Measuring q 0 Measuring the deceleration parameter, q 0, is much more difficult than measuring H 0. In order to measure the Hubble Constant, one needs to derive distances to objects at 100 Mpc; this corresponds to a redshift of z > 0.02. In terms of lookback time, this is only a couple of percent the age of the universe. Hence any universal deceleration over this time is inconsequential. To get to q 0, one needs to view objects to much greater redshift, z > 1. There are three main ways to derive q 0 : 1) Through number counts and redshift distributions. If the universe were Euclidean, then the volume of the universe, as a function of distance, would be V (r) = 4πr 2 dr (13.01) Consequently, there would be more objects at high redshift than at low redshift, just due to the increased volume. In the real universe, the same thing is true, except that the volume element is a complicated function of z and q 0 (and Λ). Specifically, from (2.28), (2.35), and (2.38), the volume element of a Λ = 0 universe is c 3 { q0 z + (q 0 1) [ (2q 0 z + 1) 1/2 1 ]} 2 dv (z) = H0 3 (1 + z)6 q0 2(2q dz 0z + 1) 1/2 (13.02) Thus, by measuring the redshift distributions of a certain class of object and by counting how many objects there are versus apparent magnitude, one can attempt to measure q 0. Of course, to do this experiment, one needs to know how the luminosity and density of the objects evolve with redshift.
2) Through identifications of gravitational lenses. As we have seen with statistical microlensing, the number of gravitational lenses changes with redshift. The density of resolved lenses has a similar dependence. Since these equations depend on angular diameter distances, which involve q 0 (and Λ), one can try to place limits on q 0 by observing the properties of a large sample of lensed systems. Of course, once again, one needs to know how the density of lenses is changing with redshift. 3) Through a Hubble diagram. For nearby objects, the observed flux is related to the emitted flux by the inverse square law, which, terms of magnitudes, is m = M + 5 log d 5 (13.03) where d is the distance of the object (in parcsecs). If we then substitute in the Hubble law, so d v H 0 cz H 0 (13.04) m = 5 log z + (M + 5 log c H 0 5) (13.05) Thus, a plot of apparent magnitude, m versus log z is a straight line with a slope of 5. (Meanwhile, the intercept gives you H 0, if the absolute magnitude of the object is known.) At cosmological distances, however, the equation is not so simple. Specifically, one must use luminosity distance in the equations, which, from (2.27) and (2.28) is d L = c H 0 q 2 0 { q 0 z + (q 0 1) [ ]} (2q 0 z + 1) 1/2 1 (13.06) (for a Λ = 0 universe). This changes the Hubble law at large distances.
The use of the Hubble diagram is the most traditional, and perhaps the most direct way of measuring q 0. To use this technique, 22 one needs a standard candle visible at large distances. SN Ia are the objects of choice, since, at least in theory, supernovae at large z should be the same as those at small z (i.e., their mass, composition, etc., should not change). Thus, one should be able to use them fairly easily to estimate q 0. The result is the evidence for an accerlating (Λ 0) universe. 26 24 22 Supernova Cosmology Project (Ω Μ, Ω Λ ) = ( 0, 1 ) (0.5,0.5) (0, 0) ( 1, 0 ) (1, 0) (1.5, 0.5) (2, 0) Flat Λ = 0 effective m B 20 18 Calan/Tololo (Hamuy et al, A.J. 1996) 16 14 0.02 0.05 0.1 0.2 0.5 1.0 redshift z FIG. 1. Hubble diagram for 42 high-redshift Type Ia supernovae from the Supernova Cosmology Project, and 18 low-redshift Type Ia supernovae from the Calán/Tololo Supernova Survey, after correcting both sets for the SN Ia lightcurve width-luminosity relation. The Note inner error that barsthe show the Tully-Fisher uncertainty due to measurement relation, errors, the whilefundamental the outer error bars show plane the total relations, indicateand supernovae even not included brightest in Fit C. The cluster horizontalgalaxies error bars represent canthealso assigned be peculiar used velocity touncertainty of uncertainty when the intrinsic luminosity dispersion, 0.17 mag, of lightcurve-width-corrected Type Ia supernovae is added in quadrature. The unfilled circles 300 estimate km s 1. The solid curves are the theoretical mb effective (z) for a range of cosmological models with zero cosmological constant: ( M ) = (0 q 0) on 0 via top, (1 a Hubble diagram. However, galaxies may evolve 0) in middle and (2,0) on bottom. The dashed curves are for a range of flat cosmological models: ( M ) = (0 1) on top, (0 5 0 5) second from top, (1 0) third from top, and (1.5,-0.5) on bottom. over time, so one needs to understand this evolution before these objects can produce a reliable q 0 measurement.
