Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F
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98 Observational Properties of the Universe
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Figure 4.9 Angular diameter versus redshift for 145 radio sources. From Gurvits et al. (1999). Picture courtesy of Leonid Gurvits.
source evolution, one must assume the source properties do not vary with cosmological time. Since there is overwhelming evidence for strong evolution with time in almost all classes of astronomical object, the prospects for using this method are highly limited.
An example is the attempt by Kellermann (1993) to resurrect this technique by applying it to compact radio sources. These sources are much smaller than the extended radio sources discussed in previous studies, so one might therefore expect them to be less influenced by, for example, the evolution of the cosmological density. Kellermann originally found a minimum in the angular-size versus distance relationship, but a subsequent analysis by Gurvits et al. (1999) found a larger scatter in the data. We must therefore conclude that the evidence from the angular size data is not particularly compelling. Indeed, it is not at all obvious that there are any ‘standard metre sticks’ in sight that will be visible at high redshift and also will have well-understood evolutionary properties that could lead to a change in this situation. It is wise not to be too optimistic
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about this method yielding decisive results, although it is possible that angular size estimates of clusters of sources, or measurements of angular separation of similar objects, could eventually give the statistical data needed for this test.
4.7.3 Number-counts
An alternative approach is not to look at the properties of objects themselves but to try to account for the cumulative number of objects one sees in samples that probe larger and larger distances. A first application of this idea was by Hubble (1929); see also Sandage (1961). By making models for the evolution of the galaxy luminosity function one can predict how many sources one should see above an apparent magnitude limit and as a function of redshift. If one accounts for evolution of the intrinsic properties of the sources correctly, then any residual dependence on redshift is due to the volume of space encompassed by a given interval in redshift; this depends quite strongly on Ω0. The considerable evolution seen in optical galaxies, even at moderately low redshifts, as well as the large K-corrections and uncertainties in the present-day luminosity function, renders this type of analysis prone to all kinds of systematic uncertainties. One of the major problems here is that one does not have complete information about the redshift distribution of galaxies appearing in the counts. Without that information, one does not really know whether one is seeing intrinsically fainter galaxies relatively nearby, or relatively bright galaxies further away. This uncertainty makes any conclusions dependent upon the model of evolution assumed.
Controversies are rife in the history of this field. A famous application of this approach by Loh and Spillar (1986) yielded a value Ω0 = 1+−00..75. This is, of course, consistent with unity but cannot be taken as compelling evidence. A slightly later analysis of these data by Cowie (1988) showed how, with slightly di erent assumptions, one can reconcile the data with a much smaller value of Ω0. Further criticisms of the Loh–Spillar analysis have been lodged by other authors (Bahcall and Tremaine 1988; Caditz and Petrosian 1989). Such is the level and apparent complexity of the evolution in the stellar populations of galaxies over the relevant timescale that we feel that it will be a long time before we understand what is going on well enough to even try to disentangle the cosmological and evolutionary aspects of these data. There has been significant progress, however, with number-counts of faint galaxies, beginning in the late 1980s (Tyson and Seitzer 1988; Tyson 1988) and culminating with the famous ‘deep field’ image taken with the Hubble Space Telescope, which is shown in Figure 4.10. The ‘state-of-the-art’ analysis of number-counts (Metcalfe et al. 2001) is shown in Figure 4.11, which displays the very faint number-counts from the HST in two wavelength bands, together with ground-based observations from other surveys. The implications of these results for cosmological models are unlikely to be resolved unless and until there are major advances in the theory of galactic evolution.
100 Observational Properties of the Universe
Figure 4.10 Part of the HST deep field image, showing images of galaxies down to limiting visual magnitude of about 28.5 in blue light. By extrapolating the local luminosity function of galaxies, one concludes that a large proportion of the galaxies at the faint limit have z > 2. Picture courtesy of the Space Telescope Science Institute.
4.7.4Summary
The problem with most of these tests is that, if the Big Bang is correct, objects at high redshift are younger than those nearby. One should therefore expect to see evolutionary changes in the properties of galaxies, and any attempt to define a standard ‘rod’ or ‘candle’ to probe the geometry will be very prone to such evolution. Indeed, as we shall see, many of these tests require considerable evolution in order to reconcile the observed behaviour with that expected in the standard models. It is worth mentioning these problems at this point in order to introduce the idea of evolution in galaxy properties, which we shall return to in Section 19.4.
Direct observations of gravitational lensing may prove to be a more robust diagnostic of spatial curvature and hence of the cosmological model. The statistics of the frequency of occurrence of multiply lensed quasars can, in principle, be used to measure q0. This method is in its infancy at the moment, however, and no strong constraint on the spatial geometry has yet emerged; see Chapter 20 for more details of this.
