Background dynamics and observational constraints of flat and non-flat $Λ(t)$CDM models from $H(z)$ and DESI DR2 BAO measurements

This study utilizes the latest DESI DR2 BAO and $H(z)$ cosmic chronometer data to constrain time-varying vacuum $\Lambda(t)$CDM models, finding that observations strongly favor the standard $\Lambda$CDM limit while significantly reducing parameter degeneracies and alleviating the Hubble tension.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: Fixing a Cosmic Riddle

Imagine the universe is a giant, expanding balloon. For decades, scientists have had a very successful "instruction manual" for how this balloon inflates, called the $\Lambda$CDM model. It says the balloon is filled with invisible "Dark Matter" (the glue holding galaxies together) and "Dark Energy" (the mysterious force pushing the balloon to expand faster).

However, there's a problem. When we measure how fast the balloon is expanding right now using two different methods, we get two different answers.

1. Method A (The Baby Picture): Looking at the Cosmic Microwave Background (the "baby photo" of the universe), we predict a speed of about 67.
2. Method B (The Adult Photo): Measuring nearby exploding stars (Supernovae), we get a speed of about 73.

This disagreement is called the "Hubble Tension." It's like measuring your height as a child and as an adult and getting two numbers that don't match the growth chart. Scientists are worried that our "instruction manual" might be missing a page.

The New Idea: A Vacuum That Changes Its Mind

The author of this paper, Olga Avsajanishvili, asks: What if the "Dark Energy" isn't a constant, unchanging force?

In the standard model, Dark Energy is like a stiff, unyielding spring inside the balloon. It pushes with the exact same strength forever.
In this new $\Lambda(t)$CDM model, Dark Energy is more like a breathing lung or a smart thermostat. It can change its strength over time. It interacts with the matter in the universe, perhaps giving energy to it or taking it away, depending on the era.

The paper tests a specific version of this "breathing" vacuum using a new parameter called $\alpha$ (alpha).

• If $\alpha = 0$: The vacuum is stiff (Standard Model).
• If $\alpha > 0$: The vacuum is "quintessence-like" (it changes in a specific way).
• If $\alpha < 0$: The vacuum is "phantom-like" (it gets stronger and stronger).

The Experiment: Using the Latest Data

To test this, the author didn't just guess. She used the two most powerful tools available in modern astronomy:

1. Cosmic Chronometers ($H(z)$): These are like "cosmic stopwatches." By looking at old galaxies, scientists can measure how fast the universe was expanding at different times in the past.
2. DESI DR2 BAO: This is the "Dark Energy Spectroscopic Instrument." Think of it as a massive 3D map of the universe. It measures the "sound waves" frozen in the distribution of galaxies (Baryon Acoustic Oscillations). It's like measuring the spacing between trees in a forest to figure out how the forest grew.

The author combined these two datasets to see if the "breathing vacuum" model fits the data better than the "stiff spring" model.

The Results: The "Stiff Spring" Still Wins

Here is the punchline, explained simply:

1. The "Breathing" Vacuum is a Nice Idea, But...
The math showed that the "breathing vacuum" model can work. It changes the expansion history of the universe in interesting ways. For example, if the vacuum interacts with matter, it can delay the moment when the universe started accelerating.

2. The Data Says "No Change, Please"
When the author ran the numbers (using a super-computer method called MCMC), the results were surprising. The data strongly preferred the value $\alpha \approx 0$.

• Translation: The universe doesn't seem to need a "breathing" vacuum. The "stiff spring" (the standard model) fits the new data just fine. The "breathing" idea doesn't make the prediction significantly better.

3. The Tension is Reduced, But Not Solved
One good thing happened: When the author added the new DESI data (the 3D map), the uncertainty in the measurements got much smaller.

• The gap between the "Baby Picture" (67) and the "Adult Photo" (73) didn't disappear, but it got smaller. It went from a huge, angry disagreement to a more manageable, "maybe we're just slightly off" disagreement.
• The new data suggests the universe is flat (like a sheet of paper), not curved like a bowl or a saddle.

The Verdict: Stick to the Basics

The paper concludes that while the idea of a time-varying vacuum is fascinating and theoretically possible, the current observational evidence doesn't support it.

