One-loop power spectrum corrections in interacting dark energy cosmologies

This paper computes one-loop corrections to the matter power spectrum in Interacting Dark Energy models, reinterprets them within the Effective Field Theory of Large-Scale Structure framework, and uses BOSS DR12 data to constrain a novel velocity-dependent interaction term, finding results consistent with the standard $\Lambda$CDM model.

The Gist

The Big Picture: The Universe's Invisible Tug-of-War

Imagine the universe as a giant, expanding party. For decades, physicists have been trying to figure out the rules of this party. The current "rulebook" is called ΛCDM (Lambda Cold Dark Matter). It says there are two invisible guests at the party:

1. Dark Matter: The "glue" that holds galaxies together.
2. Dark Energy: The "push" that is making the party get bigger and bigger faster.

In the standard rulebook, these two guests never talk to each other. They just do their own thing. But recently, measurements of the universe have started to show some weird glitches (called "tensions"). It's like if you measured the size of a room with a tape measure and a laser, and they gave you two different answers.

This paper asks a simple question: What if Dark Matter and Dark Energy are talking to each other? What if they are exchanging energy, like two friends passing a ball back and forth? This is called Interacting Dark Energy (IDE).

The Problem: The "Mildly Nonlinear" Zone

To test this, scientists look at how galaxies are clustered together.

• The Linear Zone (Easy Mode): On very large scales, the universe is smooth and predictable. It's like watching a calm ocean; you can easily predict the waves.
• The Nonlinear Zone (Hard Mode): On smaller scales, galaxies clump together, crash into each other, and form complex structures. It's like a mosh pit at a concert. Things get chaotic, messy, and hard to calculate.

Most previous studies only looked at the "calm ocean" (Linear Zone). But the real clues about Dark Energy interactions might be hidden in the "mosh pit" (the mildly nonlinear zone). The problem is, doing the math for the mosh pit is incredibly difficult.

The Solution: A New Mathematical Toolkit

The authors of this paper did two main things:

1. They built a better calculator for the "Mosh Pit."
They used a method called Perturbation Theory (think of it as a way to add small corrections to a simple prediction to make it more accurate). They calculated the "one-loop corrections."

• Analogy: Imagine you are trying to predict the path of a leaf falling in a windstorm. A simple model says it falls straight down. A "one-loop" model adds the swirling winds and the leaf's wobble. The authors added these "wobbles" specifically for a universe where Dark Matter and Dark Energy are interacting.

2. They invented a new way to describe the interaction.
They looked at two old ways of describing the interaction (like "Dark Energy gives energy to Dark Matter at a steady rate") and found they didn't change the "mosh pit" math much. So, they invented a new interaction term:

• The New Idea: They proposed that the interaction depends on how fast the Dark Matter is "screaming" (moving/accelerating).
• Analogy: Imagine the interaction isn't a steady handoff of a ball, but a high-five that only happens when the players are running fast. This new term, called $\Gamma$, links the strength of the interaction to the velocity of the Dark Matter.

The Experiment: Checking the Data

The team took their new math and compared it against real data from the BOSS survey (a massive map of millions of galaxies). They used a technique called Full-Shape Analysis, which looks at the entire shape of the galaxy distribution, not just a few specific points.

They asked: "Does the data look better with the standard rulebook (no talking), or does it look better with our new 'High-Five' rulebook (interacting)?"

The Results: The "High-Five" is Quiet

Here is the punchline: The data didn't show any strong evidence that Dark Matter and Dark Energy are talking.

• They measured the interaction strength ($\Gamma$) and found it to be 0.0039, with a "fuzziness" (error bar) of 0.0082.
• Because the error bar is bigger than the number itself, the result is effectively zero.
• Analogy: It's like trying to hear a whisper in a hurricane. You might think you heard something, but the noise level is so high that you can't be sure. The data is perfectly happy with the standard model where the two dark sectors ignore each other.

Why This Paper Matters (Even if the Answer is "No")

You might think, "If they found nothing new, why write a paper?"