4.8 The Cosmic Microwave Background
The discovery of the microwave background by Penzias and Wilson in 1965, for which they later won the Nobel Prize, provided one of the most impor-
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The Cosmic Microwave Background |
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Figure 4.11 Compilation of number-count data in the B (blue) band, from Metcalfe et al. (2001). Picture courtesy of Tom Shanks.
tant pieces of evidence for the hot Big Bang model. In fact this discovery was entirely serendipitous. Penzias and Wilson were radio engineers investigating the properties of atmospheric noise in connection with the Telstar communication satellite project. They found an apparently uniform background ‘hiss’ at microwave frequencies which could not be explained by instrumental noise or by any known radio sources. After careful investigations they admitted the possible explanation that they had discovered a thermal radiation background such as that expected to be left as a relic of the primordial fireball phase. In fact, the existence of a radiation background of roughly the same properties as that observed was predicted by George Gamow in the mid-1940s, but this prediction was not known to Penzias and Wilson. A group of theorists at Princeton University, including Dicke and Peebles, soon saw the possible interpretation of the background ‘hiss’ as relic radiation, and their paper (Dicke et al. 1965) was published alongside the Penzias and Wilson (1965) paper in the Astrophysical Journal.
102 Observational Properties of the Universe
The cosmic microwave background is a source of enormous observational and theoretical interest at the present time, so we have devoted the whole of Chapter 17 to it. For the present we shall merely mention two important properties.
First, the CMB radiation possesses a near-perfect black-body spectrum. The theoretical ramifications of this result are discussed in Chapter 9 and Section 19.3; the latest spectral data are also shown later, in Figure 9.1. At the time of its discovery the CMB was known to have an approximately thermal spectrum, but other explanations were possible. Advocates of the steady state proposed that one was merely observing starlight reprocessed by dust and models were constructed which accounted for the observations reasonably well. In the past 30 years, however, continually more sophisticated experimental techniques have been directed at the measurement of the CMB spectrum, exploiting ground-based antennae, rockets, balloons and, most recently and e ectively, the COBE satellite. The COBE satellite had an enormous advantage over previous experiments: it was able to avoid atmospheric absorption, which plays havoc with ground-based experiments at microwave and submillimetric frequencies. The spectrum supplied by COBE reveals just how close to an ideal black body the radiation background is; the temperature of the CMB is now known to be 2.726±0.005 K. Attempts to account for this in a steady-state model by non-thermal processes are entirely contrived. The CMB radiation really is good evidence that the Big Bang model is correct.
The second important property of the CMB radiation is its isotropy or, rather, its small anisotropy. The temperature anisotropy is usually expressed in terms of the quantity
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which gives the temperature fluctuation as a fraction of the mean temperature T0 as a function of angular position on the sky. Penzias and Wilson (1965) were only able to give rough constraints on the departure of the sky temperature of the CMB from isotropy. Theorists soon realised, however, that if the CMB actually did originate in the early stages of a Big Bang, it should bear the imprint of various physical processes both during and after its production. However, attempts to detect variations in the temperature of the CMB on the sky have, until recently (with the exception of the dipole anisotropy; see below), been unsuccessful. The observed level of isotropy of the cosmic microwave background radiation is important because:
1.it provides strong evidence for the large-scale isotropy of the Universe;
2.it excludes any model in which the radiation has a galactic origin or is produced by a random distribution of sources, also on the grounds of its nearperfect black-body spectrum; and
3.it can provide important information on the origin, nature and evolution of density fluctuations which are thought to give rise to galaxies and large-scale structures in the Universe.
Let us mention some of the possible sources of anisotropy here, though we shall return to the CMB in much more detail in Chapter 17. First, there is known to be
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a dipole anisotropy (a variation on a scale of 180◦)
T(ϑ) = T0 1 + ∆TD cos ϑ , (4.8.2)
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which is due to the motion of the observer through a reference frame in which the CMB is ‘at rest’, meaning the frame in which the CMB appears isotropic; notice that there is no dependence upon φ in this expression. The amplitude and direction of the dipole anisotropy have been known for some time: the amplitude is around ∆TD/T0 10−3 v/c, where v is the velocity of the observer. After subtracting the Earth’s motion around the Sun, and the Sun’s motion around the galactic centre, this observation can be used to determine the velocity of our Galaxy with respect to this ‘cosmic reference frame’. The result is a rather large velocity of v 600 km s−1 in the direction of the constellations of Hydra-Centaurus (l = 268◦, b = 27◦). This velocity can be used in an ingenious determination of Ω0, as we describe later in Chapter 18.
On smaller scales, from the quadrupole (90◦) down to a few arcseconds, there are various possible sources of anisotropy as follows.
1.If there are inhomogeneities in the distribution of matter on the surface of last scattering, described in Section 9.5, these can produce anisotropies by the redshift or blueshift of photons from regions of di erent gravitational potential, the Sachs–Wolfe e ect (Sachs and Wolfe (1967)).
2.If material on the last scattering surface is moving, then it will induce temperature fluctuations by the Doppler e ect (material moving towards the observer will be blueshifted, that moving away will be redshifted).
