A Summing Up — Question 10

Question 10

Temperature and distance data can jointly test an expansion model

A source has redshift z=1. Assume the standard relations $T(z)=T_0(1+z)$ and $D_L=(1+z)^2D_A$, with T0=2.725 K and DA=1.50 Gpc known exactly for this exercise. Independent observations give T=$5.60\pm.10$ K and DL=$6.30\pm.20$ Gpc.

1. Calculate both predicted values and normalized residuals.

2. Calculate their joint chi-squared and explain why the two observables test different parts of the framework.

Question 10 - Solution

Temperature and distance data can jointly test an expansion model

(a) Method and calculation

$$\boxed{T_{\mathrm{pred}}=5.45\ {\mathrm{K}},\quad D_{L,\mathrm{pred}}=6.00\ {\mathrm{Gpc}}}.$$$$\boxed{z_T=(5.60-5.45)/.10=1.5,\quad z_D=(6.30-6.00)/.20=1.5}.$$

(b) Interpretation and further calculation

$$\boxed{\chi^2=z_T^2+z_D^2=4.50\quad(2\text{ independent residuals})}.$$$$P(\chi_2^2\ge4.5)=e^{-4.5/2}=\boxed{.1054}.$$

Physical check

Temperature scaling probes thermal redshifting; distance duality also relies on photon propagation and conservation. Shared calibration errors would require correlated residuals.