C osm ology E q uations

As one moves to higher redshifts the effects of cosmology and evolution become important. Below are the cosmological equations used throughout this thesis. These are derived from the Fi'iedmann equations and taken from Peacock (1999). The general formula for the co-moving (proper) distance and the redshift is given in Eqn 2.1.

c dz'

~ k f oJo ^/(l — Q)(1 ± z')^ + Dy + f^m(l ± -2^0^ ± (1 + z')^

^ _C_ r _________ dz^__________ (2.1)

Ho JO Y [(l + z'), Q, Qy, Q,.]

where 0 = 12^ + + ^2,., c is the speed of light and Ho is Hubble’s constant. Throughout this thesis we use H q — /i 100kms“ ^Mpc“ b Q is the total energy density of the Universe, Dm

is the matter density of the Universe, Ü,. is the radiation density of the Universe and Dy is the vacuum energy density of the Universe. The luminosity and angular diameter distances are related to the proper distance by Eqns 2.2 & 2.3

dh{z) = (1 + z)dp(z) (2.2)

d A (^ = ^ (2.3)

To calculate the density of galaxies, it is essential to be able to compute the volume as a function of the redshift. The volume between two redshift limits is calculated using Eqn 2.4.

_ CO- d \ dz

Chapter 2: A Set of Useful Tools For Producing and Using the BBD. 31 where <j is the area on the sky in steradians. X is defined in Eqn 2.1.

Cosmological dimming occurs because of the universal expansion. In a non-expanding Universe, the surface brightness of an object would remain constant, irrespective of distance. As the distance doubles, the luminosity is reduced by a factor of 4, but at the same time, the area is reduced by a factor of 4, keeping the surface brightness constant. In contrast, in an expanding Universe, the luminosity is reduced by a factor of [dp{l + z)]^, see Eqn 2.2, due to the reduction of energy of each photon by (1 + z) and the time dilation effect increasing the

time between the emission of each photon by (1 + z). The surface brightness on the other hand is reduced by [dp/{l + z)]^; see Eqn 2.3.

; L [d,Ki + z ) f L

TV 6“^ [dp(l+z)]^ Trr^ 7rr^(l + z)"^

where S is the apparent surface brightness in lum arcsec“ ^. I is the apparent luminosity, 0 is the apparent radius, L is the absolute luminosity and r is the intrinsic radius.

Most of the analysis was done using an Einstein-de Sitter cosmology Dm = 1.0, Q = 1.0 for reasons of efficiency. The Einstein-de Sitter equations can be solved analytically whereas more complicated cosmologies have to be solved numerically. However some analysis has been done using a A-CDM cosmology. Dm = 0.3, 12^ = 0.7, f2 = 1.0, the favoured cosmology today (Efstathiou et al. 1999; Tegmark, 1999).

As the redshift of a galaxy increases, two other effects, the K-correction and evolution become important. The K-correction comes from the change in wavelength of the emission with redshift. [For simplicity consider a perfect filter, which passes all photons with Ai < A < A2

and none outside this range. Radiation emitted at Ai by a galaxy at redshift zi will be detected at A = Ai(l + Zi). This will no longer be at the short wavelength edge of the filter. Instead shorter wavelength emission from the galaxy (A = Ai/(1 + z%)) will be detected at Ai.] The difference in magnitude can be calculated using Eqn 2.6, Oke Sz Sandage (1968).

-|q0.4JC(z) _ I T(A)/iog(A)dlnA ;T(A)/iog(A/[l + z])dlnA

where /log(A) is the logarithmic fiux density of the galaxy and T(A) is the filter transmission function, i.e. the efficiency of the filter at each wavelength. Both are complicated functions. However,K(z) can be approximated by simple parametric equations such as:

K{z) = ki z + k2Z^ + 0{z^) (2.7)

The K-correction can either dim or brighten a galaxy. In the B-band for instance it tends to dim galaxies because stars emit more light as 400nm photons than as 300nm photons. However, in the near infra-red, the situation is reversed because more photons are emitted at 800nm than 900nm. K-corrections can vary widely between galaxy types.

The magnitude of a galaxy will evolve with time due to varying star formation rates, the aging of the stellar population or growth of the galaxy. When mergers trigger a starburst the

Chapter 2: A Set of Useful Tools For Producing and Using the BBD. 32 change in magnitude can be extremely large and very rapid. Other galaxies such as elliptical galaxies evolve slowly as their stellar populations age.

We will use the (K+e) corrections determined by Norberg et al. (2002) based on the Bruzual &: Chariot (1993) stellar population synthesis code. These correct the magnitudes to a z = 0 magnitude, assuming purely luminosity evolution, and are not dependent on the

cosmology. These have been calculated for each of the spectral types defined by Madgwick et al. (2 0 0 1).

Madgwick et al. (2001) use a Principal Components Analysis (PCA) to identify the com­ ponents of the spectral data that are the most discriminatory between each galaxy. They find that two thirds of the variance comes from the first two components. A combination of these two components, 77 — apc\ —peg (with a = 0.5T0.1) describes the average absorption/emission

from each galaxy. A small value of 77 indicates a strongly absorbing spectrum and a high value

of 77 denotes strong emission. Madgwick et al. (2001) split the distribution of 77 into four broad

bins - the four spectral types used in Chapters 3, 4 and 5. The divisions are determined by the shape of the histogram of 77 values, shown in Fig. 4 of Madgwick et al. 2001. 77-type 1 galaxies

include all galaxies within the strong peak in this histogram. 77-type 2 galaxies are all those

along the flat part of the histogram, 77-type 3 galaxies are all those in the rapidly decreasing

part of the histogram and 77-type 4 galaxies are all those along the tail of the histogram. 77-type

1 galaxies are predominately ellipticals and lenticulars, 77-type 2 galaxies are mainly Sa and 8b

galaxies, with 77-type 3 and 77-type 4 galaxies being mainly late type spirals and irregulars.

The (K+e) corrections for each type are: Type 1, {K + e)(z) = (2z + 2.8z^)/(l + 3.8z^);

Type 2, (R' + e)(z) = (0.6z + 2.8z:^)/(l + 19.6z3);Type 3, (AT + e)(z) = (z + 3.6z^)/(l + 16.6z3);

Type 4, {K + e)(z) = (1.6z + 3.2z^)/(l + 14.6z^). For galaxies without a spectral classification the general correction is {K + e)(z) = (z + 6z^)/(l + 20z^).

Using the equations above we can calculate the absolute magnitude M and intrinsic surface brightness /lz at a given redshift z.

M = m — 5 log 1Q — 25. — {K + c) (z) (2.8)

p — p^^^ — 101og4o(l + ^) — (R + s)(z) (2.9)

where m and Papp are the apparent magnitude and surface brightness respectively.