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3.1. Flux

The flux within some specific wavelength range can be calculated if we consider equations (3), (5) and (6). Then we have (see Ellis 1971),

Equation 13   (13)

Therefore, the flux measured in the frequency range nu, nu + dnu by the observer may be written as

Equation 14   (14)

Fnu is also called specific flux of the radiation.

3.2. Magnitude

The apparent magnitude in a specific observed frequency bandwidth is obtained from a different form than given by equation (7), which may written as

Equation 15   (15)

where W(nu) is the function which defines the spectral interval of the observed flux (the standard UBV system, for instance). This is a sensitivity function of the atmosphere, telescope and detecting device.

From equations (14) and (15) the apparent magnitude in a specified spectral interval W may be written as

Equation 16   (16)

Some remarks about this equation are important to mention. Firstly, equation (16) calculates the apparent magnitude of a source whose intrinsic luminosity at a specific redshift is somehow known. Secondly, in a similar manner this equation can also be used to calculate the intrinsic luminosity of a cosmological source whose redshift and apparent magnitude are known from observations. Finally, since cosmological sources do evolve, the intrinsic luminosity L changes according to the evolutionary stage of the source, and therefore, L is actually a function of the redshift; L = L(z). Hence, in order to use equation (16) to obtain the apparent magnitude evolution of the source, some theory for luminosity evolution is also necessary. For galaxies, L(z) is usually derived taking into consideration the theory of stellar evolution, from where some simple equations for luminosity evolution can be drawn (see Binney & Tremaine 1987, p. 552; Peebles 1993, p. 330, and references therein). (4) Finally, since J[nu(1 + z)] is a property of the source at a specific redshift, this function must be known in order to calculate the apparent magnitude, unless the K-correction approach is used (see below).

For magnitude limited catalogues, the luminosity distance and the observer area distance have both an upper cutoff, which is a function of the apparent magnitude, the frequency bandwidth used in the observations and the luminosity of the sources. Considering equation (1), the luminosity distance of flux limited sources may be written as

Equation 17   (17)

3.3. K-Correction

The relations above demand the knowledge of both the source spectrum and the redshift. However, when the source spectrum is not known, it is necessary to introduce a correction term in order to obtain the bolometric flux from observations. This correction is known as the K-correction, and it is a different way for allowing the effect of the source spectrum.

The method that will be presented next for deriving the K-correction follows the classical work of Humason, Mayall and Sandage (1956, appendix B; see also Oke & Sandage 1968, Sandage 1988, 1995). We start by calculating the difference in magnitude produced by the bolometric flux F and the flux FW measured by the observer, but at the bandwidth W(nu) in any redshift z. Therefore, I shall write both quantities as F(z) and FW(z) respectively. Since, by definition, we know that

Equation 18   (18)

the difference in magnitude Delta m(z) will be given by

Equation 19   (19)

The rate between the observed flux FW(z) at a given redshift and at z = 0 defines the K-correction. Then, considering equation (19), we have that

Equation 20   (20)

where we have defined

Equation 21   (21)

Then it follows that

Equation 22   (22)

which means that once we know the K-term and the observed magnitude mW, the bolometric magnitude is know within a constant Delta m(0). If we now substitute equation (14) into equation (20), it is easy to show that

Equation 23   (23)

Remembering that by equation (5) we know that we can have the source spectrum transformed from the rest frame of the source to the rest-frame of the observer by a factor of (1 + z), that is, J[nu(1 + z)] dnu = [J(nuG) dnuG] / (1 + z), then we may also write equation (23) as

Equation 24   (24)

Note that the equations above allow us to write theoretical K-correction expressions for any given spacetime geometry, provided that the line element dS2 is known beforehand. These theoretical expressions for observables like the K-correction could, in principle, be directly compared with observations.

As a final remark, it is obvious that if the source spectrum is already known, all relevant observational relations can be calculated without the need of the K-correction.

3.4. Colour

With the calculations above we can obtain the theoretical expression for the colour of the sources for any given spacetime. Let us consider two bandwidths W and W'. From equation (16) we can find the difference in apparent magnitude for these two frequency bands in order to obtain an equation for the colour of the source in a specific redshift. Let us call this quantity CWW'. Thus,

Equation 25   (25)

Considering that cosmological sources do evolve, they should emit different luminosities in different redshifts due to the different evolutionary stages of the stellar contents of the sources, and this is reflected in the equation above by the source spectrum function which may be different for different redshifts. Note, however, that in the equation above the source is assumed to have the same bolometric luminosity in a specific redshift and, therefore, we can only use equation (25) to compare observation of objects of the same class and at similar evolutionary stages in certain z, since L = L(z). This often means galaxies of the same morphological type. In other words, equation (25) is assuming that a homogenous populations of cosmological sources do exist, and hence, the evolution and structure of the members of such a group will be similar.

Equation (25) also gives us a method for assessing the possible evolution of the source spectrum. For instance, by calculating B - V and V - R colours for E galaxies with modern determinations of the K-correction, Sandage (1995, p. 50) reported that no colour evolution was found to at least z = 0.4. However, for z geq 0.3 it was found that rich clusters of galaxies tend to be bluer (the Butcher-Oemler effect) than at lower redshifts (Peebles 1993, p. 202; see also Kron 1995, p. 299). Therefore, if we start from a certain metric, we can calculate the theoretical redshift range where colour evolution would be most important for the assumed geometry of the cosmological model.

