Cosmic dynamics in the era of Extremely Large Telescopes

Abstract

The redshifts of all cosmologically distant sources are expected to experience a small, systematic drift as a function of time due to the evolution of the Universe's expansion rate. A measurement of this effect would represent a direct and entirely model-independent determination of the expansion history of the Universe over a redshift range that is inaccessible to other methods. Here we investigate the impact of the next generation of Extremely Large Telescopes on the feasibility of detecting and characterising the cosmological redshift drift. We consider the Lyman alpha forest in the redshift range 2 < z < 5 and other absorption lines in the spectra of high redshift QSOs as the most suitable targets for a redshift drift experiment. Assuming photon-noise limited observations and using extensive Monte Carlo simulations we determine the accuracy to which the redshift drift can be measured from the Ly alpha forest as a function of signal-to-noise and redshift. Based on this relation and using the brightness and redshift distributions of known QSOs we find that a 42-m telescope is capable of unambiguously detecting the redshift drift over a period of ~20 yr using 4000 h of observing time. Such an experiment would provide independent evidence for the existence of dark energy without assuming spatial flatness, using any other cosmological constraints or making any other astrophysical assumption. Comment: Accepted for publication in MNRAS, 27 pages, 19 figures

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arXiv:0802.1532v1 [astro-ph] 11 Feb 2008
Mon. Not. R. Astron. Soc. 000, 1–27 (2007)Printed 11 February 2008(MN LATEX style file v2.2)
Cosmic dynamics in the era of Extremely Large Telescopes
J. Liske,1⋆A. Grazian,2E. Vanzella,3M. Dessauges,4M. Viel,3,5L. Pasquini,1
M. Haehnelt,5S. Cristiani,3F. Pepe,4G. Avila,1P. Bonifacio,6,3F. Bouchy,7,8
H. Dekker,1B. Delabre,1S. D’Odorico,1V. D’Odorico,3S. Levshakov,9C. Lovis,4
M. Mayor,4P. Molaro,3L. Moscardini,10,11M.T. Murphy,5,12D. Queloz,4P. Shaver,1
S. Udry,4T. Wiklind13,14and S. Zucker15
1European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany
2INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monteporzio Catone (Roma), Italy
3INAF – Osservatorio Astronomico di Trieste, Via Tiepolo 11, 34143 Trieste, Italy
4Observatoire de Gen` eve, 51 Ch. des Maillettes, 1290 Sauverny, Switzerland
5Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA
6CIFIST Marie Curie Excellence Team, GEPI, Observatoire de Paris, CNRS, Universit´ e Paris Diderot, Place Jules Janssen 92190 Meudon, France
7Laboratoire d’Astrophysique de Marseille, Traverse du Siphon, 13013 Marseille, France
8Observatoire de Haute-Provence, 04870 St Michel l’Observatoire, France
9Department of Theoretical Astrophysics, Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
10Dipartimento di Astronomia, Universit` a di Bologna, via Ranzani 1, 40127 Bologna, Italy
11INFN – National Institute for Nuclear Physics, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
12Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
13Space Telescope Science Institute, 3700 San Martin Drive, Baltimore MD 21218, USA
14Affiliated with the Space Sciences Department of the European Space Agency
15Department of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Accepted ...... Received .....
ABSTRACT
The redshifts of all cosmologically distant sources are expected to experience a small, sys-
tematic drift as a function of time due to the evolution of the Universe’s expansion rate. A
measurement of this effect would represent a direct and entirely model-independent deter-
mination of the expansion history of the Universe over a redshift range that is inaccessible
to other methods. Here we investigate the impact of the next generation of Extremely Large
Telescopes on the feasibility of detecting and characterising the cosmological redshift drift.
We consider the Lyman α forest in the redshift range 2 < z < 5 and other absorption lines in
the spectra of high redshift QSOs as the most suitable targets for a redshift drift experiment.
Assuming photon-noise limited observations and using extensive Monte Carlo simulations
we determine the accuracy to which the redshift drift can be measured from the Lyα forest
as a function of signal-to-noise and redshift. Based on this relation and using the brightness
and redshift distributions of known QSOs we find that a 42-m telescope is capable of unam-
biguously detecting the redshift drift over a period of ∼20 yr using 4000 h of observing time.
Such an experiment would provide independent evidence for the existence of dark energy
without assuming spatial flatness, using any other cosmological constraints or making any
other astrophysical assumption.
Key words: cosmology: miscellaneous – intergalactic medium – quasars: absorption lines.
1INTRODUCTION
The universal expansion was the first observational evidence that
general relativity might be applicable to the Universe as a whole.
SinceHubble’s(1929) discovery much effort has been invested into
completing the basic picture of relativistic cosmology. The central
⋆E-mail: jliske@eso.org
question is: what is the stress-energy tensor of the Universe? As-
suming homogeneity and isotropy reduces this question to: what is
the mean density and equation of state of each mass-energy com-
ponent of the Universe? Since these parameters determine both
the evolution with time and the geometry of the metric that solves
the Einstein equation, one can use a measurement of either to in-
fer their values. Over the past decade the successes on this front
have reached their (temporary) culmination: observations of the

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J. Liske et al.
Cosmic Microwave Background (CMB; Spergel et al. 2003, 2007),
type Ia supernovae (SNIa; Riess et al. 2004; Astier et al. 2006),
the large-scale galaxy distribution (Peacock et al. 2001; Cole et al.
2005; Eisenstein et al. 2005) and others now provide answers of
such convincing consistency and accuracy that the term ‘precision
cosmology’ is now commonplace (e.g. Primack 2005).
By far the most unexpected result of this campaign was the
discovery that the expansion of the Universe has recently begun
accelerating (Riess et al. 1998; Perlmutter et al. 1999). The phys-
ical reason for this acceleration is entirely unclear at present.
Within relativistic cosmology it can be accommodated by mod-
ifying the stress-energy tensor to include a new component with
negative pressure. In its simplest incarnation this so-called dark en-
ergy is the cosmological constant Λ (e.g. Carroll, Press & Turner
1992), i.e. a smooth, non-varying component with equation of
state parameter w = −1. It could also be time variable (e.g.
Overduin & Cooperstock 1998), either because of a specific equa-
tion of state (xCDM, e.g. Turner & White 1997; phantom energy,
Caldwell 2002) or because the component consists of a dynami-
cal scalar field evolving in a potential (e.g. Ratra & Peebles 1988).
More generally it may be an inhomogeneous, time varying com-
ponent with −1 ? w ? 0 (where w may also vary with time),
sometimes called quintessence (e.g. Caldwell, Dave & Steinhardt
1998), or even a component with an exotic equation of state
(e.g. Chaplygin gas, Kamenshchik, Moschella & Pasquier 2001).
Alternatively, instead of modifying the stress-energy tensor
one can also modify gravity itself to explain the acceleration
(e.g. Deffayet, Dvali & Gabadadze 2002; Freese & Lewis 2002;
Carroll et al. 2005). Although all current observations are consis-
tent with a cosmological constant (e.g. Seljak, Slosar & McDonald
2006) many more unfamiliar models are not ruled out (e.g.
Capozziello et al. 2005).
Probably the best way to probe the nature of the accelera-
tion is to determine the expansion history of the Universe (Linder
2003; Seo & Eisenstein 2003). Observables that depend on the ex-
pansion history include distances and the linear growth of density
perturbations (Linder & Jenkins 2003; Linder 2005), and so SNIa
surveys, weak lensing (Heavens 2003; Jain & Taylor 2003) and
baryon acoustic oscillations in the galaxy power spectrum (BAO;
Seo & Eisenstein 2003; Wang 2006) are generally considered to be
excellent probes of the acceleration.
In practice, however, extracting information on the expansion
history from weak lensing and BAO requires a prior on the spa-
tial curvature, a detailed understanding of the linear growth of den-
sity perturbations and hence a specific cosmological model. Given
the uncertain state of affairs regarding the source of the accelera-
tion, and given that even simple parameterisations of dark energy
properties can result in misleading conclusions (Maor et al. 2001;
Bassett et al. 2004), theseareconceptually undesirable features and
several authors have pointed out the importance of taking a cosmo-
graphic, model-independent approach todetermining theexpansion
history (e.g. Wang & Tegmark 2005; John 2005; Shapiro & Turner
2006). Using SNIa to measure luminosity distances as a function
of redshift is conceptually the simplest experiment and hence ap-
pears to be the most useful in this respect. The caveats are that dis-
tance is ‘only’ related to the expansion history through an integral
over redshift and that one still requires a prior on spatial curvature
(Caldwell & Kamionkowski 2004).
Here we will revisit a method to directly measure the expan-
sion history that was first explored by Sandage (1962, with an ap-
pendix by McVittie). He showed that the evolution of the Hubble
expansion causes the redshifts of distant objects partaking in the
Hubble flow to change slowly with time. Just as the redshift, z, is
in itself evidence of the expansion, so is the change in redshift, ˙ z,
evidence of its de- or acceleration between the epoch z and today.
This implies that the expansion history can be determined, at least
in principle, by means of a straightforward spectroscopic monitor-
ing campaign.
The redshift drift is a direct, entirely model-independent mea-
surement of the expansion history of the Universe which does not
require any cosmological assumptions or priors whatsoever. How-
ever, the most unique feature of this experiment is that it directly
probes the global dynamics of the metric. All other existing cos-
mological observations, including those of the CMB, SNIa, weak
lensing and BAO, are essentially geometric in nature in the sense
that they map out space, its curvature and its evolution. Many of
these experiments also probe the dynamics of localised density per-
turbations but none actually measure the global dynamics. In this
sense the redshift drift method is qualitatively different from all
other cosmological observations, offering a truly independent and
unique approach to the exploration of the expansion history of the
Universe.
Following the original study by Sandage (1962), the redshift
drift and its relevance to observational cosmology were also dis-
cussed by McVittie (1965), Weinberg (1972), Ebert & Tr¨ umper
(1975), R¨ udiger (1980), Peacock (1999), Nakamura & Chiba
(1999), Zhu & Fujimoto (2004), Corasaniti, Huterer & Melchiorri
(2007) and Lake (2007). Lake (1981) gave equations expressing the
deceleration and matter density parameters, q0and ΩM, in terms of
z, ˙ z and ¨ z. An excellent expos´ e of the equations relevant to red-
shift evolution was also presented by Gudmundsson & Bj¨ ornsson
(2002). In addition, these authors investigated the redshift drift in
the presence of quintessence, while other non-standard dark en-
ergy models were considered by Balbi & Quercellini (2007) and
Zhang et al. (2007). The case of Dicke-Brans-Jordan cosmologies
wasscrutinised by R¨ udiger (1982) and Partovi & Mashhoon (1984)
studied the effects of inhomogeneities. Seto & Cooray (2006) sug-
gested that a measurement of the anisotropy of ˙ z could be used to
constrain the very low-frequency gravitational wave background.
Several of these authors have pointed out the superior redshift ac-
curacy achieved in the radio regime compared to the optical, and
Davis & May (1978) entertained the possibility of using a 21 cm
absorption line at z = 0.69 in the radio spectrum of 3C 286 for a
˙ z measurement. The detrimental effects of peculiar velocities and
accelerations, which may swamp the cosmic signal, were discussed
by Phillipps (1982), Lake (1982) and Teuber (1986). Finally, Loeb
(1998) first proposed the Lyman α (Lyα) forest as an appropri-
ate target for a ˙ z measurement (acknowledging D. Sasselov for the
idea) and assessed the prospects for a successful detection in the
context of currently existing observational technology. All except
the last of these studies concluded that a ˙ z measurement was be-
yond the observational capabilities of the time.
The purpose of the present paper is to examine the impact
of the next generation of 30–60-m Extremely Large Telescopes
(ELTs) on the feasibility of determining ˙ z. The key issue is ob-
viously the accuracy to which one can determine redshifts. In the
absence of systematic instrumental effects this accuracy depends
only on the intrinsic sharpness of the spectral features used, the
number of features available and the signal-to-noise ratio (S/N) at
which they are recorded (assuming that the features are resolved).
In a photon-noise limited experiment the latter in turn depends only
on the flux density of the source(s), the size of the telescope, the
total combined telescope/instrument efficiency and the integration
time.In thispaper wewill investigatethisparameter space in detail,