Galaxy Classification There are only a few types of objects capable of probing the z > 1 universe. Quasars, AGN, and γ-ray bursts come to mind immediately, but these objects are not standard candles, are extremely difficult to physically understand, and exhibit extremely strong density (or luminosity) evolution. To date, supernovae (especially SN Ia, which presumably come from a homogeneous class of white dwarfs) are close to ideal, but these transient objects are rare and require a large observing infra-structure to find and study. That leaves galaxies and galaxy clusters. Much of modern extragalactic astronomy deals with studying how galaxies evolve with time. If galactic evolution can be understood, then parameters such as q 0, Ω 0, and Λ will follow, along with data on the initial conditions for the formation of the universe. Before we can study evolution, however, we have to define what a galaxy is, and how it is structured. Hence, the first step to understanding galaxies is to classify them. The most famous classification scheme, of course, is the Hubble tuning-fork system, which starts out with spherical ellipticals, evolves to the transitional S0 (lenticular) galaxies, and then branches out into Sa, Sb, and Sc spirals and barred spirals. Over the years, this system has been built up, so that lenticulars are now subdivided into S0 1, S0 2, and S0 3 (depending on the smoothness of the luminosity profile, and the amount of gas present in the galaxy). In addition, there are now the additional symbols (r), which indicates the presence of an inner ring, R, which signifies the presence of an outer ring, and (s), which says that the spiral arms begin at the end of a bar or are traced to the galaxy s center, rather than the galaxy s inner ring. Needless to say, this scheme isn t very elegant.
A complementary classification system, which seeks to make a parallel with stellar evolution, is the DDO (David-Dunlap Observatory) Luminosity Classification system of van den Bergh. In this method, supergiant galaxies with well-developed bright spiral arms and bars have the Roman numeral I (like supergiant stars), and small, low-surface-brightness, irregular galaxies have the roman numeral V. Of course, since one doesn t usually know a galaxy s distance, it is somewhat difficult to estimate its true luminosity. The system therefore assumes that the galaxies with the most well-developed arms are also the most luminous. In the Revised Shapley-Ames Galaxy Catalog (which lists the 1300 brightest galaxies in the sky), Sandage adds the DDO luminosity classification onto his Hubble classification, so, for example, the galaxy NGC 1097 is given the type RSBbc(rs)I-II. (In other words, NGC 1097 is a very large barred Sbc spiral with an outer ring, an inner ring, and arms that begin at the end of the galaxy s bar.) A more computer-friendly system was devised for the 2nd Reference Catalog of Bright Galaxies by de Vaucouleurs. In his system, galaxies are given a numerical T designation based on compactness. The most compact elliptical galaxies are assigned T = 6; normal ellipticals have T = 5, and lenticular galaxies have negative numbers near zero. Spiral galaxies start at T = +1 for Sa, and proceed to T = +11 for those blue, irregular galaxies that are essentially extragalactic H II regions. In de Vaucouleurs scheme, there is no difference between a normal spiral and a barred-spiral galaxy.
In the 1970 s van den Bergh noticed that spiral galaxies in clusters seemed different from those in the field. In particular, many cluster spirals seemed to have less gas and less star-formation than their counterparts in low-density environments. (The S0 galaxies, which are spiral disks without arms or gas, are the extreme example of this phenomenon.) Van den Bergh therefore defined a system where Anemic Spirals occupied the transition between regular spirals and lenticulars. In this system, sequences would be Sa Aa S0a, Sb Ab S0b, etc. For these Anemic galaxies, it is as if something is quenching their active star formation.
An interesting system that is still (partially) with us and has (some) interesting uses is the classification system of Morgan. The system has two components, a concentration component, and a form component. The concentration part of the system is the observed correlation between the types of stars present in a galaxy, and how compact the galaxy is. Elliptical galaxies have mostly old stars, and their integrated light is dominated by K supergiants. These objects are also the most condensed systems; i.e., they are highly concentrated. Irregular galaxies with no central mass condensation tend to have younger stars and a corresponding earlier spectral type. Thus, Morgan defined an a-f-g-k concentration index based on the spectral classification of stars. For the form index, Morgan chose the (capital) letters S for spiral, B for barred spiral, E for elliptical, I for irregular, E p for elliptical peculiar (with dust), D for a rotationally symmetry without elliptical structure (i.e., a diffuse system), L for low-surface brightness, and N for any system with a small, brilliant nucleus (like a Seyfert galaxy). On top of this, Morgan then added a number from 1 to 7 based on apparent inclination: face-on spirals were S1, while highly elongated systems could be S7. Thus, a Morgan class might be ks4, fs1, fgb1, or gks7. The Morgan system lives on today principally in the designation of N galaxies, which are sometimes used to refer to small galaxies with an active galactic nucleus, and through the identification of some galaxies as cd. The cd classification (which was actually defined about 5 years after the original Morgan paper) refers to galaxies in the centers of clusters which have an elliptical galaxy-like core surrounded by a huge amorphous envelope of stars. These systems are probably the largest collections of stars in the universe; since some cd galaxies have multiple nuclei, they have sometimes been described as galaxies at lunch.