⚛️ general relativity

Non-Markovian Cosmology with Causal Memory Kernels

This paper investigates non-Markovian cosmological models with causal memory kernels, finding that while an oscillatory kernel improves the fit to Hubble parameter data, it violates causality and is disfavored by Bayesian criteria, whereas other kernels are either consistent with zero amplitude or degenerate with a simple rescaling of the expansion rate that is less favored than the standard $\Lambda$ CDM model.

Original authors: Aritra Sanyal, Prabir Rudra

Published 2026-10-26

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CC BY 4.0

Original authors: Aritra Sanyal, Prabir Rudra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The story of our universe is usually told as a smooth, predictable expansion, a cosmic balloon inflating at a rate determined solely by the energy and matter present at any given moment. In this standard view, the universe has no memory; the speed at which space stretches today depends only on what is inside it right now, not on how it behaved a billion years ago. This idea, known as the Markovian description, has served astronomers well, providing a clean framework to map the history of the cosmos. However, just as a person's current mood might be influenced by events from their past rather than just the immediate surroundings, physicists have long wondered if the universe's expansion might also carry the faint imprint of its history. If the universe does remember its past, the equation governing its growth would need to include a "memory term," a mathematical way of saying that the present rate of expansion is shaped by the journey it took to get there.

A team of researchers in India set out to test this possibility by building a new version of cosmic history that allows for such memory. They proposed that the expansion rate of the universe is not just a snapshot of the present energy density but is also influenced by a causal memory kernel, a mechanism that retains information about the universe's previous states. To explore this, they developed five different mathematical shapes for how this memory might fade over time, ranging from a simple linear decline to complex patterns that oscillate like a wave. They grounded these shapes in real physics, showing that they could arise from hidden sectors of the universe we cannot yet see, or from the way cosmic fluids resist flow. The researchers then took these theoretical models and put them to the test against the most precise measurements of the universe's expansion available today, using data from cosmic chronometers and the Dark Energy Spectroscopic Instrument.

The results of this rigorous test were surprisingly definitive. When the researchers compared their memory-filled models against the standard, memory-less model, they found that the universe appears to have very little memory after all. For the simplest types of memory, where the influence of the past fades steadily, the data showed that the memory effect is consistent with zero. In other words, the universe behaves exactly as if it has no memory at all. For the more complex shapes, the data suggested that if a memory effect exists, it is so short-lived that it is indistinguishable from a simple, uniform adjustment to the current expansion rate, rather than a true historical influence. The models that included these memory terms did not fit the data better than the standard model; in fact, the standard model remained the most reliable description of the cosmos.

One specific type of memory model, one that oscillates back and forth like a pendulum, initially seemed to offer a better fit to the data from the Dark Energy Spectroscopic Instrument. However, a closer look revealed a fatal flaw: for this model to work, the memory would have to grow stronger as we look further back in time, a behavior that contradicts the fundamental laws of physics governing how energy dissipates. When the researchers added data from cosmic chronometers to the mix, this oscillating model lost its advantage entirely. The evidence suggests that while the universe might allow for a tiny, percent-level influence from its past, there is no compelling reason to believe such an influence actually exists. The standard, memory-less description of the universe remains the most accurate map we have, and the search for a cosmic memory that shapes our expansion has, for now, come up empty.

Technical Summary: Non-Markovian Cosmology with Causal Memory Kernels

Problem Statement
The standard cosmological model, based on the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, is inherently Markovian: the Hubble parameter $H(z)$ depends solely on the instantaneous energy density, ignoring the past evolution of the Universe. This assumption, rather than a strict consequence of General Relativity, arises from coarse-graining short-wavelength fluctuations and unresolved components. Recent observational tensions (Hubble and $S_8$ ) and hints of evolving dark energy in DESI data motivate investigating whether the instantaneous relation between $H(z)$ and energy density is too rigid. The authors propose a non-Markovian extension where the present expansion rate retains information about past cosmic evolution via a causal memory integral.

Methodology
The study generalizes the Friedmann equation by introducing a causal memory kernel $K(z-z')$ :
$H^2(z) = H_M^2(z) + \int_0^z K(z-z') H_M^2(z') dz'$
where $H_M(z)$ is the standard Markovian ( $\Lambda$ CDM) rate. The authors investigate five specific continuous kernels: Linear, Power-law, Exponential, Gaussian, and Oscillatory.

