxies’Lyman-αforest transmissivities.Even if these errors were correlated from one galaxy to the next,and did not tend to cancel in the averaged transmissivity in each spatial bin,

10Measuring the Radiative Histories of QSOs

their size would be no larger than the cosmic variance.In practice continuum errors from the Croft (2004)approach should cancel signi?cantly,however.They arise,to a large extent,because the Lyman-αforest is not completely uni-form on ~100?A scales and any deviations from uniformity are incorrectly interpreted as features of the continuum.The resulting r.m.s.continuum error averaged over a spa-tial bin can be calculated with the approach of §5:

σcont ?

N ?1

N

σ′2

l

1/2

(22)

where σ′2

V and σ′2

l are given by equations 11and 10with

sin(k z l z /2)/(k z l z /2)replaced by exp[?k 2z σ2

z /2],where σz is the comoving distance corresponding to the 50?A smooth-ing length.Since σz ?l z ,σ′2V will be smaller than σ2V ,σ′2l will be smaller than σ2l ,and the uncertainty from this source of continuum errors will never dominate the total uncertainty.This is true even in the low signal-to-noise limit,since in this case the uncertainty in the smoothed continuum will be dwarfed by the uncertainty in a single ~7?A

bin.

Fig.9.—Comparison of galaxy and QSO continua in the Lyman-αforest region.The top panel shows the gravitationally lensed galaxy MS1512-cB58(Pettini et al.2002);the bottom panel shows the QSO Q1623KP78(Adelberger et al.2004,in preparation).Both spectra were taken with the same instrument,though the signal-to-noise ratio for the bright QSO is signi?cantly higher.Asterisks in the top panel mark the locations of the interstellar absorption lines discussed in ?gure 10,below.

6.3.Interstellar absorption lines

Although galaxies’continua appear to be mostly fea-tureless between Lyman-αand Lyman-β,there are some important exceptions:at a handful of wavelengths the galaxies’interstellar absorption lines are too strong to be ignored.These wavelengths are apparent in ?gure

10,

Fig.10.—Mean absorption vs.wavelength for galaxies at redshift

z ~3.The shaded curve shows the mean spectrum of 811galaxies at z ~3;the data are taken from Shapley et al.(2003).Lyman-αforest ?uctuations cancel out in this average,producing a smooth shelf

with mean transmissivity ˉf

~0.67between Lyman-αand Lyman-β.Absorption lines in this shelf are produced by material intrinsic to the galaxies.This interstellar absorption can be safely ignored when its rest-frame equivalent width is less than ~0.3?A (see text).Stronger lines have been marked with vertical labels naming the ion responsible for the absorption.Data at these wavelengths should be excluded when calculating the mean transmissivity in di?erent spatial bins.Note that the resolution of this spectrum (~12?A )is

somewhat lower than the 7?

A resolution advocated in the text.which shows average observed absorption in region be-tween Lyman-αand Lyman-βin a sample of 811Lyman-break galaxies (Shapley et al.2003).Since the r.m.s.vari-ation in Lyman-αforest transmissivity in 7?A bins is

σl ?0.17(Figure 5),interstellar absorption lines will have a non-negligible e?ect on the estimated transmissivity if their rest-frame equivalent width exceeds 7?A ×σl /(1+z )?0.3?A .In the mean spectrum of Shapley et al.(2003)there are 7such lines between Lyman-αand Lyman-β.The ex-istence of these interstellar absorption lines will not have a disastrous e?ect on the analysis,since each background galaxy will lie at a slightly di?erent redshift,but one might as well eliminate their e?ect completely by masking the relevant portions of the spectrum.This will reduce the e?ective number of background galaxies by too modest an amount to signi?cantly alter the signi?cance of any con-clusions.

7.SYNTHESIS

We can now estimate how easily we will be able to de-tect ionization gradients produced by changes in a QSO’s luminosity.This section works through the details for a single case,an isotropically radiating QSO of (t =0)mag-nitude m 912=18whose ionizing luminosity varied with time according to the curve L (t )shown in the top panel of ?gure 11.An ?M =0.3,?Λ=0.7,h =0.65cosmology with the uniform ionizing background of equation 2will be assumed throughout this section.

