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6.1. The angular size test

The first of the classical cosmological tests we will consider is the angular size test. Here one measures the angular size of a standard meter stick (hopefully) as a function of redshift; different FRW cosmologies make different predictions, but basically, for all FRW models theta(z) first decreases as 1 / z (as would be expected in a Euclidean universe) and then increases with z. This is because the angular size distance is given by DA = r / (1 + z) but the radial comoving coordinate approaches a finite value as z -> infty. The angular size distance reaches a maximum at a redshift between 1 and 2 and then decreases again.

When giant radio galaxies at large redshift were discovered in the 1960's there was considerable optimism that these could be used as an angular size cosmological probe. Radio galaxies typically have a double-lobe structure with the radio emitting lobes straddling the visible galaxy; these lobes can extend hundreds of kpc beyond the visible object. Such a linear structure may be oriented at any angle to the observer's line-of-sight, so one needs to measure the angular sizes of a number of radio galaxies in a given redshift bin and only consider the largest ones, i.e., those likely to be nearly perpendicular to the line-of-sight.

The result of all this work was disappointing. It appeared that the angular size of radio sources kept decreasing with redshift just as one would expect for a pure Euclidean universe [38]. The obvious problem, that plagues all classical tests, is that of evolution. Very likely, these radio galaxies are not standard meter sticks at all, but that they were actually smaller at earlier epochs than now. This would be expected, because such objects are thought to result from jets of relativistic particles ejected from the nucleus of the parent galaxy in opposite directions. The jets progress through the surrounding intergalactic medium at a rate determined by the density of that medium, which, of course, was higher at larger redshift.

But there is another class of radio sources that would be less susceptible to such environmental effects: the compact radio sources. These are objects, on a scale of milli-arc-seconds, typically associated with distant quasars, that are observed with radio interferometers having global baselines. The morphology is that of a linear jet with lengths typically less than 30 or 40 pc, so these would presumably be emission from the jets of relativistic particles deep in the galactic nucleus near the central engine producing them. The intergalactic medium, and its cosmological evolution, would be expected to have no effect here [39].

The result of plotting the median angular size of about 150 of these sources as a function of redshift is shown on a log-log plot in Fig. 3 [40]. Also shown are the predicted relations for three flat cosmologies (Omegak = 0) with Omegam = 0.9, 0.3, 0.1, the remainder being in a cosmological constant (the middle curve is the concordance model). In each case the linear size of the compact radio sources was chosen to achieve the best fit to the data.

Figure 3

Figure 3. The median angular size vs. redshift (log-log plots) for 145 compact radio sources in 12 redshift bins. The curves are the three flat cosmological models: dashed, OmegaLambda = 0.9; solid, OmegaLambda = 0.7 (concordance), dotted, OmegaLambda = 0.1. The physical size of the sources (20-40 pc) has been chosen for the best fit.

It is evident that the general property of FRW models (that the angular size of a standard meter stick should begin to increase again beyond a redshift of about 1.5) is present in this data. However, no statistical test or maximum likelihood analysis is necessary to see that all three models fit the data equally well. This is basically an imprecise cosmological test and cannot be improved, particularly considering that these objects may also evolve in some unknown way with cosmic time. Looking at the figure, one may notice that measurement of angular sizes for just a few objects at lower redshift might help distinguish between models. However, there are very few such objects at lower redshift, and these have a much lower intrinsic radio power than those near redshift one. It is dangerous to include these objects on such a plot because they are probably of a very different class.

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