## Transcribed Text

1. Marginalizing a Systematic: Hubble’s Law and the Distance to a Galaxy
Hubble’s Law states that a galaxy located a distance d from Earth will recede at velocity v
given by
v = H0d.
H0 is called Hubble’s constant. By measuring the redshift of a galaxy, astronomers can calculate
v and therefore d using H0. However, H0 is itself an experimentally determined quantity,
so uncertainties in H0 lead to systematic uncertainties in d. Given a measurement vm of v, H0
is a nuisance parameter in the calculation of d.
Suppose a galaxy has a measured recession velocity vm = (30 1) 103 km s1. Assume a
Gaussian uncertainty on vm, i.e., the error bar 1 103 km s1 can be treated as the width of a
Gaussian with mean 30 103 km s1.
Using the measured velocity, calculate and plot the posterior PDF p(djvm, I) of the distance
to the galaxy for the cases listed below. In all cases assume a flat prior p(djI) in the range
d 2 [0, 800] Mpc.
(a) (10 points) Let H0 = 75 km s1 Mpc1 exactly.
(b) (10 points) Using the PDF from part (a), compute the best estimate dˆ and its central 68%
credible interval.
(c) (10 points) Let H0 be uniformly distributed between 50 and 90 km s1 Mpc1, the state
of measurements prior to the 1990s.
(d) (10 points) Using the PDF from part (c), compute the best estimate dˆ and its central 68%
credible interval.
(e) (10 points) Let H0 be distributed between 50 and 90 km s1 Mpc1 using the Jeffreys
prior.
(f) (10 points) Using the PDF from part (e), compute the best estimate dˆ and its central 68%
credible interval.
(g) (10 points) Assuming Gaussian uncertainties in H0, plot the posterior PDF of d on the
same plot for the following cases:
1. H0 = 74.031.42 km s1 Mpc1, estimated in 2019 using Cepheid variable stars and
Type Ia supernovae (Riess et al., ApJ 876:85, 2019).
2. H0 = 67.66 0.42 km s1 Mpc1, estimated in 2018 using the cosmic microwave
background (Planck 2018 results: VI. Cosmological parameters).
(h) (10 points) Compute two best estimates dˆ for the two values of H0 used in part (g). In
spite of the 4.5s disagreement between the reported values of H0, do the distance estimates
still agree given their 68% credible intervals?

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.