According to equation 6,the radiation intensity as a function of position in the observed frame (i.e.,the radia-tion intensity that was present when the photons from the background sources passed through each point)is

J (R,z )= 1+L (t I (R,z ))

r 2 J bg (23)

where t I (R,z )and r eq are given by equations 7and 8.The left panel of ?gure 12shows this function for r eq =8.27

Adelberger

11

Fig.11.—Top panel:the QSO’s ionizing luminosity as a function of time.This curve,generated for the example of section 7,is not intended to be a realistic model of a QSO’s output.It features were chosen to illustrate a number of points made in the text.Bottom panel:the recovered mean transmissivity in bins whose edges align with contours of the time-delay surface.Points mark the mean trans-missivity calculated from ?gure 12;error bars show the 1σrange that would be observed if ?gure 12included cosmic variance,the dom-inant noise source.The solid curve shows the mean transmissivity than would have been observed if the QSO’s luminosity were con-stant,L (t )=L (0).The dotted curve shows the mean transmissivity in the primed “control”bins (?gure 12);it di?ers from the solid line due to the increase in the QSO’s luminosity from t =0to t =?10,though this e?ect would presumably be removed by a maximum like-lihood ?t of the data to a surface.Decreases of the luminosity to 0and subsequent increases back to L (0)can be detected with mod-erate signi?cance if they happened within the interval ?50<~t <

~0Myr.Only extremely large increases in the QSO’s luminosity can be detected at earlier times.

proper Mpc,which is appropriate to the QSO described in the preceding paragraph.Since the ionization and recom-bination times (§2)are short compared to the time for signi?cant changes in L (t ),the neutral fraction will be in-versely proportional to J (R,z ),and the mean correspond-ing transmissivity f (R,z )can be derived from the curve shown in the top panel of ?gure 2.The result is shown in the right panel of ?gure 12.The panel shows the mean transmissivity that would be observed if one averaged re-sults from many identical QSOs;the actual transmissivity surrounding a single QSO would have signi?cant variations about this mean,due primarily to random ?uctuations in the density of intergalactic matter.

Changes in the QSO’s ionizing luminosity could be de-tected in various ways,but for now I will assume that one is aiming to detect the changes by looking for di?erences in

the mean transmissivity ˉf bin among bins whose edges trace

contours of the time delay surface t I (R,z ).One set of such bins is shown in the left panel of ?gure 12.Also indicated are symmetrically distributed “control”bins.These bins have shapes identical to the others,but are located on the opposite side of the QSO,in a region that is

illuminated

Fig.12.—Left panel:variations in the intensity of ionizing radiation in the observed frame for a QSO at redshift z =3with m 912=18that has the history of outbursts shown in ?gure 11.Also shown are sample bins used in the analysis at larger lookback times.Right panel:variations in the expected mean transmissivity implied by the radiation intensity of the left panel.The mean transmissivity would hew to these values only if results from numerous identical QSOs were averaged.A more realistic single realization would show large ?uctuations in the transmissivity due to random variations in the density of intergalactic matter.The labeled regions surrounding the QSO are sample bins for the analysis at smaller lookback times.

by the QSO’s radiation at short time delays 0>t >?10Myr.The mean transmissivity in these bins provides an indication of how the mean transmissivity for z >0would vary with position if the QSO’s luminosity were always equal to its observed (t =0)value(§4).Note that the requirement t >?10Myr for the control bins limits the size of their partner bins at smaller radii.

Our ability to detect changes in the QSO’s luminosity will depend on the uncertainty in the binned transmissiv-ity ˉf bin .For a reasonable number of background sources (>~few dozen)this uncertainty will be nearly equal to the cosmic variance σV (§5).As discussed in §5,the size of σV can be roughly estimated by (1)approximating each bin as a cylinder of radius R bin and depth l bin ,(2)calculating r.m.s.transmissivity ?uctuation along a skewer of length l bin by interpolating from ?gure 5and scaling according to the local expected transmissivity f (R,z )with equation 21,

and (3)multiplying by β1/2,where β(R bin ,l bin )≡σ2V /σ2

l is a constant that can be determined for each bin by nu-merically integrating equations 10,19,and 17.The bot-tom panel of ?gure 11shows the expected mean transmis-sivity in each bin along with the estimated 1σuncertainty calculated in this way.

