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3.4 Observations in Cosmology

The various distances that we have discussed are of course not directly observable; all that we know about a distant object is its redshift. Observers therefore place heavy reliance on formulae for expressing distance in terms of redshift. For high-redshift objects such as quasars, this has led to a history of controversy over whether a component of the redshift could be of non-cosmological origin:

Equation 3.75 (3.75)

For now, we assume that the cosmological contribution to the redshift can always be identified.

DISTANCE-REDSHIFT RELATION The general relation between comoving distance and redshift was given earlier as

Equation 3.76 (3.76)

For a matter-dominated Friedmann model, this means that the distance of an object from which we receive photons today is

Equation 3.77 (3.77)

Integrals of this form often arise when manipulating Friedmann models; they can usually be tackled by the substitution u2 = k (Omega - 1) / [(Omega(1 + z)]. This substitution produces Mattig's formula (1958), which is one of the single most useful equations in cosmology as far as observers are concerned:

Equation 3.78 (3.78)

A more direct derivation could use the parametric solution in terms of conformal time, plus r = eta now - eta emit. The generalisation of the sine rule is required in this method: Sk (a - b) = Sk (a) Ck (b) - Ck (a) Sk (b).

Although the above is the standard form for Mattig's formula, it is not always the most computationally convenient, being ill defined for small Omega. A better version of the relation for low-density universes is

Equation 3.79 (3.79)

It is often useful in calculations to be able to convert this formula into the corresponding one for Ck (r). The result is

Equation 3.80 (3.80)

remembering that R0 = (c / H0) [(Omega - 1) / k]-1/2.

It is possible to extend this formula to the case of contributions from pressureless matter (Omegam) and radiation (Omegar):

Equation 3.81 (3.81)

There is no such compact expression if one wishes to allow for vacuum energy as well (Dabrowski & Stelmach 1987). The comoving distance has to be obtained by numerical integration of the fundamental dr / dz, even in the k = 0 case. However, for all forms of contribution to the energy content of the universe, the second-order distance-redshift relation is identical, and depends only on the deceleration parameter:

Equation 3.82 (3.82)

[problem 3.4]. The sizes and flux densities of distant objects therefore determine the geometry of the universe only once an equation of state is assumed, so that q0 and Omega0 can be related.

CHANGE IN REDSHIFT Although we have talked as if the redshift were a fixed parameter of an object, having a status analogous to its comoving distance, this is not correct. Since 1 + z is the ratio of scale factors now and at emission, the redshift will change with time. To calculate how, we differentiate the definition of redshift and use the Friedmann equation. For a matter-dominated model, the result is [problem 3.2]

Equation 3.83 (3.83)

(e.g. Lake 1981; Phillipps 1982). The redshift is thus expected to change by perhaps 1 part in 108 over a human lifetime. In principle, this sort of accuracy is not completely out of reach of technology. However, in practice these cosmological changes will be swamped if the object changes its peculiar velocity by more than 3 m s-1 over this period. Since peculiar velocities of up to 1000 km s-1 are built up over the Hubble time, the cosmological and intrinsic redshift changes are clearly of the same order, so that separating them would be very difficult.

AN OBSERVATIONAL TOOLKIT We can now assemble some essential formulae for interpreting cosmological observations. Since we will mainly be considering the post-recombination epoch, these apply for a matter-dominated model only. Our observables are redshift, z, and angular difference between two points on the sky, d psi. We write the metric in the form

Equation 3.84 (3.84)

so that the comoving volume element is

Equation 3.85 (3.85)

The proper transverse size of an object seen by us is its comoving size d psi Sk (r) times the scale factor at the time of emission:

Equation 3.86 (3.86)

