Cosmology · Clusters and gravitational lensing

Mass calibration and shared systematic uncertainty

Compare two conditional gas-support and lensing-calibration accounts with the same reported mass ratio, then examine a finite shared-error construction.

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

Match mass definitions and geometry before comparing estimates.
Calculate how pressure support and a supplied calibration factor can share one mass ratio.
Explain why a common error does not average down like independent errors.

Two mass labels can refer to different quantities

Cluster masses can be inferred through gas dynamics, gravitational lensing and calibrated observable relations. A comparison is meaningful only when the mass definition, aperture, reference density, geometry and selected objects are matched.

Raw lensing constraints concern projected matter. For the example below, explicitly supply an exact spherical deprojection and the same three-dimensional aperture convention before defining a reported lensing mass. This step is an assumption, not a property of every lensing measurement.

Gas-based estimates can use a pressure gradient and an equilibrium model. Additional support, acceleration or geometry can affect the relation between the inferred thermal mass and the true gravitating mass.

A numerical ratio between two reported estimates therefore does not automatically identify one physical cause. Calibration, support and selection may enter together.

A pressure gradient gives a conditional mass relation

Choose a finite radial interval 1 ≤ q = r/R₀ ≤ 4. Supply an isothermal tracer gas with ρ_g = ρ₀q⁻² and c_T² = k_BT/(μm_p), where μm_p is the stated mean particle mass. Then P_th = ρ_gc_T².

Under steady spherical isotropic support, M_th = −r²(dP_th/dr)/(Gρ_g) = 2c_T²r/G. Define M₀ = 2c_T²R₀/G, giving M_th/M₀ = q.

Now impose a constant nonthermal pressure fraction f, so P_tot = P_th/(1−f). Under these specific local conditions M_true = M_th/(1−f). The tracer normalization may be chosen small enough for a positive dominant total gravitating density.

No rotation, anisotropic stress, acceleration or radial variation of f is included. If f varies with radius, its derivative enters the pressure gradient; the factor 1−f is not a universal mass-bias theorem.

The same reported ratio can hide different true masses

After the supplied matched deprojection, define M_lens,reported = bM_true. The aggregate factor b is a teaching calibration parameter, not a direct conversion of one particular shear or redshift error into mass.

In account A, choose f = 0.20 and b = 1. In account B, choose f = 0.12 and b = 1.10. Both give M_th/M_lens,reported = (1−f)/b = 0.8.

Their true masses differ: account A gives 1.25qM₀, while B gives qM₀/0.88 ≈ 1.136364qM₀. Yet both reported lensing masses equal 1.25qM₀ over the supplied interval.

One matching ratio therefore does not choose between the two accounts. The numbers are supplied alternatives, not observed support fractions, measured lensing biases or fitted cluster masses.

Two chosen conditional accounts with the same matched reported lensing mass; all masses are divided by M₀
q	Thermal mass	True mass A	Reported lensing A	True mass B	Reported lensing B
1	1	1.25	1.25	1.136364	1.25
2	2	2.5	2.5	2.272727	2.5
4	4	5	5	4.545455	5

Repeated estimates can retain a common uncertainty

Consider a separate additive-error construction in an arbitrary mass unit U, unrelated to the multiplicative accounts above. Let a common error c be −0.1U or +0.1U, and each individual error e₁ and e₂ be −0.05U or +0.05U.

Choose the three binary signs independently with equal probability. Their eight joint outcomes are exactly equiprobable. Each reported error is c+e_i, with variance 0.0125U², while their covariance is 0.01U² from the shared c.

The mean of the two reported errors is c+(e₁+e₂)/2. Its variance is 0.01U²+(0.0025U²+0.0025U²)/4 = 0.01125U², giving standard deviation about 0.106066U.

Incorrectly treating the two full errors as independent would give variance 0.00625U² and standard deviation about 0.0790569U. The common contribution survives averaging. This finite supplied distribution is not a survey covariance estimate, Gaussian likelihood or automatic confidence interval.

