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

4.3 The Distance Ladder

The value of H0 found by Hubble in 1929 was around 500 km s−1 Mpc−1, much larger than the values currently accepted. This discrepancy was due to errors in the calibration of distance indicators that he used, which were only corrected many years later. In the 1950s, Baade derived a value of H0 of order 250 km s−1 Mpc−1, but this was also a ected by a calibration error. A later recalibration by Sandage in 1958 brought the value down to between 50 and 100 km s−1 Mpc−1; present observational estimates still lie in this range. This demonstrates the truth of the comment we made above: Hubble’s ‘constant’ is not actually constant because it has changed by a factor of 10 in only 50 years! Joking apart, the term ‘constant’ was never intended to mean constant in time, but constant in the direction in which one observes the recession of a galaxy. As far as time is concerned, the Hubble constant changes in a period of order H−1.

One simple way to estimate the Hubble constant is to determine the absolute luminosity of a distant source and to measure its apparent luminosity l. From these two quantities one can calculate its luminosity distance

which, together with the redshift z which one can measure via spectroscopic observations of the source, provides an estimate of the Hubble constant through Equation (4.2.3) (in the appropriate interval of z). The main di culty with this approach is to determine L. The usual approach, which is the same as that developed by Hubble, is to construct a sort of distance ladder: relative distance measures are used to establish each ‘rung’ of the ladder and calibrating these measures against each other allows one to measure distances up to the top of the ladder. A modern analysis might use several rungs, based on di erent distance measures, in the following manner.

First, one exploits local kinematic distance measures to establish the length scale within the galaxy. Kinematic methods do not rely upon knowledge of the absolute luminosity of a source. Nearby distances can be derived using the trigonometric parallax Q of a star, i.e. the change in angular position of a star on the sky in the course of a year due to the Earth’s motion in space. Measuring Q in arcseconds is convenient here because the distance in parsecs is then just d = Q−1, as we mentioned in Section 4.1. Until recently this direct technique was limited to distances of order 30 pc or so, but the astrometric satellite Hipparcos has established a distance scale based on parallax to kiloparsec scales.

The secular parallax of nearby stars is due to the motion of the Sun with respect to them. For stellar binaries one can derive distances using the dynamical parallax, based on measurements of the angular size of the semi-major axis of the orbital ellipse, and other orbital elements of the binary system. Another method is based on the properties of a moving cluster of stars. Such a cluster is a group of stars which move across the Galaxy with the same speed and parallel trajectories; a perspective e ect makes these stars appear to converge to a point on the sky. The

80 Observational Properties of the Universe

position of this point and the proper motion of the stars lead one to the distance. This method can be used on scales up to a few hundred parsecs; the Hyades cluster is a good example of a suitable cluster. With the method of statistical parallax one can derive distances of order 500 pc or so; this technique is based on the statistical analysis of the proper motions and radial velocities of a group of stars. Taken together, such kinematic methods allow us to establish distances up to the scale of a few hundred parsecs, much smaller even than the scale of our Galaxy.

Once one has determined the distances of nearby stars with a kinematic method, one can then calculate their absolute luminosities from their apparent luminosities and their (known) distances. In this way it was learned that most stars, the so-called main sequence stars, follow a strict relationship between spectral type (an indicator of surface temperature) and absolute luminosity: this is usually visualised in the form of the HR (Hertzsprung–Russell) diagram. Using the properties of this diagram one can measure the distances of main sequence stars of known apparent luminosity and spectral type. With this method, one can measure distances up to around 30 kpc.

Another important class of distance indicators contains variables stars of various kinds, including RR Lyrae and Classical Cepheids. The RR Lyrae all have a similar (mean) absolute luminosity; a simple measurement of the apparent luminosity su ces to provide a distance estimate for this type of star. These stars are typically rather bright, so this can extend the distance ladder to around 300 kpc. The classical Cepheids are also bright variable stars which have a very tight relationship between the period of variation P and their absolute luminosity: log P log L. The measurement of P for a distant Cepheid thus allows one to estimate its distance. These stars are so bright that they can be seen in galaxies outside our own and they extend the distance scale to around 4 Mpc. Errors in the Cepheid distance scale, due to interstellar absorption, galactic rotation and, above all, a confusion between Cepheids and another type of variable star, called W Virginis variables, were responsible for Hubble’s large original value for H0. Other distance indicators based on novae, blue supergiants and red supergiants allow the ladder to be extended slightly to around 10 Mpc. Collectively, these methods are given the name primary distance indicators.

