Questions: Robertson-Walker Metric

Question 1

In the Robertson-Walker metric, the spatial coordinates of galaxies participating in the Hubble flow remain constant as the universe expands. What changes to produce the observed recession of distant galaxies?

• A. The galaxies accelerate through space away from us
• B. The scale factor a(t) increases, stretching the metric distance between fixed comoving coordinates
• C. The speed of light decreases over time, making distant objects appear to recede
• D. The curvature parameter k changes with time

Answer: (B)

In comoving coordinates, galaxies at rest in the Hubble flow have fixed coordinate positions. The physical (proper) distance between two comoving galaxies is d(t) = a(t) × Δr, where Δr is the fixed coordinate separation. As a(t) increases, d(t) increases — galaxies recede from each other not because they move through space but because the space between them expands. This distinction is subtle but important: the expansion is a property of the metric (the geometry of space), not a motion of galaxies through a pre-existing space.

Question 2

The curvature parameter k in the Robertson-Walker metric determines whether the spatial geometry of the universe is finite or infinite.

• T. True
• F. False

Answer: False

The curvature parameter k determines the local geometry (spherical, flat, or hyperbolic), but not necessarily the global topology. A flat (k = 0) or hyperbolic (k = -1) universe could be infinite (the simplest topology) or finite with a non-trivial topology (e.g., a flat torus or a compact hyperbolic manifold). A positively curved (k = +1) universe is finite in volume in the standard topology (a 3-sphere). The Robertson-Walker metric constrains local geometry, not global topology.

Question 3

Explain what cosmological redshift is in terms of the Robertson-Walker metric, and how it differs from Doppler redshift.

Model answer: In the RW metric, a photon emitted at time t_e with wavelength λ_e is observed at time t_o with wavelength λ_o = λ_e × a(t_o)/a(t_e). The redshift z = (λ_o - λ_e)/λ_e = a(t_o)/a(t_e) - 1 arises because the photon's wavelength stretches with the expanding space — the metric scale factor a(t) increases during the photon's travel. This differs from a Doppler shift, which is caused by the relative motion of source and observer through space. Cosmological redshift is a property of the expanding metric, not of the relative velocity of galaxies. For small redshifts (z << 1), the two descriptions approximately coincide (v ≈ cz ≈ H₀d), but for large z they diverge — galaxies at z > 1.5 have recession velocities exceeding c, which is allowed because it is the metric expanding, not objects moving through space faster than light.

The distinction between cosmological redshift and Doppler shift is subtle and partly semantic. In GR, there is no unique way to define the 'velocity' of a distant galaxy; the redshift is the observable, and it directly measures the ratio of scale factors at emission and observation.

Question 4

Derive the relationship between the Hubble parameter H(t) and the scale factor a(t).

Model answer: The Hubble parameter is defined as H(t) = ȧ(t)/a(t), where ȧ = da/dt. For a nearby comoving galaxy at fixed coordinate distance Δr, the proper distance is d = a(t)Δr. The recession velocity is v = ḋ = ȧ(t)Δr = (ȧ/a)(aΔr) = H(t)d. This is Hubble's law: v = Hd. The present value H₀ ≈ 70 km/s/Mpc is the Hubble constant. Note that H(t) generally changes with time — it is not truly a 'constant' except at the present epoch. In a matter-dominated universe, H decreases; in a dark-energy-dominated universe, H approaches a constant asymptotically.

The Hubble parameter is the most directly measurable cosmological quantity. Its present value H₀, combined with the density parameters Ω_m, Ω_r, Ω_Λ, determines the entire expansion history a(t) through the Friedmann equations.