Lecture 05 Notes: Baryon Acoustic Oscillations as a Standard Ruler

Preparation: algebra, introductory mechanics, and the previous module's stated prerequisites. This is a 60-minute instructional module; use the slide timings as a guide and pause at the active check.

Terms Developed in Context

These terms organize the chain from recorded signal to calibrated observable, likelihood, and bounded cosmological claim.

Baryon acoustic oscillation

The late-time clustering imprint of sound waves in the pre-recombination photon-baryon plasma. In this module it is used to answer How do galaxy clustering measurements encode distance?

Observable/model connection: The term links Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data to Sound horizon and the quantitative relation \(D_V(z)=[D_M(z)^2 c z/H(z)]^(1/3)\). State which part is measured, calibrated, derived, or model-dependent.

Limit: Do not compare D_V in Mpc directly to D_V r_d,fid/r_d unless explicitly setting r_d=r_d,fid; BAO does not measure an absolute ruler without early-universe calibration. That boundary must remain attached when the term is used in a cosmological claim.

Sound horizon

The comoving distance an acoustic wave traveled before baryon drag ended. In this module it is used to answer How do galaxy clustering measurements encode distance?

Observable/model connection: The term links Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data to Standard ruler and the quantitative relation \(D_V(z)=[D_M(z)^2 c z/H(z)]^(1/3)\). State which part is measured, calibrated, derived, or model-dependent.

Standard ruler

A feature with calibrated physical length whose observed angular or redshift extent constrains geometry. In this module it is used to answer How do galaxy clustering measurements encode distance?

Observable/model connection: The term links Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data to Volume-averaged distance and the quantitative relation \(D_V(z)=[D_M(z)^2 c z/H(z)]^(1/3)\). State which part is measured, calibrated, derived, or model-dependent.

Volume-averaged distance

The isotropic BAO combination D_V that mixes transverse area and radial depth. In this module it is used to answer How do galaxy clustering measurements encode distance?

Observable/model connection: The term links Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data to Anisotropic BAO and the quantitative relation \(D_V(z)=[D_M(z)^2 c z/H(z)]^(1/3)\). State which part is measured, calibrated, derived, or model-dependent.

Anisotropic BAO

Separate use of transverse and radial clustering to constrain D_M/r_d and H r_d. In this module it is used to answer How do galaxy clustering measurements encode distance?

Observable/model connection: The term links Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data to Baryon acoustic oscillation and the quantitative relation \(D_V(z)=[D_M(z)^2 c z/H(z)]^(1/3)\). State which part is measured, calibrated, derived, or model-dependent.

Physical and observational argument

Before baryon drag, pressure in the photon-baryon fluid launches acoustic waves. The maximum propagation distance becomes a preferred comoving separation in later matter clustering. An isotropic analysis averages radial and transverse dilation into D_V, while anisotropic clustering preserves D_M/r_d and H(z)r_d. The ruler's physical length is set by early-universe densities; a late-time distance in Mpc is therefore conditional on its calibration. When using published BOSS summaries, carry the r_d,fid/r_d factor explicitly and do not compare it as an unscaled absolute distance by accident.

The pre-recombination sound horizon imprints a preferred clustering separation. Isotropic analyses compress transverse and radial information into D_V/r_d; anisotropic analyses constrain D_M/r_d and H(z)r_d separately.

Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data.

Derivation and interpretation

\[D_V(z)=[D_M(z)^2 c z/H(z)]^(1/3)\]

Begin by naming observables, coordinate convention, and model assumptions; each algebraic transformation must preserve units and the distinction between measured and inferred quantities. Test a limiting case, then perturb one input and state which observable changes.

Do not compare D_V in Mpc directly to D_V r_d,fid/r_d unless explicitly setting r_d=r_d,fid; BAO does not measure an absolute ruler without early-universe calibration.

Worked numerical anchor

The flat-LCDM model gives D_V(0.38)=1477.922 Mpc; comparison requires the explicitly stated r_d=r_d,fid convention.

Reproduce the value with the course computation module, retain full precision in code, and round only for communication. Report units and all conditional assumptions with the result.

Evidence, systematics, and limits

Do not compare D_V in Mpc directly to D_V r_d,fid/r_d unless explicitly setting r_d=r_d,fid; BAO does not measure an absolute ruler without early-universe calibration.

Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data.

Separate the directly recorded observable, its calibration, the derived intermediate quantity, and the model-dependent parameter. An uncertainty bar is not a complete likelihood when shared covariance or selection is omitted.

Study and retrieval practice

Derive D_V from transverse and radial distance information.
Distinguish D_V/r_d from D_M/r_d and H r_d.
State the sound-horizon convention when comparing a model to a published BAO point.

Checkpoint: Explain the degeneracy between late-time distances and the calibrated physical ruler.

Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.

Three published BOSS DR12 consensus points at z=0.38, 0.51, and 0.61 are included as literature-summary measurements, not raw clustering data.

Lecture 05 notes · Sources and caveats in reference-log.md.