Joint Curvature and Growth Rate measurements with Supernova Peculiar Velocities and the CMB

This paper demonstrates that combining Type Ia supernova peculiar velocity measurements with CMB data allows for the simultaneous constraint of cosmic curvature, the growth rate of structure, and matter density, revealing hints of positive curvature and a resolution of the $H_0$ tension through negative curvature and suppressed structure growth when external distance ladder data is included.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, cosmologists have been trying to figure out exactly how this balloon is shaped, how fast it's inflating, and what's inside it. They usually do this by looking at "standard candles"—stars called Type Ia supernovae that explode with a predictable brightness. By measuring how dim they look, we can tell how far away they are.

But this paper introduces a clever new twist: instead of just looking at where the stars are, the authors look at how they are moving.

Here is a simple breakdown of what they did, using some everyday analogies.

1. The "Drunk Walker" vs. The "Crowded Street"

Usually, when we look at a supernova, we assume it's just drifting along with the general expansion of the universe, like a leaf floating down a river.

However, the universe isn't empty. It has clumps of matter (galaxies, dark matter) that act like gravity magnets. These magnets pull on the supernovae, giving them a little "kick" or Peculiar Velocity.

• The Analogy: Imagine you are walking down a crowded street (the expanding universe). Most people are walking forward at the same speed. But if you see a group of people huddled together (a galaxy cluster), you might get pulled slightly toward them, or pushed away if you're near a crowd moving the other way.
• The Insight: These little "kicks" aren't random. They trace the invisible map of where all the matter is hiding. By measuring these kicks, the authors can map out the "skeleton" of the universe.

2. The Detective Team: Two Tools, One Case

The authors combined two very different detective tools to solve the mystery of the universe's shape and growth:

• Tool A: The Supernovae (The Local Scouts): They used two massive lists of exploding stars (Pantheon+ and DES-Y5). These are great at seeing what's happening in the "neighborhood" (the nearby universe).
• Tool B: The CMB (The Ancient Snapshot): They used data from the Cosmic Microwave Background (the afterglow of the Big Bang). This is like looking at a baby photo of the universe to see how it started.

Why combine them?
If you only look at the baby photo, you might guess the baby will grow into a tall person, but you aren't sure. If you only look at the neighborhood, you see the person walking, but you don't know their full history. Putting them together gives a much clearer picture.

3. The Big Questions They Asked

The team asked three big questions, which are like trying to solve a puzzle with three missing pieces:

1. Is the universe flat or curved? (Is the balloon perfectly round, or is it slightly squashed or stretched?)
2. How fast are structures growing? (Are galaxies clumping together quickly or slowly?)
3. What is the Hubble Constant? (How fast is the universe expanding right now?)

4. The Surprising Findings

Here is what they discovered, translated from "science-speak" to plain English:

• The "Flatness" Hint: Standard physics says the universe should be perfectly flat (like a sheet of paper). However, when they combined the supernova kicks with the ancient baby photo, the data hinted that the universe might actually be slightly curved (like a sphere), not flat. It's a small hint (about a 2-3% chance of being wrong), but it's exciting because it challenges our current models.
• The Growth Rate: They measured how fast cosmic structures are growing. The result matched the predictions of Einstein's General Relativity perfectly. It's like checking a recipe and finding the cake rose exactly as the instructions said it should.
• The "Hubble Tension" (The Big Argument): Right now, there is a famous fight in science. One group measures the universe's expansion speed using the "baby photo" (CMB) and gets one number. Another group measures it using local stars and gets a faster number. They disagree by a lot!
  • The Paper's Twist: When the authors added their new "kick" data and allowed the universe to be curved and grow at different rates, the gap between the two groups got smaller.
  • The Catch: The gap didn't disappear because the numbers changed; it disappeared because the uncertainty got bigger. It's like two people arguing about the temperature. One says 70°F, the other says 80°F. If you suddenly say, "Well, we don't really know the thermometer is accurate, so it could be anywhere between 60 and 90," the argument feels less intense, but you still don't know the real temperature. The paper suggests that if the universe is curved, the tension is less severe, but it doesn't fully solve the mystery.