The main conclusions from ?gure 11are (1)that only large changes in the QSO’s luminosity will leave an imprint on the IGM that is easily detectable with this approach,and (2)that the minimum size of a detectable luminosity

12Measuring the Radiative Histories of QSOs

di?erence increases rapidly towards earlier times.These conclusions are easier to apprehend in ?gure 13,which shows the 10-Myr moving-average magnitude ˉm 912a QSO would need to have had at various lookback times to for its sudden death (or revival)to leave a detectable fossil record in the spectra of background galaxies.Measuring

changes in a QSO’s luminosity will be di?cult for t <

~?50

Myr and nearly impossible for t <~?100Myr.At smaller

time delays the prognosis is

good.

Fig.13.—The 10Myr running-average luminosity required at various lookback times for the ?ux in the corresponding 10Myr bin to be Nσabove the background level.Luminosity changes among even the brightest QSOs will be di?cult to detect at moderate signi?cance

for t <

~?100Myr.

As t →0,the shape of the time-delay contours becomes increasingly incompatible with the goal of having symmet-rical control bins.For ?20<~t <~

0the best approach may be to abandon the binning altogether and simply ?nd the maximum likelihood ?t of the data to an appropriate sur-face.To give some indication of the uncertainty in the ra-diative history that would result,however,I will continue a binned analysis with the bins shown in the right panel of ?gure 12.The outermost bins (B –BBB )enclose regions with time delays ?20

wide range of distances r to the QSO.Variations in ˉf

due to changes in 1/r 2could obscure the variations of interest from changes in the QSO luminosity L (t ).My approach in this simpli?ed analysis is to divide each time-delay region into a number of bins with similar values of r (e.g.,bins B ,BB ,and BBB for ?20

does not depend sensitively on the details of this approxi-

mation.

Fig.14.—The mean transmissivity and its uncertainty in the

bins shown in the right panel of ?gure 12.Bins with the lightest shading are bathed in light emitted by the QSO at ?3.5Myr

2Mpc are unlabeled in ?gure 12for clarity.At ?xed time delay,ˉf

declines with radius due to the 1/r 2decline in ?ux from the QSO.

Comparing ˉf

radius by radius clearly reveals the decline in L (t )(?gure 11)from ?10Myr

With these bins,the decrease in L (t )from ?10

had happened at an earlier time ?10

?3.5Myr it would have been detected with higher signi?cance still.Instead the luminosity increased from ?3.5Myr

to ?10

?3.5.As the plot shows,increases in the luminosity at small look-back times are much harder to

detect than decreases,since the sensitivity of ˉf

to J Q falls as ˉf

→1(?gure 2).Fortunately decreases must be far more likely than increases:QSOs with m 912=18lie on the steep bright-end of the luminosity function.

8.LIMITS TO THE TIME RESOLUTION

In the previous sections the data were placed into spa-tial bins whose depth l z =1.5Mpc gave us sensitivity to luminosity ?uctuations time-scales of 10Myr or greater.Ideally one would be able to detect ?uctuations on any time scale.Could we have achieved signi?cantly better time resolution by placing the data in bins with smaller l z ?The answer is no;this section explains why.

If cosmic variance were the only problem,the bin depth

Adelberger13

l z could be made arbitrarily small.σV(equation19)is almost independent of l z for l z?R,the case of interest, and so a bin that is in?nitesimally thin will have nearly the same cosmic variance as the adopted bins with l z=1.5 proper Mpc.8

Unfortunately the ability to obtain reasonably accurate estimates of the mean transmissivity in in?nitesimally thin bins is not the same as the ability to measure changes in the QSO’s luminosity that happened on arbitrarily short time scales.Our estimate of a gas element’s longitudi-nal separation z from the QSO will be inaccurate for two reasons:(1)we will not know the precise redshift of the QSO,and(2)our z positions are derived from redshifts and will be distorted from their true values by peculiar velocities.As a result the time delay to the element is uncertain.When the time delays to di?erent elements are uncertain,we cannot combine elements with exactly the same delays into one bin;the various elements that make up a single bin will inevitably have a range of time de-lays.The minimum range of time delays in a bin is what limits our time resolution.The remainder of the section considers this limit in more detail.

