Cosmology of f(Q,L_m) gravity with Holographic Ricci Dark Energy: Early-Time Inflation and Late-Time Acceleration and RGUP Corrected Observables

This paper proposes a unified geometric model within f(Q,L_m) gravity that successfully describes both early-time Starobinsky-like inflation and late-time accelerated expansion via Holographic Ricci Dark Energy, while demonstrating consistency with observational data and incorporating sub-leading quantum corrections from the Relativistic Generalized Uncertainty Principle.

The Gist

Imagine the universe as a giant, expanding balloon. For a long time, scientists have used a standard set of rules (called General Relativity) to explain how this balloon inflates. However, these rules hit a snag: they struggle to explain two specific moments in the balloon's history. First, a tiny, explosive burst of growth right at the very beginning (Inflation). Second, a slow, steady speeding-up of the expansion happening right now (Dark Energy). Usually, scientists have to invent two different "magic ingredients" to explain these two different times.

This paper proposes a new, single set of rules called $f(Q, L_m)$ gravity that acts like a universal remote control. Instead of needing two different ingredients, this theory suggests that the same underlying geometric rules of the universe change their behavior depending on how "stretched" or "curved" space is.

Here is a breakdown of their findings using simple analogies:

1. The Universal Remote (The New Gravity Theory)

Think of the universe's gravity not as a fixed law, but as a dimmer switch with different settings based on the "brightness" (curvature) of the universe.

• The "High-Brightness" Setting (Early Universe): When the universe was brand new and incredibly dense (high curvature), the theory says the rules shift to a "quadratic" mode. This acts like a powerful engine that naturally pushes the universe to expand exponentially fast. This explains Inflation without needing any extra, mysterious fields. It's like the universe had a built-in "turbo button" that only works when things are super hot and dense.
• The "Low-Brightness" Setting (Late Universe): As the universe expanded and cooled down (low curvature), that "turbo button" turns off. However, a different part of the theory kicks in: a connection between the "stuff" in the universe (matter) and the "shape" of the universe (geometry). This connection acts like a gentle, persistent push, causing the universe to speed up again today. This explains Late-Time Acceleration (Dark Energy) without needing a separate "Dark Energy" particle.

2. The Holographic Mirror (Dark Energy)

To figure out exactly how this "gentle push" works today, the authors used a concept called Holographic Ricci Dark Energy.

• The Analogy: Imagine the universe is a hologram. In this view, the energy driving the expansion isn't just floating around randomly; it's tied to the "curvature" of the hologram itself. The authors treated this energy like a fluid that flows through the universe, helping to reconstruct the history of how the universe expanded from the Big Bang to today.
• The Result: When they ran the numbers, the universe expanded smoothly. It didn't crash, bounce, or stop. It just kept growing, slowly speeding up over time, exactly like we observe.

3. The "Fuzzy" Reality (Quantum Corrections)

The authors also asked: "What if we look at this through the lens of Quantum Mechanics?"

• The Analogy: Imagine taking a high-resolution photo of the universe. Standard physics gives you a sharp picture. But at the tiniest scales (like the Planck scale), the picture gets a little "fuzzy" or "pixelated" due to quantum uncertainty. This is called the RGUP (Relativistic Generalized Uncertainty Principle).
• The Effect: The authors applied this "fuzziness" to their model. They found that it didn't break the universe or change the main story. The universe still expanded the same way. However, it did add a tiny, subtle "ripple" to the details.
• The Ripple: Specifically, it changed a very specific number called the "running of the spectral index." Think of this as a tiny adjustment to the color palette of the cosmic background radiation. While current telescopes (like Planck) can't see this tiny ripple yet, the theory predicts it exists. It's like a secret signature that future, super-powerful telescopes might one day detect.

4. Checking the Receipts (Data Analysis)

The authors didn't just make up a story; they checked it against real data.

• The Test: They compared their model against three massive datasets:
  1. Supernovae: Distant exploding stars that act as cosmic mile markers.
  2. Cosmic Chronometers: The ages of old galaxies.
  3. BAO (Baryon Acoustic Oscillations): Fossil sound waves from the early universe.
• The Verdict: Their model fits the data just as well as the current standard model (Lambda-CDM). The "connection" between matter and geometry (the new ingredient) was found to be very weak, which is good because it means their theory doesn't contradict what we already know. It essentially says, "We can explain the universe's acceleration using geometry alone, and it looks just like the standard model we already trust."

Summary

This paper suggests that the universe doesn't need two different "magic wands" to explain its early explosion and its current speeding up. Instead, it proposes a single, elegant geometric theory where the rules of gravity naturally change their tune depending on the era:

1. Early Times: High curvature triggers a geometric "turbo" for inflation.
2. Late Times: Low curvature triggers a matter-geometry "nudge" for acceleration.
3. Quantum Level: Tiny quantum "fuzziness" adds a subtle, detectable signature to the details, waiting for future technology to find it.

It's a unified story where the geometry of space itself is the hero, handling both the beginning and the present of the cosmos.

Technical Summary

Technical Summary: Cosmology of f(Q, Lm) Gravity with Holographic Ricci Dark Energy and RGUP Corrections

Problem Statement
The paper addresses the challenge of unifying two distinct epochs of cosmic acceleration: the early-time inflationary phase and the late-time accelerated expansion driven by dark energy. While standard General Relativity (GR) requires ad hoc additions (such as an inflaton field or a cosmological constant $\Lambda$) to explain these phases, modified gravity theories offer a geometric alternative. Specifically, the authors investigate whether a single geometric framework based on symmetric teleparallel gravity, where gravity is described by non-metricity ($Q$) rather than curvature or torsion, can naturally account for both epochs. Furthermore, the study explores the impact of quantum-gravity-inspired corrections, specifically the Relativistic Generalized Uncertainty Principle (RGUP), on inflationary observables within this framework.

