Problem 1
Solution route: define the measured quantity, write the model, substitute shared values, and test the stated assumption.
Key check: rho_crit = 3H0^2/(8 pi G) = 1.260744e+11 solar masses Mpc^-3.
Rubric
Problems 1-4 are 25 points each. Require dimensional consistency, correct comparative direction, explicit assumptions, reproducibility, and interpretation. Accept equivalent derivations when the physical meaning is preserved.
Problem 2
Solution route: define the measured quantity, write the model, substitute shared values, and test the stated assumption.
Key check: The analysis must contain a labeled data display, uncertainty or sensitivity treatment, and a provenance statement; an unsupported scalar answer is incomplete.
Problem 3
Solution route: define the measured quantity, write the model, substitute shared values, and test the stated assumption.
Key check: The analysis must contain a labeled data display, uncertainty or sensitivity treatment, and a provenance statement; an unsupported scalar answer is incomplete.
Problem 4
Solution route: define the measured quantity, write the model, substitute shared values, and test the stated assumption.
Key check: The analysis must contain a labeled data display, uncertainty or sensitivity treatment, and a provenance statement; an unsupported scalar answer is incomplete.
Common errors
- using cz as a universal cosmological velocity;
- mixing Mpc, km, and seconds without conversion;
- reversing a ratio or calling a model-dependent quantity direct;
- ignoring covariance, selection, or the approximation domain.