8.1.Uncertainty in the QSO redshift

If the QSO’s redshift is measured from the CIV emission line,the uncertainty in its systemic recession velocity will beσv?510km s?1(Richards et al.2002),which corre-sponds to a positional uncertainty ofσz?1.75h?165proper Mpc at z=3for?M=0.3,?Λ=0.7.The uncertainty can be reduced to0.9h?165proper Mpc(i.e.,270km s?1)if MgII is used instead(Richards et al.2002),and to0.3h?165 proper Mpc(i.e.,80km s?1)if[OIII]is used(Vrtilek& Carleton1985).Radio observations of molecular emission lines could presumably reduceσz even further,but this is unlikely to bene?t us much.The time resolution achiev-able withσz=0.3proper Mpc is t~2σz/c~2Myr,and other e?ects prevent us from obtaining a resolution even this coarse.

8.2.Thermal motions

A?rm lower limit to the time resolution is set by the thermal motions of the20000K intergalactic gas. The intergalactic hydrogen at a particular true z posi-tion will have an rms range of apparent z position of σz=(kT/m H)1/2/H?0.04proper Mpc for?M=0.3,?Λ=0.7,h=0.65.We will not be able to measure changes in the QSO’s luminosity that happen much more rapidly than the corresponding time scale t~2σz/c~0.3 Myr.This is unlikely to be the limiting factor in the anal-ysis.

8.3.Streaming towards the QSO

The e?ect of larger-scale peculiar velocities is more se-vere.First there is the average streaming motion towards the QSO,which can be crudely estimated as follows.If the scale dependence of QSOs’bias b is weak,then the mean 8The reason is that a cylinder with l z?R in real space will have l z?R in Fourier space,and since the power P(k)is concen-trated near k~0the variance is dominated by contributions from wavenumbers k near the origin.There is little power at the distant ends of the cylinder with large k z,and altering the limits of the k z integral in equation19(i.e.,changing the thickness l z of the cylinder in real space)does not a?ect the variance by much.matter overdensity at a distance r from a QSO will be roughlyδ(r)=ξQ(r)/b,whereξQ is the correlation func-tion of QSOs.According to Croom et al.(2002),a corre-lation function of the formξQ(r)=(r/r0)?γ,r0?8.4h?1 comoving Mpc,γ?1.56is appropriate for the brightest QSOs at any redshift.The variance of QSO number den-sity in cells of radius r cell=8h?1comoving Mpc is there-foreσ2Q=72(r0/r cell)γ/[(3?γ)(4?γ)(6?γ)2γ]?1.3, (Peebles1980eq.59.3),which implies a QSO bias at z=3of b?4.5if?M=0.3,?Λ=0.7,and the rms linear matter-density?uctuation in r=8h?1Mpc spheres at z=0isσ8=0.9.Integrating over the correlation function shows that the mean matter overdensity within a comoving radius r c,ˉδ(r c),is roughly

ˉδ(r)?3r?γc

r p

=H?

1

dt

(25)

where H(t)is the Hubble parameter.This follows from the fact that concentric shells of matter do not cross un-til just before?nal collapse.Adopting the spherical Zel-dovich approximation for simplicity,the linear overdensity densityˉδL will be related to the true overdensity through 1+ˉδ~(1?ˉδL/3)?3,which reduces equation25to

˙r p

(R2+z2)1/2 1? 1+ˉδ(r p) 1/3 ,(27) is the rough proper distance between the a volume ele-ment’s true position(R,z)and the position we that we erroneously infer from assuming that it is at rest with re-spect to the Hubble?ow.Here f≡d ln D/d ln a??0.6M(z) where D is the linear-growth factor and a is the scale fac-tor of the universe.Figure15showsˉδand?z as a function of distance.

By itself the net streaming towards the QSO is not a major problem,at least for r>~5h?1comoving Mpc.Its primary e?ect is to make absorbing gas appear to be closer to the QSO than it actually is.Although this produces a slight systematic error in the lookback time assigned to each volume element,the error can be corrected to a large degree with simple formulae such as equation27,and in any case slight inaccuracies in the times assigned to the x axis of?gure11would not diminish its scienti?c value by much.

