Cosmology · Testing gravity and alternative cosmological models

Different scales, screening and tested regimes

Distinguish an environment-dependent scalar range from the reduced charge of a screened source, while checking the assumptions of a leading thin-shell calculation.

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

Derive leading environmental field, mass and range scalings.
Keep source screening, coupling and test-body response distinct.
Check several approximation conditions before using a force formula.

Different mechanisms suppress additional responses differently

A proposed extra field can have a weak response in one environment and a larger response in another. The mechanism must explain that dependence while remaining consistent with the regimes where it is used.

An environment-dependent mass, an environment-dependent coupling and nonlinear derivative effects are distinct screening mechanisms. One mechanism’s formula cannot be substituted for another without its assumptions.

Use an inverse-power chameleon example to separate two effects: a changing scalar range in a surrounding medium and a reduced exterior charge of a finite dense source.

This model is different from the quadratic massive scalar of the preceding example. All numbers here are hypothetical dimensionless inputs; no laboratory object, cosmological density or present experimental bound is assigned to them.

The density convention belongs to the field equation

Use c = ħ = 1 in the field equations, positive φ and Λ, potential V = Λ⁵/φ and conformal coupling A = exp(βφ/M_P), with β>0 and β² = 1/6. Define M_P = (8πG_*)^(−1/2), using the bare tensor coupling G_* in these natural units.

Let ρ be the specified conserved nonrelativistic Einstein-frame rest-energy density entering V_eff = Λ⁵/φ+ρexp(βφ/M_P). It is not an unspecified Jordan-frame density or an SI mass density inserted without conversion.

For βφ/M_P≪1, the leading minimum satisfies φ_min ≈ [Λ⁵M_P/(βρ)]^(1/2), and the leading curvature gives m_eff² ≈ 2Λ⁵/φ_min³.

Here φ,Λ,M_P and m_eff have energy units; ρ and V have energy⁴ units, while a natural-unit radius has inverse-energy units. In SI, the corresponding physical range is ħc/m_energy, not a numerical inverse mass expressed directly in metres.

A denser medium lowers the minimum and shortens the range

Take a reference environment with ρ_* = ρ_env and its leading values φ_*,m_* and range λ_*. For u = ρ/ρ_*, the scalings are φ_min/φ_* = u^(−1/2), m/m_* = u^(3/4) and λ/λ_* = u^(−3/4).

Increasing the density ratio to 16 lowers the leading field value to 1/4, increases mass to 8 and shortens the range to 1/8 of their reference values. At ratio 256, the mass is 64 times larger and the range 1/64.

These are leading small-coupling scalings. They do not solve the full exponential minimum or the spatial boundary-value problem around a finite body.

The environment comparison and the source comparison below are separate exercises. For the source examples keep the reference surrounding environment fixed rather than silently inserting a different row of this table.

Leading n = 1 environmental ratios under the stated small-coupling approximation
ρ/ρ_*	φ_min/φ_*	m/m_*	λ/λ_*
1	1	1	1
16	1/4	8	1/8
256	1/16	64	1/64

A thin source shell reduces charge without setting the coupling to zero

In the fixed reference environment, supply βφ_env/M_P = 10⁻⁸ and uniform source density ρ_c/ρ_env = 10⁸. The leading relations give φ_c/φ_env = 10⁻⁴ and m_core/m_env = 10⁶.

Compare three source radii at that same source density, with compactness Φ_N = G_*M/R equal to 10⁻⁶,4×10⁻⁶ and 1.6×10⁻⁵ in natural units. For a uniform sphere, Φ_N = ρ_cR²/(6M_P²).

For a thin source shell, ε = ΔR/R ≈ (φ_env−φ_c)/(6βM_PΦ_N). An unscreened small test mass in the long-range exterior limit has leading F_scalar/F_Newton ≈ 2β²(3ε), equal to ε for this β.

The supplied choices also give (m_envR)² = 2Φ_N. At test radius r = 2R, the table checks ε≪1, m_envr≪1 and m_coreR≫1 together; the universal β has not become zero.

