Cosmology · Complementary probes of cosmic history

Baryon acoustic oscillations as calibrated rulers

Compare radial and transverse template responses at one supplied epoch and trace how a common ruler calibration changes their absolute scale.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Describe a statistical acoustic feature rather than a literal individual rod.
Calculate distances and an expansion rate from stipulated dimensionless responses.
Identify which combinations retain a common calibration dependence.

A ruler is inferred from a pattern of clustering

Sound propagation in the early coupled photon–baryon medium can leave a characteristic comoving scale in later statistical clustering. A BAO analysis models this pattern rather than identifying one rigid object of known physical length.

Galaxy positions and redshifts enter a selected tracer catalogue. Converting its statistical feature into geometric information requires a template, a treatment of tracer and observational effects, and an assumed physical ruler calibration.

Transverse and radial responses involve different geometric combinations. Comparing them can be useful even when both inherit the same uncertain absolute ruler length.

Use a stipulated noise-free template comparison at z_* = 0.6. No measured correlation function, sound-horizon integral, galaxy pair catalogue or fitted covariance is supplied.

Separate the geometry from the calibrated length

Let r_d be the chosen comoving ruler scale and D_M(z_*) the transverse comoving distance. Define X = D_M/r_d and Y = c/[H(z_*)r_d]. Supply X = 30 and Y = 20.

These numbers are dimensionless template-response summaries. They are not exact finite-angle or finite-redshift measurements of a single rod.

Choose an arbitrary positive length L in metres and compare three calibrations r_d/L = 0.04, 0.044 and 0.05. The response summaries X and Y are held fixed across the three alternatives.

Define D_parallel = c/H(z_*), a radial distance scale at the selected epoch. It is distinct from the earlier reference notation c/H₀. Then D_M/L = Xr_d/L, D_parallel/L = Yr_d/L and HL/c = 1/[Y(r_d/L)].

One supplied epoch and fixed X = 30, Y = 20; alternatives differ only in the chosen ruler calibration
r_d/L	D_M/L	D_parallel/L	H(z_*)L/c	D_MH/c	D_V/r_d
0.04	1.2	0.8	1.25	1.5	22.104189
0.044	1.32	0.88	1.136364	1.5	22.104189
0.05	1.5	1	1	1.5	22.104189

A shared scale changes several reported quantities together

Increasing the adopted r_d by ten percent changes the first calibration to the second. D_M and D_parallel each increase by ten percent, while H is divided by 1.1.

The changes are linked consequences of one calibration choice. They do not represent three independent new pieces of geometric evidence.

The combination F_AP = D_MH/c = X/Y remains 3/2 because the ruler cancels. This cancellation belongs to the stipulated response definitions and does not eliminate every modeling or observational dependency.

A ruler calibration derived using CMB or other early-Universe information can share their physical assumptions. Combining the resulting absolute-scale measurements requires keeping that shared origin visible.

This original noise-free comparison stipulates one epoch z_* = 0.6 and fixed statistical-template responses X = D_M/r_d = 30 and Y = c/[H(z_*)r_d] = 20. L is an arbitrary positive length in metres. Neither response is an exact finite rod angle or finite redshift interval. Both panels have a linear adopted comoving-ruler axis r_d/L from 0.04 to 0.05, with markers at 0.04, 0.044 and 0.05. Panel A has a zero-based linear distance ordinate from 0 to 1.6 in units L. Solid blue circles show D_M/L = 30r_d/L, with values 1.2, 1.32, 1.5. Dashed orange squares show D_parallel/L = (c/H)/L = 20r_d/L, with values 0.8, 0.88, 1.0. D_parallel is the defined scale c/H at the chosen epoch, not the present c/H₀ or a measured finite radial separation. Panel B has a separate zero-based dimensionless ordinate H(z_*)L/c from 0 to 1.4. Dotted violet diamonds show 1/[20(r_d/L)], giving 1.25, approximately 1.136364, 1. The first to second marker is a ten-percent increase in the one shared calibration: each distance increases by ten percent, while H is divided by 1.1. These linked changes are alternatives, not three independent observations, fitted uncertainties or error bands. The text records the invariant F_AP = D_MH/c = X/Y = 1.5 and the derived isotropized summary D_V/r_d = [z_*X²Y]^(1/3) = 10800^(1/3) ≈ 22.104189, equal for all three calibrations. That summary is computed from the same X,Y and adds no independent datum. All axis quantities are dimensionless ratios of lengths or rate times length divided by speed. A real BAO feature is statistical clustering with a physical early-time calibration, selection, tracer and template assumptions; CMB-derived calibration can share physical inputs with it. No sound-horizon production integral, measured ruler, detection significance, curvature inference or full expansion history is supplied. Distinct circles, squares, diamonds and solid, dashed, dotted curves identify the quantities without color. Original Ark teaching model.

Optional mathematical path: distinguish a derived combination

Define the isotropized geometric combination D_V = [z_*D_M²c/H(z_*)]¹ᐟ³. Under the supplied ratios, D_V/r_d = [z_*X²Y]¹ᐟ³.

Substitution gives [0.6×30²×20]¹ᐟ³ = 10800¹ᐟ³ ≈ 22.104189. Its equality across the rows follows algebraically from the fixed X,Y and selected epoch.

This D_V value is a derived summary of those same inputs. It is not a third independent measurement to multiply into an inference as if it added new data.

The logarithmic calibration responses are +1 for D_M and D_parallel and −1 for H, with zero response for F_AP. These derivatives describe the specified common rescaling, rather than a complete multi-parameter cosmological fit.

A calibrated geometric response remains conditional

Real analyses must specify observational selection, redshift-space effects, tracer response and the statistical template. The feature’s amplitude and its geometric dilation are related to different assumptions.

A physical prediction for r_d requires an early-time model and its parameters. The present exercise instead supplies the calibration alternatives so their geometric consequences can be followed directly.

The comparison is at one chosen epoch; it does not reconstruct a full expansion history or infer spatial curvature. No real ruler length, detection significance or current parameter estimate follows from the displayed arithmetic.

BAO information is useful when its measured response, physical calibration and shared assumptions remain distinct. Then one can compare radial and transverse geometry without treating a derived summary as additional independent evidence.

Worked reasoning

Increase the adopted ruler from 0.04L to 0.044L

What happens when X and Y remain fixed?

D_M increases from 1.2L to 1.32L, and D_parallel from 0.8L to 0.88L.
HL/c decreases from 1.25 to 1.25/1.1 ≈ 1.136364.
D_MH/c stays 1.5 because the common ruler cancels.

The two distances increase together while H decreases reciprocally; this is one shared calibration change.

Try the reasoning

Does the model contain one observed rigid rod whose exact finite angular size is measured?
Show the answer and reasoning
No.
X and Y are stipulated dimensionless summaries of a statistical-template response.
Does calculating D_V from the same X and Y create independent data?
Show the answer and reasoning
No.
It is a deterministic combination of those inputs and the chosen epoch.
Is Y alone sufficient to specify H in inverse seconds without a ruler calibration?
Show the answer and reasoning
No.
H = c/(Yr_d) still needs the physical length r_d.

What to carry forward

BAO uses a calibrated statistical feature with distinct radial and transverse responses.
A common ruler scale shifts derived distances and H together.
Calibration-free combinations and derived summaries do not remove all shared physical assumptions.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.