• The Analogy: Imagine you are trying to fix a car that is making a weird noise. You try a new, complex engine part (the $\Lambda(t)$ model). You test it, and the car runs okay, but the original, simple engine part (the standard $\Lambda$CDM model) runs just as well and is much cheaper. The new data (the DESI map) confirms that the simple engine is still the best choice.

Summary for the General Public

• The Problem: We have a disagreement about how fast the universe is expanding.
• The Hypothesis: Maybe Dark Energy changes over time, interacting with matter like a dynamic fluid.
• The Test: We used the latest, most precise maps of the universe (DESI) and cosmic stopwatches to check this.
• The Result: The universe seems to prefer the "old school" model where Dark Energy is constant. The new, complex model doesn't fit the data any better.
• The Silver Lining: The new data helps narrow down the numbers, making the "Hubble Tension" less severe, even if it doesn't solve it completely yet.

In short: The universe is still a bit of a mystery, but for now, the simplest explanation remains the most accurate one.

Technical Summary

Here is a detailed technical summary of the paper "Background dynamics and observational constraints of flat and non-flat $\Lambda(t)$CDM models from $H(z)$ and DESI DR2 BAO measurements" by Olga Avsajanishvili.

1. Problem Statement

The standard spatially flat $\Lambda$CDM model, while successful, faces significant observational tensions, most notably the Hubble tension (the discrepancy between early-universe $H_0$ inferred from Planck CMB data and late-universe $H_0$ from SH0ES local distance-ladder measurements) and the coincidence problem (why dark energy and dark matter densities are comparable today).

To address these issues, the paper investigates the time-varying vacuum $\Lambda(t)$CDM model. In this framework, the vacuum energy density is not constant but evolves with cosmic time and interacts with the matter sector. This model is phenomenologically equivalent to a Generalized Chaplygin Gas (GCG), unifying Dark Matter (DM) and Dark Energy (DE) into a single fluid. The study aims to:

• Quantify how vacuum dynamics affect the expansion history and effective equations of state.
• Constrain the vacuum dynamics parameter ($\alpha$) using the latest observational data.
• Determine if this model can alleviate the Hubble tension compared to the standard $\Lambda$CDM model.

2. Methodology

Theoretical Framework

The author analyzes a spatially flat and non-flat universe described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The vacuum energy density is parameterized as $\Lambda(t) \propto H^{-2\alpha}$, where $H$ is the Hubble parameter and $\alpha$ is the dimensionless vacuum dynamics parameter.

• $\alpha = 0$: Recovers the standard $\Lambda$CDM model.
• $\alpha > 0$: Corresponds to a quintessence-like regime (vacuum energy decays into matter).
• $\alpha < 0$: Corresponds to a phantom-like regime (matter decays into vacuum).

The paper derives explicit analytical expressions for:

1. Normalized Hubble Parameter $E(a)$: Describing the expansion history.
2. Total Effective Equation of State (EoS) $w_{tot}(a)$: Characterizing the unified cosmic fluid.
3. Effective DE EoS $w_{DE}(a)$: Treating DE as a distinct component.
4. Deceleration Parameter $q(a)$: Determining the transition from deceleration to acceleration.

Observational Data

The study utilizes two primary datasets:

1. Cosmic Chronometers ($H(z)$): 32 measurements in the redshift range $z \in [0.07, 1.965]$.
2. Baryon Acoustic Oscillations (BAO): 13 measurements from the DESI Data Release 2 (DR2) in the range $z \in [0.295, 2.330]$.

Statistical Analysis

• Parameter Estimation: Full Markov Chain Monte Carlo (MCMC) analysis using the Cobaya software package to constrain parameters ($H_0, \Omega_{m0}, \Omega_{k0}, \alpha$) for both flat and non-flat models.
• Model Comparison: The performance of $\Lambda(t)$CDM is compared against standard $\Lambda$CDM using:
  • Goodness-of-fit ($\chi^2_{min}$, reduced $\chi^2$).
  • Information Criteria: Akaike Information Criterion (AIC), corrected AIC (AICc), and Bayesian Information Criterion (BIC).
  • Akaike weights ($w_i$) to quantify relative model probability.

