Cosmology · Evidence for accelerated expansion

Calibration, dust and population evolution

Follow a supplied magnitude calibration and luminosity residual into a distance bias, distinguishing a common scale shift from a redshift-dependent shape change.

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

Explain why brightness calibration and source luminosity affect inferred distances.
Calculate a distance ratio from a supplied magnitude residual.
Identify observations that can test dust, selection and population differences.

Predict the measured light before interpreting distance

An observed source can appear faint because it is distant, intrinsically faint, attenuated along the path or measured with a shifted calibration. These possibilities enter different parts of a brightness prediction.

A supernova distance analysis uses relations among light-curve shape, color and luminosity rather than assuming every event has identical raw luminosity. Comparing populations across redshift requires checking that the relevant calibration remains applicable.

Dust can change both the amount and wavelength distribution of received light. Its interpretation requires a wavelength-dependent response and an intrinsic source-color model; a single faintness measurement does not separate them.

The following construction supplies an unmodeled residual directly in magnitudes. It does not assign a measured dust law, an actual population-evolution coefficient or a correction that should be applied to real observations.

Define the sign before converting to a distance

Let μ_true = 5log₁₀(D_L,true/10 pc) be the true distance modulus for a positive luminosity distance. In an ideal bolometric example, supply the true standardized absolute magnitude M_true(z) = M_cal + ΔM(z).

A positive ΔM means the source is intrinsically fainter than the assumed calibrated value M_cal. Also supply an additive positive measurement offset b = 0.05 magnitudes, making the reported apparent magnitude larger.

Choose ΔM(z) = 0.2z/(1+z) magnitudes. If the analysis ignores ΔM and b and subtracts M_cal from the reported apparent magnitude, it infers μ_inf = μ_true + ΔM + b.

The inferred-to-true distance ratio is therefore D_L,inf/D_L,true = 10^[(ΔM+b)/5]. All inputs are stipulated and noise-free. The z = 0 row below specifies the formal residual’s intercept, not a ratio of zero distances.

Supplied residual law and its conditional effect on a positive inferred distance
z	ΔM, magnitudes	b+ΔM, magnitudes	D_L,inf/D_L,true
0, formal intercept	0	0.05	1.023293
0.5	0.066667	0.116667	1.055196
1	0.100000	0.150000	1.071519
2	0.133333	0.183333	1.088095

Different residuals affect different combinations

The constant b alone multiplies all inferred positive distances by 10^0.01 ≈ 1.023293. It changes their common normalization without changing their ratios at different redshifts.

The chosen ΔM(z) varies with redshift. Its effect therefore cannot be removed at all redshifts by one constant distance rescaling.

At z = 1, the total supplied residual is 0.15 magnitudes and the distance factor is about 1.071519. Interpreting that residual solely as a geometry change would misattribute the specified luminosity and calibration effects.

This exact construction establishes a possible degeneracy of the ideal brightness equation. It is not evidence that this particular residual explains a real acceleration measurement or survives independent observational checks.