8.4.Random peculiar velocities

More troubling are random deviations around the net streaming motion.As a result of them,the gas that is illuminated by the QSO’s luminosity at time t will lie on a complicated surface that wanders randomly around the parabolic time-delay surface shown in?gure3.Since par-ticles maintain their linear velocities long after the density

14Measuring the Radiative Histories of

QSOs

Fig.15.—The mean interior overdensity (solid line)and ap-proximate infall velocity (dashed line)as a function of distance from a bright QSO at z =3.The infall velocity is expressed as ?r/r ≡(r apparent ?r true )/r true ,the relative shift in an object’s position that the infall velocity would produce if the velocity were directed exactly towards the observer.The infall velocity is esti-mated with a simple version of the Zeldovich approximation that

becomes inaccurate for ˉδ

?1;see text.?eld itself has left the linear regime,and since most of in-tergalactic space should be occupied by low density (i.e.,

uncollapsed)gas that is not far from the linear regime,lin-ear perturbation theory should provide a rough estimate of the typical size of these excursions.Let σv z (r )be the rms di?erence in the z comoving peculiar velocities of two points separated by the vector r .In the linear regime the

Fourier transform ?v

(k )of the comoving peculiar-velocity ?eld v (r )is related to the Fourier transform ?δ

(k )of the co-moving density ?eld δ(r )through ?v

(k )=?iHf ?δ(k )k /k 2,(e.g.,Peebles 1980equation 27.22)where H is the Hubble

parameter and f ??0.6

M (z ).Convolving v z (r )by the sum of a positive delta function at r ′/2and a negative delta function at ?r ′/2produces a new random ?eld whose value at each point is equal to the z -velocity di?erence between the points r ′/2and ?r ′/2.The variance of this ?eld is

equal to σ2

v z

(r ),which can therefore be written,according to equation 12,as

σ2

v z

(r )=

H 2f 2

k 4

P L (k )sin

2

k ·r

2π2

dk P L (k )

sin(kr )

(kr )3

?

cos(kr )

2π2

dk P L (k )

sin(kr )

(kr )2

.(31)

This is a special case of a result derived by G′o rski (1988).If the wavenumbers in the integrals are comoving,as is the convention,the rms error in comoving z separation due to peculiar velocities is σz (r )=σv z (r )/H .Inserting the Γ=0.2linear powerspectrum of Bardeen et al.(1986)into these equations,normalizing to σ8=0.9at redshift z =0(i.e.,to σ8?0.29at z =3,appropriate for ?M =0.3,?Λ=0.7),and integrating numerically,I ?nd the values of σz shown in ?gure

16.

Fig.16.—The peculiar velocity ?eld.Peculiar velocities add

random o?sets to the apparent z positions of intergalactic gas and systematically shift them towards the QSO.Black points mark the true spatial positions of selected intergalactic volume elements;error bars show the expected 1σrange of each volume element’s apparent position.The mean o?sets were calculated with equation 27and the sizes of the ranges with equation 29.Contours in the background show the time delay to each position,starting with 1,3,10,20Myr and increasing in steps of 20Myr thereafter.Because of peculiar velocities,gas with a range of time delays is mixed together in any spatial bin that is de?ned in the observed (redshift-space)frame.This limits the time resolution of the method.For time delays greater

than ~20Myr,the time resolution is limited to >

~10Myr.At very small time delays the resolution is far better.Peculiar velocities are unlikely to make material with a delay of ~1Myr appear to have a delay of 3Myr,for example,so we should be able to use the method to detect QSO lifetimes of order 1Myr even though we will not be able to distinguish a life time of 30Myr from one of 31Myr.

8.5.Upshot

Adelberger15

Random peculiar velocities are likely to be the dominant source of uncertainty in a volume element’s distance to the QSO.The uncertainty in the element’s proper z position isσz~1proper Mpc,which corresponds to a time-scale of t~2σz/c~7Myr.It might be possible,with very high signal-to-noise spectra for a very large number of back-ground sources,to trace and correct these distortions to the time delay surface.A number of interesting applica-tions would then be possible.With current technology, however,we will have accept that the mean transmissivity in any bin we devise will be sensitive to the QSO’s lumi-nosity over a~10Myr range of lookback times.Time resolution signi?cantly better than this does not appear to be achievable.9