3. Key Contributions

1. First DESI DR2 Analysis of $\Lambda(t)$CDM: This is the first observational analysis of both flat and non-flat $\Lambda(t)$CDM models using the latest DESI DR2 BAO data.
2. Reconstruction of Dynamics: The paper provides explicit reconstruction formulas for the total effective EoS, DE EoS, and deceleration parameters within the $\Lambda(t)$CDM framework, clarifying the distinction between the unified fluid behavior and the specific DE component behavior.
3. Resolution of Degeneracies: It demonstrates that while $H(z)$ data alone suffers from strong parameter degeneracies (especially in non-flat models), the inclusion of DESI BAO data significantly tightens constraints and breaks these degeneracies.

4. Key Results

Theoretical Dynamics

• Expansion History:
  • For $\alpha > 0$, the expansion rate $E(a)$ decreases faster than in $\Lambda$CDM (quintessence-like).
  • For $\alpha < 0$, $E(a)$ decreases slower (phantom-like).
• Equation of State:
  • The total effective EoS ($w_{tot}$) transitions from 0 (matter domination) to -1 (vacuum domination).
  • Positive $\alpha$ delays the transition from matter domination to DE domination.
  • Crucially, at the present epoch ($a=1$), $w_{DE} = -1$ (matching $\Lambda$CDM), but $w_{tot} = -\Omega_{DE0}$ due to residual matter contributions. The model only asymptotically approaches the standard $\Lambda$CDM scenario in the future.
• Acceleration: Stronger vacuum-matter interaction (higher $\alpha$) delays the onset of cosmic acceleration.

Observational Constraints

• Parameter Constraints:
  • $H(z)$ alone: Results in large uncertainties and strong degeneracies, particularly for the non-flat $\Lambda(t)$CDM model ($H_0 \approx 61 \pm 9$ km/s/Mpc).
  • $H(z) + \text{DESI BAO}$: Significantly tightens constraints.
    • Flat $\Lambda(t)$CDM: $H_0 = 67.87 \pm 0.91$ km/s/Mpc, $\alpha = -0.127 \pm 0.097$.
    • Non-flat $\Lambda(t)$CDM: $H_0 = 67.76 \pm 1.66$ km/s/Mpc, $\alpha = -0.081 \pm 0.133$, $\Omega_{k0} \approx 0$.
• Vacuum Dynamics Parameter ($\alpha$): In all scenarios, $\alpha$ is statistically consistent with zero ($\alpha \simeq 0$) within $1\sigma$to$2\sigma$confidence levels. This implies no significant deviation from the standard$\Lambda$CDM model is detected.
• Hubble Tension:
  • The inclusion of BAO data reduces the tension with Planck CMB results to < 1$\sigma$ for all models.
  • The tension with SH0ES local measurements is reduced to $\sim 2.6–3.2\sigma$.
• Model Selection:
  • While non-flat $\Lambda(t)$CDM sometimes achieves a slightly lower $\chi^2_{min}$, the Akaike weights and BIC/AIC strongly favor the simpler spatially flat $\Lambda$CDM model.
  • The flat $\Lambda$CDM model carries 44–58% of the total statistical support, whereas extended models receive significantly less weight due to the penalty for the extra parameter $\alpha$.

5. Significance and Conclusion

The paper concludes that while the $\Lambda(t)$CDM model is a physically viable extension that unifies DM and DE and offers a mechanism to potentially alleviate the Hubble tension, current observational data (specifically DESI DR2 combined with $H(z)$) do not provide evidence for vacuum dynamics.

• The data strongly drive the vacuum dynamics parameter $\alpha$ toward zero, effectively recovering the standard $\Lambda$CDM model.
• The addition of DESI BAO data is critical; it resolves parameter degeneracies inherent in $H(z)$ data alone and reinforces the preference for a spatially flat universe with a cosmological constant.
• The study highlights that while dynamical vacuum models can theoretically ease the Hubble tension, the specific $\Lambda(t)$CDM parametrization tested here is indistinguishable from standard $\Lambda$CDM given current precision, suggesting that the Hubble tension may require different theoretical modifications or that the tension is statistical/systematic in nature rather than a sign of new vacuum physics.