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Constraining dark energy with complementary probes of large-scale structure

This paper demonstrates that combining redshift space distortions, $3\times2$ pt measurements, and Integrated Sachs-Wolfe cross-correlations with standard expansion history probes significantly improves constraints on dark energy perturbations within the Effective Field Theory framework, revealing strong complementarity and a notable $\sim3\sigma$ deviation from the $\Lambda$ CDM model.

Original authors: Neel Shah, Kazuya Koyama, Johannes Noller, Lanyang Yi

Published 2026-07-01

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CC BY 4.0

Original authors: Neel Shah, Kazuya Koyama, Johannes Noller, Lanyang Yi

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

Imagine the universe is a giant, expanding balloon. For decades, scientists have been trying to figure out what is inside that balloon pushing it to expand faster and faster. They call this mysterious pushing force "Dark Energy."

For a long time, the best guess was that this force is a constant, unchanging "cosmological constant" (like a fixed amount of air in the balloon). But recent clues suggest it might be something more dynamic, like a fluid that changes its behavior over time.

This paper is like a team of detectives (Neel Shah, Kazuya Koyama, and colleagues) trying to solve the mystery of Dark Energy. They didn't just look at one piece of evidence; they combined a massive array of different "probes" (clues) to get a clearer picture.

Here is the breakdown of their investigation in simple terms:

1. The Toolkit: A Detective's Bag of Tricks

To understand Dark Energy, you need to look at two things:

The Background Expansion: How fast the universe is stretching (like watching the balloon get bigger).
The Structure Growth: How galaxies clump together or spread apart (like watching how the paint on the balloon stretches and cracks).

The team used a sophisticated mathematical framework called EFTofDE (Effective Field Theory of Dark Energy). Think of this as a "universal translator" that allows them to test many different theories of Dark Energy at once, rather than testing them one by one.

They combined data from four major sources:

The Cosmic Microwave Background (CMB): The "baby picture" of the universe, showing us the afterglow of the Big Bang.
Redshift Space Distortions (RSD): Measuring how fast galaxies are moving toward or away from us, which tells us how gravity is pulling them together.
Weak Gravitational Lensing (3x2pt): Looking at how the gravity of galaxy clusters bends light from distant galaxies, acting like a cosmic magnifying glass.
The Integrated Sachs-Wolfe (ISW) Effect: A subtle signal where light gains or loses energy as it travels through the changing gravitational landscape of the expanding universe.

2. The Big Discovery: "The Whole is Greater Than the Sum of Its Parts"

The paper's main finding is about complementarity. Imagine trying to solve a puzzle where some pieces are blurry. If you only look at the "expansion" pieces, you get a vague idea. If you only look at the "clumping" pieces, you get a different vague idea.

But when they combined all the clues:

The "Figure of Merit" (a score for how well they can pin down the answer) jumped by nearly 3 times.
Specifically, adding the new data (from the DESI and DES surveys) made their constraints on Dark Energy's behavior 2.69 times tighter than before.
When they translated these results into the "language" of how gravity changes (using functions called $\mu$ and $\Sigma$ ), the improvement was even better: 3.37 times tighter.

It's like going from looking at a blurry photo of a suspect to having a high-definition, 3D scan.

3. The Twist: The "Stability Rule"

One of the most interesting parts of the paper is a discovery about how the universe's expansion and its structure are secretly linked.

In their mathematical model, there is a rule called the Gradient Stability Condition. Think of this as a "safety valve" for the universe. If the Dark Energy behaves in a certain way, the universe would become unstable and collapse or explode in a way that doesn't match reality.

The team found that this safety rule creates a "shadow" on their data. Even if the data itself didn't explicitly rule out the standard "Cosmological Constant" theory (where Dark Energy is just a fixed number), the mathematical requirement for stability pushed the results away from that simple theory.

The Analogy: Imagine you are trying to balance a broom on your hand. If you just look at your hand's movement (the data), you might think you can balance it perfectly still. But if you know the physics of the broom (the stability rule), you realize that to keep it from falling, you must move your hand in a specific way. The "rules of physics" forced the answer to be different from the "simplest guess."

