What question did this study set out to answer?

To propose a framework called Discrete Symplectic Cosmology that models cosmic evolution and time-dependent vacuum energy.

April 7, 2026Open Access

Discrete Symplectic Cosmology: A Phenomenological Framework for Time-dependent Vacuum Energy from Planck-Lattice Spectral Statistics

Key Points

To propose a framework called Discrete Symplectic Cosmology that models cosmic evolution and time-dependent vacuum energy.
Developed a framework based on discrete spacetime and symplectic evolution.
Introduced an adiabatic cooling law and a dynamical cosmological term linked to vacuum fluctuations.
Compared predictions with data from quasar absorbers and performed a Fisher matrix analysis against cosmological data.
Conducted numerical simulations to validate the cooling schedule and symplectic preservation.
Framework predictions align with current observational data, particularly with 1/ln² corrections being small at observable epochs.
Indicates possible detection of effects at γ_eff ≳ 500 with future BAO surveys.
Verifies numerical simulations confirming cooling behavior across extensive lattice time.

Abstract

We propose a phenomenological framework — *Discrete Symplectic Cosmology* (DSC) — in which cosmic evolution is modeled as a non-autonomous discrete dynamical system on a Planck-scale lattice Z³ × N, equipped with a symplectic structure. From three assumptions (discrete spacetime, symplectic evolution, and a Berry–Keating spectral correspondence between the quantized lattice and the Riemann zeta function), we motivate: (i) an adiabatic cooling law μ (n) = μc + (αF/2) /ln² (n) whose 1/ln² functional form is suggested by the asymptotic density of Riemann zeros; (ii) a dynamical cosmological term Λ (t) = Λ_∞ + γ/ln² (t/tP) arising from residual vacuum fluctuations of the not-yet-frozen lattice modes; and (iii) a Hubble relaxation law H (t) = H_∞ + β/ln² (t/tP) obtained by substituting Λ (t) into the standard Friedmann equation. The framework provides a candidate mechanism for modifying the early expansion rate without introducing new scalar fields. We present comparisons with 293 real quasar absorbers from King et al. (2012) jointly constrained by atomic-clock bounds, a CAMB-based Fisher matrix analysis against Planck and BAO data, and a late-time stability check. The framework is consistent with all current observations — the 1/ln² correction is perturbatively small at all observable epochs — but its predictions lie below present experimental sensitivity for γₑff ≲ 1. The current cosmological comparison is a phenomenological proxy (effective wₐ mapping), not a self-consistent DSC perturbation treatment. Stage IV BAO surveys may begin to probe γₑff ≳ 500 at the order-of-magnitude level. Numerical simulations using a Störmer–Verlet integrator verify consistency of the assumed cooling schedule over 60 decades of lattice time while preserving the symplectic form to machine precision. The framework makes falsifiable predictions, including an asymptotic Hubble limit H_∞ testable by next-generation BAO surveys.

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