Scalaron Screening Mechanism - Dark Matter from Field Dynamics

RFT Research Team

🌌 Scalaron Screening Mechanism

Scalaron gradients renormalize Newton's G at large r, yielding a logarithmic tail that screens the 1/r² fall-off.

📐 From Field Equation to Modified Gravity

\\[ \\Box\\Phi - m_s^2\\Phi = 8\\pi G\\rho \\]

Klein-Gordon equation for the scalaron field Φ with mass m_s

\\[ \\nabla^2\\Phi = 4\\pi G\\rho + \\alpha\\nabla^2\\ln\\left(\\frac{r}{r_0}\\right) \\]

Weak-field limit gives modified Poisson equation with logarithmic screening

Show 4-step derivation

Start from \\(f(R) = R + \\beta R^2\\) → introduce scalar field via \\(e^{2\\kappa\\phi}\\tilde{g}_{\\mu\\nu}\\)
Linear weak-field limit: \\(g_{\\mu\\nu} = \\eta_{\\mu\\nu} + h_{\\mu\\nu}\\), retain terms to \\(\\mathcal{O}(h,\\phi)\\)
Identify scalaron mass: \\(m_s^{-2} = 6\\beta\\) ⇒ Klein-Gordon \\((\\Box - m_s^2)\\phi = -\\kappa T\\)
Static Green's function in 3-D: \\(\\phi(r) = -\\frac{\\kappa M}{4\\pi r}\\left[1 + \\alpha\\ln\\left(\\frac{r}{r_0}\\right)\\right]\\) with \\(\\alpha = \\frac{\\kappa^2 M}{4\\pi m_s^2}\\)

Substituting \\(\\Phi = \\phi + \\Phi_N\\) yields the modified force law quoted above.

The full algebra is omitted here for clarity — see Math Ref §5 or open the Jupyter notebook for the complete derivation.

Source keys: 13.2 §B eq (17) → 13.99 §C eq (3)

🔄 Modified Force Law & Gravitational Potential

\\[ F(r) = \\frac{Gm_1m_2}{r^2}\\left[1 + \\alpha\\ln\\left(\\frac{r}{r_0}\\right)\\right] \\]

Modified Gravitational Force

\\[ \\Phi(r) = \\Phi_N + \\alpha\\ln\\left(\\frac{r}{r_0}\\right) \\]

Screened Potential

Think gelatin: near a spoon (high density) it flows, far away it stiffens. Scalaron's curvature-coupling makes the effective G run with scale, so gravity 'fat-tails' instead of fading.

🎮 Interactive Rotation Curve Explorer

Select a galaxy or enter a flat velocity to see how scalaron screening creates the rotation curve

Galaxy

Flat Velocity (km/s)

Scale r₀ (kpc)

Calculated α

                    5.4 × 10⁻⁷
                

📊 Galaxy Parameters

Galaxy	v_flat (km/s)	α	r₀ (kpc)	Reference
Milky Way	220	5.4 × 10⁻⁷	1.1	SPARC Database
M33	180	3.8 × 10⁻⁷	0.9	SPARC Database
NGC 2403	135	2.0 × 10⁻⁷	0.8	SPARC Database
M31 (Andromeda)	250	6.9 × 10⁻⁷	1.5	Corbelli et al. 2010
NGC 3198	155	2.7 × 10⁻⁷	1.0	SPARC Database
NGC 2841	230	5.9 × 10⁻⁷	1.2	SPARC Database
UGC 2885	300	1.0 × 10⁻⁶	2.0	Rubin et al. 1985
NGC 6503	120	1.6 × 10⁻⁷	0.7	SPARC Database
IC 2574	70	5.4 × 10⁻⁸	0.5	SPARC Database
NGC 7331	245	6.7 × 10⁻⁷	1.3	SPARC Database