1. They opened the door: This is the first time anyone has successfully applied this complex "mosh pit" math to Interacting Dark Energy models. Before this, we couldn't test these models on small scales. Now we can.
2. Future Proofing: Current telescopes (like BOSS) aren't sensitive enough to hear the whisper. But future telescopes (like DESI and Euclid) will be much louder and clearer. This paper provides the mathematical "ear" needed to listen to those future, more precise data.
3. Ruling out possibilities: By showing that the "High-Five" model fits the data just as well as the "Silent" model, they prove that if there is an interaction, it must be very weak.

Summary in One Sentence

The authors created a sophisticated new mathematical tool to test if Dark Matter and Dark Energy are secretly exchanging energy in the chaotic, clumpy parts of the universe; they applied it to current galaxy maps and found no evidence of interaction, but they have now paved the way for future telescopes to find out for sure.

Technical Summary

1. Problem Statement

The standard $\Lambda$CDM cosmological model faces significant tensions, most notably the Hubble tension ($H_0$) and the $S_8$ tension (clustering amplitude). Interacting Dark Energy (IDE) models, which postulate a non-gravitational energy exchange between Dark Matter (DM) and Dark Energy (DE), offer a potential dynamical solution to these discrepancies.

However, previous constraints on IDE models have largely relied on linear perturbation theory (very large scales) or N-body simulations. Linear analyses often suffer from parameter degeneracies and cannot fully exploit the information contained in the mildly nonlinear regime ($k \sim 0.1 - 0.3 \, h/\text{Mpc}$). Furthermore, standard Perturbation Theory (SPT) breaks down in this regime, and applying Effective Field Theory of Large-Scale Structure (EFTofLSS) to IDE models requires deriving specific corrections to the power spectrum that account for the modified continuity and Euler equations.

The core problem: There is a lack of a consistent theoretical framework to compute one-loop corrections to the matter power spectrum in IDE scenarios, specifically to test these models against high-precision "full-shape" (FS) galaxy clustering data.

2. Methodology

The authors develop a perturbative framework within the EFTofLSS formalism to calculate one-loop corrections for IDE models.

• Theoretical Framework:

  • They start with the modified conservation equations where the interaction term $Q$ allows energy transfer between DM and DE.
  • They analyze two standard linear couplings ($Q = \xi H \rho_m$ and $Q = \xi H \rho_{DE}$) and introduce a novel nonlinear coupling:
    $$Q = \Gamma \rho_m \rho_{DE} \theta_m$$
    where $\Gamma$ is a nonlinear coupling constant and $\theta_m = \nabla \cdot v_m$ is the velocity divergence of dark matter. This specific form is chosen because it vanishes at the background level ($Q=0$) and only affects perturbations, isolating the IDE effects on mildly nonlinear scales.
  • They derive the modified continuity equation in Fourier space. Crucially, they show that while the Euler equation remains standard (under the assumption of no momentum transfer orthogonal to the fluid velocity), the continuity equation acquires a source term dependent on the interaction.

• Perturbative Calculations:

  • Using Standard Perturbation Theory (SPT), they expand the density contrast $\delta$ and velocity divergence $\theta$ into perturbative kernels ($F_n$ and $G_n$).
  • They derive the modified kernels $F_n^{IDE}$ and $G_n^{IDE}$ which deviate from standard $\Lambda$CDM kernels due to the interaction source term.
  • They compute the one-loop power spectrum contributions ($P_{22}$ and $P_{13}$), explicitly calculating the deviations $\Delta P_{22}$ and $\Delta P_{13}$ induced by the nonlinear coupling $\Gamma$.
  • The full one-loop power spectrum is expressed as:
    $$P_{1\text{-loop}}(k) = P_{1\text{-loop}, \Lambda\text{CDM}}(k) + \Delta P_{IDE}(k) + P_{ctr}(k)$$
    where $P_{ctr}$ represents the EFT counterterms.