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Cosmic dynamics
3
expanding on previous work in several ways: in Section 3 we con-
firm the usefulness of the Lyα forest by quantifying the peculiar
motions of the absorbing gas using hydrodynamic simulations of
the intergalactic medium (IGM). In Section 4 we use Monte Carlo
simulations of the Lyα forest to quantify how its properties trans-
late to a radial velocity accuracy, and we consider the usefulness of
other absorption lines in Section 5. These results are then used in
Section 7 where we explore the observational parameter space and
realistically assess the feasibility of a ˙ z experiment with an ELT.
Finally, we summarise our findings in Section 8.
2MEASURING THE DYNAMICS
In any metric theory of gravity one is led to a very specific form of
the metric by simply assuming that the Universe is homogeneous
and isotropic. The evolution in time of this so-called Robertson-
Walker metric is entirely specified by its global scale factor, a(t).
The goal is to measure or reconstruct this function. Recall that we
observe the change of a with time by its wavelength-stretching ef-
fect on photons traversing the Universe. A photon emitted by some
object at comoving distance χ at time tem and observed by us at
tobssuffers a redshift of
1 + z(tobs,tem) =a(tobs)
a(tem)
(1)
(where, obviously, only two of the three variables χ, tem and tobs
are independent). If it were possible to measure not only a pho-
ton’s redshift but also its tem, then the problem would be solved by
simply mapping out the present-day relation between redshift and
look-back time, i.e. z(tobs= t0,tem), where t0denotes today.
A different approach to the problem is to consider how the
redshift of an object at a fixed comoving distance χ evolves with
tobs, i.e. to consider the function z|χ(tobs). For a given χ, tobs
determines temand so we have dropped the dependence on tem. In
principle, it is possible to map out z|χ(tobs) (at least for tobs> t0),
and this would be the most direct determination of a(t), no matter
which object is used. However, a full characterisation of z|χ(tobs)
would require observations over several Gyr. Over a much shorter
timescale, ∆tobs, one can at most hope to determine the first order
term of the Taylor expansion
dz|χ
dtobs(tobs) ≈z|χ(tobs+ ∆tobs) − z|χ(tobs)
∆tobs
.
(2)
As we will see presently, it turns out that measuring dz|χ/dtobs
is in fact sufficient to reach our goal of reconstructing a(t). By
differentiating equation (1) with respect to tobswe find
dz|χ
dtobs(tobs) = [1 + z|χ(tobs)]H(tobs) − H(tem),
(3)
where we have used that dtobs= [1 + z(tobs)]dtemfor a fixed χ,
and H = ˙ aa−1. Evaluating at tobs = t0, replacing the unknown
temwith its corresponding redshift, and dropping the reminder that
we are considering the redshift of an object at a fixed distance χ we
simply obtain (McVittie 1962):
˙ z ≡
˙ z is a small, systematic drift as a function of time in the redshift
of a cosmologically distant source as observed by us today. This
effect is induced by the de- or acceleration of the expansion, i.e.
by the change of the Hubble parameter H. Since H0 is known
dz
dtobs(t0) = (1 + z)H0− H(z).
(4)
Figure 1. Redshift evolution of three objects with present-day redshifts
z(t0) = 0.5,3 and 8 as a function of time of observation, for three dif-
ferent combinations of ΩMand ΩΛas indicated. For each case the dotted
lines indicate the Big Bang. tobsis shown relative to the present day, t0.
(Freedman et al. 2001), this drift is a direct measure of the expan-
sion velocity at redshift z. Measuring ˙ z for a number of objects
at different z hence gives us the function ˙ a(z). The point is that
given a(z) and ˙ a(z), one can reconstruct a(t). A measurement of
˙ z(z) therefore amounts to a purely dynamical reconstruction of the
expansion history of the Universe.
Predicting the redshift drift ˙ z(z) requires a theory of gravity.
Inserting the Robertson-Walker metric into the theory’s field equa-
tion results in the Friedman equation, which specifically links the
expansion history with the densities, Ωi, and equation of state pa-
rameters, wi, of the various mass-energy components of the Uni-
verse. In the case of general relativity the Friedman equation is
given by:
H(z) = H0
??
i
Ωi(1 + z)3(1+wi)+ Ωk(1 + z)2
? 1
2
,
(5)
where Ωk = 1 −?Ωi. Here we consider only two components:
form of a cosmological constant with wΛ = −1. In Fig. 1 we plot
z|χ(tobs) for three different objects [chosen to have z(t0) = 0.5,3
and 8] and for three different combinations of ΩM and ΩΛ, where
we have also assumed H0 = 70 h70 km s−1Mpc−1. For each
object, the redshift goes to infinity at some time in the past when
the object first entered our particle horizon. If ΩΛ = 0 the redshift
continually decreases thereafter as the expansion is progressively
slowed down by ΩM. Hence, in this case the redshifts of all objects
are decreasing at the present time.
However, if ΩΛ ?= 0 the initial decrease is followed by a
subsequent rise due to Λ relieving matter as the dominant mass-
energy component and causing the expansion to accelerate. The
turn-around point may lie either in the past or in the future, de-
pending on the object’s distance from us; i.e. if ΩΛ ?= 0 an object’s
redshift may be either increasing or decreasing at the present time.
For distant (nearby) objects, the Universe was mostly matter (Λ)-
dominated during the interval [tem,t0] and hence underwent a net
deceleration (acceleration), resulting in ˙ z < (>) 0 at the present
time (see Gudmundsson & Bj¨ ornsson 2002 for a more detailed dis-
cussion).
cold (dark) matter (CDM) with wM = 0 and dark energy in the

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J. Liske et al.
Figure 2. The solid (dotted) lines and left (right) axis show the redshift drift
˙ z (˙ v) as a function of redshift for various combinations of ΩMand ΩΛas
indicated. The dashed line shows ˙ z for the case of dark energy having a
constant wX= −2
3(and ΩM,ΩX= 0.3,0.7).
These features are evident in Fig. 2, where we plot the ex-
pected present-day redshift drift, ˙ z(z), for various values of ΩM
and ΩΛ(solid lines), and for a case where the dark energy w ?= −1
(dashed line). The redshift drift is also shown in velocity units (dot-
ted lines), where ˙ v = c ˙ z (1 + z)−1. As noted above, the existence
of a redshift region where ˙ z > 0 is the hallmark of ΩΛ ?= 0. Note
also the scale of Fig. 2. At z = 4 the redshift drift is of order 10−9
or 6 cm s−1per decade. For comparison, the long-term accuracy
achieved in extra-solar planet searches with the high-resolution
echelle spectrograph HARPS on the ESO 3.6-m Telescope is of
order 1 m s−1(e.g. Lovis et al. 2005).
3CHOOSING AN ACCELEROMETER
A priori, it is not at all obvious which spectral features of which
set or class of objects might be best suited for a ˙ z measurement.
Clearly though, potential candidate targets should boast as many
of the following desirable attributes as possible. (i) They should
faithfully trace the Hubble flow. Although peculiar motions are
expected to be random with respect to the Hubble flow, the addi-
tional noise introduced by them could potentially conceal the cos-
mic signal (Phillipps 1982; Teuber 1986; but see also Appendix A).
(ii) The targets should have the sharpest possible spectral features
to minimise the error on individual redshift measurements. (iii)The
number of useful spectral features per target should be as high as
possible in order to maximise the amount of relevant information
per unit observing time. (iv) The targets should be as bright as pos-
sible and finally, (v) they should exist over a wide redshift range,
and particularly at high z, where the signal isexpected tobe largest.
Clearly, several of these features are in conflict with each
other. Requirements (i) and (ii) are conflicting because sharp spec-
tral features require cold material which isgenerally found in dense
regions inside deep potential wells, which in turn generate large
peculiar accelerations. Similarly, point (i) clashes with point (iv)
because high intrinsic luminosities require very massive and/or
highly energetic processes, again implying deep potential wells.
Since very dense regions are relatively rare there is likewise ten-
sion between requirements (ii) and (iii). Finally, the demands for
brightness and high redshift are also difficult to meet simultane-
ously.
Hence, it seems that it is impossible to choose a class or set
of objects that is the ideal target, in the sense that it is superior to
every other class or set in each of the above categories. However,
there is one class of ‘objects’ that meets all but one of the criteria.
3.1The Lyman α forest
The term ‘Lyα forest’ refers to the plethora of absorption lines
observed in the spectra of all quasi-stellar objects (QSOs) short-
wards of the Lyα emission line.1Almost all of this absorption
arises in intervening intergalactic H I between us and the QSO (see
Rauch 1998 for a review). Since the absorbing gas is physically
unconnected with the background source against which it is ob-
served we elegantly avoid the conflict between requirements (i)
and (iv) above. However, as the gas is in photoionization equi-
librium with the intergalactic ultraviolet (UV) background, its
temperature is of order 104K (Theuns, Schaye & Haehnelt 2000;
Schaye et al. 2000). Consequently, the absorption lines are not par-
ticularly sharp and the typical line width is ∼30 km s−1(e.g.
Kim, Cristiani & D’Odorico 2001). On the other hand, QSOs are
among the brightest sources in the Universe and exist at all red-
shifts out to at least ∼6 (Fan et al. 2006). Furthermore, each QSO
spectrum at z ? 2 shows on the order of 102absorption features.
In the following we will consider the question whether, apart
from the Hubble expansion, other evolutionary processes acting on
the absorbing gas might also significantly affect the measured po-
sitions of the absorption lines over the timescale of a decade or
so. First and foremost is the issue to what extent the Lyα forest is
subject to peculiar motions.
3.1.1 Peculiar motions
Cosmological
Zhang, Anninos & Norman
Miralda-Escud´ e et al. 1996; Cen & Simcoe 1997; Charlton et al.
1997; Theuns et al. 1998; Zhang et al. 1998), analytic mod-
elling(e.g.Bi & Davidsen
observationsofthesizesand
(Bechtold et al. 1994; Dinshaw et al. 1994; Smette et al. 1995;
Charlton, Churchill & Linder1995;
Liske et al. 2000; Rollinde et al. 2003; Becker, Sargent & Rauch
2004; Coppolani et al. 2006; see also Rauch & Haehnelt 1995) all
suggest that the Lyα forest absorption occurs in the large-scale,
filamentary or sheet-like structures that form the cosmic web.
These are at most mildly overdense and are participating, at least
in an average sense, in the general Hubble expansion. However,
at least some fraction of the absorbing structures (depending on
redshift) must be expected to have broken away from the Hubble
flow and to have begun collapsing under the influence of local
gravitational potential wells.
This exact issue was investigated by Rauch et al. (2005) who
studied thedistributionof velocity shear between high-redshift Lyα
absorption common to adjacent lines of sight separated by 1 to
hydrodynamic simulations
1995;
(Cen et al.1994;
1996;Hernquist et al.
1997;Viel et al.
shapes
2002)
the
and
ofabsorbers
D’Odorico et al.1998;
1Since the Lyα forest is only observable from the ground for z ? 1.7
we only consider the high-z Lyα forest in this paper. We also exclude the
higher column density Ly limit and damped Lyα (DLA) absorbers from the
discussion.

