Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

418 Gravitational Lensing

19.4 Applications

19.4.1 Microlensing

Even if a gravitational lens is not strong enough to form two distinct images of a background source it may still amplify its brightness to an observable extent (e.g. Paczynski 1986a,b). This phenomenon is called microlensing. If a star or other object approaches to within an angle θE of a lens, then it will be magnified and will consequently brighten. Inside the galactic halo stars will move across the line of sight to a distant source, such as a star in the Large Magellanic Cloud (LMC). As it traverses the lensing region it will brighten and diminish in a symmetrical fashion. Moreover, because gravitational lensing is achromatic, the variation in brightness can be distinguished from intrinsic stellar variability, which is usually di erent at di erent wavelengths. The timescale for a microlensing event in our Galaxy is

where v is the transverse velocity of the lens with respect to the source. For solar mass lenses at a distance Dd of order 10 kpc and v of order 200 km s−1 this timescale is of order a few months. Continuous monitoring of stars over this timescale is necessary to detect microlensing. Because the probability of a lens crossing the Einstein radius is small, many millions of stars need to be monitored.

The idea that galactic-halo dark matter might lens the light from distant stars has recently born fruit with convincing evidence for microlensing of stars in the LMC by sub-stellar mass objects in the halo of the Milky Way (Alcock et al. 1993; Aubourg et al. 1993). Although these do not strongly constrain the total amount of dark matter in our Galaxy, the relatively small number of microlenses detected does constrain the contribution to the mass of the halo in brown dwarfs; see Carr (1994).

A more exotic claim by Hawkins (1993) to have observed microlensing on a cosmological scale by looking at quasar variability is much less convincing. To infer microlensing from quasar light curves requires one to exclude the possibility that the variability seen in the light curves be intrinsic to the quasar. One might naively expect the timescale of intrinsic variability to increase to increase with QSO redshift as a consequence of cosmological time dilation. This increase is not seen in the data, suggesting the variation is not intrinsic, but time dilation is only one of many e ects that could influence the timescale of intrinsic variability in either direction. For example, the density of cosmological material surrounding a QSO increases by a factor of eight between z = 1 and z = 3. Alexander (1995) gives arguments that suggest that observational selection e ects may remove the expected correlation and replace it with the inverse e ect that is actually observed. It is not inconceivable therefore that a change in fuelling e ciency could change the timescale of variability in the opposite direction to the time dilation e ect. In any case, the classic signature of microlensing is that the variability

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Figure 19.2 William Herschel Telescope images (taken by Geraint Lewis and Michael Irwin) of the ‘Einstein cross’, a multiply imaged quasar. The two images were taken three years apart and the variation in brightness may be due to microlensing within our Galaxy.

be achromatic: even this is not known about the variability seen by Hawkins. QSOs, and active galaxies in general, exhibit variability on a wide range of timescales in all wavelength regions from the infrared to X-rays. If the lensing interpretation is correct, then one should be able to identify the same timescale of variability at all possible observational wavelengths. An independent analysis of QSO variability by Dalcanton et al. (1994) has also placed Hawkins’ claim in doubt, so we take the evidence that extragalactic microlensing has been detected to be rather tenuous.

19.4.2 Multiple images

The earliest known instance of gravitational lensing by anything other than the Sun was the famous double quasar 0957+561, which upon close examination was found to be a single object which had been lensed by an intervening galaxy (Walsh et al. 1979). As time has gone by, searches for similar such lensed systems have yielded more candidates, but the total number of candidate lens systems known is still small.

It has been known for some time that the predicted frequency of quasar lensing depends strongly on the volume out to a given redshift (Turner et al. 1984; Turner 1990; Fukugita and Turner 1991) and that the number of lensed quasars observed can consequently yield important constraints on cosmological models. Compared with the Einstein–de Sitter model, both flat cosmologies with a cosmological constant and open low-density (Ω0 < 1) models predict many more lensed systems. The e ect is particularly strong for the flat Λ models: roughly ten times as many lenses are expected in such models than in the Ω0 = 1 case. Of course, the number of lensed systems also depends on the number and mass of inter-

