Classical Cosmological Tests:

Conceptually, the simplest distance is the proper distance, i.e. the ruler-measured distance between two points. This has little practical value since there is no way to measure it directly.

Instead, consider a source which emits N photons isotropically at a time t. These are observed at time to. We would like to calculate the observed bolometric flux (total energy per unit time per unit area). This allows us to define a luminosity distance.

The apparent brightness of objects as a function of redshift changes under different cosmologies. This is called a redshift-distance test, since the apparent brightness tells you the distance (if you know the absolute magnitude).

The key to determining luminosity distance is finding a standard candle which one can observe at various redshifts. A standard candle has the properties of 1) being clearly identified at large redshifts, 2) being stable over time. Condition 1 is fairly easy, condition 2 may be impossible due to the large amounts of time involved.

Age Tests:

Another indirect test of cosmology is to determine the age of the Universe from the oldest objects in the Universe (i.e. lower limit).

Globular clusters arguably the best clocks, with ages estimated from MS fitting to be between 9 and 13 Gyrs.

White dwarf cooling gives and age of 7.3±1.5 Gyr for the age of the Galactic disk and 12.7 ± 0.7 (95% CL) for the age of the globular cluster M4.

Decay of radioactive isotopes with long half-lives can be used to age-date stars. Observation of 238U in a single old star gives it an age of 12.5±3 Gyr.

Geometric Tests of Cosmology:

Another method is to attempt to measure the curvature, k, value directly. Size based tests: The apparent size of objects changes as a function of redshift in different cosmologies, and actually increases at high redshift! (Think of lines of longitude on a sphere...) Note that even if the Universe is flat, angular size increases at high z -- the Universe is acting sort of like a gravitational lens... Okay, there's another test. Think of something that has a fixed size -- a standard rod. Look at it at different redshifts and see how big it looks.

Volume based tests:

Think about the analogy of surface area on a ball or saddle... Since there is more volume at high redshift in an negatively curved universe, and things at high redshift are faint (they are far away!), a negatively curved universe should have more faint galaxies than a positively curved one So go out and count the number of things you see as a function of apparent magnitude and compare to models. It turns out there are LOTS of faint things -- a negatively curved universe? Actually there are TOO MANY faint things -- no cosmology fits it! What went wrong? Galaxy evolution: galaxies had younger, brighter stars back then, and probably were smaller, so the intrinsic brightness of galaxies is different, which screws up the model predictions. The predicted differences in the numbers of faint galaxies (between different cosmologies) is much smaller than the predicted scatter due to galaxy evolution. This test no longer used much in cosmology.