5. The "Lens" Analogy

The paper also checked if the universe acts like a weird lens that bends light more than expected (called CMB lensing). They found that the universe behaves exactly as standard physics predicts—it's not a "funhouse mirror." This makes their other findings (about curvature and growth) even more trustworthy.

The Bottom Line

This paper is like upgrading from a blurry black-and-white photo to a high-definition color video. By listening to the "whispers" of moving stars (peculiar velocities) and combining them with the "echo" of the Big Bang, the authors showed that:

1. We can measure the universe's growth and shape simultaneously.
2. The universe might be slightly curved, not perfectly flat.
3. The famous disagreement about the universe's expansion speed might be explained by the universe's shape, but we need more data to be sure.

It's a reminder that even with our current tools, the universe still has a few secrets left to tell us, and sometimes you just need to look at how things are moving to understand where they are going.

Technical Summary

1. Problem Statement

Current cosmology faces significant tensions, most notably the Hubble tension (discrepancy between local $H_0$ measurements and CMB-inferred values) and potential anomalies in the growth of structure ($\sigma_8$) and spatial curvature ($\Omega_k$).

• Standard Limitations: Type Ia Supernovae (SN) are traditionally used as distance indicators to constrain background expansion parameters ($\Omega_m, \Omega_k, w$). However, their Peculiar Velocities (PV)—deviations from the Hubble flow caused by gravitational infall into large-scale structures—have historically been treated as noise.
• The Opportunity: SN peculiar velocities trace the underlying matter distribution and growth rate of cosmic structures. While next-generation surveys (e.g., LSST, ZTF) promise high-precision PV data, current catalogs (Pantheon+ and DES-Y5) are limited in size.
• The Gap: There is a need to determine if current SN catalogs, when combined with CMB data, can simultaneously constrain the growth index ($\gamma$), spatial curvature ($\Omega_k$), and matter fluctuation amplitude ($\sigma_8$) to break degeneracies inherent in CMB-only analyses.

2. Methodology

The authors employ a Bayesian statistical framework to analyze SN peculiar velocities in conjunction with CMB data.

• Datasets:
  • Supernovae: Two catalogs are used: Pantheon+ (1550 SNe, $0.001 < z < 2.3$) and DES-Y5 (1800 SNe, $0.025 < z < 1.13$).
  • CMB: Planck PR4 data (HiLLiPoP and LoLLiPoP likelihoods for TT, TE, EE spectra, plus lensing).
  • Local $H_0$: SH0ES prior ($H_0 = 73.30 \pm 1.04$ km/s/Mpc) is used in specific tests to probe the Hubble tension.
• Likelihood Modeling:
  • The analysis uses a Gaussian likelihood for the distance modulus residuals ($\delta_m$).
  • The covariance matrix ($C$) is decomposed into three components:
    1. $C_{PV}$: Linear peculiar velocity correlations derived from linear perturbation theory (using CAMB-GammaPrime).
    2. $C_{nonlin}$: A diagonal matrix modeling non-linear velocity dispersion (nuisance parameter $\sigma_v$).
    3. $C_{cat}$: Catalog-specific errors (photometry, calibration, intrinsic scatter).
  • Redshift Cutoff: To ensure the validity of linear perturbation theory, the PV analysis is restricted to low redshifts: $z < 0.1$ for Pantheon+ and $z < 0.2$ for DES-Y5.
• Parameter Space:
  • Base Model: Flat $\Lambda$CDM with fixed growth index $\gamma = 0.55$.
  • Extended Models: Simultaneous variation of spatial curvature ($\Omega_k$), growth index ($\gamma$), and CMB lensing amplitude ($A_L$).
  • Priors: Flat/uninformative priors for cosmological parameters; BBN priors on $\omega_b$ for SN-only analyses.
• Inference: Markov Chain Monte Carlo (MCMC) sampling using Cobaya and emcee.

3. Key Contributions

1. Simultaneous Constraint of $\Omega_k, \gamma, \sigma_8$: The paper demonstrates that SN peculiar velocities and CMB data are highly complementary. While CMB data suffers from strong degeneracies between $\Omega_k$ and $\gamma$, SN PV data provides orthogonal constraints, allowing for the simultaneous measurement of all three parameters.
2. Validation of Current Catalogs: The study confirms that current catalogs (Pantheon+ and DES-Y5) contain sufficient PV information to yield competitive constraints on $\sigma_8$ and $\gamma$, despite their limited low-redshift sample sizes compared to future surveys.
3. Robustness to Lensing Anomalies: The analysis tests the impact of the CMB lensing amplitude parameter ($A_L$), showing that the results for $\Omega_k$ and $\sigma_8$ are robust even when $A_L$ is left as a free parameter.
4. Reframing the Hubble Tension: The authors investigate how introducing free curvature and growth parameters affects the Hubble tension when local $H_0$ data is included.

