Exercise 2: Integration/Interpolation for Cosmological Ages/Magnitudes

Solutions (partial)

Part 2: The analytic solution for the (1,0) model is

Part 3: At z=5 the lookback times for the models are

(0.4,0): 0.71174, (0.4,0.4): 0.77567, (0.4,0.6): 0.81632, (1,0): 0.62131

The lookback time plot shows the redshift vs. time curves out to z=5 for the four models. The triangles near the right axis give the extrapolation to infinite redshift, i.e. the total ages. Physically, one sees that increasing the matter density decreases the lookback time at a given redshift while adding a cosmological constant has the opposite effect. Basically, ordinary matter decelerates the expansion while exotic matter accelerates it.

For choosing the boundary conditions for the spline fit, use your knowledge that as z approaches 0, H0T(z) goes to z. Therefore the derivative, k0 on page 9 of Neal Katz' Computation I spline notes, equals 1 and the second derivative q"(0)=T"(0)=0. At z=infinity we expect the time to converge to the total age and again the second (and first) derivative should be zero, so for fitting this function the "natural boundary" conditions are appropriate and in fact match the first case of "known first derivatives".

Part 4: Numerically, it is not generally a good idea to integrate over an infinite interval. It is best to transform the variable of integration to create a finite interval; the original form of the lookback time relation given on page 5 of Eric Linder's Cosmology II notes in terms of scale factor a satisfies this, with the limits being 0 to 1. Taking a factor of a-3 out of the integrand bracket makes the integrand well behaved as a goes to 0.

The total ages for the four models are plotted as triangles near the right axis of the lookback time plot. The numerical values for the ages are

(0.4,0): 0.77870, (0.4,0.4): 0.84563, (0.4,0.6): 0.88796, (1,0): 0.66667

The analytic result for the (1,0) case is 2/3. The components have the same effect on the age as on the lookback time discussed in Part 3. Note that the values should be multiplied by
H0-1=9.78 x 109h-1 years

Part 5:

Note that increasing the ordinary matter density decreases the magnitude at fixed redshift, corresponding to intrinsically more luminous objects (since the magnitude scale is reversed). The cosmological constant "dims" objects - basically it makes the luminosity distance larger for fixed redshift since the universe was expanding more rapidly. From the deviation plot one clearly sees that exotic models can have a dimming effect recently and a brightening effect in the more distant past. Perhaps the most important conclusion is that even at moderately high redshifts, e.g. z=2, models can differ by less than 0.2 mag and therefore be hard to decide between.