The secondary distance indicators include HII regions (large clouds of ionised hydrogen surrounding very hot stars) and globular clusters (clusters of around 105–107 stars). The former of these has a diameter, and the latter an absolute luminosity, which has a small scatter around the mean. With such indicators one can extend the distance ladder out to about 100 Mpc.

The tertiary distance indicators include brightest cluster galaxies and supernovae. Clusters of galaxies can contain up to about a thousand galaxies. One finds that the brightest galaxy in a rich cluster has a small dispersion around the mean value (various authors have also used the third, fifth or tenth brightest cluster galaxy as a distance indicator). With the brightest galaxies one can reach distances of several hundred Mpc. Supernovae are stars that explode, producing a luminosity roughly equal to that of an entire galaxy. These stars are therefore

easily seen in distant galaxies, but the various indicators that use them are not too precise.

More recently, much attention has been paid to observed correlations of intrinsic properties of galaxies themselves as distance indicators. In spiral galaxies, one can use the empirical Tully–Fisher relationship:

where L is the absolute luminosity of the galaxy and Vc is the circular rotation velocity (most massive spirals have rotation curves which are constant with radial distance from the centre). The index α 3, but depends on the waveband within which L is measured. The correlation is so tight that the measurement of Vc allows the luminosity to be determined to an accuracy of about 40%. Since the apparent flux can be measured accurately, and this depends on the square of the distance to the galaxy, the resulting distance error is about 20%. This can be reduced further by applying the method to a number of spirals in the same cluster.

The situation is somewhat more complicated for elliptical galaxies because the correlation involves three parameters: the characteristic size of the galaxy R; its surface brightness Σ; and the central velocity dispersion σ. (Recall that elliptical galaxies do not have ordered motions, but random ones characterised by a dispersion rather than a mean value.) These three parameters are correlated in such a way that they occupy the so-called fundamental plane defined by a relation of the form

where C is a constant. Before the fundamental plane was established there were attempts to find relations of the form (4.3.2), such as the Faber–Jackson relation,

where Dn is the radius within which the mean surface brightness of the galaxy image exceeds a certain threshold value. The problem with these two-parameter correlations is that they suppress one variable in the relation (4.3.3). The Faber– Jackson relation does not take account of varying Σ and consequently has a large scatter. On the other hand, the relation (4.3.5) is close to an edge-on view of the fundamental plane and is almost as good as (4.3.2). The value of α needed to fit the objections in this case is α 4. The use of these distance measures, together with redshift, to map the local peculiar velocity field is described in Section 4.6 and in Chapter 18.

So there seems to be no shortage of techniques for measuring H0. Why is it then that observational limits constrain H0 so poorly, as in Equation (4.2.2)? One problem is that a small error in one ‘rung’ of the distance ladder also a ects higher levels of the ladder in a cumulative way. At each level there are actually many corrections to be made, some of them well known, others not. Some such corrections are as follows.

82 Observational Properties of the Universe

Galactic rotation: the Sun rotates around the galactic centre at a distance of around 10 kpc and with a velocity around 215 km s−1. This motion can produce spurious systematic shifts towards the red or the violet in observed spectra.

Aperture e ects: it is necessary to refer all the measurements regarding galaxies to a standard telescope aperture. At di erent distances the aperture may include di erent fractions of the galaxy.

K-correction: the redshift distorts the observed spectrum of a source in the sense that the luminosity observed at a certain frequency was actually emitted at a higher frequency. To correct this, one needs to know the true spectrum of the source.

Absorption: our Galaxy absorbs a certain fraction of the light coming to it from an extragalactic source. In fact the intensity of light received at the Earth varies as exp(−λ cosec b), where λ is a positive constant and b is the angle between the line of sight and the galactic plane, i.e. the galactic latitude.