Before comparing mass labels, supply an exact spherical deprojection and common three-dimensional aperture convention for lensing. On the finite physical radial interval 1 ≤ q = r/R₀ ≤ 4, impose isothermal tracer gas density ρ_g = ρ₀q⁻², c_T² = k_BT/(μm_p), and thermal pressure P_th = ρ_gc_T². Steady spherical isotropic support gives M_th = −r²(dP_th/dr)/(Gρ_g) = 2c_T²r/G. With mass unit M₀ = 2c_T²R₀/G, thermal mass is qM₀. A constant nonthermal pressure fraction f supplies P_tot = P_th/(1−f), giving M_true = M_th/(1−f) only under these local assumptions. Rotation, acceleration, anisotropic stress and a radial derivative of f are excluded; if f varies its derivative changes the pressure relation. Tracer normalization can be small enough for positive dominant total gravitating density over the finite interval; no infinite global atmosphere is specified. Account A has f = 0.20 and aggregate lensing calibration b = one; account B has f = 0.12 and b = 1.10, with reported lensing mass bM_true after the supplied deprojection. Panel A uses linear radius q from one to four and mass/M₀ from zero to five. Solid blue circles show M_th/M₀ = q. Dashed orange squares show true B = q/0.88. A single dotted green curve with diamonds honestly represents all coincident values true A = reported A = reported B = 1.25q, without artificial offsets. At q = one, two and four, thermal masses are 1, 2, 4; true B masses 1.136364, 2.272727, 4.545455; all shared green masses 1.25, 2.5, 5. Both thermal-to-reported ratios are 0.8, although their true masses differ. These are supplied alternatives, not observed support fractions or fitted biases, and b does not identify one particular shear or redshift error. Panel B is a separate additive-error distribution in arbitrary mass unit U, unrelated to M₀. Let common c = ±0.1U and individual e₁, e₂ = ±0.05U with three independent equally probable binary signs. All eight joint states are equiprobable. The mean of reported errors is c+(e₁+e₂)/2. Six discrete stems, not a continuous probability density or uncertainty bars on panel A, show its value divided by U: −0.15, −0.1, −0.05, +0.05, +0.1, +0.15 with probability masses 1/8, 1/4, 1/8, 1/8, 1/4, 1/8. There is no probability mass at zero. A single reported error has variance 0.0125U² and the covariance between the two is 0.01U². Their mean variance is 0.01125U² and standard deviation about 0.106066U. Incorrect independence would give variance 0.00625U² and standard deviation about 0.0790569U. The shared c remains with coefficient one. No Gaussian likelihood, confidence interval, observed survey covariance or time evolution is inferred. Original Ark teaching model.

Agreement needs a physical and statistical explanation

Independent information about gas support, geometry, source calibration and population selection can help distinguish compatible mass accounts. Its value depends on whether it constrains different combinations of the same uncertain inputs.

More objects can reduce some sampling fluctuations while leaving a common calibration uncertainty. A joint analysis must carry correlations instead of assuming every estimate or every cluster supplies independent calibration information.

Mass proxies also interact with the underlying population slope and detection rule. A calibration inferred from a selected follow-up sample may not transfer unchanged to the full catalogue without modelling that selection.

Report the matched mass convention, deprojection, pressure-support model, calibration relation and covariance together. A mass-ratio disagreement then becomes a specific conditional question, while an apparent agreement does not silently certify every underlying assumption.

Worked reasoning

Check the two supplied accounts

How do accounts A and B both produce a thermal-to-reported-lensing ratio of 0.8?

Account A gives (1−0.20)/1 = 0.8.
Account B gives (1−0.12)/1.10 = 0.88/1.10 = 0.8.
Their true masses differ because M_true = M_th/(1−f), while their reported lensing factors compensate in the ratio.

The same reported ratio is compatible with different support and calibration inputs under the stated finite model.

Try the reasoning

Does M_th/M_true = 1−f remain a general identity if f varies with radius?
Show the answer and reasoning
No.
The pressure-gradient calculation then includes the radial derivative of f and may require other support terms.
May raw projected lensing mass be inserted directly into the same spherical three-dimensional mass ratio without a geometry model?
Show the answer and reasoning
No.
The example explicitly supplies matched deprojection and aperture conventions before defining its lensing mass.
In the separate additive model, does averaging two estimates remove the common error c?
Show the answer and reasoning
No.
Their mean still contains c with full coefficient one, giving the nonzero shared variance contribution.

What to carry forward

Mass comparisons require matched definitions, geometry and calibration.
A single reported ratio can mix support assumptions with measurement calibration.
Shared uncertainty survives averaging and belongs in a joint population and mass inference.

References

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