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.
- Acoustic scale
The angle subtended by the sound horizon at last scattering. In this module it is used to answer
Why is the microwave background a distance probe?
Observable/model connection: The term links
The official Planck Legacy Archive binned TT/TE products were requested during production but could not be retrieved (TT extraction failure; TE HTTP 401). Only the published parameter summary is used; the plotted peak sequence is explicitly theoretical
to Sound horizon at recombination and the quantitative relation \(theta_* = r_s(z_*)/D_M(z_*)\). State which part is measured, calibrated, derived, or model-dependent.Limit: The plotted schematic is not Planck data, and theta_* is not simply the angular size of a rigid object with no recombination physics. That boundary must remain attached when the term is used in a cosmological claim.
- Sound horizon at recombination
The comoving acoustic propagation distance evaluated near photon decoupling. In this module it is used to answer
Why is the microwave background a distance probe?
Observable/model connection: The term links
The official Planck Legacy Archive binned TT/TE products were requested during production but could not be retrieved (TT extraction failure; TE HTTP 401). Only the published parameter summary is used; the plotted peak sequence is explicitly theoretical
to Last scattering and the quantitative relation \(theta_* = r_s(z_*)/D_M(z_*)\). State which part is measured, calibrated, derived, or model-dependent.Limit: The plotted schematic is not Planck data, and theta_* is not simply the angular size of a rigid object with no recombination physics. That boundary must remain attached when the term is used in a cosmological claim.
- Last scattering
The epoch when the photon mean free path became large enough for the CMB to free-stream. In this module it is used to answer
Why is the microwave background a distance probe?
Observable/model connection: The term links
The official Planck Legacy Archive binned TT/TE products were requested during production but could not be retrieved (TT extraction failure; TE HTTP 401). Only the published parameter summary is used; the plotted peak sequence is explicitly theoretical
to Multipole and the quantitative relation \(theta_* = r_s(z_*)/D_M(z_*)\). State which part is measured, calibrated, derived, or model-dependent.Limit: The plotted schematic is not Planck data, and theta_* is not simply the angular size of a rigid object with no recombination physics. That boundary must remain attached when the term is used in a cosmological claim.
- Multipole
An angular Fourier index whose scale is approximately inverse to angle on the sky. In this module it is used to answer
Why is the microwave background a distance probe?
Observable/model connection: The term links
The official Planck Legacy Archive binned TT/TE products were requested during production but could not be retrieved (TT extraction failure; TE HTTP 401). Only the published parameter summary is used; the plotted peak sequence is explicitly theoretical
to Recombination model and the quantitative relation \(theta_* = r_s(z_*)/D_M(z_*)\). State which part is measured, calibrated, derived, or model-dependent.Limit: The plotted schematic is not Planck data, and theta_* is not simply the angular size of a rigid object with no recombination physics. That boundary must remain attached when the term is used in a cosmological claim.
- Recombination model
The ionization and thermal history needed to predict visibility and acoustic features. In this module it is used to answer
Why is the microwave background a distance probe?
Observable/model connection: The term links
The official Planck Legacy Archive binned TT/TE products were requested during production but could not be retrieved (TT extraction failure; TE HTTP 401). Only the published parameter summary is used; the plotted peak sequence is explicitly theoretical
to Acoustic scale and the quantitative relation \(theta_* = r_s(z_*)/D_M(z_*)\). State which part is measured, calibrated, derived, or model-dependent.Limit: The plotted schematic is not Planck data, and theta_* is not simply the angular size of a rigid object with no recombination physics. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
The pre-recombination photon-baryon fluid oscillates under gravity and pressure. The angular peak spacing measures a ratio of the sound horizon at last scattering to the comoving angular-diameter distance. The precision of that ratio does not mean every ingredient is late-time or model-independent: baryon loading, radiation content, recombination, curvature and expansion history enter the conversion. A bandpower spectrum has a full covariance and foreground model; a reported parameter summary is not interchangeable with the raw spectra. This package uses the former only as a cited literature result and labels its curve theoretical.
Acoustic peak spacing compares the sound horizon at last scattering to its angular-diameter distance. The observed angular scale is precise, but converting it to late-time parameters depends on the pre-recombination model and correlated spectra.
The official Planck Legacy Archive binned TT/TE products were requested during production but could not be retrieved (TT extraction failure; TE HTTP 401). Only the published parameter summary is used; the plotted peak sequence is explicitly theoretical.
Derivation and interpretation
\[theta_* = r_s(z_*)/D_M(z_*)\]
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.
The plotted schematic is not Planck data, and theta_* is not simply the angular size of a rigid object with no recombination physics.
Worked numerical anchor
Planck reports 100 theta-star=1.0411 +/- 0.0003; conversion gives theta-star=0.596506 degrees.
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
The plotted schematic is not Planck data, and theta_* is not simply the angular size of a rigid object with no recombination physics.
The official Planck Legacy Archive binned TT/TE products were requested during production but could not be retrieved (TT extraction failure; TE HTTP 401). Only the published parameter summary is used; the plotted peak sequence is explicitly theoretical.
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
- Explain acoustic oscillations and the sound horizon at last scattering.
- Convert 100 theta_* to an angular scale with correct units.
- Identify the assumptions needed to infer H0 or curvature from the CMB.
Checkpoint: Identify it as a schematic oscillator with no observational bandpowers or likelihood attached.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.