Mapping the redshift drift at various redshifts through cosmography

This paper investigates the redshift drift within a cosmographic framework by constraining expansion parameters using Pantheon+, SH0ES, GRB, and DESI data to construct a mock Sandage-Loeb catalog, ultimately assessing the internal consistency of the reconstructed background against $\Lambda$CDM and $\omega_0\omega_1$CDM scenarios.

The Gist

Imagine the universe as a giant, expanding loaf of raisin bread baking in an oven. As the dough rises, the raisins (galaxies) move away from each other. For decades, astronomers have been trying to figure out how the dough is rising: Is it rising at a steady pace? Is it speeding up? Is it slowing down?

This paper is like a team of cosmic detectives trying to solve that mystery using a new, ultra-precise stopwatch called Redshift Drift.

Here is the breakdown of their investigation, explained simply:

1. The Mystery: The "Cosmic Speedometer"

Usually, when we look at distant galaxies, we see them as they were billions of years ago. We measure their "redshift" (how much their light has stretched), which tells us how fast they were moving back then. But that's like looking at a car's speedometer from yesterday; it doesn't tell us if the driver is currently pressing the gas or the brake.

Redshift Drift is different. It's like watching the car in real-time. If the universe is accelerating, the light from a distant galaxy will slowly stretch even more over the next few decades. By measuring this tiny change in speed over time, we can directly see if the universe is speeding up or slowing down right now.

2. The Tools: Two Different Maps

To interpret these measurements, the scientists needed a map of the universe's history. They tried drawing this map in two different ways:

• The Taylor Expansion (The Straight Line): Imagine trying to draw a winding road by connecting dots with straight lines. It works well for short distances, but if you try to draw a long, curvy road, the lines start to look jagged and inaccurate. This is the "Taylor" method. It's simple but gets messy when looking very far back in time (high redshift).
• The Padé Approximant (The Smooth Curve): This is like using a flexible ruler or a smooth spline to connect the dots. It bends naturally with the road. The scientists found this method was much better at handling the "curves" of the universe when looking at very distant objects.

3. The Evidence: Gathering Clues

The team gathered clues from three different sources to build their map:

• Supernovae (The Standard Candles): Exploding stars that act like lightbulbs of known brightness. By seeing how dim they look, we know how far away they are.
• Gamma-Ray Bursts (The Flashlights): These are incredibly bright explosions from the early universe. They are like flashlights we can see from the "deep past" (very high redshift), but they are tricky to calibrate.
• DESI (The Ruler): A massive survey measuring the "fingerprint" of the early universe (Baryon Acoustic Oscillations) to measure distances.

4. The Experiment: The "Mock" Test

Here is the clever part of the paper. The scientists knew that real Redshift Drift data won't be available for another decade or so (from telescopes like the E-ELT). So, they created a "Mock" (Fake) Catalog.

• Step 1: They used their best guesses from the Supernovae and Gamma-Ray data to predict what the Redshift Drift should look like.
• Step 2: They treated these predictions as if they were real data and fed them back into their computer models.
• Step 3: They checked if the model stayed consistent. It was like a chef tasting a soup they are cooking, adding a pinch of salt, and then tasting it again to see if the flavor profile holds up.

5. The Findings: What Did They Discover?

• The "Hubble Tension": There is a famous disagreement in astronomy. Some measurements say the universe is expanding at a speed of 67 km/s/Mpc, while others say ~73. The team found that when they used the "Taylor" (straight line) method, their results leaned toward the faster speed (73). When they added the DESI data, the results got messy and didn't match either side perfectly.
• The Smooth Curve Wins: When they used the "Padé" (smooth curve) method, their results for the universe's acceleration (deceleration parameter) matched the standard "Lambda-CDM" model (the current best theory) much better.
• The "Jerk" Problem: In physics, "jerk" is the rate of change of acceleration. The team found that when they included the new DESI data, the "jerk" parameter started acting weirdly, suggesting the universe might not be behaving exactly as the standard model predicts.
• The Mock Data's Role: Adding the fake Redshift Drift data didn't change the center of their answers much, but it made the error bars (the uncertainty) much smaller. It tightened the net, giving them a more precise picture of how the universe is accelerating.