Theoretical Derivation: The paper establishes that these kernels are not merely phenomenological but arise from the microscopic physics of an unresolved "hidden sector."
- Open-System Derivation: Using the Zwanzig–Mori projection-operator formalism, the kernels are identified as cosine transforms of the spectral densities $J(\omega)$ of a bath of oscillators coupled to the Ricci scalar.
- Causal Thermodynamics: The Exponential kernel is independently derived from the Israel–Stewart theory of causal bulk viscosity.
- Non-local Gravity: The Linear kernel emerges as the short-lag limit of the retarded Green's function in Deser–Woodard non-local gravity.
Data Analysis: The models are constrained using:
- 30 Cosmic Chronometer (CC) measurements ( $0.07 \le z \le 1.965$ ).
- DESI Baryon Acoustic Oscillation (BAO) data from Data Release 1 (DR1) and Data Release 2 (DR2), converted to $H(z)$ values.
Statistical Approach: The authors employ multi-start $\chi^2$ minimization followed by Markov Chain Monte Carlo (MCMC) sampling to obtain posterior distributions. Model performance is evaluated against the standard $\Lambda$ CDM model using the Akaike (AIC) and Bayesian (BIC) information criteria.

Key Contributions

Microscopic Origin of Kernels: The paper provides a unified physical framework showing that the five tested kernels correspond to specific spectral densities of a hidden sector, endowing their parameters (amplitude, decay rates, frequency) with physical meaning (e.g., coupling strength, correlation time, resonance frequency).
Short-Memory Limit: The authors demonstrate that for sharply peaked kernels, the memory term reduces to a uniform rescaling of the Hubble constant ( $H_0 \to H_0\sqrt{1+A_{\text{eff}}}$ ), explaining the observational degeneracy between different kernel shapes.
Comprehensive Constraints: The study provides the first joint constraints on these non-Markovian extensions using the latest DESI DR1 and DR2 data combined with Cosmic Chronometers.

Results

Linear Kernel: The amplitude is consistent with zero ( $A = 0.002 \pm 0.030$ for CC+DR2), indicating no evidence for a linear memory term. It is mildly disfavoured by information criteria.
Degenerate Kernels (Power-law, Exponential, Gaussian): These three kernels are observationally degenerate. The data constrain only an integrated amplitude $A_{\text{eff}} \simeq 0.01\text{--}0.03$ , which corresponds to a uniform rescaling of $H(z)$ by $0.5\text{--}1.5\%$ . The data select sharply peaked kernels (large decay rates), effectively pushing the models into the short-memory limit. These models are disfavoured relative to $\Lambda$ CDM ( $\Delta$ AIC = +2.5 to +3.6).
Oscillatory Kernel: This is the only kernel that significantly improves the fit to DESI data alone ( $\Delta\chi^2 = 4.5\text{--}8.1$ ), with a preferred frequency $\omega \simeq 4\text{--}5$ . However, the best fits require a negative damping rate ( $\alpha < 0$ ), implying a memory kernel that grows into the past. This is incompatible with the causal derivations (where $\alpha$ must be positive) and suggests over-fitting. When Cosmic Chronometer data are included, the preference for the Oscillatory kernel vanishes ( $\Delta$ BIC = +4.4 to +6.2).

Significance and Claims
The paper concludes that current expansion-history data allow for cosmic memory at the percent level in $H(z)$ but do not require it. The standard $\Lambda$ CDM model remains the preferred description across all datasets.

The authors assert that the non-Markovian model serves as a "physically motivated, testable extension" of $\Lambda$ CDM that is currently constrained rather than detected. The recurring preference for an oscillatory frequency in DESI data alone is noted as a potential signature of a resonant hidden sector, but the unphysical requirement of a negative damping rate and the lack of robustness when combining datasets prevent a definitive detection. The study emphasizes that the memory term is best viewed as a constrained extension where the signal is currently indistinguishable from a slight rescaling of the Hubble constant or bulk viscosity, rather than a confirmed deviation from Markovian dynamics.

Technical Summary: Non-Markovian Cosmology with Causal Memory Kernels

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