3.The coupling between matter and radiation at last scattering may mean that dense regions are actually intrinsically hotter than underdense regions.
4.An inhomogeneous distribution of material between the observer and the last scattering surface may induce anisotropy by inverse Compton scattering of CMB photons by free electrons in a hot intergalactic plasma (the Sunyaev– Zel’dovich e ect (Sunyaev and Zel’dovich 1969); see Section 17.7 for the possible use of this e ect in determining H0).
5.Photons travelling through a time-varying gravitational potential field also su er an e ect similar to (i) (usually called the Rees–Sciama e ect (Rees and Sciama 1968), but actually it is simply a version of the Sachs–Wolfe phenomenon).
As we shall see in Chapter 17, the COBE satellite has recently detected anisotropy on the scale of a few degrees up to the quadrupole. This detection, with an amplitude of ∆T/T 10−5, has been independently confirmed by an experiment on Tenerife. The characteristics of this signal are consistent with it being due to the Sachs–Wolfe e ect (i). If the primordial fluctuations giving rise to this e ect are indeed the seeds of galaxies and clusters, then this observation has profound implications for theories of galaxy and cluster formation. Attempts are currently being made to measure the anisotropy on smaller scales than this.
104 Observational Properties of the Universe
The balloon-borne experiments MAXIMA and Boomerang have mapped the smallscale structure of the cosmic microwave background over small patches of the sky. Soon, the US satellite MAP (Microwave Anisotropy Probe) will map the whole sky and around 2007 a European mission called the Planck Surveyor will do likewise with even higher resolution. As we shall see in Chapter 17, angular scales of a degree or less are a sensitive diagnostic of the form of fluctuations present in the early Universe as well as the geometry of the background Universe.
Bibliographic Notes on Chapter 4
More detailed discussions of galaxy properties can be found in Binney and Tremaine (1987) and Binney and Merrifield (1998). For historical interest, Zwicky (1952) is also worth consulting, as is Faber and Gallagher (1979).
Historically important papers on the development of cosmography are Abell (1958); Bahcall (1988); Rood (1988); Shane and Wirtanen (1967); Shapley and Ames (1932) and Zwicky et al. (1961–1968). The classic reference on the expansion of the Universe is Hubble (1929), but readers should be aware that much of the data upon which Hubble based his arguments were obtained by Slipher (1914). Rowan-Robinson (1985) gives a detailed overview of the distance ladder; a more recent paper is that by Fukugita et al. (1992). Interesting sources on the density parameter are Peebles (1986), Trimble (1987) and Sciama (1993). Arguments in favour of a Universe with Ω0 < 1 can be found in Coles and Ellis (1994, 1997).
Problems
1.Show that the Hubble profile of surface brightness (4.1.5) leads to an infinite total luminosity, while the law
I = I0 exp[−(r/a)1/4],
with a a constant, does not. In the second case, estimate (in units of a) the value of r that encloses half the total light and compare your answer for an exponential disc (4.1.6).
2.The half-life of Uranium-235 is 0.7 × 109 years, while that of Uranium-238 is 4.5 × 109 years. A rock has an observed abundance ratio
235U
238U = 0.00723,
while these isotopes are thought to be produced in supernovae explosions with a relative abundance of 1.71. Assuming all the material in the rock was produced in a single supernova event, estimate the time that has elapsed since this event took place.
3.Calculate the rotation curve, v(R), for test particles in circular orbits of radius R:
(a) around a point mass M; (b) inside a rotating spherical cloud with uniform density; and (c) inside a spherical halo with density ρ(r) 1/r2.
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4.The Tully–Fisher relation (4.3.2) usually has an index α 3. Show that, in a simple model of a galaxy in which stars undergo circular motions in a disc of constant thickness and in which the mass-to-light ratio is constant, a value α = 4 would be expected.
5.Assuming all elliptical galaxies have the same central surface brightness and that they are in virial equilibrium, derive the Faber–Jackson relation (4.3.4).
6.Assume that the mass-to-light ratio, M/L, for the Galaxy is, and always has been, 10 in solar units. What is the maximum fraction of the total mass that could have been burnt into helium from hydrogen over 1010 years? (The mass deficit for the reaction 4H → 4He is 0.7%.)
7.If the luminosity function of galaxies is given by the Schechter function (4.5.13), show that when α = 1.5 the total luminosity of all galaxies is approximately
1.77Φ L . The volume through which a galaxy of luminosity L can be seen above a fixed magnitude limit is proportional to L3/2. Hence show that in a magnitudelimited survey of galaxies with a luminosity function of this form, about half will have luminosity exceeding about 0.7L but less than about 5% will have luminosity greater than 3L .
8.Prove the virial theorem (4.5.15) for a system of self-gravitating masses in statistical equilibrium.
PART 2
The Hot Big Bang Model