9.SUMMARY AND DISCUSSION

This paper showed that changes over time in the lumi-nosity of a QSO at redshift z~3will produce ionization gradients in the IGM and alter the Lyman-αforest ab-sorption spectra of background galaxies in an observable way.Because the density of detectable galaxies(R<~25) at z~3is high,~1arcmin?2,their absorption spectra can provide a detailed view of the ionization gradients.If an isotropically radiating QSO has an AB magnitude at 912?A of m912=18,signi?cant decreases in its luminosity at larger look-back times will be detectable if they happen 1<~t<~50Myr before the time of observation.The time limits expand for brighter QSOs and shrink for fainter.In-creases in the QSO’s luminosity over this time period will be harder to detect than decreases,but since m912=18 corresponds to the steep bright end of the QSO luminos-ity distribution,they must be much rarer.§7sketches out the method and presents the uncertainties for a simu-lated QSO with known radiative history L(t);the section is aimed at those who want more detail but are reluctant to read the entire paper.

The method gives us sensitivity to changes in a QSO’s luminosity over a useful range of times.Statistical ar-guments mentioned in the introduction show that QSOs must change their luminosities signi?cantly on time-scales t<~100Myr.If these changes happen on time-scales t<~0.1–1Myr,a handful of QSOs in large(SDSS-sized) samples will show major brightness changes from one decade or century to the next(Martini&Schneider2003). The method I have described cannot detect luminosity changes that happen on time-scales so short(§8),but is sensitive changes throughout the rest of the allowed range (1<~t<~100Myr).Taken together,the two methods will be able to pin down the typical QSO lifetime in a robust and direct way.

A number of other authors(e.g.,Crotts1989;Dobrzycki &Bechtold1991;Moller&Kjaergaard1992;Fern′a ndez-Soto et al.1995;Liske&Williger2001;Jakobsen et al. 2003;Schirber,Miralda-Escud′e,&McDonald2004;Croft 2004)have attempted to measure QSO lifetimes with a similar approach.Their results were ambiguous.The rea-son is that they used the absorption lines in a single back-ground QSO(or,in some cases,a handful)to search for ionization gradients around the foreground QSO.When 9Except at very early times,0>t>?20Myr,when the posi-tional uncertainties become increasingly aligned with the time-delay contours;see?gure16.the QSO pair had a small projected separation,the analy-ses were confused by the high densities and large peculiar velocities near the foreground QSO(see,e.g.,?gure15); when the projected separation was large the foreground QSO’s weak e?ect on the IGM could not stand out above the cosmic variance.This paper’s method sidesteps these di?culties by using numerous faint galaxies rather than a small number of bright QSOs as the background sources. As shown in§5,the signal-to-noise ratio of the background objects’spectra does not a?ect the?nal result by much, but the number of background sources does.Choosing galaxies as the background sources is therefore the sensible approach.With numerous background sources,the weak e?ect of a QSO on distant intergalactic matter can be de-tected with reasonable signi?cance,the complicated region closest to the QSO can be ignored altogether,and the pe-culiar velocity and density gradients at slightly larger dis-tances can be compensated by comparing to the amount of Lyman-αabsorption in the large“control region”that is illuminated by light emitted by the QSO at t~0(§4). The approach I have described should therefore be a sig-ni?cant improvement over previous work.

Observers who would like to apply this approach in prac-tice should be aware that the optimal proper size of the observed region can be very di?erent from the naive guess ct max with t max the maximum time delay of interest.As t max→0intergalactic absorption at radii R?ct max be-comes increasingly important for the analysis,as?gures3 and14show.Two results derived in§5are also relevant. They are discussed more fully in§5.2,but may be sum-marized as follows:(A)Galaxy spectra with S:N~4per 7?A bin will be su?cient for measuring the QSO’s radia-tive history with the stated precision.Obtaining better spectra will not improve the result by much.(B)Cosmic variance places a fundamental limit on the accuracy of the method.Only a small part of the universe is bathed in the light that the QSO emitted at a particular moment in its history,and the HI content of this region will stray from the global average for reasons that have nothing to do with the luminosity of the QSO.The only detectable luminosity changes are those large enough to alter the in-tergalactic HI content by more than its intrinsic random ?uctuationsσV.A corollary is that the number of back-ground sources does not need to be incredibly high;one needs only enough to measure the mean transmissivity of a region with a precision similar toσV,and this is eas-ily achieved with spectra of a few dozen galaxies(but see below).