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Cosmic dynamics
5
300 h−1
were indeed in good agreement with the absorbing structures un-
dergoing large-scale motions dominated by the Hubble flow. In
fact, the distributions could be reproduced very convincingly by ar-
tificial pairs of spectra created from a hydrodynamical simulation
of the IGM. This result instills us with further confidence that such
simulations accurately capture the kinematics of the gas responsi-
ble for the Lyα forest.
Thus assured, we will now explicitly examine the peculiar
velocities and accelerations of the absorbing gas in a hydrody-
namic simulation of the IGM produced by the parallel TreeSPH
code GADGET-2 (Springel 2005). We have used this code in its
TreePM mode in order to speed up the calculation of long-range
gravitational forces. The simulations were performed with periodic
boundary conditions and with 4003dark matter and 4003gas par-
ticles in a box of 60 h−1
and heating processes were followed using an implementation sim-
ilar to that of Katz, Weinberg & Hernquist (1996) for a primordial
mix of hydrogen and helium. The UV background was taken from
Haardt & Madau (1996). In order to further increase the speed of
the simulation we applied a simplified star formation criterion: all
of the gas at overdensities > 103times the mean density and with
temperature < 105K was turned into stars. The cosmological pa-
rameters were set to ΩM = 0.26, ΩΛ = 0.74, Ωb = 0.0463,
ns = 0.95, σ8 = 0.85 and H0 = 72 km s−1Mpc−1, where
Ωbis the baryonic density parameter, and ns and σ8 are the spec-
tral index and amplitude of the linear dark matter power spectrum,
respectively. These values are in excellent agreement with recent
joint analyses of CMB, SNIa, galaxy clustering and Lyα forest data
(Spergel et al. 2007; Seljak et al. 2006; Viel, Haehnelt & Lewis
2006). The ΛCDM transfer function was computed with CMBFAST
(Seljak & Zaldarriaga 1996). The above corresponds to the simula-
tion series B2 of Viel, Haehnelt & Springel (2004) which has been
widely used for cosmological studies. Moreover, this is the exact
same simulation that successfully reproduced the velocity shear
distributions observed by Rauch et al. (2005).
We pierced our simulation box with 1000 random lines of
sight (LOS) and noted the physical properties of the absorbing
gas along these lines at three different redshifts (z = 2,3 and
4). In Figs. 3 and 4 we plot the distributions of peculiar velocity
and acceleration, respectively. The acceleration of the gas is com-
puted from its velocity by dividing the latter by a dynamical time
tdyn = (Gρ)−1/2(e.g. Schaye 2001), where ρ is the total matter
density. Evidently, the distributions do not evolve rapidly with red-
shift. Since a given parcel of gas cannot be decelerated, peculiar
velocities cannot decrease with time and hence their distribution
shifts to slightly larger values at lower redshifts. The average pecu-
liar acceleration, on the other hand, decreases with time because by
volume the Universe is dominated by low-density regions in which
the gas density decreases as the Universe expands.
Peculiar motions are expected to be randomly oriented with
respect to the LOS. Hence, when averaging over a large num-
ber of individual ˙ z measurements, peculiar motions will only in-
troduce an additional random noise component but no systematic
bias. In Appendix A we explicitly derive an expression for the ob-
served redshift drift in the presence of peculiar motions. This ex-
pression can be used to translate the distributions in Figs. 3 and 4
into the corresponding error distribution on ˙ z. We find that for a
decade-long experiment the error due to peculiar motion is of or-
der ∼10−3cm s−1. This must be compared to the error induced by
photon noisefor an individual ˙ z measurement from asingleabsorp-
tion line. Clearly, if we wish to be able to detect the redshift drift
70kpc. They showed that the observed shear distributions
100Mpc comoving size. Radiative cooling
0200400600
vpec (km/s)
0.000
0.001
0.002
0.003
0.004
0.005
pdf (vpec)
z=2
z=3
z=4
Figure 3. Probability distribution functions (PDFs) of the peculiar velocity
of the absorbing gas along 1000 random lines of sight through our simula-
tion box at three different redshifts as indicated. Note that we are not using
the value of the velocity’s component along the LOS, but rather the modulus
of the full 3-dimensional velocity.
-14-13-12-11-10
log10 apec (cm/s2)
10-5
10-4
10-3
10-2
10-1
100
z=2
z=3
z=4
pdf log10apec
Figure 4. As Fig. 3 for the peculiar acceleration.
over a decade or so the overall accuracy of the whole experiment
hastobeof order ∼1cms−1.Sincethiswillbeachieved usinghun-
dreds of absorption lines the error on an individual ˙ z measurement
will be at least a factor of ∼10 larger. The error due to peculiar mo-
tions is therefore sub-dominant by at least ∼4 orders of magnitude.
Hence we conclude that peculiar motions will have no detrimental
impact whatsoever on a redshift drift experiment targeting the Lyα
forest.

Page 6

6
J. Liske et al.
3.1.2Galactic feedback
So far we have only considered peculiar motions induced by grav-
ity. In principle, there are of course a number of non-gravitational
ways of imparting kinetic energy to the absorbing gas, mostly
involving galactic feedback. The main reason to believe that
galactic feedback must have had a far-reaching impact on the
Lyα forest is its early and widespread low-level metal enrich-
ment, even at fairly low column densities (e.g. Cowie et al. 1995;
Ellison et al. 2000; Schaye et al. 2000, 2003; Aguirre et al. 2004;
Simcoe, Sargent & Rauch 2004). Although very low-level enrich-
ment could be achieved in situ with Population III stars (e.g.
Gnedin & Ostriker 1997) the general consensus is that some sort
of mechanism is required to transport metals from (proto-)galaxies
into the IGM. In contrast, no consensus has been reached as to
which of the possible mechanisms (or combination thereof) is the
correct one. Candidates are mergers and tidal interactions (e.g.
Gnedin 1998), ram pressure stripping, radiation pressure on dust
grains (e.g. Aguirre et al. 2001), photo-evaporation during reion-
ization (Barkana & Loeb 1999) and SN-driven winds, either from
low-mass, (pre-)galactic halos at z ≈ 10 (e.g. Madau et al. 2001)
or from massive starbursting galaxies at z < 5 (e.g. Aguirre et al.
2001; Adelberger et al. 2003). From our point of view the last pos-
sibility is the most worrying as it has the highest potential of sig-
nificantly altering the kinematic structure of the IGM at the time of
observation.
There exists persuasive evidence of the existence of galactic
superwinds from Lyman break galaxies at z ≈ 3 (e.g. Pettini et al.
2001) and it is likely that some fraction of strong metal absorp-
tion lines are connected with these structures (e.g. Adelberger et al.
2003; Simcoe et al. 2006). On the other hand, there is no obser-
vational evidence at all that superwinds are significantly stirring
up the high redshift Lyα forest at the time we observe it. Using
column density and optical depth differences across a close pair
of lines of sight Rauch et al. (2001) found no indication in the
forest’s small-scale density structure for widespread recent distur-
bances. Similarly, as discussed above, the velocity shear between
adjacent lines of sight is entirely explained by the Hubble flow and
gravitational instability (Rauch et al. 2005), leaving little room for
non-gravitationally induced motion. Indeed, the simple fact that
the aforementioned hydrodynamical simulations – which did not
include any galactic feedback – were so successful in reproduc-
ing the observed properties of the Lyα forest, including its line
broadening distribution and clustering, raises the question of how
significant amounts of feedback could be integrated without up-
setting the existing agreement between the models and the data
(Theuns, Mo & Schaye 2001). It seems that the volume filling fac-
tor of galactic superwinds is limited to a few per cent (Theuns et al.
2002; Desjacques et al. 2004; Pieri & Haehnelt 2004; Cen et al.
2005; Bertone et al. 2005). We conclude that, whatever the process
of metal enrichment may be, there is currently no reason to believe
that it has a wholesale effect on the kinematics of the general IGM
(as probed by the Lyα forest) at the time of observation.
3.1.3 Optical depth variations
Consider the gas responsible for a given Lyα forest absorption fea-
ture. If the physical properties of the gas change over the timescale
of a decade or so this will cause a variation of the feature’s optical
depth profile. Potentially, this could in turn lead to a small shift in
the feature’s measured position and hence mimic a redshift drift.
The precise magnitude of this additional ˙ z error component will
depend on the method used to extract the signal but we can gain
an impression of the relevance of the effect by comparing the ex-
pected optical depth variation to the apparent optical depth change
(at a fixed spectral position) induced by the redshift drift.
The gas properties we consider here are density, temperature
and ionization fraction. In the fluctuating Gunn Peterson approxi-
mation (e.g. Hui, Gnedin & Zhang 1997) the Lyα optical depth is
related to these quantities by τ ∝ (1 + δ)2T−0.7Γ−1, where δ
and T are the gas overdensity and temperature, respectively, and Γ
is the photoionization rate. The interplay between photoionization
heating of the gas and adiabatic cooling leads to a tight relation be-
tween temperature and density, which can be well approximated by
T = T0(1 + δ)γ(Hui & Gnedin 1997). Hence we obtain
τ ∝ (1 + δ)2−0.7γT−0.7
How do these quantities evolve with time? According to linear the-
ory density perturbations grow as (1+z)−1. Although exactly true
only for an Einstein-de Sitter Universe, this represents an upper
limit in open and flat models with a cosmological constant. Hence
we will err on the side of caution by adopting this growth factor in
thefollowing. Theevolutionof thetemperature-density relationcan
be gleaned from figure 6 of Schaye et al. (2000): dγ/dz = −0.1
and dT0/dz = 0.25T0 (where we have conservatively used the
steep evolution between redshifts 2 and 3). We take the evolution
of Γ from figure 7 of Bolton et al. (2005): dΓ/dz = −0.3Γ. Tak-
ing the appropriate derivatives of equation (6), inserting the above
values and translating a decade in our reference frame to a red-
shift difference dz at z = 3 we find an optical depth variation of
dτ = 3 × 10−9fτ, where f ≈ 0.2,0.2,0.05,0.3 for the variation
of δ, T0, γ and Γ, respectively. As we will see in Section 4.4, these
values for dτ are at least ∼2 orders of magnitude smaller than the
typical optical depth changes due to the redshift drift. We therefore
conclude that changes in the physical properties of the absorbing
gas are not expected to interfere with a ˙ z measurement from the
Lyα forest.
0
Γ−1.
(6)
3.2Molecular absorption lines
Before we move on to investigate the details of a redshift drift ex-
periment using ELT observations of the Lyα forest, let us brieflydi-
gress here to consider a very different ˙ z experiment using another
future facility. We have explored in some detail the possibility of
using the Atacama Large Millimeter Array (ALMA) to measure
˙ z from rotational molecular transitions seen in absorption against
background continuum sources. Going to the (sub-)mm regime has
the advantages of potentially very high resolution and less photon
noise for a given energy flux. Furthermore, molecular absorption
lines can be very sharp, with line widths as low as ? 1 km s−1.
However, the molecular gas in nearby galaxies is strongly concen-
trated towards the central regions. Hence we must expect the gas to
be subject to peculiar accelerations similar in magnitude to the cos-
mological signal. That in itself would not necessarily be problem-
aticaslong as wehad many individual ˙ z measurements from differ-
ent objects over which to average. Unfortunately, it seems unlikely
that thiswill be the case. At present, rotational molecular lines have
been detected in only four absorption systems (with redshifts 0.25–
0.89; Wiklind & Combes 1999, and references therein), despite in-
tensive searches (e.g. Curran et al.2004).Based on the incidence of
these lines and the number of continuum sources with fluxes larger
than 10 mJy at 90 GHz we estimate that the number of molecular
absorption systems observable with ALMA will be ∼50, with only
5–10 of these showing narrow lines – not enough to overcome the

