In a multilayer model, H(z)H(z) acquires directional dependence due to layer-curvature inhomogeneities and fractal embedding:
H(z,)=ifi()Hi(z),H(z, \Omega) = \sum_i f_i(\Omega) H_i(z),
where fi()f_i(\Omega) represents the layer-contribution weights along angular direction \Omega. This could produce angular variations in redshift drift:
Deviations may appear at low redshift due to local underdense/overdense layer crossings,
At high redshift, effective averaging smooths the drift curve but may still leave small-scale oscillations.
Data sources: Future ELT-HIRES, SKA, and CODEX-like spectrographs are crucial.
4. Gravitational Wave Constraints on Tensor Modes and Echoes
Tensor perturbations in this model emerge from layer boundary scatterings and initial tunneling anisotropies, rather than from an inflaton field. This modifies the expected tensor power spectrum:
Suppressed primordial tensor-to-scalar ratio rr due to the absence of inflaton-driven amplification,
Possible anisotropic tensor patterns due to asymmetric tunneling paths,
And, more strikingly, echoes in gravitational wave signals from geodesic scattering at layer interfaces.
The latter produces late-time echoes in the signal from mergers or cosmogenic background bursts:
h(t)=hGR(t)+nnhecho(tnt),h(t) = h_{\text{GR}}(t) + \sum_n \epsilon_n h_{\text{echo}}(t - n\Delta t),
where t\Delta t corresponds to interlayer light-travel times and n\epsilon_n the reflectivity at each boundary.
Data sources:
LIGO/Virgo/KAGRA: Search for merger echoes.
LISA: Probes primordial GW background and low-frequency echoes.
PTA (Pulsar Timing Arrays): Tests large-scale tensor anisotropies.
5. Viability Within Current Bounds
