Cosmological Spacetime Elasticity: A Zero-Mode Effective Framework for Dynamical Dark Energy

We present a phenomenological effective framework for dynamical dark energy based on the macroscopic elastic response of the cosmological background. By formulating an effective scalar field action \(S_\phi\) under a spatial zero-mode constraint (\(\nabla_i \phi = 0\)), we show that the temporal strain mode \(\phi(t) = \ln(a/a_0)\) emerges as an exact solution to the Euler--Lagrange equations. Within the homogeneous and isotropic approximation, the strain variable is restricted to a temporal mode with vanishing spatial gradients (\(\nabla_i \phi = 0\)). Consequently, the construction introduces no additional propagating spatial scalar degrees of freedom at the background level, avoiding long-range fifth forces without invoking screening mechanisms. Cosmologically, the kinetic strain rate drives a smooth transition toward an effective dark energy equation of state \(w_{\mathrm{el}} \to -1\) at late times. By matching the stored strain energy within the causal boundary to the Bekenstein--Hawking horizon scaling, we adopt the natural normalization \(\kappa = 1\). This framework provides a physically transparent, background-effective alternative to standard scalar-tensor theories, offering dynamic signatures testable by next-generation cosmological surveys.