Separate source comparison at fixed reference environment and an unscreened test body
Φ_N	ε ≈ F_scalar/F_Newton	m_env(2R)	m_coreR
10⁻⁶	0.009999	0.002828	1414.214
4×10⁻⁶	0.00249975	0.005657	2828.427
1.6×10⁻⁵	0.0006249375	0.011314	5656.854

Two separate supplied chameleon comparisons use a positive inverse-power potential Lambda^5/phi and beta² = 1/6 in natural units. The upper logarithmic plot changes environmental density u = rho/rho*: mass ratio m/m* = u^(3/4) increases while range lambda/lambda* = u^(−3/4) decreases; the table also retains phi_min/phi* = u^(−1/2). The lower plot holds the reference environment fixed and changes three uniform-source radii at fixed density. Its three thin-shell extra-force ratios are epsilon = 10^(−8)(1−10^(−4))/Phi_N, for an unscreened test body. The source table checks small epsilon, small ambient m_env(2R) and large core m_core R together. Connecting source segments guide the eye through three supplied tests; they are not full boundary-value solutions. The coupling does not become zero. No interpolation to epsilon of order one, present bound, real object or total force-error guarantee is supplied. Leading environmental scalings; not a full exponential minimum. Columns: u; φ_min/φ*; m/m*; λ/λ*. Rows: 1; 1; 1; 1 / 16; 1/4; 8; 1/8 / 256; 1/16; 64; 1/64. Separate fixed-environment source tests; all entries dimensionless. Columns: Φ_N; ε (extra force); m_env(2R); m_core R. Rows: 10⁻⁶; 0.009999; 0.002828; 1414.214 / 4×10⁻⁶; 0.00249975; 0.005657; 2828.427 / 1.6×10⁻⁵; 0.0006249375; 0.011314; 5656.854.

A limiting formula is different from a complete force model

For the middle source, ε = 10⁻⁸(1−10⁻⁴)/(4×10⁻⁶) = 0.00249975. Its reduced scalar charge follows from the source profile approximation, while the test mass is assumed unscreened.

A screened test body or an appreciable ambient Yukawa suppression would need additional response factors. The quoted source formula does not calculate those cases or a smooth transition between screened and unscreened limits.

In particular, inserting ε = 2 into a thin-shell expression is invalid. The separate leading unscreened-source and unscreened-test limit has force ratio 2β² = 1/3, but those two limiting statements do not define an interpolation.

Checking the dimensionless conditions is useful without claiming a total force-error bound. A full theory comparison still needs the spatial field solution, appropriate coupling convention, stable applicable dynamics and tests in other regimes.

Worked reasoning

Check the middle source and its test radius

What leading extra force follows for Φ_N = 4×10⁻⁶?

Use φ_c/φ_env = 10⁻⁴ and βφ_env/M_P = 10⁻⁸ to find ε = 10⁻⁸(1−10⁻⁴)/(4×10⁻⁶) = 0.00249975.
Since 2β²×3 = 1, the leading extra-force ratio equals ε.
The additional checks are m_env(2R) ≈ 0.005657 and m_coreR ≈ 2828, consistent with the selected limiting regime.

The leading ratio is approximately 0.00249975 for this thin-shell source and unscreened test, without a claimed full boundary-solution error bound.

Try the reasoning

At density ratio 256, what are the leading mass and range ratios?
Show the answer and reasoning
m/m_* = 64 and λ/λ_* = 1/64.
Use the stated powers u^(3/4) and u^(−3/4).
May ε = 2 be inserted into the thin-shell expression as a valid enhanced charge?
Show the answer and reasoning
No.
The derivation assumes ε≪1; the distinct unscreened limit does not justify extrapolating the thin-shell formula.
Are source screening and screening of the test body the same input?
Show the answer and reasoning
No.
The supplied force assumes an unscreened test body; changing its response needs a separate calculation.

What to carry forward

State which screening mechanism and density convention are being used.
Environmental range and effective source charge are distinct effects.
Check the simultaneous approximation conditions before interpreting a suppressed force.

References

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