Probably the most important relation for observational cosmology is that between monochromatic flux density and luminosity. Start by assuming isotropic emission, so that the photons emitted by the source pass with a uniform flux density through any sphere surrounding the source. We can now make a shift of origin, and consider the RW metric as being centred on the source; however, because of homogeneity, the comoving distance between the source and the observer is the same as we would calculate when we place the origin at our location. The photons from the source are therefore passing through a sphere, on which we sit, of proper surface area 4pi [R0 Sk (r)]2. But redshift still affects the flux density in four further ways: photon energies and arrival rates are redshifted, reducing the flux density by a factor (1 + z)2; opposing this, the bandwidth d nu is reduced by a factor 1 + z, so the energy flux per unit bandwidth goes down by one power of 1 + z; finally, the observed photons at frequency nu0 were emitted at frequency nu0 (1 + z), so the flux density is the luminosity at this frequency, divided by the total area, divided by 1 + z:

Equation 3.87 (3.87)

A word about units: Lnu in this equation would be measured in units of W Hz-1. Recognizing that emission is often not isotropic, it is common to consider instead the luminosity emitted into unit solid angle - in which case there would be no factor of 4pi, and the units of Lnu would be W Hz-1 sr-1.

The flux density received by a given observer can be expressed by definition as the product of the specific intensity Inu (the flux density received from unit solid angle of the sky) and the solid angle subtended by the source: Snu = Inu d Omega. Combining the angular size and flux-density relations thus gives the relativistic version of surface-brightness conservation. This is independent of cosmology (and a more general derivation is given in chapter 4):

Equation 3.88 (3.88)

where Bnu is surface brightness (luminosity emitted into unit solid angle per unit area of source). We can integrate over nu0 to obtain the corresponding total or bolometric formulae, which are needed e.g. for spectral-line emission:

Equation 3.89 (3.89)

Equation 3.90 (3.90)

The form of the above relations lead to the following definitions for particular kinds of distances:

Equation 3.91 (3.91)

At least the meaning of the terms is unambiguous enough, which is not something that can be said for the term effective distance, sometimes used to denote R0 Sk (r). Angular-diameter distance versus redshift is illustrated in figure 3.7.

Figure 3.7

Figure 3.7. A plot of dimensionless angular-diameter distance versus redshift for various cosmologies. Solid lines show models with zero vacuum energy; dashed lines show flat models with Omegam + Omegav = 1. In both cases, results for Omegam = 1, 0.3, 0 are shown; higher density results in lower distance at high z, due to gravitational focusing of light rays (see chapter 4).

The last element needed for the analysis of observations is a relation between redshift and age for the object being studied. This brings in our earlier relation between time and comoving radius (consider a null geodesic traversed by a photon that arrives at the present):

Equation 3.92 (3.92)

So far, all this is completely general; to complete the toolkit, we need the crucial input of relativistic dynamics, which is to give the distance-redshift relation. For almost all observational work, it is usual to assume the matter-dominated Friedmann models with differential relation

Equation 3.93 (3.93)

which integrates to

Equation 3.94 (3.94)

PREDICTING BACKGROUNDS The above machinery can be given an illustrative application to obtain one of the fundamental formulae used for calculations involving background radiation. Suppose we know the emissivity jnu for some process as a function of frequency and epoch, and want to predict the current background seen at frequency nu0. The total spectral energy density created during time dt is jnu (nu0[1 + z], z) dt; this reaches the present reduced by the volume expansion factor (1 + z)-3. The redshifting of frequency does not matter: nu scales as 1 + z, but so does d nu. The reduction in energy density caused by lowering photon energies is then exactly compensated by a reduced bandwidth, leaving the spectral energy density altered only by the change in proper photon number density. Inserting the redshift-time relation, and multiplying by c / 4pi to obtain the specific intensity, we get

Equation 3.95 (3.95)

This may seem a bit of a cheat: we have considered how the energy density evolves at a point in space, even though we know that the photons we see today originated at some large distance. This approach works well enough because of large-scale homogeneity, but it may be clearer to obtain the result for the background directly. Consider a solid angle element d Omega: at redshift z, this area on the sky for some radial increment dr defines a proper volume

Equation 3.96 (3.96)

This volume produces a luminosity V jnu, from which we can calculate the observed flux density Snu = Lnu / [4pi (R0 Sk)2 (1 + z)]. Since surface brightness is just flux density per unity solid angle, this gives

Equation 3.97 (3.97)

which is the same result as the one obtained above.

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