Malmquist bias: there are various versions of this e ect, which is basically due to the fact that the properties of samples of astronomical objects limited by apparent luminosity (i.e. containing all the sources brighter than a certain apparent flux limit) are di erent from the properties of samples limited in distance because the objects in distant regions will have to be systematically brighter in order to get into the sample.

Scott e ect: there is a correlation between the luminosity of the brightest galaxy in a cluster and the richness (i.e. number of galaxies) of the cluster. At large distances one tends to see only the richest clusters, which biases the brightest galaxy statistics.

Baunt–Morgan e ect: in fact, clusters are divided into at least five classes in each of which the luminosity of the brightest galaxy is di erent from the others.

Shear: there is an apparent rotation in the Local Supercluster, as well as of the Local Group and the Virgo cluster.

Galactic evolution: the luminosity of the most luminous cluster galaxies is a function of time and, therefore, of the distance between the galaxy and us. The main reason for this is that the stellar populations of such galaxies are modified as the central cluster galaxy swallows smaller galaxies in its vicinity in a sort of ‘cannibalism’.

Given this large number of uncertain corrections, it is perhaps not surprising that we are not yet in a position to determine H0 with any great precision. We should mention at this point, however, that some methods have recently been proposed to determine the distance scale directly, without the need for a ladder. One of them is the Sunyaev–Zel’dovich e ect, which we discuss in Section 17.7. The Hubble Space Telescope (HST) is able to image stars directly in galaxies within the Virgo cluster of galaxies, an ability which bypasses the main sources of uncertainty in the calibration of the traditional distance ladder approaches. This ‘key’

project is now more-or-less complete, and has produced a value of h 0.7 with an error of about 10%.

4.4 The Age of the Universe

We now turn to the determination of the characteristic timescale for the evolution of the Universe with the ultimate aim of determining t0, the time elapsed from the Big Bang until now. The quantity we call the Hubble time is defined in Section 2.7, and is simply the reciprocal of the Hubble constant. It is interesting to note – we shall demonstrate this later – that this timescale is in rough order- of-magnitude agreement with the ages of stars and galaxies and of the nuclear timescale obtained from the radioactive decay of long-lived isotopes.

4.4.1 Theory

In a matter-dominated Friedmann model, the age of the Universe is given to a good approximation by

where, as a reminder, the density parameter Ω0 is the ratio between the present total density of the Universe ρ0 and the critical density for closure ρ0c,

in the cases Ω0 > 1, Ω0 = 1 and Ω0 < 1. These results can be compared with Equations (2.4.10), (2.2.6 e) and (2.4.3), respectively. The results (4.4.3 a) and (4.4.3 c) are well approximated by the relations

Some illustrative values are F = 1, 0.90, 0.67, 0.5 and 0 for Ω0 = 0, 0.1, 1, 10 and ∞, respectively; for values of Ω0 which are reasonably in accord with observations, as we shall discuss shortly, the age is always of order 1/H0.

their decay products. Most long-lived radioactive nuclei are synthesised in the socalled r-process reactions involving the rapid absorption of neutrons by heavy nuclei such as iron. Such processes are generally thought to occur in supernovae explosions. Given that the stars that become supernovae are very short lived (of order 107 years), nucleocosmochronology is a good way to determine the time at which stars and galaxies were formed. If the origin of our Galaxy was at t 0, at which time there occurred an era of nucleosynthesis of heavy elements lasting for some time T, and this was followed by a time ∆ in which the Solar System became isolated from the rest of the galaxy, and after which there was a period ts corresponding to the age of the Solar System, then the age estimate of the Universe one would produce is tn = T + ∆ + ts.

The age of the Solar System can be deduced in the following way. The isotope 235U decays into 207Pb with a mean lifetime τ235 = 109 years; 238U produces 206Pb with τ238 = 6.3 × 109 years; the isotope 204Pb does not have radioactive progenitors. Let us indicate the abundances of each of these elements by their atomic symbols and the su ces ‘i’ and ‘0’ to denote the initial and present time, respectively. We have

from which, dividing by the abundance of 204Pb0 = 204Pbi, we obtain

Measuring R207 and R206 in two di erent places, for example in two meteorites, which we indicate with ‘I’ and ‘II’, one can easily get

from which one can recover ts. In this way one finds an age for the Solar System of order 4.6×109 years. Analogous results can be obtained with other radioactive nuclei such as 87Rb, which decays into 87Sr with τ87 = 6.6 × 1010 years.

