REDSHIFTS of GALAXIES
Content: Extragalactic distance scale, Photometric redshifts
Adapted from The Astronomy and Astophysics Encyclopedia and  B.F. Madore, W.L. Freedman, S.D.J. Gwyn et al.
1. EXTRAGALACTIC DISTANCE SCALE 

Calibration - STARS, CEPHEID VARIABLE, PERIOD-LUMINOSIRY RELATION

In the early decades of this century, Henrietta S. Leavitt recogned that there is a statistical relationship between the average brightness of a Cepheid variable star and the period of pulsation. The periods of these stars range from a few days to a few hundred days, and their luminosities (all of wliich are intrinsically much greater than that of the Sun) span factors of several hundred in brightness. In general, the longer the period of a Cepheid, the brighter is its average thtnninc luminosity.

Because the period of pulsation can be measured independently of distance and because the apparent luminosity of a Cepheid depends on the square of the distance, one can use the intnnsic luminosity, as predicted by the period, in combination with the apparent luminosity to derive a distance. As tcols in the extragalactic distance scale, Cepheid variables provide one of the most accurate means of determining distances to the nearby spiral and irregular-tppe galaxies of the Local Group and somewhat beyond.

• THE BASIC PHYSICS

One very appealing attribute of Cepheid variables as a class of distance indicators is that the physics for the Cepheid period-luminosity relation is well understood. Put simply, all self-luminous objects, including Cepheids, give off light in proportion to both their area and the surface brightness over that area. For spherical bodies of radius R and surface temperature T, the total luminosity L (integrated over all wavelengths) is found using Stefan's law where L = 4 R² T⁴ and the Stefan-Boltsmann constant. Furthermore, the fundamental period of oscillation P of any mechanical system depends only on one thing, the mean density . Low-density systems have longer periods of oscillation than high-density systems, with the equation relating these two quantities having the following form, P^1/2 = Q, where Q is a constant. Combining Stefan's law, the P^1/2 law, and an assumed mass-luminosiry relation Indicates that the largest stars will have the Inghest luminosities and the longest periods. And that is exactly what is observed.

Of course, with a more rigorous (and more complex) application of physics, we can relax these simplifying constraints and redo the calculations for stellar models havIng a reasonable range of mass and with a variety of surface brightnesses. But these are refinements, certainly necessary for detailed calculations, but not absolutely necessary for an understanding of the underlying process leading to a period-luminosity relation itself. The refinements do tell us that the relation between period and luminosity will have scatter, and that the scatter will have a physical origin due to differences in surface temperature between stars of the same density (i.e., same mass and radius), for it is temperature that determines the surface brightness of sell-luminous objects. Ultimately, Cepheids are well described by a PLC (period-luminosity-color) relation.

It is of interest to note here that the physics that applies to the ensemble of Cepheid variables (that is to the periodluminosity relation as a whole) also applies to the individual stars as they each cycle through their oscillations. Radius variations in a single star give rise to changes in area, which result in changes in luminosity; surface temperature variations around the cycle drive changes in the surface brightness which also affect the total luminosity. There are again two parameters to the problem: The area is a geometrical property of the star and therefore has the same effect on the luminosity nearly independent of wavelength, whereas the surface brightness, radiation theory tells us, is very sensitive to where in the spectrum one observes the star. In the infrared, temperature variations make only a slight contribution to luminosiry differences, whereas in the blue and ultraviolet, the same temperature variations can dominate the luminosiry variations. These changing effects with wavelength can be seen in Fig. l where the cyclical luminosity variation for a Cepheid is shown at different wavelengths. In the near ultraviolet the amplitudes are large and driven primarily by surface temperature variations. In the far infrared almost all of the luminosity change is a reflection of the radius variation which is known to be out of phase and distinctly different in shape with respect to the temperature variations.

Figure 1. Typical light variations as a function of phase for a Cepheid variable as observed at wavelengths rangng from the ultraviolet (top light curve, labeled U) through the blue, visual and red parts of the spectrum (labeled B, V, and R, respectively) out to the near infrared (ending with the bottom light curve at K = 2.2 µm). Note the decreasing amplitude of the light variation, as well as the In shape and the shfft In the phase of rnaxirnurn brightness as longer and longer wavelengths are examilned.

The absolute calibration of the Cepheid periodluminosity relation is based on a small number of Cepheids found in galactic star clusters. These clusters have independent distances, obtained from main-sequence fitting techniques. Additionally, the same main-sequence stars can be used to independently estimate the amount of Interstellar dust obscuring and reddening the light of the Cepheids. Unfortunately, the statistiqs are poor and the intrinsic luminosities and colors of many of the cluster Cepheids are still uncertain. Moreover, most field Cepheids are too far away to have direct parallax measurements made with the present technology. However, an independent calibration will be forthcoming when the refurbished Hubble Space Telescope provides improved direct determinations of the distance to the Large and Small Magellanic Clouds in which hundreds of Cepheids, all essentially at the same distance, will enter the calibration.

