Cosmology · Complementary probes of cosmic history

Galaxy clustering and cosmic shear

Compare selected tracer amplitudes with two ideal source-weighted mass projections, retaining the weak-lensing and geometry assumptions.

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

Distinguish a selected galaxy pattern from a projected mass response.
Calculate two source-plane shear amplitudes for supplied foreground modes.
Identify what an ideal two-plane inversion does and does not determine.

Supply the distance weights and tracer response

Take foreground comoving distances χ₁ = L and χ₂ = 2L, with angular-diameter distances D₁ = L/2 and D₂ = 2L/3. Two source planes lie at χ_A = 3L and χ_B = 4L in a stipulated flat geometry.

Define each projected mass-fluctuation amplitude as x_i = (4πG/c²)D_iδΣ_i. The surface-density amplitude δΣ_i has units kg/m²; x_i is dimensionless and describes a signed nonzero mode about an already-accounted background.

The flat geometry gives W_si = D_is/D_s = (χ_s−χ_i)/χ_s. The two rows are W_A = (2/3,1/3) and W_B = (3/4,1/2). Angular-diameter distances themselves are not obtained by subtracting observer-to-plane angular-diameter distances.

Supply projected tracer amplitudes d_gi = b_ix_i, with effective dimensionless projected gains b_i. These chosen gains are not a universal three-dimensional galaxy-bias law.

Equal tracer amplitudes need not imply equal projected mass

Case A has x₁ = 0.01, x₂ = 0.02 and gains b₁ = 2, b₂ = 1.5. Case B exchanges the mass amplitudes, with x₁ = 0.02, x₂ = 0.01 and gains b₁ = 1, b₂ = 3.

Both produce d_g1 = 0.02 and d_g2 = 0.03. If the gains are not independently fixed, those two tracer amplitudes alone do not select between the cases.

At leading weak/Born order, each source-plane convergence amplitude is κ_s = Σ_iW_six_i. For the specified nonzero angular mode along θ₁, the shear amplitudes obey γ₁ = κ_s and γ₂ = 0.

The two cases therefore give different ideal source-weighted shear amplitudes. Their values follow the supplied geometry and weak-mode model, rather than observed image shapes.

One stipulated nonzero projected mode; shear columns give leading γ₁ amplitudes, with γ₂ = 0
Case	x₁	x₂	b₁	b₂	d_g1	d_g2	γ₁,A	γ₁,B
A	0.01	0.02	2	1.5	0.02	0.03	0.013333	0.0175
B	0.02	0.01	1	3	0.02	0.03	0.016667	0.02

This original two-foreground-plane comparison concerns one specified nonzero angular Fourier mode along θ₁ in a chosen flat geometry. Foregrounds have comoving distances χ₁ = L and χ₂ = 2L, and angular-diameter distances D₁ = L/2 and D₂ = 2L/3. Sources A and B have χ_A = 3L, χ_B = 4L. L is an arbitrary positive length. The signed projected fluctuation amplitudes x_i = (4πG/c²)D_iδΣ_i are dimensionless, with δΣ_i in kg per square metre about an already-accounted background; they are not three-dimensional contrasts. Solid blue circles denote case A, with (x₁,x₂,b₁,b₂) = (0.01,0.02,2,1.5). Dashed orange larger hollow squares denote case B, with (0.02,0.01,1,3). The native table displays those ten cells. Panel A has discrete foreground-plane categories 1 and 2, and a zero-based linear dimensionless d_g ordinate from 0 to 0.035. Both cases have d_gi = b_ix_i = (0.02,0.03); circles and squares intentionally coincide and the square corners remain visible. Panel B uses two discrete source categories A and B, and a zero-based linear dimensionless leading shear γ₁ ordinate from 0 to 0.025. The known weights W_si = D_is/D_s = (χ_s−χ_i)/χ_s are W_A = (2/3,1/3), W_B = (3/4,1/2), not differences of observer angular-diameter distances. At leading weak/Born order κ_s = Σ_iW_six_i, and this specific nonzero mode gives γ₁ = κ_s, γ₂ = 0. Case A has (γ₁,A,γ₁,B) = (1/75,0.0175), approximately (0.013333,0.017500); case B has (1/60,0.02), approximately (0.016667,0.020000). Connecting segments in both panels are visual guides between the displayed discrete categories, not intermediate measured planes or a full spatial field. Equal tracer amplitudes do not fix mass-mode amplitudes when the gains are unknown. The weights have determinant 1/12; an ideal inverse could recover just the two selected modes, not a uniform sheet or complete mass distribution. Real shapes respond to reduced shear γ/(1−κ), and using γ is only the explicitly stated leading weak approximation. The projected gains are not a universal three-dimensional galaxy-bias relation. No full sky, mass map, deprojection, measured shape calibration, independent errors or real precision claim is supplied. Shapes, dash styles, discrete labels and the native parameter table carry the comparison without color. Original Ark teaching model.