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Cosmic dynamics
7
uncertainties due to peculiar motions. Hence we have decided not
to pursue the case for molecular absorption lines any further.
4 SENSITIVITY OF THE LYα FOREST TO RADIAL
VELOCITY SHIFTS
In Section 2 we have seen that the redshift drift is a very small ef-
fect. In order to detect it, an experiment must achieve an overall ac-
curacy with which radial velocity shifts can be determined of order
∼1 cm s−1. In this section we will investigate how the properties of
the Lyα forest translate to a radial velocity accuracy, σv, and how
σvdepends on the instrumental characteristics of the spectra and on
redshift. Specifically, we would like to know how many Lyα forest
spectra of which resolution and S/N are needed at a given redshift
to achieve the required σv.
We will investigate these issues using artificial spectra. High-
resolution observations have demonstrated that, to first approxima-
tion, the Lyα forest can be decomposed into a collection of individ-
ual absorption lines (e.g. Kim et al. 2001). These are usually taken
to be Voigt profiles and so each line is characterised by three pa-
rameters: redshift, z, H I column density, NHI, and velocity width,
b. Here, we will reverse this decomposition process and generate
(normalised) spectra with the desired instrumental characteristics
from given lists of absorption lines. We will use two types of line
lists. First, we will generate line lists from Monte Carlo (MC) sim-
ulations based on the statistics of the largest available samples of
absorption lines. Secondly, to validate our simulations, we will use
8 line lists available in the literature that have previously been de-
rived from high-resolution observations.
4.1 Simulated absorption line lists
We form simulated MC line lists by simply randomly drawing val-
ues for the absorption line parameters from their observed distribu-
tions (e.g. Hu et al. 1995; Lu et al. 1996; Kirkman & Tytler 1997;
Kim et al. 1997, 2001, 2002):
f(z,NHI,b) ∝ (1 + z)γN−β
HI
exp
?
−(b −¯b)2
2σ2
b
?
,
(7)
where γ = 2.2, β = 1.5,¯b = 30 km s−1and σb = 8 km s−1. We
impose limits of 15 < b < 100 km s−1and also restrict NHI to
the classical Lyα forest regime, excluding Lyman limit and DLA
systems: 12 < logNHI(cm−2) < 16. The above distribution is
normalised to give 102lines with 13.64 < logNHI(cm−2) < 16
per unit redshift at z = 2 (Kim et al. 2001). The actual number of
absorption lines in a given line list is drawn from a Poisson distri-
bution with a mean determined by the normalisation.
The above MC approach allows us to quickly generate large
amounts of spectra with realistic characteristics. Note, however,
that we make no assumptions regarding the underlying physics of
the IGM in which the absorption occurs. We simply use the obser-
vational fact that Lyα forest spectra can be well represented as a
random collection of Voigt profiles.
The most significant difference between our MC line lists and
the real Lyα forest is clustering: the real Lyα forest is not randomly
distributed in redshift but shows significant redshift-space correla-
tions on scales of at least 100 km s−1(e.g. Cristiani et al. 1995;
Fern´ andez-Soto et al. 1996; Liske et al. 2000). The impact of clus-
tering will be discussed in detail in Section 4.6.
Table 1. Observed Lyα forest line lists from the literature.
QSO
zQSO
λ rangea
Nb
Lyα
Reference
Q1101−264
J2233−606
HE1122−1648
HE2217−2818
HE1347−2457
Q0302−003
Q0055−269
Q0000−26
2.145
2.238
2.400
2.413
2.617
3.281
3.655
4.127
3226−3810
3400−3850
3500−4091
3550−4050
3760−4335
4808−5150
4852−5598
5380−6242
290
226
354
262
362
223
535
431
1
2
1
2
1
1
1
3
References: 1 = Kim et al. (2002), 2 = Kim et al. (2001), 3 = Lu et al.
(1996).
aWavelength range covered by the Lyα forest line lists in˚ A.
bNumber of Lyα forest lines.
4.2 Real absorption line lists
We have collected 8 QSO absorption line lists from the literature
(see Table 1). These were derived from UVES/VLT (7 objects)
or HIRES/Keck data (1 object). All spectra have a resolution of
FWHM ≈ 7 km s−1, while the typical S/N per pixel varies from
∼10 for Q0000−26 to ∼50 in the case of HE2217−2818. In all
cases the absorption line lists were derived by fitting Voigt profiles
to the spectra using VPFIT.2Details of the data acquisition and re-
duction, as well asof the line fittingand identification processes are
given by Kim et al. (2001, 2002) and Lu et al. (1996).
We note that two of the spectra, those of Q1101−264 and
Q0000−26, contain DLAs, which effectively block out parts of the
spectra. However, this is only expected to have a significant effect
at high redshift, where the Lyα forest line density is high, and so
we have excluded the affected spectral region only in the case of
Q0000−26.
4.3The second epoch
Simulating a ˙ z measurement requires a second epoch ‘observation’
of the same line of sight: we generate a second epoch absorption
line list from the original line list (both real and simulated) by sim-
ply shifting the redshift of each line according to a given cosmo-
logical model:
∆zi = ˙ z(zi;H0,ΩM,ΩΛ) ∆t0,
(8)
where ∆t0 is the assumed time interval between the first and sec-
ond epochs. Unless stated otherwise, we use the standard, general
relativistic cosmological model, assuming fiducial parameter val-
ues of H0 = 70 h70km s−1Mpc−1, ΩM = 0.3 and ΩΛ = 0.7.
4.4Generating spectra
Given an absorption line list a normalised spectrum is generated by
S(λ) = exp
?
−
Nal
?
i
τ[λα(1 + zi),NHI,i,bi]
?
,
(9)
where λ is the observed wavelength, Nalis the number of absorp-
tion lines in the spectrum, τ is the optical depth of a Voigt pro-
file and λα = 1215.67 ˚ A is the rest wavelength of the H I Lyα
2by R.F. Carswell et al., see http://www.ast.cam.ac.uk/∼rfc/vpfit.html.