The Big Picture Analogy

Imagine you are trying to predict the weather for next week.

• Old Method: You look at the temperature today and yesterday and draw a straight line to guess next week. (Taylor).
• New Method: You look at the temperature, humidity, and wind patterns and draw a smooth, curved line that accounts for storms. (Padé).
• The Twist: You don't have a weather station for next week yet, so you simulate one based on your current best guess. You then check: "If my simulation is right, does my weather model still make sense?"

The Conclusion:
The paper concludes that while the universe seems to be accelerating (speeding up), the way we calculate that acceleration matters immensely. The "smooth curve" (Padé) method seems to handle the data better than the "straight line" (Taylor) method, especially when looking at the deep past.

Most importantly, they found that the new data from DESI is causing some friction with our current theories. It's possible that Dark Energy (the invisible force pushing the universe apart) isn't a constant "cosmological constant" as we thought, but something that changes over time. The Redshift Drift test, once we get real data in the future, will be the ultimate tie-breaker to settle this cosmic debate.

Technical Summary

1. Problem Statement

The paper addresses the need to test the expansion history of the Universe and the nature of dark energy without relying on specific dynamical models (model-independent approach). While the Redshift Drift (RD) or Sandage-Loeb (SL) test offers a direct kinematic measurement of cosmic acceleration by observing the slow time variation of redshift ($\dot{z}$), no such data exists yet. Future facilities like the E-ELT (CODEX) and SKA are expected to provide this data.

The authors aim to:

• Investigate the RD within a cosmographic framework (expanding the scale factor in a Taylor series or using Padé approximants) to avoid assumptions about the dark energy equation of state.
• Assess the consistency of current cosmological probes (Supernovae, Gamma-Ray Bursts, BAO) with the expected SL signal.
• Analyze the tension between recent DESI (Dark Energy Spectroscopic Instrument) data and standard cosmological models ($\Lambda$CDM and dynamical dark energy scenarios).

2. Methodology

A. Cosmographic Framework

Instead of assuming a specific Friedmann-Lemaître-Robertson-Walker (FLRW) model with a specific dark energy fluid, the authors expand the Hubble rate $H(z)$ using kinematic parameters defined at the present epoch ($z=0$):

• Hubble Constant ($H_0$)
• Deceleration Parameter ($q_0$)
• Jerk Parameter ($j_0$)

Two parameterizations are used to model $H(z)$:

1. Taylor Expansion (Second Order): $H_T(z) \approx H_0 [1 + (1+q_0)z + \frac{1}{2}(j_0 - q_0^2)z^2]$.
2. Padé Approximant (2,1): A rational polynomial form designed to improve convergence at high redshifts ($z \geq 1$) where Taylor series often fail.

B. Data Probes

The analysis combines multiple datasets to constrain the parameters:

• Pantheon+ & SH0ES: 1,701 Type Ia Supernovae (SNeIa) calibrated with Cepheid variables to constrain $H_0$.
• Gamma-Ray Bursts (GRBs): 118 GRBs following the $E_p - E_{iso}$ (Amati) correlation. To avoid the "circularity problem" (where GRB calibration depends on a cosmological model), the authors use Bézier polynomials fitted to Observational Hubble Data (OHD) to reconstruct $H(z)$ model-independently for calibration.
• DESI BAO: Data from the second release (DR2) of the Dark Energy Spectroscopic Instrument.
• Mock Sandage-Loeb (SL) Data: A synthetic catalog of 30 velocity shift measurements ($\Delta v$) in the redshift range $2 \leq z \leq 5$, generated using the parameters derived from the preliminary fits.

C. Analysis Strategy

The authors perform a four-step Markov Chain Monte Carlo (MCMC) analysis using the MontePython sampler and modified CLASS code:

1. Analysis 1: Constrain parameters using SNeIa + GRB.
2. Analysis 2: Constrain parameters using SNeIa + GRB + DESI-BAO.
3. Analysis 1SL & 2SL: Repeat the above including the Mock SL data to test internal consistency and tighten constraints.
4. Comparison: Compare results against the $\Lambda$CDM and $\omega_0\omega_1$CDM (dynamical dark energy) reference models.