4. Key Results

A. Flat $\Lambda$CDM (Fixed $\gamma = 0.55$)

• $\sigma_8$ Constraints: Using SN PV alone, the authors find:
  • Pantheon+: $\sigma_8 = 0.73 \pm 0.22$
  • DES-Y5: $\sigma_8 = 0.87 \pm 0.31$
  • These values are consistent with CMB and galaxy survey results, showing improved precision over previous JLA catalog analyses.

B. Extended Models (Free $\Omega_k$ and $\gamma$)

• Curvature Hints: The joint CMB + SN analysis reveals a preference for positive spatial curvature (negative $\Omega_k$):
  • Pantheon+ + CMB: $\Omega_k = -0.011 \pm 0.006$ ($2.2\sigma$ significance).
  • DES-Y5 + CMB: $\Omega_k = -0.014 \pm 0.006$ ($3.0\sigma$ significance).
  • Note: This excludes a flat universe ($\Omega_k = 0$) at high significance in the combined analysis.
• Growth Index: The measured growth indices are consistent with General Relativity (GR) predictions ($\gamma \approx 0.55$):
  • Pantheon+ + CMB: $\gamma = 0.519^{+0.061}_{-0.099}$
  • DES-Y5 + CMB: $\gamma = 0.461^{+0.085}_{-0.069}$
• $f\sigma_8$ Measurements: Converted to growth rate measurements at effective redshifts:
  • $f\sigma_8(0.024) = 0.461^{+0.066}_{-0.035}$ (Pantheon+)
  • $f\sigma_8(0.038) = 0.498^{+0.045}_{-0.050}$ (DES-Y5)

C. Impact of $A_L$ (Lensing Amplitude)

• Allowing $A_L$ to vary does not significantly shift the posterior means for $\Omega_k$, $\sigma_8$, or $H_0$.
• However, it increases the uncertainty on $\gamma$ by a factor of ~5 due to strong degeneracy, highlighting the need for low-redshift probes to break this degeneracy.

D. The Hubble Tension

• Without SH0ES: The inclusion of free $\Omega_k$ and $\gamma$ in CMB-only data broadens the $H_0$ posterior, reducing the tension with local measurements from $5.0\sigma$ to roughly $2.2\sigma$ (though this is driven by increased error bars, not a shift in the peak).
• With SH0ES: When the SH0ES $H_0$ prior is added to the joint CMB+SN analysis:
  • The tension is "recast" rather than solved.
  • The model prefers positive curvature ($\Omega_k \approx 0.009$) and a higher growth index ($\gamma \approx 0.69$).
  • The standard flat GR model ($\Omega_k=0, \gamma=0.55$) is excluded at $>4.4\sigma$ (Pantheon+) or $3.1\sigma$ (DES-Y5).
  • This suggests the Hubble tension manifests as a preference for non-standard geometry and growth in this specific joint analysis.

5. Significance

• Complementarity: The work establishes that SN peculiar velocities are a powerful, independent probe that breaks the $\Omega_k$-$\gamma$ degeneracy plaguing CMB-only analyses.
• Precision: Even with current data, the joint analysis achieves ~15% precision on $\gamma$ and ~1% on $\sigma_8$, comparable to state-of-the-art galaxy clustering results.
• Future Outlook: The results validate the methodology for upcoming surveys (LSST, DESI, ZTF). As these surveys increase the number of low-redshift SNe, the precision on curvature and growth rates will improve, potentially resolving current cosmological tensions or confirming deviations from the standard $\Lambda$CDM model.
• Hubble Tension Insight: The findings suggest that simply adding curvature and modified growth parameters does not "fix" the Hubble tension but rather shifts the statistical preference toward a non-flat, non-standard growth universe, indicating the tension is robust and deep-seated.