420 Gravitational Lensing

vening objects in the volume out to the quasar, so any constraints to emerge are necessarily dependent upon assumptions about the evolution of the mass function of galaxies, or at least their massive haloes. Nevertheless, claims of robust constraints have been published (Kochanek 1993; Maoz and Rix 1993; Mao and Kochanek 1994), which constrain the contribution of a Λ term to the total density of a flat universe to ΩΛ < 0.5 at 90% confidence, which seems to contradict the results from high-redshift supernovae we discussed in Chapter 4. Constraints on open, low-density models are much weaker: Ω0 > 0.2. Unless some significant error is present in the modelling procedure adopted in these studies, the QSO lensing statistics appear to rule out precisely those flat Λ-dominated models which have been held to solve the age problem and also allow flat spatial sections, although at a relatively low confidence level and at the expense of some model dependence. If our understanding of galaxy evolution improves dramatically it will be possible to refine these limits. New large-scale QSO surveys will also help improve the statistics of the lensed objects.

19.4.3 Arcs, arclets and cluster masses

There exists a possible independent test of the dynamical and X-ray masses of rich clusters which does not depend on the assumption of virial or hydrostatic equilibrium. Gravitational lensing of the light from background objects depends on the total mass of the cluster whatever its form and physical state, leading to multiple and/or distorted images of the background object.

The possible lensing phenomena fall into two categories: strong lensing in rich clusters can probe the mass distribution in the central parts of these objects; and weak lensing distortions of background galaxies can trace the mass distribution much further out from the cluster core. The discovery of giant arcs in images of rich clusters of galaxies as a manifestation of strong gravitational lensing (Tyson et al. 1990; Fort and Mellier 1994) has led to a considerable industry in using models of the cluster lens to determine the mass profile. Smaller arcs – usually called arclets – can be used to provide more detailed modelling of the lensing mass distribution. For recent applications of this idea, see Kneib et al. (1993) and Smail et al. (1995); the latter authors, for example, infer a velocity dispersion of σ2 1400 km s−1 for the cluster AC114.

Important though these strong lensing studies undoubtedly are, they generally only probe the central parts of the cluster and say relatively little about the distribution of matter in the outskirts. They do, for example, seem to indicate that the total distribution of matter is more centrally concentrated than the gas distribution inferred from X-ray observations. On the other hand, estimates of the total masses obtained using strong lensing arguments are not in contradiction with virial analysis methods described above.

Weak lensing phenomena – the slight distortions of background galaxies produced by lines of sight further out from the cluster core – can yield constraints on the haloes of rich clusters (Kaiser and Squires 1993; Broadhurst et al. 1995).

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Figure 19.3 HST image of the rich cluster Abell 2218 showing numerous giant arcs and arclets. Picture courtesy of the Space Telescope Science Institute.

It is also possible to use fluctuations in the N(z) relation of the galaxies behind the cluster to model the mass distribution.

The technology of these methods has developed rapidly and has now been applied to several clusters. Preliminary results are generally indicative of a larger total mass than is inferred by virial arguments, suggesting that there exists even more dark matter than dynamics would suggest. However, this technique is relatively young and it is possible that not all the systematic errors have yet been ironed out, so we take these results as indicating that this is a good – indeed important – method for use in future studies, rather than one which is providing definitive results at the present time.

19.4.4 Weak lensing by large-scale structure

The idea that clusters of galaxies produce observable distortions in the weak lensing limit suggests it may be possible to observe lensing along any line of sight through the background distribution of clusters. What one will see looking through an arbitrary distribution that lacks the special symmetry of a rich cluster will be correlated distortions of the shapes of galaxies. One needs a wide field in order to see su cient galaxies to obtain a signal, because the shear of any one galaxy is small compared with the distribution of shapes that exists in

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Figure 19.4 Simulation of the weak lensing distortion induced by large-scale structure. The pattern of density perturbations is shown as a greyscale picture upon which lines are superimposed representing the size and angle of the distortions. Picture courtesy of Alex Refregier.

the unlensed galaxy population. However di cult this may be, the payo is large because one can in principle obtain, from maps of sheared galaxy images, maps of the projected dark-matter distribution. This is a new field, but feasibility studies already show that the signal is measurable (Bacon et al. 2000; van Waerbeke et al. 2000; Wilson et al. 2001; Wittman et al. 2000). When larger CCD arrays go online, we will have maps of the evolved dark-matter distribution that can complement maps of the galaxy distribution obtained from redshift surveys and maps of the primordial fluctuations obtained from the cosmic microwave background.