• INTERSTELLAR OBSCURATION

Unfortunately, the effects of interstellar reddening due to intervening dust both in our own galaxy and in the parent spiral galaxies of extragalactic Cepheids can be quite severe and are very much stronger.in the blue as compared to the red and Infrared. A multicolored approach to the application of Cepheids to determining the distance scale is therefore required. Using modern electronic detectors (such as charge-coupled devices) and a variety of filters turning from the blue to the very near infrared, this latest, hybrid approach allows for both the distance and the total reddening to individual external galaxies to be sirnultaneously solved for by a careful application of the periodluniinosity relation, given an a priori knowledge of the interstellar extinction law. Without a determination of the extinction, all other distance estirnates are upper limits, overestimating the true distance. Examples of periodluminosity relations constructed by this technique can be seen in Fig. 2 which consists of a montage of optical wavelength observations of Cepheids in the Large and Small Magellanic Clouds, followed by the dramatically narrower period-luminosity relations found at infrared wavelengths. These latter relations are so well defined and narrow that they have allowed astronomers to calculate not only the distances to the Magellanic Clouds but also the three-dimensional shape and orientation of these galaxies in space.

Figure 2. Multiwavelength period-luminosity relations for Cepheids observed in the Small Magellanic Cloud (open circles) shifted into magnitude registry with data for Cepheids in the Large Magellanic Cloud. Note the decreased scatter as one goes from the blue wavelength data (at the top left of the figure) to the near-infrared data (at the bottom right). The individual PL relations are shifted vertically in magnitude and horiitontally in period for display purposes only.

• THE LOCAL (CEPHEID) DISTANCE SCALE

With the application of modern techniques to the Cepheid periodluminosity relation, it is now widely agreed that the extragalactic distance scale is relatively secure for galaxies within, and slightly beyond, the Local Group. For galaxies with Cepheids identified and observed, the distance estimates, out to approximately 2 Mpc, are now agreed upon at the 10% level. However, a factor of 10 further away at 20 Mpc, where Cepheids have not yet been discovered, the uncertainty, as judged by rival factions, rises to a factor of 2 difference in opinion. Much of this uncertainty is expected to disappear with the optically corrected imaging phase of the Hubble Space Telescope expected to begin in late 1993 or 1994. One of the major missions that MST is committed to working on is the extragalactic distance scale. MST will be capable of discovering Cepheids in galaxies that sample a much larger volume of space than can ever be imaged from the ground (because of atmospheric turbulence). A Cepheid-based distance to one or two galaxies at 20 Mpc (for instance in the Virgo Cluster, which is probably the practical limit even for HST) will not end the controversy about the size and age of the universe; but such observations will certainly go a long way toward narrowing the divergence of opinion. On the other hand, with many more Cepheid-based distances to galaxies Inside that volume lirnit, secondary distance indicators can be calibrated with some statistical certainty, and these altemate techniques can then be pushed deep into the extragalactic space where the pure Hubble flow is expected to be revealed.

2. PHOTOMETRIC REDSHIFTS
The concept of photometric redshifts is not new. This essay provides a brief history of photometric redshifts. The list of papers provided below is not exhaustive; however, it does cover the development of the technique up to about 1996. - S.D.J. Gwyn

• Direct Shift Measurement

Baum (1962) was the first to develop a technique for measuring redshifts photometrically. He used a photoelectric photometer and 9 bandpasses spanning the spectrum from 3730Å to 9875Å. With this system he observed the spectral energy distribution (SED) of 6 bright elliptical galaxies in the Virgo cluster. He then observed 3 elliptical galaxies in another cluster (Cl 0925+2044, also known as Abell 0801). By plotting the average SED of the Virgo galaxies and the average SED of the Cl0925 galaxies on the same graph using a logarithmic wavelength scale, he was able to measure the displacement between the two energy distributions, and hence the redshift of the second cluster. His redshift value of z = 0.19 agreed closely with the known spectroscopic value of z = 0.192, so he extended his technique to a handful of clusters of then unknown redshifts out to maximum redshift of z = 0.46. He then derived a very rough value of Omega_0. Baum's technique was fairly accurate, but because of its dependence on a large 4000Å break spectral feature, it could only work on elliptical galaxies.

• Color-Color Diagrams

Koo (1985) followed a different approach. First, he used photographic plates instead of a photometer, making it possible to measure photometric redshifts for a large number of galaxies simultaneously. Second, instead of using 9 filters he used only 4: UJFN (photographic U, B_J, R_F and I_N). Third, instead of using an empirical spectral energy distribution, he used the theoretical Bruzual (1983, among others) no-evolution models for all galaxy types.