Optional mathematical path: recover two mode amplitudes

The distance-weight matrix has determinant 1/12, so the ideal noise-free system is invertible when these weights and source assignments are known.

The inverse is x₁ = 6κ_A−4κ_B and x₂ = −9κ_A+8κ_B. In case A this gives x₁ = 6(1/75)−4(0.0175) = 0.01 and x₂ = −9(1/75)+8(0.0175) = 0.02.

For the nonzero cosine mode, a lens potential ψ = −2κ_amp cos(kθ₁)/k² gives κ = γ₁ = κ_amp cos(kθ₁) and γ₂ = 0 by differentiation. This verifies the chosen mode relation; it does not measure a uniform mass sheet.

The ideal inverse recovers two specified projected fluctuation amplitudes. It does not determine a full finite-field mass distribution, a mean sheet, deprojected halo masses or three-dimensional density.

Weak shear is an approximation to a shape response

Image shapes respond to reduced shear g_red = γ/(1−κ), not exactly to γ. Replacing it by shear is the leading weak approximation used here, and must not be promoted to an exact image-response equation.

Actual inference also needs intrinsic shape statistics and alignments, instrumental image response, source redshifts and shear calibration. The toy omits those uncertainties, nonlinear corrections and lens–lens effects.

Known projection weights are calibration assumptions. Shared foreground modes, a common shape calibration or source-distance uncertainty can correlate nominally different source samples.

The comparison shows why a projected mass response can complement a tracer response under stated conditions. Assessing real complementarity requires retaining the geometry, calibration and shared-data dependencies rather than assuming that one probe supplies unconditional mass.

Worked reasoning

Invert the ideal case A projections

Do the supplied source weights recover the chosen x₁ and x₂?

Case A gives κ_A = (2/3)(0.01)+(1/3)(0.02) = 1/75 and κ_B = 0.0175.
Use x₁ = 6κ_A−4κ_B = 0.08−0.07 = 0.01.
Use x₂ = −9κ_A+8κ_B = −0.12+0.14 = 0.02.

Yes: the ideal two-projection inverse recovers the two supplied projected mode amplitudes.

Try the reasoning

Do equal d_g1 and d_g2 force equal x₁ and x₂ when the gains are unknown?
Show the answer and reasoning
No.
The two supplied mass/gain cases have identical tracer amplitudes but different mass amplitudes.
Is the exact lensing shape response always γ rather than reduced shear?
Show the answer and reasoning
No.
The reduced-shear response is γ/(1−κ); using γ alone is the stated leading weak approximation.
Does the ideal inverse determine an entire three-dimensional mass distribution?
Show the answer and reasoning
No.
It solves only the two chosen nonzero projected-mode amplitudes with known geometry.

What to carry forward

Selected galaxy patterns and projected lensing distortions have different response assumptions.
Different source distances can add distinct projections of shared foreground structure.
A conditional weak-mode inverse is not a complete mass reconstruction or calibration-free shape measurement.

References

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