19.4.5 The Hubble constant

One of the consequences of gravitational lensing is that the paths traversed by photons coming from the same source but forming di erent images may have di erent lengths. If the source happens to be variable, then one can hope to recognise a pattern in its output in more than one image at di erent times. If one understands the structure of the lens, then one can estimate the distances

Comments 423

involved. Knowing the redshift allows one to obtain an estimate of H0. This idea, of course, rests on the correct identification of the time delay. An example is the quasar 0957 + 561 which has a measured time lag of 415 days between features seen in the two images it presents to the observer. The lens seems to be dominated by a single galaxy sitting inside a cluster and the modelling is consequently fairly straightforward. Preliminary estimates by Grogin and Narayan (1996) yield a rather high value of the Hubble constant around 80 km s−1 Mpc−1 but with considerable theoretical uncertainty in the model parameters needed to reproduce the known images. In principle, such studies can yield accurate estimates of the Hubble constant but the technique is relatively young and clearly needs more work to develop it.

19.5 Comments

It is rather ironic that the oldest known observational consequence of general relativity should produce one of the newest and most dynamic areas in cosmology. Now that observational technology is so advanced and wide-field cameras are becoming increasingly available, it seems likely that weak lensing will have a particularly strong impact on cosmology in the relatively near future. In particular we should be able to understand the extent to which the large-scale structure seen in the galaxy distribution represents genuine fluctuations in the mass density and how much may be attributable to bias. Even the cosmic microwave background o ers the possibility for lensing studies.

Bibliographic Notes on Chapter 19

Schneider et al. (1992) is now the standard reference book on gravitational lensing. Other useful review articles are Blandford and Narayan (1992) and Fort and Mellier (1994). The paper by Refsdal (1964) is a classic which inspired much work in this area, long before the observational discovery of extragalactic lensed systems. Much of the material for this chapter was gleaned from the lecture notes of Narayan and Bartelmann, which are available on the internet at

http://www.mpa-garching.mpg.de/Lenses/Preprints/JeruLect.html

Problems

1.If Dd, Ds and Dds are angular-diameter distances, show that, in general, Dds ≠ Ds −

Dd.

2.Derive Equation (19.2.14).

3.Obtain estimates of the Einstein radius for (i) a point lens of mass M and D = 10 kpc, and (ii) a lens of mass 1011M and D = 1 Gpc.

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4.Show that, for a point-mass lens, the magnifications of the two images are given by

where u = β/θE. Hence show that when β = θE the total magnification of flux is 1.34.

5.A singular isothermal sphere is defined by a three-dimensional density profile of the form

σ2 ρ(r) = v .

2πGr2

Show that the deflection produced by such a lens is 4πσv2/c2 and derive an expression for the Einstein radius. Under what circumstances does this system produce multiple images?

6.Show that a circular source of unit radius is mapped into an ellipse with major and minor axes (1 − κ − γ)−1 and (1 − κ + γ)−1, respectively. Show further that the magnification is [(1 − κ)2 − γ2]−1.

20

The High-Redshift

Universe

20.1 Introduction

In the previous four chapters we have tried to explain how observations of galaxy clustering, the cosmic microwave background, galaxy-peculiar motions and gravitational lensing can be used to place constraints on theories of structure formation in the Big Bang model. In this chapter we shall discuss a number of independent pieces of evidence about the process of structure formation which can also, in principle, shed light upon how galaxies and clusters of galaxies might have formed. The common theme uniting these considerations is that they all involve phenomena occurring after recombination and before the present epoch.

Since galaxy properties are only observable at relatively small distances, and therefore at relatively small lookback times, galaxy clustering and peculiar motions give us information about the Universe here and now. On the other hand, primary anisotropies of the CMB yield information about the Universe as it was at t trec. In between these two observable epochs lies a ‘dark age’, before visible structure appeared but after matter was freed from the restraining influence of radiation pressure and viscosity. As we shall see, there are, in fact, a number of processes that can yield circumstantial evidence of various goings-on in this interval and these can, in turn, give us important insights into the way structure formation can have occurred. It should be said at the outset, however, that many of the issues we shall discuss in this chapter are controversial and clouded by observational uncertainties. We shall therefore concentrate upon the questions raised by this set of phenomena, rather than trying to incorporate them firmly in an overall picture of galaxy formation.