I have neglected an important complication in the dis-cussion so far.It is possible that the QSO’s ionizing ra-diation will not emerge isotropically but will instead be focused into a bipolar beam with opening angleα~90o (Barthel1989).If the beam were pointed towards earth there would be almost no e?ect on the analysis(see?g-ure12),but in the typical case the intergalactic volume that is a?ected by the QSO’s radiation will be only~30% as large as I have assumed.This will increase the un-certainty in the results.Shrinking the radius of one of our idealized cylindrical bins(§§5and7)until it contains only30%of its previous volume will increase its uncer-tainty due to cosmic variance,σV,by a factor of~1.5.If cosmic variance is still the dominant source of noise in the

16Measuring the Radiative Histories of QSOs

smaller bins,the error bars in theˉf vs.t curve will increase by a similar factor.Fortunately this is not enough to pre-vent us from detecting changes in the QSO’s luminosity for ?50Myr<~t(?gure11).Moreover,even with the enlarged error bars it should be easy to distinguish intergalactic volumes that are illuminated by the QSO’s ionizing radi-ation from neighboring regions with transmissivities close to the global mean ofˉf~0.67,at least at small radii where the QSO’s radiation is most intense(see,e.g.,?g-ure14,which shows that the QSO’s in?uence at small radii should be detected with high signi?cance).This statement is independent of the QSO’s radiative history as long as it has been shining for more than~10Myr,since so large a volume is(potentially)illuminated by light emitted at ?10Myr

In any case,the possibility of beaming makes it clear that the ideal number of background sources is in fact several times larger than the arguments of point(B)sug-gest(see above).We would like to be cosmic-variance lim-ited in even the possibly small fraction of the?eld that is struck by the QSO’s ionizing radiation.Several hundred background sources would be ideal.Fortunately multi-object spectrographs with the required large?eld-of-view (e.g.,IMACS;Dressler et al.2003)can obtain this many spectra with a small number of slitmasks.Achieving the necessary signal-to-noise ratio for the background galax-ies at z>~3is less daunting than it sounds,since the large ?eld-of-view allows one to pick sources from the bright end of the luminosity distribution.

I should mention in closing that the results of this pa-per could be extended or improved in a number of ways. The expected cosmic varianceσV played a large role in the analysis,yet was estimated from an imperfect model of the three-dimensional transmissivity power-spectrum. Better numbers should be derived from numerical simu-lations.It would be interesting to know if measuring the radiation intensity with something other than the mean transmissivityˉf would let us achieve tighter constraints on the QSO’s radiative history before we were limited by cosmic variance.I assumed that the QSO would have red-shift z=3.0,but in fact observers could choose to observe QSOs at any redshift2<~z<~4where the Lyman-αfor-est is visible from the ground and reasonable numbers of background galaxies can be identi?ed.Finding the QSO redshift that minimizes the required observing time would allow one to study a larger number of targets.This work is admittedly un?nished.I hope to have shown that it is worth pursuing further.

George Becker and Luis Ho were a useful sources of information about AGN and the uncertainty in di?erent QSO-redshift estimators.Esther Hu helped me?nd elec-tronic version of her Voigt-pro?le line lists.Chuck Steidel ?elded my random questions like a pro.Alice Shapley’s email responses to my questions were always prompt and enlightening,and in addition she graciously allowed me to show the data in Figure8.Rob Simcoe had helpful advice on several subjects.Paul Martini o?ered useful comments on an earlier draft.Andrew Carnegie’s generosity put the food on my table.It is a pleasure to acknowledge the hos-pitality of Las Campanas Observatory during a long stay in Dec2003when this paper was begun.