Page 8

8
J. Liske et al.
Figure 5. (a) Example of an artificial Lyα forest spectrum at z ≈ 3 gen-
erated from a Monte Carlo absorption line list. (b) Close-up of the region
around 5000˚ A. (c) Artificial spectrum generated from the observed line list
of Q0302−003. Both spectra have S/N = 100.
transition. This spectrum is then pixelised, using a pixel size of
0.0125˚ A, and convolved with aGaussian line-spread function. Un-
less stated otherwise, we will use a resolution element four times
the pixel size, corresponding to a resolution of R = 100000 at
5000˚ A. We then add random noise to the spectrum assuming Pois-
son statistics, i.e. we assume a purely photon-noise limited exper-
iment. All S/N values quoted in this paper refer to the S/N per
pixel in the continuum. To begin with, we will only consider the
spectral range between the assumed background QSO’s Lyα and
Lyβ emission lines, i.e. the classical Lyα forest region. By con-
struction this region cannot contain any H I transitions that are of
higher order than Lyα, and it covers an absorption redshift range of
λβ/λα(1 + zQSO) − 1 < z < zQSO, where zQSO is the QSO’s
redshift and λβ = 1025.72 ˚ A is the rest wavelength of the H I
Lyβ transition. In Sections 5.1 and 5.2 we will also consider other
spectral regions.
InFig.5weshow examplesofartificialspectrageneratedfrom
both simulated (panels a and b) and real line lists (panel c). The ex-
pected flux difference between two spectra of the same absorption
lines, taken a decade apart, is shown in Fig. 6. A similar plot for
the optical depth difference reveals that the variations discussed in
Section 3.1.3 are unproblematic.
4.5Defining σv
In principle, a ˙ z determination will involve the measurement of ra-
dial velocity differences between the corresponding features of a
pair of spectra of the same object taken several years apart. We
now need a method to estimate the accuracy to which these differ-
ences can be determined. This requires knowledge of how exactly
the measurement will be performed. However, a priori it is not at
all obvious what the optimal signal extraction method might be. To
proceed nevertheless we choose to base our analysis on the generic
concept of the total radial velocity information content of a spec-
trum, which was developed by Bouchy, Pepe & Queloz (2001) in
Figure 6. Flux difference between two artificial, noiseless spectra of the
same Lyα forest at z ≈ 3 simulated for two observing epochs separated by
∆t0 = 10 yr and for various combinations of ΩMand ΩΛas indicated.
The redshift drift implied by these parameters is also given.
the context of optimising radial velocity searches for extra-solar
planets.
Following Bouchy et al. (2001) we begin by expressing the
flux observed in pixel i at the second epoch as a small perturbation
on the first epoch flux in the same pixel:
S2i = S1i+dSi
dλ
∆vi
c
λi,
(10)
which defines asmall velocity shift ∆vifor each pixel. λiisthe ob-
served wavelength of the ith pixel and dSi/dλ is the spectral slope
of the flux at that pixel. To first order the slope does not change
between the two epochs and hence it carries no epoch designation.
Averaging the velocity shift over all pixels in a spectrum, using
weights wi, we have:
∆v =
?
i∆vi wi
?
iwi
.
(11)
Clearly, the weight for the ith pixel should be chosen as the inverse
variance of ∆vi. In calculating this variance we must differ from
Bouchy et al. (2001). In the case of stars one of the spectra can
be assumed to be a perfect, noiseless template, essentially because
additional informationonthesametypeofstarcanbeusedtodefine
it. However, since every Lyα forest spectrum is unique we cannot
make the same assumption here, so that in our case both spectra
have noise. Hence we find:
?2?
where σ1iand σ2iare the flux errors in the ith pixel of the first and
second epoch spectra, respectively, and σS′
of the flux at pixel i. We can see that a low weight is assigned to
noisy pixels and tothose that have a small gradient, i.e. pixels in the
continuum or in the troughs of saturated absorption lines. Finally,
with the above choice of weights the error on ∆v is given by:
σ2
vi=
?
c
λidSi
dλ
σ2
1i+ σ2
2i +
(S2i− S1i)2
?dSi
iis the error on the slope
dλ
?2
σ2
S′
i
?
,
(12)
σ2
v=
?
??
iσ2
iwi?2=
viw2
i
1
?
iσ−2
vi
.
(13)
The above process has the advantage of conveniently attach-
ing a single figure of merit to a given pair of spectra in a non-
parametric, model-independent way: σvsimply represents the fun-

Page 9

Cosmic dynamics
9
damental photon-noise limit of the accuracy to which an overall
velocity shift between the two spectra can be determined. It is es-
sentially just a measure of the ‘wiggliness’ and of the S/N of the
spectra. However, we point out that σv does not entirely capture
all of the information contained in a pair of spectra with respect
to a ˙ z measurement. From Fig. 2 it is clear that the difference be-
tween the first and second epoch spectra is not simply an over-
all velocity shift. The second epoch spectrum will also be slightly
compressed with respect to the first epoch spectrum because the
redshift drift is larger at higher redshifts (i.e. longer wavelengths)
than at lower redshifts. The additional information that is contained
in this small alteration of the spectrum’s shape is entirely ignored
by the above method [because of the simple averaging operation in
equation (11)] and hence it is clearly a sub-optimal method of es-
timating the sensitivity of a pair of spectra to ˙ z. Despite this short-
coming we will retain the above definition of σv for the sake of its
simplicity.
4.6Results
Weare now ready to derive the relevant scaling relations for σv. We
begin by making two points regarding equation (12). First, we note
that for a fixed total integration time (the sum of the integration
times spent observing the first and second epoch spectra) the sum
σ2
and second epoch integration times are equal. Hence, the smallest
possible σv is only achieved when the spectra of both epochs have
the same S/N. In the following we will assume that this is the case.
Secondly, from equation (12) it is clear that σv scales as (S/N)−1,
as expected for a photon-noise limited experiment.
Consider now a set of NQSO targets that all lie at the same
redshift, each of which has been observed at two epochs such that
all 2NQSOspectra have the same S/N. Again, since we are consid-
ering a purely photon-noise limited experiment, σvshould scale as
N−1/2
pix
(where Npixis the total number of independent data points
in the sample) and hence also as N−1/2
case, it is irrelevant how the total S/N is divided among the targets.
Hence, for simplicity we will use NQSO = 1 in the following.
We now examine the behaviour of σvas a function of redshift
using the MC absorption line lists. For various QSO redshifts in
the range 2 ? zQSO ? 5 we have generated 10 pairs of Lyα forest
line lists and spectra as described in Sections 4.1 to 4.4, where each
spectrum was given a S/N of 13000. We then measured each pair’s
σv according to equation (13). The result is shown as blue dots in
Fig. 7, where each point and error bar is the mean and ±1 r.m.s.
of the 10 individual measurements at each redshift. We stress that
we are plotting the expected accuracy of a velocity shift measure-
ment performed on a single pair of spectra of a single target at a
given redshift, where each spectrum has S/N = 13000. We are not
plotting the combined accuracy of 10 such pairs.
From Fig. 7 we can see that the radial velocity sensitivity im-
proves rather rapidly with redshift for zQSO < 4, but the decrease
is somewhat shallower at zQSO > 4. Overall, σv improves by a
factor of almost 3 when moving from zQSO = 2 to 5. Specifically,
we find:
1i+ σ2
2iin equation (12) takes on its minimum when the first
QSO. Furthermore, in this ideal
σv ∝
?
(1 + zQSO)−1.7
(1 + zQSO)−0.9
zQSO < 4
zQSO > 4
(14)
The behaviour of σvas a function of redshift is due to the combina-
tion of several factors. The first is the redshift evolution of the Lyα
forest line density (cf. equation 7). At higher redshift more spec-
Figure 7. The blue dots with error bars show the accuracy with which a ra-
dial velocity shift can be determined from a Lyα forest spectrum as a func-
tion of QSO redshift. The red line parameterises the redshift dependence as
in equation (14). Each point is the mean σvmeasured from 10 pairs of arti-
ficial spectra with S/N = 13000, generated from simulated MC absorption
line lists. The error bars show the ±1r.m.s. range of the 10 simulations. The
green triangles show the results for simulations where the redshift evolution
of the Lyα forest has been switched off, i.e. where γ = 0 (cf. equation 7).
The orange crosses show the result of also restricting the σv measurement
in each spectrum to a redshift path of constant length ∆z = 0.4, as op-
posed to using the full Lyα forest region between the QSO’s Lyα and Lyβ
emission lines.
tral features are available for determining a velocity shift and so
σv decreases. However, from z ≈ 4 the absorption lines severely
blanket each other and the number of sharp spectral features does
not increase as rapidly anymore, causing the flattening of σv at
zQSO > 4. The green triangles in Fig. 7 show the σv measure-
ments that result from simulations where the redshift evolution of
the Lyα forest has been switched off, i.e. where the evolutionary
index γ has been set to 0 (cf. equation 7). Indeed, we can see that
in this case there is no evidence of a break.
Secondly, we recall that each Lyα forest spectrum covers the
entire region between the QSO’s Lyα and Lyβ emission lines. The
redshift path length of this region is given by ∆z = 0.156(1 +
zQSO). Hence the number of independent pixels per spectrum also
increases as (1 + zQSO). Since the S/N per pixel is kept constant
this implies a larger number of photons per spectrum and hence an
improved sensitivity to radial velocity shifts. The effect of this can
be seen by comparing the green triangles in Fig. 7 with the orange
crosses which are the result of using a constant redshift path length
of ∆z = 0.4 for each σv measurement, as well as γ = 0.
Finally, with γ = 0 and ∆z = const, the sensitivity to wave-
length shifts should be constant as a function of zQSO, and so the
sensitivity to velocity shifts should go as (1 + zQSO)−1(as can be
seen from equation 12). In fact, the orange crosses in Fig. 7 de-
crease more slowly than this because the widths of the absorption
linesin wavelength space increase as(1+zQSO), making the edges
of the lines less steep and hence slightly decreasing the spectrum’s
sensitivity to wavelength shifts.
In Fig. 8 we compare these results derived from the MC line
lists to those from the real line lists. From Table 1 we can see that
thereallinelists(andhence thecorresponding spectra) donot cover
the full Lyα forest regions, with differently sized pieces missing
both at the low and high redshift ends. Therefore we must correct
the σv values derived from the real line lists in order to make them

