ASTRO 735 - Problem Set Solutions

Set 3

1. and 2.: Values for dV, V(z), D_L, and D_A for 5 sets of cosmological parameters are listed here, where

O_t = Omega_total
O_l = Omega_lambda
h65 = H₀ / 65 km/s-Mpc

Note concerning units and scales: The question was raised in class why isn't dV/dz or V of order D_L³ ? The answer is because D_L is a distance and dV/dz is a volume within and solid angle, i.e. only one of the three dimensions in dV or V is necessarily of order D_L. The transverse dimensions can be substantially smaller depending on the scaling of the units of solid angle.

To see this more formally, consider writing

D_L = (c/H₀) f(O_t,O_l,z)

where f is a function of order z; and

dV/dz = (c/H₀) S (D_L/(1+z))² g(O_t,O_l,z),

where g is anothher function of order z and S is the solid angle. We have defined f and g in detail in class. The ratio dV/dz / D_L is then:

dV/dz / D_L = (c/H₀)² * S * f * g / (1+z)²

Ignoring f * g / (1+z)² at small z ( <=1 ), this ratio is roughly (c/H₀)² S, where

c/H₀ = 3e3 h^-1 Mpc
S = 3e-4 deg^-2.

You can readily verify from the above tabulation that D's and V's -- in respective units involving Mpc -- differ by roughly a factor of 3e3 in favor of V as we have calculated here.

Trends and differences:

dV/dz and V -- At z=1 more open models have more volume, however, flat models with comparable matter density have even more volume. At z = 1000, on the other hand, the open models have more volume, and in general there is more volume for less matter density.

D_L and D_A -- These have the same qualitative behavior as volume, there are larger distances in the cases where there is more volume. The only additional surprise is for D_A which gets SMALLER at z = 1000 than at z = 1. That is, the angular size of a fixed rod is 20 to 100 times larger at z = 1000 than at z = 1.

Also note that none of these quantities, including D_A, are constant quantities at constant spatial curvature at high redshift. That is to say, a fixed rod at z = 1000 will have different apparent angular sizes for different flat universes with varying amounts of matter and vacuum energy.

3. Cosmological Leverage

This figure displays the principal results.

First note that the amplitude of |Q/Qref-1| does not continue to increase dramatically at high redshift, and in general begins to asymptote (i.e. there exists a knee in this redshift function) between z = 3 and 10.

Second note that for some pairs of cosmologies there exists intermediate redshift local maxima, roughly in the range z = 0.3 - 2. These are cases for difference curvature. Hence for testing whether there exists a non-zero cosmological constant necessary to make space flat for a given matter density, intermediate redshifts is clearly the best place to make the measurement. This is a robust statement.

Third note that in increasing order D_A, D_L², V, and then dV/dz provide more leverage at any given redshift. This follows from the fact that dV/dz = (c/H₀) S (D_L/(1+z))² g(O_t,O_l,z), i.e. g adds additional dependence on O_t and O_l. The reason V is not quite as sensitive as dV/dz is that it is an integral over redshift; at lower redshift there is less leverage so the integrated leverage is diluted.

4. Cosmological S/N

Improperly formulated -- not graded.

last update: Dec 04, 2000

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