3. Key Contributions

• Model-Independent RD Mapping: The paper provides a rigorous framework for predicting the redshift drift signal using only kinematic parameters, bypassing specific dark energy models.
• GRB Calibration Innovation: It employs a robust, model-independent calibration of GRBs using Bézier polynomials and OHD, allowing them to serve as reliable distance indicators up to $z \approx 8.2$.
• DESI-BAO Tension Analysis: The study highlights significant tensions between DESI-DR2 data and standard cosmological models when combined with other probes, specifically regarding the jerk parameter ($j_0$).
• Internal Consistency Test: By generating mock SL data based on preliminary fits and re-analyzing, the authors demonstrate how future SL data would impact current constraints, serving as a consistency check rather than a blind forecast.

4. Key Results

A. Impact of Parameterization (Taylor vs. Padé)

• Without DESI (Analysis 1):
  • Taylor: $H_0$ agrees with local measurements (Riess et al., $H_0 \approx 73$) but not Planck ($H_0 \approx 67$). $q_0$ and $j_0$ agree with the dynamical $\omega_0\omega_1$CDM model at $1\sigma$ but show tension with $\Lambda$CDM.
  • Padé: Improves agreement with $\Lambda$CDM for $q_0$ and $j_0$ at the $1\sigma$ level, suggesting Padé approximants better handle high-redshift convergence issues.
• With DESI (Analysis 2):
  • The inclusion of DESI-BAO data degrades the overall consistency.
  • $H_0$ becomes incompatible with both local and CMB measurements.
  • The jerk parameter $j_0$ deviates significantly from the $\Lambda$CDM prediction ($j_0=1$) and the dynamical model, showing no agreement at $1\sigma$.
  • The agreement for $q_0$ with $\Lambda$CDM drops to the $2\sigma$ level when using Padé.

B. Impact of Mock Sandage-Loeb Data

• Error Reduction: Adding the mock SL data significantly tightens the error bars on $q_0$ and $j_0$ (reducing uncertainties by roughly 30-40% in some cases).
• Shift in Compatibility:
  • In Analysis 1SL (SNe+GRB+SL), the Taylor reconstruction shows reduced compatibility with $\omega_0\omega_1$CDM for $q_0$ (dropping to $2\sigma$).
  • The Padé parameterization maintains $1\sigma$ compatibility for $j_0$ with $\Lambda$CDM even with SL data.
  • In Analysis 2SL (including DESI), the tension with $\Lambda$CDM for $q_0$ persists at the $2\sigma$ level when using Padé.
• Physical Insight: The SL signal primarily constrains the deceleration parameter $q_0$ (as it directly probes acceleration). However, when BAO data is removed, the SL data also influences the jerk parameter $j_0$, indicating a strong interplay between high-redshift probes and higher-order cosmographic terms.

C. Redshift Drift Behavior

The reconstructed redshift drift ($\dot{z}$) generally follows the $\Lambda$CDM and $\omega_0\omega_1$CDM predictions up to $z \approx 3$. Beyond this, deviations appear, particularly when DESI data is included or when using Padé approximants. The SNe+GRB+SL reconstruction tends to align more closely with the dynamical dark energy scenario than the standard $\Lambda$CDM.

5. Significance and Conclusion

This work demonstrates that cosmography is a viable tool for interpreting future Redshift Drift data. The study reveals that:

1. DESI-DR2 Data: Introduces significant tension with standard models, particularly regarding the jerk parameter, supporting recent literature suggesting dynamical dark energy or systematics in BAO analysis.
2. Parameterization Matters: The choice between Taylor and Padé expansions significantly affects the compatibility with $\Lambda$CDM, with Padé offering better convergence and agreement at high redshifts.
3. Future Prospects: The inclusion of future SL data will primarily tighten constraints on the acceleration history ($q_0$) and help resolve the current tensions in the jerk parameter ($j_0$).

The authors conclude that while the mock SL data improves precision, the central values remain sensitive to the dataset combination (specifically the inclusion of DESI and GRBs) and the mathematical form of the Hubble rate expansion. Future work should focus on independent BAO reconstructions and testing alternative parameterizations to fully understand the nature of dark energy.