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10
J. Liske et al.
Figure 8. Comparison ofσvmeasurements derived from simulated MC line
lists (blue dots and solid line, same as in Fig. 7) and from real line lists (red
stars). The green circles show the results derived from simulated MC line
lists that include a simple scheme for clustering absorption lines in redshift
space (see text for details).
directly comparable to the values from the simulated lists. The cor-
rection is achieved by first assigning a new, slightly different QSO
redshift to each spectrum, such that the ‘missing’ low and high red-
shift parts of the Lyα forest region are equally large. We then de-
crease the measured σv by a factor (∆zobs/∆z)1/2, where ∆zobs
is the redshift path length covered by the observed line list and ∆z
is the redshift path length of the full Lyα forest region at the new
zQSO. The correction factors range from 0.56 to 0.99.
The red stars in Fig. 8 show the corrected σv values derived
from single pairs of spectra generated from the real absorption line
lists with S/N = 13000, while the blue dots show the measure-
ments from the MC line lists (same as in Fig. 7). Overall the agree-
ment between the results from the MC and real line lists is very
good, particularly at high redshift. At zQSO ≈ 2.4 the σv values
from the real line lists are ∼15 per cent higher than those from the
MC lists. By far the most significant deviation occurs at the lowest
redshift where the σv of Q1101−264 is higher than expected by
47 per cent. However, this is not too surprising as the line of sight
towards Q1101−264 is known to pass through an unusually low
number of absorbers with logNHI(cm−2) > 14 (Kim et al. 2002).
We believe that the small differences at zQSO ≈ 2.4 are
mainly due to clustering of real absorption lines in redshift space.
Clusteringhasthe effectof reducing thenumber of spectral features
because it increases line blanketing. However, at high redshift line
blanketing is already severe because of the high line density and
so clustering has a relatively smaller effect at high redshift than at
low redshift. We demonstrate that clustering can explain the ob-
served differences by generating a new set of MC line lists which
incorporate a toy clustering scheme: first, we randomly draw the
positions of ‘cluster’ centres from the Lyα forest redshift distribu-
tion. We then populate each ‘cluster’ with n ? 0 absorbers, where
n is drawn from a Borel distribution (Saslaw 1989). Since n can be
0, 1 or > 1 this process generates voids, single ‘field’ absorbers,
as well as groups and clusters of lines. Finally, absorption lines are
distributed around their ‘host’ clusters according to a Gaussian dis-
tribution with FWHM = 120 km s−1. The σv values that result
from this new set of MC line lists are shown as open green circles
in Fig. 8. The increase of σv compared to the unclustered simu-
lations is clearly very similar to that observed for the real line lists
Figure 9. Radial velocity accuracy as a function of spectral resolution. Each
point and error bar is the mean and ±1 r.m.s. of 10 simulations at zQSO=
4 and S/N = 13000. The pixel size is kept constant and is chosen such
that a resolution element is sampled by 3 pixels at the highest resolution
considered. The labels along the bottom axis denote the resolving power at
5000 ˚ A, while the labels along the top axis show the equivalent FWHM of
an unresolved line. The vertical red line marks the mode of the absorption
lines’ b parameter distribution (cf. equation 7).
and we conclude that clustering can indeed explain the small differ-
ence between the results obtained from the observed and simulated
line lists at zQSO ≈ 2.4. In any case, the near coincidence of the
σv value of J2233−606 at zQSO = 2.23 with the expected value
demonstrates that not all lines of sight are adversely affected by
clustering. In the following we will assume that such sightlines can
be pre-selected and hence we will ignore the effects of clustering
in the rest of this section.
We now turn to the behaviour of σv as a function of spectral
resolution. For various resolving powers in the range 2500 ? R ?
1.33×105we have generated 10 pairs of line lists and spectra with
zQSO = 4 and S/N = 13000, and measured their σv values as be-
fore. The result is presented in Fig. 9, where we show the resolving
power along the bottom axis and the corresponding FWHM of an
unresolved line along the top axis. We stress that the pixel size was
the same for all spectra (= 0.0125 ˚ A) and that it was chosen such
that a resolution element was well sampled even at the highest reso-
lution considered. Hence the strong increase of σvtowards lower R
in Fig. 9 is not due to different numbers of photons per spectrum or
sampling issues. Instead, it is simply due to the loss of information
caused by convolving the spectrum with a line-spread function that
is broader than the typical intrinsic absorption line width (marked
by the vertical line in Fig. 9). Indeed, at R ? 30000 the Lyα forest
is fully resolved and in this regime σv is independent of R.
Summarising all of the above we find that the accuracy with
which a radial velocity shift can be determined from the Lyα forest
scales as:
σv = 2
?S/N
2370
?−1?NQSO
30
?−1
2?1 + zQSO
5
?−1.7
cm s−1, (15)
where the last exponent changes to −0.9 at zQSO > 4, and where
the same S/N (per pixel) is assumed for all NQSO spectra at both
epochs.

Page 11

Cosmic dynamics
11
5SENSITIVITY GAINS FROM OTHER SPECTRAL
REGIONS
In the previous section we have investigated the sensitivity of the
Lyα forest to radial velocity shifts, using only the H I Lyα transi-
tion in the region between a QSO’s Lyα and Lyβ emission lines.
However, modern echelle spectrographs are capable of covering a
much wider spectral range in a single exposure and so the question
arises whether other spectral regions, containing absorption lines
from other ions or other H I transitions, can expediently contribute
towards a ˙ z measurement.
Before applying the procedure of the previous section to more
extended spectra, we briefly follow-up on the discussion at the end
of Section 4.5, where we pointed out that σv does not capture the
information that is contained in the ˙ z-induced change of the shape
of a spectrum. In the case of a Lyα-only spectrum this change of
shape consisted of a compression of the spectrum. If weallow addi-
tional transitions with different rest wavelengths then we no longer
have a one-to-one correspondence between absorption redshift and
wavelength, and so the redshift drift will in general induce a much
more complex change of the shape of a spectrum. Hence it is clear
that any attempt to harness this additional information must involve
the complete identification and modelling of all absorption features
used in the analysis.
Indeed, the complete identification of all metal absorption
lines will be necessary in any case, even if one endeavours to mea-
sure the redshift drift only from the Lyα forest. The point is that the
Lyα forest may of course be ‘contaminated’ by metal lines. Since
these lines may arise in absorption systems that lie at completely
different redshifts compared to the Lyα lines, their redshift drift
may also be very different. Hence, any lines wrongly assumed to
be Lyα could potentially lead to erroneous results. The only way to
avoid such biases is to completely identify all absorption features.
5.1The Lyβ forest
Each H I absorption line in the Lyα forest has corresponding coun-
terparts at shorter wavelengths that result from the higher order
transitions of the Lyman series. Obviously, all the arguments con-
cerning the suitability of the Lyα forest for a ˙ z measurement also
apply to these higher order lines, and so, in principle, they should
be almost as useful as the Lyα lines (where the qualifier ‘almost’ is
owed to the decreasing optical depth with increasing order). How-
ever, in practice one has to contend with (i) confusion due to the
overlap of low-redshift, low-order lines with high-redshift, high-
order lines, and (ii) with an increased uncertainty in the placement
of theQSO’scontinuum. Bothofthesedifficultiesareaggravated as
one proceeds up the series towards shorter wavelengths. For these
reasons we will not consider any H I transitions that are of higher
order than Lyβ.
We now extend our simulated spectra by adding the region be-
tween a QSO’sLyβ and Lyγ emission lines immediately bluewards
of the Lyα forest. This region extends the redshift path for Lyα
lines by a factor of 1.28 towards lower redshifts and also contains
Lyβ lines in the redshift range λγ/λβ(1+zQSO)−1 < z < zQSO,
where λγ = 972.54 ˚ A. Note, however, that the real Lyα line lists
do not extend below the QSOs’ Lyβ emission lines (cf. Table 1).
Hence we are forced to supplement the real Lyβ absorption lines in
this region with Lyα lines drawn from the MC simulations.
In Fig. 10 we compare the σv measurements derived from the
extended MC line lists (blue dots) to the Lyα-only results (solid
line, same as in Figs. 7 and 8). We find that including the Lyβ for-
Figure 10. Comparison of σv measurements derived from Lyα-only line
lists and various extended line lists covering additional spectral regions and
including other absorption lines. The solid line and solid stars show, respec-
tively, the results from the Lyα-only MC and real line lists as in Fig. 8. The
blue dots and open stars show the corresponding improved σv values that
result from the addition of the Lyβ forest. For the real line lists, the open
squares show the effect of further adding the available metal lines in the
Lyα+Lyβ forest region. Finally, for two of these QSOs, we show as solid
triangles the outcome of using all lines accessible in existing spectra.
est improves σv by a factor of 0.88 ± 0.006. The corresponding
comparison for the real line lists (open and solid stars) yields im-
provement factors of 0.82 to 0.92, with a mean of 0.88, in very
good agreement with the MC results.
5.2 Metal lines
Inaddition tothe H I absorption, every high-redshift QSO spectrum
shows absorption lines from a number of other ions, such as C IV,
Si IV or Mg II. Although these metal lines are far less numerous
than the H I lines, they are also much narrower: their widths are of
order a few km s−1. In fact, many metal lines are unresolved in cur-
rent spectra and so the widths of the narrowest lines are unknown.
This suggests that metal lines may considerably increase a spec-
trum’s sensitivity to radial velocity shifts and may hence supply
valuable additional constraints on ˙ z.
Could peculiar motions vitiate this supposition? This ques-
tion is difficult to answer with certainty because the exact ori-
gin of many of the various classes of metal absorption lines is
still under debate. It is nevertheless evident that the structures
responsible for the absorption form an inhomogeneous set (e.g.
Churchill et al. 2000, 2007) and represent a large range of environ-
ments, depending on the absorbing ion, column density and red-
shift. For example, low column density C IV absorbers probe the
IGM (e.g. Simcoe et al. 2004; Songaila 2006) while strong Mg II
lines are associated with galaxies (e.g. Bergeron & Boiss´ e 1991;
Steidel, Dickinson & Persson 1994; Zibetti et al. 2005). However,
we note that absorbers associated with galaxies are generally found
to have absorption cross-sections on the order of tens of kpc (e.g.
Steidel 1995; Churchill et al. 2000; Adelberger et al. 2005). For
such distances and for halo masses of ∼1012M⊙ (Steidel et al.
1994; Bouch´ e et al. 2004, 2006) one derives accelerations of a
few × 10−8cm s−2, while absorber kinematics point to pe-
culiar velocities of no more than a few hundred km s−1(e.g.
Churchill et al. 1996). According to Appendix A, peculiar motions