We have already mentioned, in Chapter 17, some ways of probing the postrecombination Universe, by exploiting secondary anisotropies in the CMB radi-

426 The High-Redshift Universe

ation such as the Sunyaev–Zel’dovich e ect. We shall raise some of these issues again here in the context of other observations and theoretical considerations. For the most part, however, this chapter is concerned with early signatures of galaxy formation, sources of radiation at high redshift and constraints on the properties of the intergalactic medium (IGM) at moderate and high redshifts.

20.2 Quasars

The most obvious way to acquire information about the Universe at early times is to locate objects with high redshifts. To be detectable, such objects must be very luminous at frequencies that get redshifted into the observable range of some earthly detector.

The objects with largest known redshifts are the quasars. The current record holder has z = 6.28, but quasars with redshifts as high as this are very di - cult to detect and/or identify. As we shall see, even the observation of a single high-redshift quasar can place strong direct constraints on models of structure formation. There are many more quasars at z 2 than at the present epoch. Efstathiou and Rees have estimated that the comoving number density of quasars

at this epoch (i.e. scaled to the present epoch), with luminosity greater than LQ 2.5 × 1046 erg s−1, is

At higher redshifts the luminosity function of quasars is very poorly known. It seems unlikely that the number density given in (20.2.1) rises drastically and there is also little evidence that it falls sharply before z 3.5. The existence of the record holder shows that there are at least some quasars with redshifts of order 5.

The usual model for a quasar is that its luminosity originates from matter accreting onto a central black hole embedded within a host galaxy. The central mass required depends on the luminosity, the lifetime of the quasar tQ (which is poorly known) and the e ciency H with which the rest-mass energy is released as radiation. For quasars with the luminosity given above, the required mass is

The number density of quasars given in (20.2.1) is, of course, very much less than the present value for galaxies. In a hierarchical clustering model, however, the number of bound objects on a given mass scale decreases at earlier times. It is an interesting exercise therefore to see if the existence of objects on the mass scale required to house a quasar contradicts theories of galaxy formation. To do this we first need to calculate how big the parent galaxy of a quasar has to be. There are three factors involved: the fraction fb of the matter in baryonic form which is subject to the constraints discussed in Chapter 8; the fraction fr of the baryons retained in a halo and not blown out by supernova explosions when star formation begins; the fraction fh of the baryons which participate in the fuelling

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of the quasar. All these factors are highly uncertain, so one can define a single quantity F = fbfrfh to include them all. It is unlikely that F can be larger than 0.01.

To model the formation of haloes we can use the Press–Schechter theory discussed in Section 14.5 (Efstathiou and Rees 1988). The z-dependence of the mass function of objects can be inserted into equation (14.5.7) by simply scaling the RMS density fluctuation by the factor 1/(1 + z) coming from linear theory. Recall that the parameter δc in equation (14.5.7) specifies a kind of threshold for collapse and that δc 1.68 is the appropriate value for isolated spherical collapse; numerical experiments suggest this analytic formula works fairly well, but with a smaller δc 1.33. Anyway, the number density of quasars is

The lower limit of integration Mmin is the minimum mass capable of housing a quasar, which is estimated to be

and tmin is either 0 or [t(z) − tQ ], whichever is the larger. Using equation (14.5.7) with δc = 1.33 and defining

Efstathiou and Rees (1988) obtained, for a spectrum with n −2.2 (appropriate to a CDM model on the relevant scales),

(20.2.6)

in the same units as Equation (20.2.1). Notice above all that this falls precipitously at high z because of the exponential term. This can place strong constraints on models where structure formation happens very late, such as in the biased CDM picture. The result (20.2.6) is not, however, incompatible with (20.2.1) for this model. A similar exercise could be attempted for clusters of galaxies and absorption-line systems in quasar spectra, but we shall not discuss this possibility here.

As we explained in Chapter 4 there are also other types of active galaxy that can be observed at high redshifts, although not as high as quasars. One of these types is particularly interesting in the present context: steep-spectrum radio sources. In recent years, samples of these objects have been studied in the optical wavelength region. Many of them are associated with galaxies having redshifts greater than two, and one, called 4C41.17, has a redshift of 3.8, which is the largest known redshift of a galaxy. These objects may yield important clues about the relationship