The most important difference, however, was the way the colours were used. Instead of converting the photometric colours into a kind of low resolution spectrum, he converted the Bruzual templates into colours, and plotted lines of constant redshift and varying spectral type, known as iso-z lines, on a colour-colour diagram. Finding that the most normal colour-colour diagrams (e.g. U-J versus J-F and J-F versus F-N) were degenerate in a range of redshifts, he invented what he called colour-shape diagrams. The shape measured whether the SED turned up or down at both ends, that is, whether the spectrum was bowl shaped or humped. Another way to put it is that the colour measured the first derivative with respect to wavelength of the spectrum and the shape measured the second derivative. For colour he used either 2U-2F or U+J-F-N , both of which span a large wavelength range. For shape, he used either U+2J-F or -U+J+F-N . Following this method to measure the redshift of a galaxy, Koo calculated the colour and the shape from the UJFN magnitudes and plotted them on the colour-shape diagram. The redshift of the galaxy was then found by finding the iso-z line closest to the point representing the galaxy. Koo tested this method on a sample of 100 galaxies with known spectroscopic redshifts ranging from z = 0.025 to z = 0.700.

This method is similar to that used by Pello et al. (1996) and Miralles, Pello & Le Borgne (1996). They used the colours of galaxies to determine ``permitted redshifts'' in the following manner: The colors of galaxies are plotted as a function of redshift from the Bruzual & Charlot (1993) models. Each available color (with its associated uncertainty) of a galaxy defines a ``permitted'' redshift range on the corresponding color-redshift diagram. The intersection of the permitted redshift ranges for all the colours determines the redshift. This method was used by Pello et al. (1996) to discover a cluster of galaxies at z > 0.75 by looking for an excess in the redshift distribution in the field of a gravitationally lensed quasar. Miralles et al. (1996) used the method to determine the redshift distribtion of the Hubble Deep Field.

The ``ultra-violet dropout'' techniques of Steidel et al. (1996) and Madau et al. (1996) are similar if simpler. All galaxy spectra have a large Lyman break; shortward of 912Å, the continuum drops dramatically. When this break is redshifted into and past the U filter, the U flux is greatly reduced or non-existant, resulting in very red ultra-violet colours.

In the ultra-violet dropout techniques, an exact redshift of a galaxy is not determined. Rather, the redshift is determined to be in the redshift range where the Lyman break is in or just past the U filter. Since U filters typically have a central wavelength of 3000Å, this works out to a redshift of z > 2.25. In practical terms, redshifted template galaxy spectra are used to determine a locus on a colour-colour plot where most galaxies lie in a particular redshift range. Those galaxies whose measured colours lie within the locus are deemed to be in that redshift range. Clearly, this method is a lot simpler than that of Pello et al. (1996) as only two colours are considered. It is also a lot less precise as the redshift is not very constrained. For both these reasons it is ideally suited for pre-selecting galaxies at high redshift for spectroscopic confirmation. Steidel et al. (1996) did exactly this using the UGR filters. Madau et al. (1996) applied this technique to the Hubble Deep Field using the F300W, F450W, F606W and F814W filters. The technique was extended by using F450W dropouts to find galaxies of redshifts z = 4.

• Template Fitting

The template fitting technique developped by Loh & Spillar (1986b) more closely resembles that of Baum (1962) than that of Koo (1985). Loh & Spillar (1986b) observed 34 galaxies of known redshift in the galaxy cluster 0023+1654 through 6 non-standard filters to test their method. The standard deviation of the redshift differences (z_spec - z_phot) was 0.12. They went on to use their technique to measure photometric redshifts for 1000 field galaxies in order to determine a value for the density parameter, ₀ (Loh & Spillar 1986a).

Gwyn (1995) tested this method using BVRI photometry of the Colless et al. (Colless et al., 1990; Colless et al., 1993) galaxies. The larger uncertainty in photmetric redshifts thus derived (_z = 0.18) was attributed to the lack of a U filter. Numerous authors (Gwyn and Hartwick 1996; Lanzetta et al., 1996; Mobasher et al., 1996; Sawicki et al., 1996; Cowie et al., 1996) have used this technique to determine redshifts in the Hubble Deep Field.

• Linear Regression

Prehaps the simplest and certainly the most empirical photometric redshift technique yet is that of Connolly et al. (1995a). This method requires a ``training set'' of a large number of galaxies with multi-color photometry and spectroscopic redshifts. Redshift, z, is assumed to be a linear or quadratic function of the magnitudes (M_i) of the galaxies, i.e. if N is the number of filters:

(1)
or

(2)

The constants, a_i and a_ij, are found by linear regression. Connolly et al. (1995a) used a UJFN plus redshift data set extending to z = 0.5 of 370 galaxies. They showed that this method could determine redshifts with uncertainties of _z = 0.057 with a linear fit and _z = 0.047 with a quadratic fit. There is little or no loss of accuracy if colors (Ci = M_i - M_i+1) are used instead of magnitudes. Using this technique they were able to measure the luminosity function out to J = 24 (SubbaRao et al., 1996).

The advantage of the linear regression technique is its extreme simplicity. It has a few disadvantages: (1) A substantial collection of spectroscopic redshifts must have been measured before the technique can used. (2) Extension to fainter magnitudes or deeper redshifts is not possible.

Compiled by G.T.Petrov, 2004