Spatiotemporal Topological Order D, Cosmological Observational Imprints of Spatiotemporal Topological Order: From CMB Modulation to Nanohertz Gravitational Wave Background

This paper explores the potential observational manifestations of spatiotemporal topological order theory in the early universe. Based on the framework ofspatiotemporally dual topological quantum field theory (TQFT), we propose thatthe quantum gravitational fluctuations in the early universe were dominated bya ”temporal barcode” (birth-death intervals in persistent homology) fluctuationfield, which was stretched during inflation and left unique periodic modulationimprints on the angular power spectrum of the cosmic microwave background radiation (CMB), while simultaneously generating a stochastic gravitational wavebackground (SGWB) with a peak in the nanohertz frequency band. We derivethe analytical form of the CMB modulation template, predicting an approximatelyperiodic oscillatory structure on large angular scales, and assess its detectabilityin CMB-S4 experiments. Furthermore, we calculate the SGWB energy spectrumgenerated by barcode merging events, with its peak frequency (10−9–10−8 Hz) lying precisely within the detection window of pulsar timing arrays (PTA) such asthe Square Kilometre Array (SKA). This paper provides concrete theoretical predictions and an observational roadmap for testing the fundamental assumptions ofspatiotemporal topological order using next-generation multi-messenger cosmological observations.