4. The Verdict: Is Dark Energy Different from the Standard Model?

The standard model of cosmology ( $\Lambda$ CDM) assumes Dark Energy is a constant. The team asked: "How sure are we that Dark Energy is not just a constant?"

Using the standard method (ignoring the complex structure of Dark Energy), the data suggested a deviation of 3.1 sigma (a statistical way of saying "very likely, but not quite certain").
Using their new, more complex method (EFTofDE) which accounts for the stability rules, the deviation was 2.9 sigma.

The Surprising Result: Even though their new model was more complex and allowed for more freedom, the result didn't change much. The "safety valve" (stability rule) pulled the expansion results back toward the standard model, while the structure results pushed them away. These two effects cancelled each other out, leaving the final "significance" of the deviation almost the same as the simpler model.

Summary

This paper didn't find a new type of Dark Energy, but it did something arguably more important: It showed how to listen to the universe more clearly.

By combining different types of cosmic data and respecting the fundamental rules of physics (stability), they tightened the constraints on what Dark Energy can be. They proved that while the universe is still a bit of a mystery, we are now much better at narrowing down the list of suspects. The "simple" constant theory is still a strong contender, but the universe is pushing back, hinting that the truth might be slightly more dynamic than we thought.

Technical Summary: Constraining Dark Energy with Complementary Probes of Large-Scale Structure

Problem Statement
The nature of dark energy (DE) remains a central pursuit in cosmology. While the background expansion history (constrained by CMB, BAO, and SNe) provides limits on dynamical DE, signatures on the evolution of cosmological perturbations contain further microphysical information and are closely linked to modified gravity. Existing analyses often focus on background constraints or specific perturbation probes in isolation. This work addresses the need to consistently model cosmological perturbations within the Effective Field Theory of Dark Energy (EFTofDE) framework alongside background expansion, utilizing a comprehensive array of complementary large-scale structure (LSS) probes to constrain the joint theory space.

Methodology
The authors employ the EFTofDE framework, specifically working in the $\alpha$ -basis which parametrizes the theory via the DE equation of state $w(z)$ and two free functions describing the braiding ( $\alpha_B$ ) and the running of the Planck mass ( $\alpha_M$ ). These are parametrized as:
$w(a) = w_0 + w_a(1-a)$
$\alpha_B = c_B \Omega_{DE}(z), \quad \alpha_M = c_M \Omega_{DE}(z)$
The study focuses on the perturbation parameters $\{c_B, c_M\}$ and background parameters $\{w_0, w_a\}$ .

Datasets and Likelihoods
The analysis supplements the standard "minimal" combination of CMB anisotropies (Planck PR4), CMB lensing, BAO (DESI DR2), and SNe (DES Y5) with several LSS probes:

Redshift Space Distortions (RSD): DESI DR1 full-shape measurements (combined with BAO to account for cross-covariance).
Weak Lensing (3 $\times$ 2pt): DES Y3 measurements of cosmic shear, galaxy clustering, and galaxy-galaxy lensing.
Integrated Sachs-Wolfe (ISW) Effect: Two cross-correlation likelihoods:
- ISW-Tg: CMB temperature anisotropies cross-correlated with galaxy number counts.
- ISW-T $\kappa$ : CMB temperature anisotropies cross-correlated with CMB lensing.

Covariance Handling: The authors adopt a cautious approach regarding dataset combinations. They avoid combining probes with non-negligible, unmodelled covariances (e.g., 3 $\times$ 2pt is not combined with CMB lensing or ISW-T $\kappa$ ; ISW-Tg and ISW-T $\kappa$ are not combined).

Quantification of Constraints:
Due to the highly non-Gaussian nature of the posteriors for $\{c_B, c_M\}$ , traditional Gaussian-based metrics are insufficient. The authors define:

Posterior Width: The difference between upper and lower 68% limits for 1D parameters.
Figure of Merit (FoM): The inverse area of the 68% credible region for the 2D posterior of $\{c_B, c_M\}$ , normalized to the minimal dataset combination.