• Observational Analysis:

  • They utilize the CLASS-PT code (a Boltzmann solver incorporating perturbative corrections) interfaced with MontePython for Markov Chain Monte Carlo (MCMC) sampling.
  • Data: They use full-shape measurements of the galaxy power spectrum multipoles (monopole and quadrupole) from BOSS DR12 (CMASS and LOWZ samples) at effective redshifts $z_{eff} = 0.38$ and $0.61$.
  • Parameters: They constrain the physical cold dark matter density ($\Omega_c h^2$), Hubble constant ($H_0$), primordial amplitude ($A_s$), and the nonlinear coupling parameter $\Gamma$. They also marginalize over bias parameters ($b_1, b_2, b_{G2}$) and EFT counterterms.

3. Key Contributions

1. Derivation of One-Loop Corrections: The paper provides the first systematic derivation of one-loop corrections to the matter power spectrum for IDE models, specifically addressing how the interaction modifies the mode-coupling kernels ($F_2, F_3, G_2, G_3$).
2. Novel Nonlinear Coupling: The introduction of the interaction term $Q = \Gamma \rho_m \rho_{DE} \theta_m$ is a significant innovation. Unlike linear couplings that affect background evolution, this term is purely perturbative and nonlinear, making it an ideal probe for isolating IDE effects in the mildly nonlinear regime without altering the linear growth history significantly.
3. EFTofLSS Integration: The authors successfully map IDE modifications into the EFTofLSS framework, demonstrating that the IDE corrections can be absorbed into or treated alongside standard bias operators and counterterms, ensuring a consistent comparison with observational data.
4. Full-Shape Constraints: They demonstrate that full-shape galaxy clustering data alone (without CMB or BAO priors) is sensitive enough to constrain the nonlinear coupling parameter $\Gamma$.

4. Results

• Constraint on Interaction: Using BOSS DR12 data, the authors constrain the nonlinear coupling constant to be:
  $$\Gamma = 0.0039 \pm 0.0082$$
  This result is consistent with zero at the $1\sigma$ level, implying no statistically significant evidence for this specific type of interaction in the current dataset.
• Cosmological Parameters: The inferred cosmological parameters ($H_0, \Omega_m, S_8$) for the IDE model are consistent with the standard $\Lambda$CDM results derived from the same dataset.
  • $H_0 = 67.95 \pm 0.82 \, \text{km/s/Mpc}$
  • $S_8 = 0.749 \pm 0.054$ (slightly lower than Planck, closer to weak lensing, but within uncertainties).
• Impact of Corrections: The analysis shows that while the IDE terms introduce new correlations (specifically a negative correlation between $\Gamma$ and $S_8$), the magnitude of the one-loop corrections for the allowed range of $\Gamma$ is small. The deviations from the standard power spectrum shape are subtle but detectable in principle by future high-precision surveys.
• Model Dependence: The paper highlights that the power spectrum is highly sensitive to the specific parametrization of $Q$. For instance, the linear coupling $Q=\xi H \rho_{DE}$ introduces different loop corrections compared to the nonlinear coupling studied here.

5. Significance

• Bridging the Gap: This work bridges the gap between theoretical IDE models and observational large-scale structure data by providing the necessary theoretical machinery (one-loop corrections) to test these models in the mildly nonlinear regime.
• Methodological Advancement: It establishes a robust pipeline for testing "beyond-$\Lambda$CDM" physics using full-shape analyses, moving beyond simple BAO or distance-ladder constraints.
• Future Prospects: While current data (BOSS) is consistent with $\Lambda$CDM, the methodology presented is ready for next-generation surveys like DESI, Euclid, and LSST. These future datasets, with larger volumes and better control over systematics, will significantly tighten the constraints on $\Gamma$, potentially reaching sensitivities an order of magnitude better than the current $\sim 10^{-2}$ level.
• Theoretical Insight: The study confirms that IDE models can be consistently treated within the EFTofLSS framework, validating the approach of using perturbative expansions to capture the effects of dark sector interactions on structure formation.

In conclusion, the paper successfully extends the theoretical toolkit for cosmological analysis to include interacting dark energy at the one-loop level, providing a rigorous method to test these models against current and future galaxy clustering data.