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Information-theoretic astrophysical uncertainties in the effective theory of dark matter direct detection

This paper quantifies how astrophysical uncertainties in the dark matter velocity distribution impact direct detection rates across all non-relativistic effective field theory operators by employing Kullback-Leibler divergence to reveal that uncertainties can vary from less than one order of magnitude to as much as three orders of magnitude, depending on the specific operator's dependence on velocity-weighted moments.

Original authors: Gonzalo Herrera

Published 2026-07-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gonzalo Herrera

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, invisible ocean, and we are tiny fish swimming in it. Most of the water in this ocean isn't made of the stuff we can see—stars, planets, or you and me. It's made of "dark matter," a mysterious substance that doesn't shine or reflect light, but we know it's there because its gravity pulls on the things we can see. Scientists have built giant, ultra-sensitive detectors deep underground, like underwater listening posts, hoping to catch a dark matter particle bumping into an atom in their tank. If that happens, it would be a huge discovery, telling us what this invisible ocean is made of.

But there's a catch. To know if a bump is really a dark matter particle, scientists have to guess how fast these invisible particles are swimming past Earth. They usually assume the dark matter ocean is calm and smooth, with particles moving at speeds that follow a predictable pattern, like a bell curve. However, the real ocean might be choppy. There could be currents, whirlpools, or streams of dark matter that don't fit the smooth pattern. If the real ocean is different from the smooth guess, the scientists might miscount the bumps, thinking they found a particle when they didn't, or missing a real one entirely. This paper asks: "How much does our guess about the ocean's currents mess up our search for dark matter?"

The authors of this paper, Gonzalo Herrera and colleagues, decided to tackle this problem using a clever mathematical tool called "information theory." Instead of guessing exactly what the choppy currents look like, they asked a simpler question: "How different can the real ocean be from our smooth guess before our detectors get completely confused?" They used a measure called the Kullback-Leibler (KL) divergence, which is like a "confusion meter." It tells you how much information you lose if you pretend a messy, real distribution is a neat, smooth one.

They applied this confusion meter to a wide variety of ways dark matter might interact with atoms. Think of these interactions as different types of "bumps." Some bumps happen when a dark matter particle just gently taps an atom (like a slow swimmer). Others happen when the particle hits hard, or spins, or interacts in complex ways that depend heavily on how fast it's moving. The team ran computer simulations for two famous detectors, XENONnT and PICO60, to see how much the "confusion" about the dark matter currents would change the results for each type of bump.

Here is what they found: It depends entirely on how the bump happens. For the simple, slow taps (called operators O1 and O4), the confusion meter didn't matter much. Even if the dark matter ocean was quite different from the smooth guess, the results only changed by a small amount—maybe a factor of a few. It's like trying to hear a slow drumbeat; even if the wind is blowing weirdly, you can still tell the rhythm.

However, for the more complex bumps that depend heavily on speed (like operators O5, O8, and O14), the story changed dramatically. These interactions are like trying to hear a high-speed jet engine; if you get the wind speed wrong, you can't hear the engine at all. For these fast-moving interactions, the uncertainty in the dark matter currents could make the scientists' limits on what they can detect swing wildly. The paper suggests that near the edge of what the detectors can see, the uncertainty could be as large as three or even four orders of magnitude. That means if the smooth guess says a particle is impossible to find, the real, choppy ocean might actually make it possible, or vice versa.

The authors also discovered that different types of bumps are sensitive to different "moments" of the speed distribution. Some bumps care about the average speed (the mean), some care about how spread out the speeds are (the variance), and some care about the extreme outliers—the super-fast particles in the tail of the distribution. The more a bump relies on those rare, super-fast particles, the more the scientists' results get thrown off by not knowing the exact shape of the dark matter ocean.

In short, this paper doesn't tell us what dark matter is, but it gives us a map of how shaky our ground is while we look for it. It shows that for some types of dark matter, we can be pretty confident in our search, but for others, the unknown currents of the dark matter ocean could be hiding the answer right in front of our noses, or making us think we found something when we didn't. As we get closer to the "neutrino floor"—where signals from dark matter might get mixed up with signals from solar neutrinos—understanding these uncertainties becomes crucial. The authors suggest that using their "confusion meter" approach could help future experiments interpret their data more accurately, ensuring that when we finally catch a dark matter particle, we know exactly what we've found.