Page 12

12
J. Liske et al.
Table 2. Observed lists of metal absorbers.
Name
λ rangea
Nb
m
Reference
Q1101−264
3050−5765,
5834−8530
3400−3850
3500−4091
3550−4050
3760−4335
4808−5150
4852−5598
4300−6450,
7065−8590,
8820−9300
225
this work
J2233−606
HE1122−1648
HE2217−2818
HE1347−2457
Q0302−003
Q0055−269
Q0000−26
49
18
59
48
5
14
100
2
1
2
1
1
1
this work
References: 1 = Kim et al. (2002), 2 = Kim et al. (2001).
aWavelength range(s) searched for metal lines in˚ A.
bNumber of metal absorbers, defined as the number of
unique sets of {ion, z, N, b}.
of this magnitude are not problematic. The main caveat here is the
unknown fraction of strong metal absorbers arising in starburst-
driven outflows (e.g. Adelberger et al. 2003; Simcoe et al. 2006;
Bouch´ e et al. 2006, see also Section 3.1.2) which must be expected
to experience much larger accelerations. However, it may be possi-
ble to identify these systems kinematically (Prochaska et al. 2008)
and hence to exclude them from the analysis. In any case, in this
section we will proceed on the assumption that peculiar motions do
not generally invalidate the use of metal lines for a ˙ z measurement.
Equation (7) gave a succinct parameterisation of the proper-
ties of the Lyα forest. Unfortunately, equivalents for all the various
metal line species do not exist, and hence we are unable to gauge
the effects of metal lines using MC simulations.
Kim et al. (2001, 2002) and Lu et al. (1996), from whom we
obtained the real Lyα forest line lists used in the previous sections,
also identified metal lines in their spectra. Unfortunately, they only
published measured parameters for lines lying in the Lyα forest
region. Although these lines will also have transitions elsewhere
(which were probably used in the identification process) the pub-
lished line lists do not provide a complete view of the metal line
population outside of this region.
Toimproveonthissituationwewentback tothespectraoftwo
of our QSOs, Q1101−264 and Q0000−26, and derived our own
metal line lists covering the entire spectral range available to us. In
the case of Q1101−264 the spectrum was the same as that used by
Kim et al. (2002), whereas for Q0000−26 we used a UVES/VLT
spectrum of similar quality as the HIRES/Keck spectrum used by
Lu et al. (1996). Inthe following we will refer to these twoline lists
as the ‘complete’ metal line lists.
Table 2 summarises the wavelength ranges that were searched
for metal lines and the total number of metal absorbers that were
found in each QSO spectrum. The simulated spectra are generated
in exactly the same way as in Section 5.1, except that we now add
all available metal lines, and extend the spectra of Q1101−264 and
Q0000−26 to the red limits given in Table 2.
Fig. 10 shows the effect of the metal lines on σv. The green
open squares show the result of only using the Lyα+Lyβ forest re-
gion as in Section 5.1 but adding in all the available metal lines in
this region. Comparing this to the H I-only results of the previous
section (open red stars) we find that the metal lines improve σv
by factors of 0.58 to 0.99, with a mean of 0.85. We point out that,
strictlyspeaking, the derived σvvalues are only upper limitsfor the
six line lists taken from the literature because of their incomplete
coverage of the Lyβ forest region. Indeed, the best improvement is
achieved for one of the complete line lists, Q1101−264 (the low-
est redshift QSO), which has a particularly rich metal absorption
spectrum (cf. Table 2).
Finally, for our two complete line lists we show as solid green
triangles the effect of also adding in the accessible spectral regions
redwards of the Lyα forest. The additional metal lines further im-
prove the σv values of Q1101−264 and Q0000−26 by factors of
0.55 and 0.90, respectively. With only two values it is obviously
impossible to draw a firm conclusion regarding the average im-
provement offered by the metal lines redwards of the Lyα forest.
We therefore choose to be conservative and adopt the larger of the
two as a typical value.
Summarising Sections 5.1 and 5.2, the above experiments
have shown that the normalisation of equation (15) can be reduced
by a factor of 0.88 × 0.85 × 0.9 = 0.67 by considering not just
the Lyα forest but all available absorption lines, including metal
lines, over the entire accessible optical wavelength range down to a
QSO’s Lyγ emission line. Hence we now obtain:
σv = 1.35
?S/N
2370
?−1?NQSO
30
?−1
2?1 + zQSO
5
?−1.7
cm s−1.(16)
6 MULTIPLE EPOCHS
Fundamentally, a redshift drift experiment consists of simply mea-
suring the velocity shift between two spectra of the same QSO(s)
taken at two distinct epochs separated by some time interval ∆t0.
This is the view we took in the previous two sections where we de-
termined the fundamental photon-noise limit of the accuracy with
which this shift can be measured, and its scaling behaviour. How-
ever, inpracticethisnotion istoo simplistic.Firstof all,it implicitly
assumes that the total integration time, tint, required to achieve the
necessary S/N would be negligible compared to ∆t0, so that the
two epochs of observation are well-defined. As we will see in the
next section this assumption is not valid. Secondly, for a variety of
reasons it may be desirable to spread the observations more evenly
over thewhole period∆t0insteadof concentrating theminjust two
more or less well-defined epochs at the interval’s endpoints. Hence
the question arises how the accuracy of a redshift drift experiment
is affected by distributing the total available tint over multiple ob-
serving epochs within the interval ∆t0.
Let us assume then that observations take place at Nedifferent
epochs, where the jth epoch is separated from the first by ∆tj, so
that ∆t1 = 0 and ∆tNe= ∆t0. We can straightforwardly gener-
alise the framework developed in Section 4.5 by turning equation
(10) into a continuous equation for the expected normalised flux in
the ith pixel at time ∆t:
Si(∆t) = S0i+dSi
dλ
λi
˙ vi
c
∆t ≡ S0i+ mi∆t.
(17)
The idea is now to fit this linear relation to the observed fluxes
Sji at times ∆tj with errors σji, yielding an estimate of the slope
mi and hence of ˙ vi for each pixel. (Note that S0i is a nuisance
parameter. It represents the ‘true’ flux at the first epoch as opposed
to the observed value S1i.) The maximum likelihood estimator for
miis
mi =Si∆t − Si ∆t
∆t2− ∆t
where the bar denotes the weighted average over all epochs:
2
,
(18)

Page 13

Cosmic dynamics
13
x =
?Ne
j=1x σ−2
?Ne
??
ji
j=1σ−2
ji
.
(19)
The variance of miis given by
σ2
mi=
j
σ−2
ji
?
∆t2− ∆t
2??−1
.
(20)
With miand its variance in place we can write down the equivalent
of equation (12):
σ2
˙ vi=
?
c
dSi
dλ
λi
?2?
σ2
mi+
m2
i
?dSi
dλ
?2σ2
S′
i
?
,
(21)
which in turn allows us to compute ˙ v averaged over all pixels and
its error, σ˙ v, corresponding to equations (11) and (13). Finally, we
re-define σv ≡ σ˙ v∆t0. This new version of σv now includes the
effect of multiple observing epochs and an arbitrary distribution of
the total integration time among them. It is straightforward to show
that for Ne = 2 one recovers exactly the original σvof Section 4.5.
We are now in a position to amend equations (15) and (16)
to include the effect of multiple epochs. Recall that these scaling
relations were derived for the case of Ne = 2 and for equal S/N
in the spectra of both epochs. Since the variance of the normalised
flux in pixel i scales as the inverse of the integration time we now
write σjias
σ2
ji= σ2
i
0.5
fj,
(22)
where fj is the fraction of the total tint used at the jth epoch
(?
∆tj ≡ hj∆t0, we can re-write equation (20) as:
?
j
= σ2
jfj = 1), and σi denotes the flux error (in the ith pixel) that
one would obtain if half of the total tintwere used. Further defining
σ2
mi=2σ2
i
∆t2
0
4
??
h2
jfj−
??
j
hjfj
?2??−1
mi(Ne = 2,f1 = f2 = 0.5) g2(Ne,h1...Ne,f1...Ne).(23)
The first term above is just the variance of mi that one obtains in
the case of Ne = 2 and equal splitting of tint. The second term is
a ‘form factor’ that only depends on the distribution of tintwithin
∆t0. Again, it is straightforward to show that g(Ne = 2,f1 =
f2 = 0.5) = 1. Since the form factor is the same for all pixels, and
since σ2
modification of the σvscaling relation amounts to simply applying
the form factor:
?−1
× g(Ne,f1...Ne) cm s−1.
Note that the symbol ‘S/N’ now refers to the total S/N per object
accumulated over all epochs, in contrast to equations (15) and (16)
where it referred to the S/N achieved in each of two epochs.
Note also that we have dropped the dependence of the form
factor on h1...Neby considering every night within the period ∆t0
as a potential epoch. This fixes Ne and hj = (j − 1)/(Ne − 1),
while fj is constrained to lie in the range 0 ? fj ? l/tint, where l
isthe length of a night (which wewill assume tobe 9 h on average).
Thus we find:
miis the dominant term in equation (21), the sought-after
σv = 1.35
?S/N
3350
?−1?NQSO
30
2?1 + zQSO
5
?−1.7
(24)
g(Ne,f1...Ne) =Ne− 1
2
?Ne−1
j=1
?
j2fj−
?Ne−1
j=1
?
jfj
?2?−1
2
.(25)
Clearly, fj will be 0 for most nights. Nevertheless, there are
obviously a large number of different possible distributions for
the fjs. The best distributions are those that are symmetric and
peaked towards the endpoints of ∆t0. A flat distribution, withequal
observations taking place on n equally spaced nights, results in
g =
?
rate throughout the period ∆t0 comes with a rather severe penalty
attached. A priori, it is difficult to estimate the best g value that
can be realistically achieved in practice. From now on we will as-
sume, perhaps somewhat arbitrarily, that all observations occur as
much as possible towards the beginning and end of ∆t0 with the
constraint that the observing rate averaged over some intermediate
timescale of, say, a month cannot exceed 1/3, i.e. that no more than
a third of any month’s telescope time is used for the redshift drift
experiment. Depending on the ratio of tintand ∆t0this results in g
values of ∼1.1. Essentially, this configuration simply shortens the
effective length of the experiment by the amount of time it takes to
complete the observations at either end of ∆t0.
3(n − 1)/(n + 1) ≈ 1.7 for n ≫ 1. Thus the otherwise
quite desirable arrangement of observing at a more or less constant
7 CAN WE COLLECT ENOUGH PHOTONS?
InSections4,5and6welearnt whatS/Nratioisrequired toachieve
a given sensitivity to radial velocity shifts using QSO absorption
spectra. In aphoton-noise limitedexperiment the attainableS/N de-
pends only on four quantities: the brightness of the source, the size
of the telescope’s collecting area, the total integration time and the
total efficiency. By ‘total efficiency’ we mean the ratio of the num-
ber of detected photo-electrons to the number of source photons at
the top of the atmosphere, i.e. it comprises atmospheric absorption
and all losses occurring in the combined telescope/instrument sys-
tem, including entrance aperture losses and the detector’s quantum
efficiency.
In this Section we will investigate in detail the 5-dimensional
parameter space that is spanned by the above four quantities and
redshift, in order to determine whether a feasible combination ex-
ists that would allow a meaningful ˙ z measurement.
7.1S/N formula
We begin by writing down the relation between the S/N per pixel
and the above four parameters for the photon-noise limited case:
S
N= 700
?ZX
Zr
100.4(16−mX)?D
42m
?2tint
10h
ǫ
0.25
? 1
2
,
(26)
where D, tint and ǫ are the telescope diameter, total integration
time and total efficiency, ZX and mX are the zeropoint and appar-
ent magnitude of the source in the X-band, respectively, and Zr =
(8.88 × 1010) s−1m−2µm−1is the AB zeropoint (Oke 1974) for
aneffectivewavelength of6170˚ A[corresponding totheSloanDig-
ital Sky Survey (SDSS) r-band]. The normalisation of the above
equation assumes a pixel size of 0.0125 ˚ A (see Section 4.4) and a
central obscuration of the telescope’s primary collecting area of 10
per cent. D = 42 m corresponds to the Baseline Reference Design
for the European ELT (E-ELT; Gilmozzi & Spyromilio 2007).
7.2High-redshift QSOs
The photon flux from QSOs is of course not a free parameter that
can be varied at will. Instead we will have to content ourselves

