But how, concretely, does dark matter excite this gigantic membrane? Each dark matter particle crossing a funnel follows a precise ballet that creates the cosmic symphony we observe.
The Dark Matter Dance
Each dark matter particle crossing a gravitational funnel follows three precise steps:
- Departure: It temporarily leaves the brane, carrying its momentum
- Journey: It travels a short geodesic in the bulk
- Return: It re-impacts the brane near another funnel
This return deposits a momentum “hit” δp ~ m_MN × v_⊥ radially opposite to the outgoing flux. The surface density of these impacts, summed over all black holes, creates a periodic pressure:
\[Π(t) = \sum_i \dot{N}_i m_{MN} v_⊥ ≃ f_{osc} ρ_{DM} v_⊥^2\]The Miracle of Synchronization
The miracle: In the limit where the bulk crossing time is very short compared to period T, this pressure Π(t) becomes quasi-sinusoidal. Even more remarkable, it selectively couples to the fundamental mode (ℓ = 0) because all funnels share the same topology toward the bulk-point—the phase is identical across the entire surface!
It’s as if millions of tiny hammers were striking the membrane in perfect synchrony, creating a global standing wave rather than a chaos of ripples.
The Universal Spring Constant
The beauty of this approach lies in its simplicity. The second derivative of energy gives:
\[k_{eff} = \frac{∂^2E}{∂z^2} = \frac{τ_0 A}{R_H^2} ≈ τ_0\]Dimensional miracle: The spring constant is simply the tension itself!
Stability and Resonances
A membrane can vibrate in an infinity of modes, like a bell ringing with its harmonics. Why does our universe favor the fundamental mode?
Higher modes (ℓ ≥ 2) have frequencies:
\[ω_ℓ ≃ \sqrt{ℓ(ℓ+1)} × ω_0\]For ℓ = 2, the frequency is already √6 ≈ 2.5 times higher. Since the source Π(t) is quasi-monochromatic at ω₀, coupling to higher modes decreases as δω⁻², naturally damping them.
Guaranteed stability: The predicted maximum amplitude δτ/τ₀ ~ 10⁻⁴ remains far below the fragmentation threshold (δτ/τ₀ > 1). The membrane can oscillate eternally without risk of tearing.
However, secondary local resonances are possible around superclusters, where mass concentration creates “hard points.” These micro-oscillations could generate tiny gravitational anisotropies (δg/g ~ 10⁻⁸), a subtle but potentially detectable signature.
Primordial Black Holes: The Cosmic Pushpins
Beyond stellar and supermassive black holes, a hidden population could play a crucial role: primordial black holes (PBH). A PBH of mass 10⁻¹¹ M_☉ has a Schwarzschild radius r_s ≈ 30 nm, creating a funnel comparable in size to our extra dimension L.
If these PBHs represent a fraction Ω_PBH ~ 10⁻⁴ of cosmic density, they form a dense network of small-scale entry points. Like thousands of needles piercing fabric, they increase the oscillating fraction f_osc without changing the macroscopic dark matter density.
Consequence: a possible enhancement of the dark energy oscillation amplitude A_w, offering an additional signature to search for in future observations.