Another point worth mentioning, from equation (25) we see that colour is directly related to the intrinsic characteristics of the source, its evolutionary stage, as given by the redshift and the assumptions concerning the real form of the source spectrum function at a certain z. However, this reasoning is valid for point sources whose colours are integrated and, therefore, we are not considering here structures, like galactic disks and halos, which in principle may emit differently and then will produce different colours. If we remember that cosmological sources are usually far enough to make the identification and observation of source structures an observational problem for large scale galaxy surveys, this hypothesis seems reasonable at least as a first approximation.

As a final remark, it is clear that in order to obtain a relationship between apparent magnitude and redshift we need some knowledge about the dependence of the intrinsic bolometric luminosity L and the source spectrum function J with the redshift. It seems that such a knowledge must come from astrophysically independent theories about the intrinsic behaviour and evolution of the sources, and not from the underlying spacetime geometry.

3.5. Number Counts

In any cosmological model if we consider a small affine parameter displacement dy at some point P on a bundle of past null geodesics subintending a solid angle dOmega0, and if n is the number density of radiating sources per unit proper volume at P, then the number of sources in this section of the bundle is (Ellis 1971, p. 159)

Equation 26   (26)

where, as before, ka is the propagation vector of the radiation flux and ua is the 4-velocity of the observer. Equation (26) considers the counting of all sources at P with number density n. Consequently, if we want to consider the more realistic situation that only a fraction of galaxies in the proper volume dV = (r0)2 dOmega0 dl = (r0)2 dOmega0 (- kaua)dy is actually detected and included in the observed number count, we have to write dN in terms of a selection function psi which represents this detected fraction of galaxies. Then equation (26) becomes (Ellis et al. 1985)

Equation 27   (27)

where dN0 is the fraction number of sources actually observed in the unit proper volume dV with a total of dN sources.

In principle psi can be estimated from a knowledge of the galactic spectrum, the observer area distance, the redshift, and the detection limit of the sample as given by the limiting flux in a certain frequency bandwidth. The other quantities in equation (27) come from the assumed cosmological model itself, and inasmuch as equation (27) is general, it is valid for any cosmological model, either homogeneous or inhomogeneous.

In order to determine psi we need to remember that in any spacetime geometry the observed flux in bandwidth W is given by equations (14) and (18),

Equation 28   (28)

Then, if a galaxy at a distance r0 is to be seen at flux FW, its luminosity L(z) must be bigger than {4pi (r0)2 (1+z)3 FW} / {int0infty W(nu) J[nu(1 + z)] dnu}. Therefore, the probability that a galaxy at a distance r0 and with redshift z is included in a catalog with maximum flux FW is,

Equation 29   (29)

where this integral's lower limit is

Equation 30   (30)

L* is a parameter, and phi(w) is the luminosity function. model L* is a characteristic luminosity at which the luminosity function exhibits a rapid change in its slope.

Now, if we assume spherical symmetry, then equation (27) becomes

Equation 31   (31)

Thus, the number of galaxies observed up to an affine parameter y at a point P down the light cone, may be written as

Equation 32   (32)

All quantities in the integrand above are function of the past null cone affine parameter y, and, in principle, they must be explicitly calculated before they can be entered into equation (32). In some cases one may avoid this explicit determination and use instead the radial coordinate, a method which turns out to be easier than finding these expressions in terms of y (Ribeiro 1992). Then, once N0(y) is obtained, it becomes possible to relate it to other observables, since they are all function of the past null cone affine parameter. For example, if one can derive an analytic expression for the redshift in a given spacetime, say z = z(y), and if this expression can be analitically inverted, then we can write N0 as a function of z.

It is important to mention that the local number density n is given in units of proper density and, therefore, in order to take a proper account of the curved spacetime geometry, one must relate n to the local density as given by the right hand side of Einstein's field equations. If, for simplicity, we suppose that all sources are galaxies with a similar rest-mass MG, then

Equation 33   (33)

An indication on how to use the expressions above can be grasped for the Einstein-de Sitter model, where it is well-known that the local density rho may be written as

Equation 34   (34)


Equation 35   (35)

If we remember that from a relativistic viewpoint astronomical observations are actually made along the past light cone, where dS2 = 0, we must calculate a(t) and find its expression along the backward null cone,

Equation 36   (36)

before we can use equation (33) back into equation (32).

From the discussion above it is clear that the theoretical determination of N0 depends critically on the spacetime geometry and the luminosity function phi. For the latter, in the Schechter (1976) model it has the form

Equation 37   (37)

where phi* and alpha are constant parameters. One must not forget that this luminosity function shape was originally determined from local measurements (Schechter 1976), and there is now a controversy about the change of shape and parameters of the luminosity function in terms of evolution (Lonsdale & Chokshi 1993; Gronwall & Koo 1995; Ellis et al. 1996), that is, as we go down the light cone. In any case the Schechter's function above can be used at least as a starting point. In addition, since the luminosity function is being used as a probability in equation (29), it must be properly normalized. However, considering equation (32) one can choose the number density n to agree with the normalization of psi.

As a final remark, one must note that gravitational lensing magnification can also affect the counting of point sources, because weak sources with low flux might appear brighter due to lensing magnification. Such an effect will not be treated here, since its full treatment demands more detailed information about the sources themselves, such as considering them as extended ones, and is considered to be most important for QSO's (see Schneider et al. 1992).

4 Note that equation (16) also indicates that the source spectrum function J might evolve and change its functional form at different evolutionary stages of the source. Back.

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