Page 14

14
J. Liske et al.
with what will be offered by the population of real QSOs known
at the time of a hypothetical ˙ z experiment. Here we do not wish to
speculate on possible future discoveries of QSOsand hence we will
restrict ourselves to the ones known already today. In the following
we will extract a list of potential targets for a ˙ z experiment from
existing QSO catalogues. For each candidate target QSO we will
need a reliable magnitude that can be used to estimate its photon
flux, as well as its redshift.
The largest QSO catalogue with reliable, homogeneous pho-
tometry and redshifts currently available is the fourth edition of
the SDSS Quasar Catalog (Schneider et al. 2007). Being based on
the fifth data release of the SDSS, it yields 16913 QSOs with
zQSO ? 2. The catalogue provides PSF magnitudes in the ugriz
bands which we do not correct for Galactic extinction (as is ap-
propriate for S/N calculations). Since we are interested in the con-
tinuum flux we will use, for each QSO, the magnitude of the
bluest filter that still lies entirely redwards of the QSO’s Lyα emis-
sion line. Specifically, for objects with zQSO < 2.2 we will use
the g-band magnitude; for 2.2 ? zQSO < 3.47 the r-band; for
3.47 ? zQSO < 4.61 the i-band; and for 4.61 ? zQSOthe z-band.
Wethen apply a small correction to theselected magnitude totrans-
form the observed flux to that expected at the centre of the Lyα for-
est assuming a power-law spectral shape of the form fν ∝ ν−0.5
(Francis 1993).
Unfortunately, the SDSS catalogue does not cover the whole
sky. The largest collection of QSOs covering the entire sky is the
12th editionof theCatalogue of Quasars and ActiveNuclei recently
published by V´ eron-Cetty & V´ eron (2006), which contains many
additional bright QSOs not included in the SDSS catalogue. How-
ever, since the V´ eron catalogue is a compilation of data from many
different sources its photometry is very inhomogeneous and cannot
readily be converted to a photon flux. We will dispense with this
inconvenience by matching the V´ eron catalogue to the all-sky Su-
perCOSMOS Sky Survey (SSS; Hambly et al. 2001). Although the
photographic SSS photometry is not endowed with reliable abso-
lute calibrations either, at least it is homogeneous and covers three
bands (BJ, R and I), allowing us to synthesise approximate SDSS
magnitudes.
We proceed by first selecting all QSOs from the V´ eron cata-
logue with zQSO ? 2 and not flagged as unreliable, resulting in
21895 objects. For each of these we then identify the nearest SSS
object, allowing a maximum separation of 5 arcsec, and retrieve the
corresponding SSS catalogue data. 938 V´ eron objects have no SSS
counterpart, presumably because their coordinates are unreliable.
We then use those 11744 objects that are in common to the SDSS
and combined V´ eron-SSS catalogues to derive linear relations be-
tween the SDSS and SSS magnitudes, allowing for zeropoint off-
sets and colour terms. Such relations are reasonable representations
of the data and we find that the distributions of the residuals have
r.m.s. values of less than 0.3 mag in all cases. Finally, we purge the
common objects from the V´ eron-SSS catalogue and use the above
relations to synthesise an SDSS magnitude in the appropriate band
(see above) for each of the remaining QSOs.
For those QSOs in the initial V´ eron catalogue that have no
match in the SSS, or which are missing an SSS band needed to
synthesise the required SDSS magnitude, we will simply use the V
or R-band magnitude as listed in the V´ eron catalogue, provided it
is non-photographic.
In summary, the final combined sample of 25974 QSOs is
constructed fromthreesub-sets: (i)SDSS;(ii)objectswithredshifts
from the V´ eron catalogue and photometry from the SSS (converted
to the SDSS system); and (iii) objects where both the redshifts and
Figure 11. The dots show the known, bright, high-redshift QSO population
(separated by sub-sets as indicated, see text) as a function of redshift and
estimated photon flux at the centre of the Lyα forest. Along the right-hand
vertical axis we have converted the photon flux to a corresponding Johnson
V -band magnitude. The background colour image and solid contours show
the value of σv that can be achieved for a given photon flux and redshift,
assuming D = 42 m, ǫ = 0.25 and tint = 2000 h. The contour levels
are at σv = 2,3,4,6,8 and 10 cm s−1. The dotted contours show the
same as the solid ones, but for D = 35 m or, equivalently, for ǫ = 0.17 or
tint= 1389 h.
the photometry are taken from the V´ eron catalogue. We remind the
reader that the quality and reliability of the photometry decreases
rapidly from (i) to (iii).
7.3 Achievable radial velocity accuracy
In Fig. 11 we plot our QSO sample, split by the above sub-sets, in
the Nphot-zQSOplane, where Nphotis a QSO’s photon flux at the
top of the atmosphere and at the centre of the QSO’s Lyα forest,
as implied by the appropriate magnitude described above. Using
equations (24) and (26) and assuming values for D, ǫ and tint we
can calculate, for any given combination of Nphot and zQSO, the
value of σv that would be achieved if all of the time tint were in-
vested into observing a single QSO with the given values of Nphot
and zQSO. The background colour image and solid contours in
Fig. 11 show the result of this calculation, where we have assumed
D = 42 m, ǫ = 0.25 and tint = 2000 h. Note that we have in-
cluded both the improvement of σv afforded by the Lyβ forest and
the metal lines as well as the deterioration caused by spreading tint
over a 0.9 yr period at either end of a ∆t0 = 20 yr interval.
From Fig. 11 we can see that, although challenging, a rea-
sonable measurement of ˙ z(z) is within reach of a 42-m telescope.
There exist a number of QSOs that are bright enough and/or lie
at a high enough redshift to provide reasonable values of σv. We
find 18 objects at σv < 4 cm s−1and 5 objects at σv < 3 cm s−1,
withgood coverage of theredshift range2–4. Oneobject evengives
σv = 1.8 cm s−1. However, for asmaller telescope withD = 35 m
the number of objects with σv < 4 cm s−1reduces to only 7 (cf.
dotted contours).
In Fig. 12 we show more comprehensively how the number of
QSOs with σvsmaller than a given value depends on the telescope

Page 15

Cosmic dynamics
15
Figure 12. The colour image and the contours show the number of QSOs
for which the σvvalue on the ordinate or better can be achieved for a given
combination of telescope size, efficiency and integration time. The contour
levels are at NQSO= 3,5,10 and 30.
parameters and integration time, which we summarise into a single
‘normalised observational setup parameter’, O:
O =
?D
42m
?2
ǫ
0.25
tint
2000h
(27)
(cf. equation 26). For a given value of O the colour image and con-
tours in Fig. 12 show the number of QSOs that would give a σv
equal to or smaller than the value along the ordinate if all of tint
was spent on any one of them. For example, if we wanted to be
able to choose our targets from 30 QSOs bright enough and/or at a
high enough redshift to be able to achieve σv = 3 cm s−1or better
on each object individually, then we would require O ≈ 2.1. Note
however, that the ordinate of Fig. 12 does not give the overall value
of σv for a ˙ z experiment using the best NQSOtargets and setup O.
The reason is of course that the total value of σv of such an exper-
iment depends on how the total integration time is split up among
the NQSOtargets (see below).
7.4Target selection and observing strategies
The question of how to select targets for a ˙ z experiment depends
on what exactly one wishes to achieve. For example, we could sim-
ply rank the candidate targets by their achievable σv for a fixed
observational setup O, as suggested by Fig. 11, and select the ob-
jects with the smallest values. This would be the correct selection
strategy if the goal were to obtain the smallest possible overall σv
of the combined sample. Another possible goal might be to max-
imise the significance with which ˙ z ?= 0 can be detected. In this
case we would select the targets with the largest values of |˙ v|/σ˙ v,
which would give much larger weight to the high redshift objects
and would practically deselect all objects at zQSO ? 2.5 where the
redshift drift is too small to be detected. Yet another option would
be to maximise the sensitivity to ΩΛ(at a fixed, given value of ΩM)
by selecting the objects with the largest values of
clear that the choice of selection method will also depend on the
possible adoption of priors from other cosmological observations.
Once the exact goal and hence the target selection strategy has
been defined we must also choose the number of targets to include
d˙ v
dΩΛ/σ˙ v. It is
in the experiment. Including more than just the ‘best’ target (ac-
cording to a given selection strategy) is desirable for several prac-
tical reasons, including the ability to observe for as large a fraction
of the total available telescope time as possible (to ease schedul-
ing), and to be able to identify any potential directional systematics
caused by the atmosphere, the telescope, the instrument, the cal-
ibration procedures or the transformation of redshifts to the cos-
mological reference frame. Obviously though, the more objects are
included into the experiment the worse the final result will be (in
the absence of systematics) because observing time will have to be
redistributed from the ‘best’ object to the less suited ones.
Finally, we must also decide how to divide the total avail-
able integration time among the objects included in the experiment.
Again, there are several possibilities, including: the time could be
split equally among the targets or in such a way as to ensure equal
S/N, equal σv or an equal value of the selection parameter. (Note
that in practice it is likely that operational and scheduling con-
straints would limit the available choices.)
From the above it is clear that there are a large number of dif-
ferent possibilities of implementing a ˙ z experiment. In particular,
the decision on what precisely the objective of such an experiment
should be does not appear to be straightforward. We will now focus
on three alternatives which may be perceived to be representative
of three different approaches.
1. The simplest possible goal is to aim for the most precise
˙ z measurement possible, i.e. the smallest overall σv. This may be
considered a ‘pure experimentalist’ approach where virtually no
prior observational information or theoretical expectation is used
to (mis-)guide the design of the experiment.
2. Another approach is to emphasise the most basic and yet
most unique (and perhaps most captivating) feature of the redshift
drift experiment, which is being able to watch, literally and in real
time, the Universe changing its rate of expansion. In this case the
aim is to prove the existence of a dynamical redshift drift effect
and hence to measure a non-zero value of ˙ z with the highest possi-
ble significance.3However, selecting targets according to this ap-
proach requires the adoption of, and hence dependence on, a spe-
cific model for ˙ z(z), and is therefore not free of prior assumptions,
in contrast to our first example.
3. As explained in the introduction, the discovery and the un-
known physical source of the acceleration of the universal expan-
sion provide a strong motivation for any observation seeking to de-
termine H(z), and this is also true for the direct and dynamical
method of measuring ˙ z. From this point of view the goal should
be to place the strongest possible constraints on the acceleration.
Translating this goal to a target selection strategy again requires a
model of ˙ z(z), including the acceleration.
7.5Simulations of example measurements and constraints
on model parameters
We now proceed to implement each of the above approaches on
the QSO sample shown in Fig. 11 in order to illustrate what can
be achieved with each of them. However, we stress that the above
3The value of this approach is perhaps best appreciated by picturing the
PI of a future ˙ z experiment angrily muttering “And yet it moves!” under
her/his breath as s/he leaves a seminar room after having tried to convince
a sceptical audience that (i) an effect has indeed been observed and (ii) that
it is not an instrumental artifact.