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中英文对照版合同翻译样本

1.Sales Agreement The agreement, (is) made in Beijing this eighth day of August 1993 by ABC Trading Co., Ltd., a Chinese Corporation having its registered office at Beijing, the People’ Repubic of China (hereinafter called “Seller”) and International Tradi ng Co., Ltd., a New York Corporation having its registered office at New York, N.Y., U.S.A. (hereinafter called “Buyer”). 2.WITNESSETH WHEREAS, Seller is engaged in dealing of (product) and desires to sell (product)to Buyer, and WHEREAS, Buyer desires to purchase(product) from Sellers, Now, THEREFORE, it is agreed as follows: 3.Export Contract This Contract is entered into this 5th day of August 1993 between ABC and Trading Co., Ltd. (hereinafter called “Seller”) who agrees to sell, and XYZ Trading Co., Ltd. (hereinafter called “Buyer”) who agrees to buy the following goods on the following terms and condition. 4.Non-Governmental Trading Agreement No. __This Agreement was made on the_day of_19_, BETWEEN _ (hereinafter referred to as the Seller) as the one Side and _ (hereinafter referred to as the Buyer) as the one other Side. WHEREAS, the Seller has agreed to sell and the buyer has agreed to buy _ (hereinafter referred to as the Goods ) the quantity, specification, and price of which are provided in Schedule A. IT IS HEREBY AGREED AS FOLLOWS: 5.Contract For Joint-Operation Enterprise __ COMPANY LTD., a company duly organized under the Law of __ and having its registered office at (hereinafter called “Party A”) AND __ COMPANY LTD., a company duly organized under the Law of __ and having its registere d office at (hereinafter called “Party B”) Party A and Party B (hereinafter referred to as the “Parties”) agree to jointly form a Co-operation Venture Company (hereinafter referred to as the “CVC”) in accordance with “the Laws of the People’s Republic of C hina on Joint Ventures Using Chinese and Foreign Investment” and the “Regulations for the Implementation of the Laws of the People’s Republic of China on Joint Ventures Using Chinese and Foreign Investment” and other applicable laws and regulations. 6.MODEL CONTRACT Contract No. Date: Seller: Signed at: Address: Cable Address: Buyer: Address: Cable Address: The Seller and the Buyer have agreed to conclude the following transactions according to the terms and conditions stipulated below: https://www.sodocs.net/doc/219766541.html, of Commodity: 2.Specifications: 3.Quantity: 4.Unit Price: 5.Total Price: U.S.$: 6.Packing: 7.Time of Shipment: days after receipt of L/C. 8.Loading Port & Destination Port: From via to . 9.Insurance:

对翻译中异化法与归化法的正确认识

对翻译中异化法与归化法的正确认识 班级:外语学院、075班 学号:074050143 姓名:张学美 摘要:运用异化与归化翻译方法,不仅是为了让读者了解作品的内容,也能让读者通过阅读译作,了解另一种全新的文化,因为进行文化交流才是翻译的根本任务。从文化的角度考虑,采用异化法与归化法,不仅能使译文更加完美,更能使不懂外语的人们通过阅读译文,了解另一种文化,促进各民族人们之间的交流与理解。翻译不仅是语言符号的转换,更是跨文化的交流。有时,从语言的角度所作出的译文可能远不及从文化的角度所作出的译文完美。本文从翻译策略的角度,分别从不同时期来说明人们对异化法与归化法的认识和运用。 关键词:文学翻译;翻译策略;异化;归化;辩证统一 一直以来,无论是在我国还是在西方,直译(literal translation)与意译(liberal translation)是两种在实践中运用最多,也是被讨论研究最多的方法。1995年,美籍意大利学者劳伦斯-韦努蒂(Lawrence Venuti)提出了归化(domestication)与异化(foreignization)之说,将有关直译与意译的争辩转向了对于归化与异化的思考。归化与异化之争是直译与意译之争的延伸,是两对不能等同的概念。直译和意译主要集中于语言层面,而异化和归化则突破语言的范畴,将视野扩展到语言、文化、思维、美学等更多更广阔的领域。 一、归化翻译法 Lawrwnce Venuti对归化的定义是,遵守译入语语言文化和当前的主流价值观,对原文采用保守的同化手段,使其迎合本土的典律,出版潮流和政治潮流。采用归化方法就是尽可能不去打扰读者,而让作者向读者靠拢(the translator leaves the reader in peace, as much as possible, and moves the author towards him)。归化翻译法的目的在于向读者传递原作的基本精神和语义内容,不在于语言形式或个别细节的一一再现。它的优点在于其流利通顺的语言易为读者所接受,译文不会对读者造成理解上的障碍,其缺点则是译作往往仅停留在内容、情节或主要精神意旨方面,而无法进入沉淀在语言内核的文化本质深处。 有时归化翻译法的采用也是出于一种不得已,翻译活动不是在真空中进行的,它受源语文化和译语文化两种不同文化语境的制约,还要考虑到两种文化之间的

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