By analogous reasoning to that above, one finds that T + ts (0.6–1.5) × 1010 years and that ∆ (1–2) × 108 years T + ts, from which the age of the Universe must be

It is worth remarking that the time deduced for the isolation of the Solar System ∆ is of the same order as the interval between successive passages of a spiral arm through a given location in a galaxy.

86 Observational Properties of the Universe

In summary, we can see that the theoretical age of the Universe t0, the ages of globular clusters tc and the nuclear timescale tn are all in rough agreement with each other. This does not necessarily mean that the Universe was ‘born’ at a time t0 in the past, in the sense that it must have been created with a singularity at t = 0. Some ways of avoiding this kind of ‘creation’ are discussed in Chapter 6.

4.5 The Density of the Universe

Let us now give some approximate estimates of the total energy density of the Universe. We shall see that this is also uncertain by a large factor. More sophisticated methods for measuring the density parameter are discussed in Chapter 18.

4.5.1 Contributions to the density parameter

The evolution of the Universe depends not only on the total density ρ but also on the individual contributions from the various components present (baryonic matter, photons, neutrinos). Let us denote the contribution of ith component to the present density by

For this section only we drop the zero su x on Ω that indicates the present value of this parameter. All quantities in this section are at the present time, so it should do no harm to simplify the notation. We shall estimate the contribution Ωg from the mass concentrated in galaxies a little later. Within a considerable uncertainty we have

There may, of course, be a contribution from matter which is not contained in galaxies, but is present, for example, in clusters of galaxies. The size of this contribution is even more uncertain. We shall see later that a reasonable estimate for the total amount of mass contributing to the gravitational dynamics of large-scale objects is around

The discrepancy between the two values of Ω given by Equations (4.5.2) and (4.5.3) is attributed to the presence of non-luminous matter, called dark matter, which may play an important role in structure formation, as we shall see in Section 4.6 and, in much more detail, later on.

As well as matter, the Universe is filled with a thermal radiation background, called the cosmic microwave background (CMB) radiation. This was discovered in 1965, and we shall discuss it later in Section 4.9 and Chapter 17. The radiation

has a thermal spectrum and a well-defined temperature of T0r = 2.726 ± 0.005 K. The mass density corresponding to this radiation background is

(σr = π2k4B/15 3c3 is the so-called black-body constant; the Stefan–Boltzmann constant is just σc/4), so that the corresponding density parameter is

As we shall see in Section 8.5, there is also expected to be a contribution to Ω from a cosmological neutrino background which, if the neutrinos are massless, yields

where Nν indicates the number of massless neutrino species (Nν 3, according to modern particle physics experiments). The resulting ρ0ν is comparable with ρ0r expressed by (4.5.4). If the neutrinos have mean mass of order 10 eV, as used to be thought in the 1980s, then

which is much larger than that implied by Equation (4.5.2); if neutrinos have a mass of this order, then they would dominate the density of the Universe. However, more recent experimental measurements of neutrino oscillations suggest they have a much smaller mass than this, much less than one electronvolt. Such light neutrinos have some e ect on cosmic evolution, but they do not dominate.

As far as the contribution to Ω from relativistic particles in general is concerned, there is a good argument, which we shall explain in Section 11.7, why such particles should not dominate the matter component. If this were the case, then fluctuations would not be able to grow in order to generate galaxies and large-scale structure by the present epoch.

Upper and lower limits on the contribution Ωb from baryonic material can be obtained by comparing the observed abundances of light elements (deuterium, 3He, 4He and 7Li) with the predictions of primordial nucleosynthesis computations. The latest results, described in more detail in Chapter 8, give

if we allow the historical lower limit for the Hubble constant, h 0.5, then the largest allowed upper limit on Ωb becomes 0.08 and, if h 1, the lower limit is just 0.01. For small h it is therefore clear that Ωb may be compatible with Ωg, but not with Ωdyn.