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SCIENCE · INVESTIGATE AND EXPLAIN

Separate a local expansion estimate from velocity offsets

At sufficiently low redshift, a Hubble–Lemaître relation approximates recession velocity as H 0d. Galaxies also have peculiar motions relative to the large-scale expansion, and observer-frame corrections matter. A nearby object’s velocity divided by distance can therefore differ substantially from the underlying expansion slope.

Learning outcome: Compare single-source, forced-origin and free-intercept slopes. State which nuisance was modeled and which real distance/velocity uncertainties remain outside the packet.

Starting recap

Keep galaxy mass, source calibration, expansion history and independent cosmological evidence distinct.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

With H=70 and u=300, the 20 Mpc velocity is 1700 km/s, giving a single-source slope 85. The four-point forced-origin fit gives about 75.2941, while a free-intercept fit recovers slope 70 and intercept 300 exactly.

Open the experiment

Predict

Can a nonzero intercept bias a fit even when every supplied distance and velocity is exact?

Experiment

Compare single-source, forced-origin and free-intercept slopes. State which nuisance was modeled and which real distance/velocity uncertainties remain outside the packet.

Separate expansion slope and local velocity

Worked teaching model · synthetic values · Original low-redshift distances 10, 20, 40, 80 Mpc, velocities H d+offset. Compare a single 20 Mpc estimate, zero-intercept fit and slope with free intercept. The shared offset is a teaching nuisance, not an actual coherent flow measurement or high-z velocity law.

Reset keeps your prediction, captures and practice history.

A local velocity or offset can bias a low-distance or forced-origin expansion estimate.

Original low-redshift distances 10, 20, 40, 80 Mpc, velocities H d+offset. Compare a single 20 Mpc estimate, zero-intercept fit and slope with free intercept. The shared offset is a teaching nuisance, not an actual coherent flow measurement or high-z velocity law.

59000080Distance / Mpc; velocity / km/s, vertical scale varies
Distance / MpcSupplied velocity / km/s
101000
201700
403100
805900
Original conditional model ledger
QuantityValueUnit
Velocity at 20 Mpc1700km/s
Single-distance inferred slope85km/s/Mpc
Forced-zero-intercept fitted slope75.29412km/s/Mpc
Free-intercept fitted slope70km/s/Mpc
Free fitted intercept300km/s

These supplied models are not current measured cosmological parameters, a new galaxy discovery or independent actual observing evidence.

A local velocity or offset can bias a low-distance or forced-origin expansion estimate.

Original low-redshift distances 10, 20, 40, 80 Mpc, velocities H d+offset. Compare a single 20 Mpc estimate, zero-intercept fit and slope with free intercept. The shared offset is a teaching nuisance, not an actual coherent flow measurement or high-z velocity law.

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the cosmic evidence

At sufficiently low redshift, a Hubble–Lemaître relation approximates recession velocity as H 0d. Galaxies also have peculiar motions relative to the large-scale expansion, and observer-frame corrections matter. A nearby object’s velocity divided by distance can therefore differ substantially from the underlying expansion slope.

2. Connect mechanisms and alternatives

The original distances are 10, 20, 40 and 80 Mpc, with velocities H d+u. H is supplied between 60 and 80 km/s/Mpc, and u is a shared offset of−300, 0 or 300 km/s. The common offset is an instructional nuisance term, not a claim that all real peculiar velocities have the same value.

A single 20 Mpc estimate gives H+u/20. A least-squares line forced through zero gives H+uΣd/Σd², where Σd=150 and Σd²=8500. Allowing an intercept recovers H and u for these exact supplied records. A real fit would also need distance errors, sample selection and correlated velocity structure.

The low-redshift approximation must not be extended by treating every high-redshift cosmological shift as a special-relativistic peculiar speed. Large-scale expansion has no preferred central galaxy under the homogeneous model, while gravitationally bound systems do not simply expand internally with the background scale factor.

3. Work the supplied comparison

With H=70 and u=300, the 20 Mpc velocity is 1700 km/s, giving a single-source slope 85. The four-point forced-origin fit gives about 75.2941, while a free-intercept fit recovers slope 70 and intercept 300 exactly.

4. Predict and investigate

Can a nonzero intercept bias a fit even when every supplied distance and velocity is exact?

Compare single-source, forced-origin and free-intercept slopes. State which nuisance was modeled and which real distance/velocity uncertainties remain outside the packet.

Apply the evidence to a changed case

Use H=70 and u=−300. Find the 20 Mpc ratio and forced-origin slope.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

The velocity is 1100 km/s and the single-source ratio 55 km/s/Mpc. The forced-origin slope is 70−300×150/8500≈64.7059. The underlying supplied slope remains 70.

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.

Apply

Why do the three slope estimates differ in the offset case?

Public practice checks a specific claim. It does not establish independent mastery.

Use H=70 and u=−300. Find the 20 Mpc ratio and forced-origin slope.

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Propagate a correlated calibration through a distance ladderNext: Connect redshift to expansion ratios without a speed shortcut
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