Key Results

Complementarity of Probes:
- RSD: Primarily improves constraints on $c_M$ (reducing posterior width by 22%) by breaking degeneracies in the growth rate direction.
- 3 $\times$ 2pt: Significantly stronger than CMB lensing for DE perturbations, reducing the posterior width of $c_B$ by 27% and $c_M$ by 20%. It effectively cuts off the posterior region of low $c_M$ and high $c_B$ .
- ISW-Tg: Shifts the means of both parameters and reduces the $c_M$ width by 24%. It uniquely extends the 95% credible region in the direction of positively correlated $\{c_B, c_M\}$ while cutting off the lower-right tail.
- Combined Constraints: The most constraining combination (Minimal + DESI DR1 RSD + DES Y3 3 $\times$ 2pt + ISW-Tg) improves the FoM for $\{c_B, c_M\}$ by a factor of 2.69 compared to the minimal dataset.
Phenomenological Implications:
Mapping constraints to the phenomenological functions $\{\mu(z), \Sigma(z)\}$ (modifications to the Newtonian and Weyl potentials), the combined dataset yields an FoM improvement of 3.37. The constraints firmly place the present-day values in the regime $\mu_{today} > 1$ and $\Sigma_{today} < 1$ , disfavouring strong enhancements to matter clustering or lensing. Specifically:
- RSD disfavors $\mu(z) \gtrsim 0.2$ at $z \lesssim 2$ .
- 3 $\times$ 2pt disfavors $\Sigma(z) \gtrsim 0.1$ at $z \lesssim 2$ .
- ISW-Tg constrains the time evolution of $\Sigma(z)$ at very low redshifts ( $z \lesssim 0.5$ ).
Interplay of Background and Perturbations:
- Gradient Stability Prior: A theoretical prior arising from the requirement of positive sound speed ( $c_s^2 > 0$ ) couples the background $\{w_0, w_a\}$ and perturbations $\{c_B, c_M\}$ . This prior creates a boundary in the 4D parameter space that disfavors the General Relativity (GR) limit ( $c_B=c_M=0$ ) even when data has no direct sensitivity to perturbations, provided the background expansion deviates from $\Lambda$ CDM.
- Background Dependence: Assuming a fixed $\Lambda$ CDM background weakens constraints on $\{c_B, c_M\}$ compared to the dynamical $w_0w_a$ background, despite the latter having more free parameters.
Significance of Deviation from $\Lambda$ CDM:
The authors compute the significance of deviation from $\Lambda$ CDM using the $\Delta\chi^2_{MAP}$ metric.
- For the standard $w_0w_a$ CDM model (using PPF formalism), the deviation is 3.1 $\sigma$ .
- For the EFTofDE model (consistently modelling subhorizon perturbations), the deviation is 2.9 $\sigma$ .
  The authors note this result is "nontrivially similar," indicating that the competing effects of the gradient stability prior (pulling $\{w_0, w_a\}$ toward $\Lambda$ CDM) and the perturbation constraints (pushing $\{c_B, c_M\}$ away from GR) largely compensate for each other.

Significance and Claims
The paper claims to demonstrate that combining complementary LSS probes (RSD, 3 $\times$ 2pt, and ISW) significantly tightens constraints on DE perturbations within the EFTofDE framework, improving the Figure of Merit by nearly a factor of 3 compared to background-only probes.

Crucially, the work highlights the interdependence between background expansion constraints and perturbation constraints mediated by the gradient stability condition. This theoretical prior is shown to be a dominant factor in shaping the posteriors near the GR limit, often driving the constraints away from GR even when observational data alone might suggest consistency. The authors conclude that consistently modelling DE perturbations at the Lagrangian level does not drastically alter the observational significance of dynamical DE over $\Lambda$ CDM compared to standard phenomenological approaches, finding a deviation of 2.9 $\sigma$ versus 3.1 $\sigma$ .

The study emphasizes the necessity of careful covariance handling when combining diverse LSS probes and provides a physical interpretation of how different probes (clustering vs. lensing) constrain specific regimes of modified gravity phenomenology.