This original ideal bolometric teaching construction specifies a reported apparent-magnitude offset b = +0.05 magnitudes and a true standardized absolute-magnitude residual ΔM(z) = 0.2z/(1+z) magnitudes relative to M_cal. A positive ΔM means intrinsically fainter, and positive b increases reported apparent magnitude. If analysis ignores both, μ_inf−μ_true = b+ΔM and D_L, inf / D_L, true = 10^[(b+ΔM)/5] for every supplied positive true distance. Panel A uses linear redshift z from 0 to 2 and zero-based magnitude-residual ordinate from 0 to 0.2 magnitudes. Dotted green diamonds show the common b = 0.05; solid blue circles show ΔM; dashed orange squares show their sum. At z = 0, 0.5, 1, 2, the intrinsic residuals are 0, 1/15, 0.1, 2/15 magnitudes, approximately 0, 0.066667, 0.100000, 0.133333. Total residuals are 0.05, 7/60, 0.15, 11/60 magnitudes, approximately 0.05, 0.116667, 0.150000, 0.183333. At zero, b and total coincide intentionally. Panel B has the same linear redshift domain but a separate linear positive-distance-factor ordinate from 1 to 1.1; this ordinate deliberately starts at 1, not 0, and shows no filled area or zero-baseline bars. Dotted green diamonds show the common b-only factor 10^0.01 ≈ 1.023293. Dashed orange squares show the total inferred-to-true factors approximately 1.023293, 1.055196, 1.071519, 1.088095 at those four redshifts. The two factors coincide intentionally at the formal intercept. Every z = 0 marker evaluates the residual law and its limiting multiplicative calibration effect on a positive distance; none evaluates a ratio of zero cosmological distances. At z = 1, ΔM = 0.1 and b = 0.05 produce 0.15 magnitudes and factor 10^0.03 ≈ 1.071519. A constant b rescales all positive inferred distances together; this chosen changing residual cannot be removed at all redshifts by one common scale. Magnitudes are logarithmic brightness differences with dimensionless distance ratios, not physical distance units. Distinct circles, squares and diamonds, solid/dashed/dotted lines, direct legends and separate axis labels identify the quantities without color. The supplied residual is not a fitted dust law, measured source evolution, observational uncertainty band or an explanation shown to survive real acceleration tests. Real bandpass, spectra, color, source selection and shared calibration covariance require their own models. Increasing the sample does not turn one common offset into independent per-source noise. Original Ark teaching model.

Use information beyond one brightness residual

Color and spectral measurements can test a proposed attenuation or population model, with intrinsic diversity and calibration treated together. A physical dust model must predict these responses rather than only an arbitrary dimming curve.

Light-curve shape and host-population comparisons can test whether the same standardization applies across selected samples. Such comparisons need compatible observing bands and selection rules.

Redshift shifts a source spectrum relative to an observer’s filters. A band-limited analysis requires the appropriate spectral correction before comparing emitted and observed passbands; the bolometric equation avoids that issue only by assumption.

Survey selection can change the mixture of observable sources. Its forward model and calibration uncertainty should be included in the inference rather than inferred from a residual after a preferred geometry has been selected.

More objects do not automatically remove a common offset

A calibration offset shared by many sources can shift all their inferred distances together. Treating it as an independent random error for each source would misdescribe that dependence.

A joint analysis should therefore include common calibration parameters, their constraints and any correlations with source or population parameters. A larger sample does not by itself measure an unknown common zero point.

Independent distance responses can help test whether a brightness trend belongs to geometry or to a source-specific effect, while retaining shared assumptions and covariance.

The worked law teaches how a specified residual propagates. Real acceleration evidence is assessed by comparing complete competing explanations, including calibration, dust, population and selection tests.

Worked reasoning

Convert 0.15 magnitudes to a distance factor

At z = 1, how does the supplied residual change an inferred positive distance?

The chosen intrinsic residual is 0.2 × 1/(1+1) = 0.1 magnitudes.
Adding b = 0.05 gives μ_inf−μ_true = 0.15 magnitudes.
The inferred distance is multiplied by 10^(0.15/5) ≈ 1.071519.

It is overestimated by a factor about 1.071519 under the supplied assumptions.

Try the reasoning

Does positive ΔM make the supplied source intrinsically brighter or fainter?
Show the answer and reasoning
Fainter.
A larger absolute magnitude corresponds to lower luminosity in the same magnitude convention.
Can a constant distance normalization remove the chosen ΔM(z) at every redshift?
Show the answer and reasoning
No.
The residual varies with redshift, so its distance factor varies as well.
May one common calibration offset be treated as independent noise in every source without changing the inference?
Show the answer and reasoning
No.
A shared offset introduces dependence among the predicted measurements.

What to carry forward

Brightness residuals can come from geometry, source physics or measurement.
A constant magnitude shift rescales distances; a changing residual can alter their inferred redshift dependence.
Independent responses and explicit shared calibration are needed to test competing explanations.

References

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