Paper XXVIII - Strong Lensing Time Delays as Propagation-Dominated Chronometric Probes

DOI: 10.5281/zenodo.18009452 Strong Lensing Time Delays as Propagation-Dominated Chronometric Probes Paper XXVIII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction 1 2 Geometric Origin of Lensing Time Delays 2 3 Operational Time and Null Propagation 2 4 Measured Time Delays 3 5 Insensitivity to Temporal Memory 3 6 Contrast with Other Distance Probes 3 7 Implications for Cosmological Inference 3 8 Role in the Series 4 9 Conclusion 4 trong gravitational lensing time delays provide a unique cosmological observable: direct measurement of elapsed time between null geodesics traversing different paths through spacetime. Unlike distance-ladder probes, time-delay measurements do not rely on clockbased reconstruction of periodic or standardized sources, but on differential propagation timing. Within the Ordered-Dynamics Reconstruction Program, this distinction is crucial. Earlier papers established that finite clocks reconstruct operational time with history-dependent processing overhead, quantified by αeff , and that this overhead biases clock-based observables while leaving propagation unchanged. In this paper we show that strong-lensing time delays are dominated by geometric propagation delay and are structurally insensitive to operational time reconstruction effects. This places lensing time delays in a distinct operational class from Cepheid calibration (Paper XXVI) and Type Ia supernova standardization (Paper XXVII), and establishes them as clean chronometric probes of spacetime geometry. 1 Introduction Strong gravitational lensing time delays arise when light from a variable background source follows multiple null geodesics around a foreground mass, arriving at the observer at different times. Measurement of these delays provides direct access to the so-called time-delay distance, a combination of angular diameter distances that scales inversely with the Hubble constant. Unlike standard candles or rulers, time-delay cosmography is intrinsically chronometric: it measures elapsed time rather than calibrated luminosity or length. This makes it a critical probe for testing frameworks in which the operational meaning of time differs from naive assumptions. In the Ordered-Dynamics Reconstruction Program, operational time ˜ tis reconstructed by finite clocks from bounded records and generally differs from the ordering parameter λgoverning reversible microscopic dynamics. Earlier papers established that this reconstruction – 1 – incurs processing overhead quantified by αeff (Papers XXI–XXII), accumulates as temporal memory (Paper XXIV), biases dynamical inference (Paper XXV), and affects clock-based distance calibration (Paper XXVI), while canceling identically in Type Ia supernova standardization (Paper XXVII). The purpose of the present paper is to classify strong-lensing time delays within this operational framework. Remark 1 (Scope).This paper is structural. It does not perform lens modeling or data analysis, but establishes the operational sensitivity of lensing time delays prior to numerical inference. 2 Geometric Origin of Lensing Time Delays In standard lensing theory, the time delay between two images iand jis ∆tij =D∆t c∆ϕij,(2.1) where ∆ϕij is the difference in Fermat potential and D∆t= (1 + zL)DLDS DLS (2.2) is the time-delay distance. This expression depends on: •null geodesic propagation, •gravitational potential structure, •spacetime geometry. It does not depend on periodicity, standardization, or internal clock structure of the source. 3 Operational Time and Null Propagation In the Ordered-Dynamics framework, operational time reconstruction affects clocks through processing overhead, d˜ t dλ =1 1+αeff ,(3.1) while null propagation is governed by influence-delay structure encoded in the geometric factor Z(x) (Paper XIII). Remark 2.Operational time reconstruction applies to clocks; null geodesics do not reconstruct time and therefore do not accumulate processing overhead. The arrival time difference between two images is therefore determined by differences in geometric propagation delay rather than by clock reconstruction history. – 2 – 4 Measured Time Delays Observed time delays are measured by clocks at the detector. However, the relevant observable is a difference between arrival times of signals that share the same detection environment. Let ∆λij denote the difference in ordering parameter accumulated along two null paths. The observer reconstructs both arrival times using the same clock architecture and environment, yielding ∆˜ tij =∆λij 1+αobs .(4.1) Because αobs is common to both arrivals, it factors out of differential inference. [Common-mode cancellation] Operational time reconstruction overhead at the observer cancels identically in lensing time-delay differences. Remark 3.This cancellation differs in origin from SALT2 cancellation (Paper XXVII): here it arises from common-mode detection rather than redshift normalization. 5 Insensitivity to Temporal Memory Earlier papers showed that cumulative operational lag ∆Tencodes temporal memory for clocks traversing extended histories (Paper XXIV). Lensing time delays are insensitive to this effect for two reasons: •null geodesics do not reconstruct time, •differential arrival times are measured at a single spacetime location. Strong-lensing time delays are insensitive to cumulative operational lag ∆Taccumulated along clock histories. 6 Contrast with Other Distance Probes The operational classification of late-time cosmological probes now separates clearly: •Cepheid periods depend directly on clock reconstruction (Paper XXVI), •SN Ia standardization cancels operational time exactly (Paper XXVII), •Lensing time delays probe geometric propagation (this paper). Remark 4.Disagreement among these probes therefore reflects probe-specific operational structure rather than a universal rescaling of time. 7 Implications for Cosmological Inference Because lensing time delays are operationally clean, they provide an anchor against which clock-based probes can be compared. Any structured discrepancy between lensing-based and ladder-based inferences constrains the magnitude and variability of operational time reconstruction effects. Remark 5.This role is diagnostic rather than explanatory: lensing time delays do not themselves generate bias, but reveal it elsewhere. – 3 – 8 Role in the Series This paper completes the operational classification of late-time distance probes initiated in Papers XXVI and XXVII. With Cepheids, Type Ia supernovae, and strong lensing now separated by operational sensitivity, the framework is in place for quantitative reinterpretation of dark matter and dark energy observables in subsequent papers. 9 Conclusion Strong gravitational lensing time delays measure differential null propagation and are structurally insensitive to operational time reconstruction effects. They therefore provide a clean chronometric probe of spacetime geometry within the Ordered-Dynamics Reconstruction Program. Together with Papers XXVI and XXVII, this result establishes a probe-specific operational taxonomy essential for correct interpretation of cosmological data. – 4 –