Methodology
The authors adopt the $f(Q, L_m)$ gravity framework, which introduces a direct coupling between the non-metricity scalar $Q$ and the matter Lagrangian density $L_m$.

1. Model Construction: They propose a minimal polynomial form for the gravitational action:
   $$f(Q, L_m) = -Q + \alpha Q^2 + 2L_m + \beta Q L_m$$
   Here, the linear terms recover GR, the quadratic term $\alpha Q^2$ targets high-curvature regimes (inflation), and the coupling term $\beta Q L_m$ targets late-time dynamics.
2. Late-Time Reconstruction: To model the late-time universe, Holographic Ricci Dark Energy (HRDE) is introduced as an effective fluid. The authors derive a second-order non-linear differential equation for the Hubble parameter $H(t)$ by substituting the HRDE density into the modified Friedmann equations. This equation is solved numerically to reconstruct the background evolution (scale factor, deceleration parameter, and equation of state).
3. Bayesian Inference: The model parameters are constrained using late-time observational data, including Pantheon Type Ia supernovae, cosmic chronometers, and DESI 2024 Baryon Acoustic Oscillation (BAO) data. A phenomenological parametrization of the Hubble function is employed to facilitate Markov Chain Monte Carlo (MCMC) analysis.
4. Inflationary Dynamics: In the high-curvature limit (early Universe), the matter coupling is negligible, and the model reduces to an $f(Q) \simeq -Q + \alpha Q^2$ theory. The authors map this to the Starobinsky $R + R^2$ model to derive predictions for the scalar spectral index ($n_s$) and tensor-to-scalar ratio ($r$).
5. RGUP Corrections: To account for Planck-scale effects, the authors implement RGUP as a momentum-dependent deformation of the effective spacetime metric. This deformation modifies the kinetic and potential terms of an effective scalar field propagating on the background, leading to perturbative corrections in inflationary observables, particularly the running of the spectral index ($\alpha_s$).

Key Contributions and Results

• Unified Geometric Framework: The study demonstrates that the chosen $f(Q, L_m)$ model can simultaneously describe early-time inflation and late-time acceleration without introducing separate scalar fields or dark energy components. The transition is driven by the dominance of different terms in the action based on the curvature scale: the $\alpha Q^2$ term dominates at high curvature (inflation), while the $\beta Q L_m$ coupling dominates at low curvature (late-time acceleration).
• Late-Time Dynamics: Numerical reconstruction shows a smooth, stable accelerated expansion. The effective equation of state ($\omega_{eff}$) evolves from a quintessence regime, crosses the phantom divide, and asymptotically approaches the de Sitter limit ($\omega \to -1$). The Hubble parameter decreases monotonically, consistent with observational expectations.
• Observational Constraints (Late-Time): Bayesian analysis using Pantheon, cosmic chronometer, and DESI BAO data reveals that the matter-geometry coupling parameter $\beta$ is weakly constrained. The posterior distribution accumulates at the lower boundary of the prior, indicating no statistical preference for a non-zero $\beta$. The results are consistent with the $\Lambda$CDM limit ($\beta = 0$), suggesting that current late-time data cannot tightly constrain mild matter-geometry couplings.
• Inflationary Predictions: In the high-curvature limit, the model yields Starobinsky-like predictions. For $N=50, 55, 60$ e-folds, the predicted scalar spectral index ($n_s \approx 0.960 - 0.967$) and tensor-to-scalar ratio ($r \sim 10^{-3}$) fall comfortably within the Planck 2018 confidence contours.
• RGUP Effects: The inclusion of RGUP corrections introduces a momentum-dependent deformation to the metric. While the background inflationary dynamics remain Starobinsky-like, the corrections induce a small, finite shift in the running of the spectral index ($\alpha_s$). For a deformation parameter $\beta_{gup} = 0.05$, the model predicts a distinct value for $\alpha_s$ compared to the standard attractor, though this shift remains below current Planck 2018 sensitivity. The tensor-to-scalar ratio is slightly suppressed, and the spectral index is slightly enhanced, maintaining consistency with observational bounds.

Significance and Claims
The paper claims to provide a "dynamically consistent geometric model" where the two phases of cosmic acceleration arise from distinct curvature regimes of a single underlying theory.

• Unification: Unlike models that treat inflation and dark energy as separate phenomena, this work posits that both are consequences of the same $f(Q, L_m)$ gravitational action, differentiated only by the energy scale (curvature) of the epoch.
• Geometric Origin of Inflation: The study highlights that inflation can be driven purely by geometric effects (the quadratic non-metricity term) without the need for a fundamental inflaton field, effectively reproducing the successful Starobinsky scenario within symmetric teleparallel gravity.
• Quantum-Geometric Signatures: The authors argue that while RGUP corrections are sub-leading and currently undetectable by CMB experiments, they provide a theoretically distinct signature (specifically in the running of the spectral index) that differentiates the model from classical attractors. This suggests that future high-precision experiments could potentially probe these quantum-geometric deformations.
• Extension of Previous Work: The authors position this work as an extension of the foundational $f(Q, L_m)$ theory proposed by Hazarika et al. (2025), adding specific phenomenological reconstruction via HRDE, observational constraints, and quantum-gravity corrections to explore the model's viability across the entire cosmic history.