Technical Summary: Information-theoretic astrophysical uncertainties in the effective theory of dark matter direct detection

Problem Statement
Direct detection experiments searching for dark matter (DM) rely on the scattering rate of DM particles off nuclei. This rate is determined by both particle physics microphysics (the interaction cross-section) and astrophysical inputs, specifically the local DM density and velocity distribution. While the Standard Halo Model (SHM) typically assumes a Maxwell-Boltzmann velocity distribution, cosmological simulations and kinematic tracers suggest significant deviations from this form. These astrophysical uncertainties are not uniform across different DM models; they vary significantly depending on the specific non-relativistic effective field theory (EFT) operator governing the DM-nucleon interaction. Different operators introduce distinct dependencies on the DM velocity ( $v$ ) and momentum transfer ( $q$ ), meaning that uncertainties in the velocity distribution can amplify or suppress the predicted event rate differently for each operator. Existing halo-independent methods often focus on specific operators or lack a unified framework to quantify these deviations across the full EFT basis.

Methodology
The authors address these uncertainties by applying an information-theoretic approach to the non-relativistic EFT of DM-nucleon interactions. The core methodology involves:

Information-Theoretic Metric: The deviation of the true DM velocity distribution ( $f$ ) from the reference Maxwell-Boltzmann distribution ( $f_{MB}$ ) is quantified using the Kullback-Leibler (KL) divergence, $D_{KL}[f_{MB} \parallel f]$ . This metric measures the information lost when approximating the true distribution with the SHM, providing a model-independent and basis-invariant constraint.
Optimization Framework: The authors formulate a convex optimization problem to find the most conservative and aggressive upper limits on the DM-nucleon scattering cross-section. The objective is to minimize or maximize the expected event rate ( $N_E$ ) for a given experiment, subject to the constraints that the velocity distribution is normalized, non-negative, and satisfies $D_{KL} \leq K$ .
Benchmarks: Two values for the maximal allowed KL divergence are considered: $K=0.1$ , representing a conservative benchmark motivated by cosmological simulations (e.g., Aquarius, Eagle) and Gaia-informed models, and $K=1$ , representing an aggressive scenario to bracket extreme astrophysical deviations.
Experimental Application: The method is applied to data from the XENONnT and PICO60 direct detection experiments. The analysis covers all 15 non-relativistic Galilean invariant operators ( $O_1$ through $O_{15}$ ) for DM with spin up to 1/2.
Numerical Implementation: The problem is solved using convex optimization techniques (CVXPY with the ECOS solver) after discretizing the velocity distribution.

Key Contributions and Results
The paper provides a unified quantification of how astrophysical uncertainties affect the exclusion limits for every operator in the DM-nucleon EFT basis. The results reveal a clear hierarchy in sensitivity based on the velocity and momentum dependence of each operator:

Velocity-Independent Operators ( $O_1, O_4$ ): These operators, which correspond to standard spin-independent and spin-dependent scattering, depend primarily on the bulk of the velocity distribution (analogous to the first moment). For $D_{KL}=0.1$ , the impact of astrophysical uncertainties is relatively small, varying by less than one order of magnitude near the kinematic threshold.
Velocity-Dependent Operators: Operators with explicit velocity dependence probe higher-order moments of the distribution.
- Operators like $O_3$ and $O_{12}$ , which involve quadratic velocity terms, show uncertainties of approximately two orders of magnitude near threshold for $D_{KL}=0.1$ .
- Operators with cubic velocity weighting, such as $O_5$ and $O_8$ , exhibit uncertainties of about three orders of magnitude.
- The operator $O_{14}$ displays particularly large uncertainties, approaching four orders of magnitude in the near-threshold regime.
Mass Dependence: For heavy DM masses ( $m_{DM} \gtrsim 20$ GeV), the accessible velocity range covers a large fraction of the distribution, making the impact of uncertainties more uniform across operators (roughly a factor of 3 for XENONnT and 10 for PICO60 at $D_{KL}=0.1$ ). For lighter masses, the rate is dominated by the high-velocity tail, leading to larger variations.
Analytic Insight: The authors establish a functional analogy between the velocity-weighted integrals determining the scattering rate and statistical moments (mean, variance, skewness) of the velocity distribution. Operators probing higher raw moments (associated with variance and skewness) are shown to be more sensitive to the high-velocity tail and thus more susceptible to astrophysical uncertainties.

Significance and Claims
The paper claims to provide a general, model-independent technique to assess the impact of astrophysical uncertainties on direct detection limits without assuming a specific functional form for the velocity distribution. By linking the scattering rate to the statistical moments of the velocity distribution, the work offers new analytic insight into which features of the velocity distribution are most relevant for detecting specific DM models.

The authors emphasize that their technique is general and could be applied to broader classes of physics problems where a physical observable depends on the statistical moments of an uncertain theoretical distribution. They note that this approach is particularly relevant for the "neutrino floor" era, where distinguishing DM signals from coherent elastic neutrino-nucleus scattering requires precise understanding of astrophysical uncertainties. The paper concludes that interactions generated by derivative, dipole, or higher-dimension tensor operators are intrinsically more sensitive to astrophysical uncertainties than standard contact interactions, a finding that must be accounted for when interpreting current and future experimental results.

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