Standard rulers and the sound-horizon scale

The acoustic ruler is a scale in a pattern

A standard ruler supplies a calibrated length scale that can be compared with observed angles or redshift separations. In baryon acoustic oscillation work, the ruler is a preferred statistical separation in large-scale structure, not a rigid rod connecting one chosen pair of galaxies.

Before baryon drag ends, the coupled photon–baryon system supports acoustic propagation. The comoving sound-travel scale accumulated by the drag epoch is conventionally denoted r_d. Its value depends on the early expansion and sound-speed history.

With a_today = 1, the formal comoving definition is r_d = ∫ c_s(t)dt/a(t) up to the drag epoch, or equivalently ∫ from z_d toward earlier redshifts of c_s(z)dz/H(z). The integration history and component physics determine the calibrated scale.

Baryon drag and photon last scattering are distinct definitions. The acoustic ruler is also different from a photon horizon or a material object’s diameter. The numerical value supplied next is a teaching choice, not a calculation of the actual sound horizon.

Declare the comoving and physical lengths

Choose a comoving ruler r_d = 120 Mpc at redshift z = 1. Supply transverse comoving distance D_M = 4,000 Mpc and background expansion rate H(z) = 90 km/s/Mpc there. These are hypothetical model inputs, not measurements from a named survey.

At z = 1, the corresponding emitted physical ruler length is r_d/(1 + z) = 60 Mpc. Angular diameter distance is D_A = D_M/(1 + z) = 2,000 Mpc. Their ratio is 60/2,000 = 0.03.

The same small-angle result follows from the comoving quantities: θ = r_d/D_M = 120/4,000 = 0.03 radians, about 1.719 degrees. Use either physical ruler with D_A or comoving ruler with D_M consistently.

Mixing the 120 Mpc comoving ruler with the 2,000 Mpc angular diameter distance would instead give 0.06 radians, a factor-of-two error. The length labels and emission epoch determine which pair belongs in the angular comparison.

A radial interval uses the local expansion rate

For a sufficiently narrow radial redshift interval in the smooth background, the first-order conversion is Δz ≈ H(z)r_d/c. This treats H as approximately constant across the interval and uses radial comoving separation. It is not an exact finite-pair redshift formula for every real galaxy configuration.

Insert c = 299,792.458 km/s. The supplied values give Δz ≈ (90 km/s/Mpc)(120 Mpc)/(299,792.458 km/s) ≈ 0.036025. The Mpc and speed units cancel, leaving dimensionless redshift.

Define D_H = c/H, which is about 3,331.03 Mpc for this chosen rate. The dimensionless distance ratios are D_M/r_d ≈ 33.33 and D_H/r_d ≈ 27.76. Their reciprocal relationships give θ ≈ 1/(D_M/r_d) and Δz ≈ 1/(D_H/r_d) in the stated approximations.

Actual statistical ruler analyses model positions, peculiar velocities, clustering, survey selection and measurement effects. The two orientations here isolate their leading geometric relationships; they do not reproduce a full baryon-acoustic-oscillation estimator.

Ratios alone do not fix an absolute scale

Now keep the supplied θ and Δz fixed but adopt a ruler calibration of 126 Mpc, five percent larger than 120 Mpc. The angular relation gives D_M = 126/0.03 = 4,200 Mpc, also five percent larger.

The radial relation gives H ≈ cΔz/r_d. Increasing r_d by the factor 1.05 therefore decreases H by 1/1.05, from 90 to about 85.714 km/s/Mpc. The new D_H increases by five percent, so both distance-to-ruler ratios remain unchanged.

This is another deterministic calibration sensitivity test, not a measured uncertainty or probability interval. It demonstrates why a measurement of a distance ratio does not independently fix both its numerator and its ruler calibration.

An absolute distance or expansion rate requires additional information about r_d or another absolute scale. Using an early-Universe sound-horizon prediction introduces the physical assumptions of that prediction; they should be named when comparing inferences.

Keep calibration and geometry visible

A reported D_M/r_d or D_H/r_d is valuable distance information even before an absolute ruler is selected. Converting it to Mpc or km/s/Mpc adds calibration information and its uncertainty account.

A joint analysis must also retain shared inputs and covariance. Treating two quantities derived from the same calibrated ruler as independent absolute measurements can overstate the information they add.

The example’s sound-horizon integral has not been evaluated as 120 Mpc. That number is simply supplied to make the angular, radial and calibration calculations transparent. No current acoustic scale, survey constraint or fitted cosmological parameter is claimed.

When reading a ruler result, identify the comoving scale, transverse or radial orientation, redshift interval, approximation, calibration and statistical measurement model. These details connect the geometric ratio to a defensible physical inference.

Worked reasoning

Try the reasoning

1. At z = 1, which ruler length should be paired with D_A = 2,000 Mpc?

   Show the answer and reasoning

   The emitted physical length 60 Mpc.

   The 120 Mpc comoving length instead pairs consistently with D_M = 4,000 Mpc.

2. At unchanged θ, what D_M follows if r_d increases from 120 to 126 Mpc?

   Show the answer and reasoning

   4,200 Mpc.

   D_M = r_d/θ, so the absolute distance increases by the same factor 1.05.

3. Is the acoustic feature a rigid rod between one selected galaxy pair?

   Show the answer and reasoning

   No.

   It is a statistical preferred-separation scale, extracted with a survey and clustering model.

What to carry forward

• The acoustic ruler is a calibrated comoving statistical scale, not a material rod.
• Transverse angles and radial redshift intervals constrain different distance-to-ruler ratios.
• Absolute distances and rates require calibration information beyond those ratios.