Distinguish expansion rate, distance and lookback time

1. Define the cosmic evidence

A cosmological model relates expansion to matter, radiation, spatial curvature and other energy components. Matter dilutes as the inverse cube of scale factor, while a cosmological-constant energy density remains constant. A fitted model must be tested against observations; choosing its parameters in a workbench does not measure them.

2. Connect mechanisms and alternatives

The original flat matter-plus-Λ model sets H 0=70 km/s/Mpc, ΩΛ=1−Ωm and neglects radiation over the supported redshift range zero to two. Its expansion ratio is E(z)=sqrt[Ωm(1+z)³+ΩΛ]. The deceleration parameter is[Ωm(1+z)³/2−ΩΛ]/E². Negative q denotes accelerated scale-factor expansion, not necessarily an increasing H at every time.

Radial comoving distance is(c/H 0)∫dz/E, luminosity distance is(1+z) times that distance in this flat geometry, and lookback time is(1/H 0)∫dz/[(1+z)E]. The extra factor in the time integrand matters. The numerical routine integrates the stated model; it does not substitute distance divided by c for lookback time.

The transition q=0 occurs at z=[2ΩΛ/Ωm]^(1/3)−1 within this matter-plus-constant model. Radiation becomes important earlier than this workbench’s intended regime, and nonconstant dark-energy or curvature models change these relations. The supplied parameter values are examples, not a statement of the latest measured cosmology.

3. Work the supplied comparison

With Ωm=0.3 at z=1, E²=3.1 and E≈1.76068. The deceleration parameter is(1.2−0.7)/3.1≈0.161290, while at z=0 it is−0.55. The transition redshift is about 0.67110 under these assumptions.

4. Predict and investigate

Can the same model accelerate today while having decelerated at an earlier epoch?

Compare q and H/H 0 across redshift, then distinguish comoving distance, luminosity distance and lookback time. Retain the flatness and negligible-radiation assumptions.

Source and boundaries

Primary source. Checked 31 October 2026. Original model records are not new galaxy observations, current parameter measurements or a full cosmological inference pipeline. No physical observing procedure is requested.

Additional derivation reference: Hogg, Distance Measures in Cosmology, sections 4, 7 and 10, checked 31 October 2026. The original numerical exercise specializes those distance and lookback-time integrals to a flat matter-plus-cosmological-constant model with radiation neglected; it is not a fitted expansion history.