By Edward L. Wright, Phd, UCLA
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Additional resources for Astronomy 275 Lecture Notes, Spring 2009
For the more general case we note that sin r and sinh r differ from r only in the cubic term. However, the integral on the RHS of Eqn(121) depends on a(t) and differs from a linear approximation cz/H◦ R◦ in the second order. The second order deviation of the angular size distance away from the linear approximation DA = cz/H◦ thus depends only on the time history of the scale factor a(t). We can write 1 a(t◦ + ∆t) = a(t◦ ) 1 + H◦ ∆t − q◦ (H◦ ∆t)2 + . . 5, and the Steady State model has q◦ = −1.
87. The differences between models are clear on this plot. 48 10. Number Counts One kind of cosmological observation is the number versus flux law, N(S). We can compute the expected N(S) law for various cosmological models using the distances dA and dL . The physical volume of the shell between redshift z and z + dz is given by the surface area of the sphere which is 4πdA (z)2 times the thickness of the shell which is (cdt/dz)dz. Thus if we have conserved objects, so their number density varies like n(z) = n◦ (1 + z)3 , then the number we expect to see in the redshift range is cdt dN = n◦ (1 + z)3 dA (z)2 dz dz (154) where N is the number of sources with redshift less than z per steradian.
We get dz/dto = √ H◦ (1 + z)(1 − 1 + z) = −4H◦ . This is negative because this model is decelerating, so redshifts decrease with time. Unfortunately, the velocity change associated with this redshift change is only dv/dt = c(dz/dt)/(1 + z) = −2 cm/sec/yr for H◦ = 65, so it will be very difficult to measure. 5Ωm◦ − Ωr◦ )z = −q◦ H◦ z dto 30 (82) where q◦ is the deceleration parameter. 6. Flatness-Oldness Even in the general case with radiation, matter and vacuum densities, the energy equation is still 8πGρR2 8πGρR2 = H 2 R2 − = const (83) 2Etot = v 2 − 3 3 Thus we still get const′ (84) Ω−1 − 1 = ρa2 Currently Ωr◦ ≈ 10−4 but for z > 104 the radiation will dominate the density.