CMB constraints on dark matter-proton scattering: investigating prior-volume effects using profile likelihoods

Using Planck 2018 CMB data, this paper demonstrates that Bayesian constraints on dark matter-proton scattering are biased by prior-volume effects that overestimate limits, and thus recommends supplementing them with frequentist profile-likelihood analyses to obtain prior-independent results.

The Gist

Imagine the universe is a giant, invisible ocean. For decades, scientists have been trying to figure out what's swimming in it. We know there's a lot of "Dark Matter" (the invisible stuff holding galaxies together), but we don't know exactly what it is.

The standard theory says Dark Matter is like a ghost: it has mass, it has gravity, but it doesn't bump into anything else. It just floats through the universe, ignoring everything.

But what if Dark Matter isn't a ghost? What if it's more like a swimmer who occasionally bumps into the water (protons)? This paper asks: If Dark Matter does bump into normal matter, how much does it bump?

Here is the breakdown of the paper's discovery, explained through a simple story.

The Mystery: The "Ghost" vs. The "Swimmer"

Scientists use a telescope called Planck to look at the oldest light in the universe (the Cosmic Microwave Background). It's like looking at a baby picture of the universe. If Dark Matter bumps into normal matter, it would leave a specific "bruise" or pattern on that baby picture.

The researchers wanted to measure exactly how "bumpy" the Dark Matter is. They looked at two possibilities:

1. 100% Bumpy: All Dark Matter bumps into protons.
2. Fractional Bumpy: Only some of the Dark Matter bumps (like a school of fish where only a few are wearing bumpers).

The Problem: The "Empty Room" Trap

To find the answer, scientists use a method called Bayesian Statistics. Think of this like trying to find a lost coin in a giant, dark warehouse.

• You have a flashlight (the data).
• You have a map of where the coin might be (the "Prior," or your best guess before looking).

The Trap:
Imagine the warehouse has a tiny corner where the coin could be, but it's so dark you can't see it. However, there is a massive, empty room next to it where the coin definitely isn't, but it's huge.

If you use the "Bayesian" method, your search algorithm gets confused. It thinks, "Well, I can't find the coin in the tiny corner, and the huge empty room is so big that statistically, the coin is probably just somewhere in that big empty space."

Because the "empty space" (where Dark Matter doesn't interact at all) is so vast, the math tricks itself into thinking the coin is very likely to be in the "no interaction" zone. This makes the scientists think they have found a very strict rule: "Dark Matter definitely doesn't bump!"

But this isn't because they found evidence of no bumping; it's because their map (the math) was too big and empty in the wrong places. This is called the Prior-Volume Effect.

The Solution: The "Profile Likelihood" Flashlight

The authors of this paper decided to try a different method called Profile Likelihood.

Instead of searching the whole warehouse and getting confused by the empty rooms, this method is like a laser pointer.

• You point the laser at a specific spot (e.g., "What if the bumping is this strong?").
• You ask: "If the bumping is this strong, does the data match?"
• If the data matches, great. If not, you move the laser.

This method ignores the size of the empty rooms. It only cares about: "Does the data actually support this specific idea?" It doesn't get tricked by the size of the "no interaction" zone.

The Big Discovery

When the researchers compared the two methods:

1. The Bayesian Method (The Warehouse Search): Said, "We are 99% sure Dark Matter doesn't bump at all! The limit is super strict!"
2. The Profile Likelihood Method (The Laser Pointer): Said, "Actually, Dark Matter could bump a little bit more than you think. The limit is looser."

The Result: The Bayesian method was being too strict. It was "overestimating" the constraints because it was getting lost in the math of the "empty rooms." The Profile Likelihood method showed that the universe allows for a bit more "bumping" than the other method suggested.

The Takeaway

The paper is a warning to scientists: "Don't trust your map if the map is too big and empty."

When looking for new physics (like Dark Matter interacting), if you use the standard Bayesian math, you might accidentally convince yourself that the new physics doesn't exist just because your math prefers the "nothing happens" scenario.

The Advice:
To be safe, scientists should use both methods.

• Use the Bayesian method to see what the data says.
• Use the Profile Likelihood method to make sure you aren't being tricked by the size of the "empty rooms."

In a Nutshell

Imagine you are trying to prove that a specific type of bird exists in a forest.

• Method A (Bayesian): You look at the whole forest. Since 99% of the forest is empty trees, you conclude, "The bird probably isn't here."
• Method B (Profile Likelihood): You look at the specific branch where the bird might be. You say, "I can't prove the bird is here, but I also can't prove it's not here yet. Let's keep looking."

This paper says: Method B is more honest. We need to be careful not to declare "No new physics found" just because our math got lost in the size of the forest.

Technical Summary

1. Problem Statement

The paper addresses a critical methodological issue in cosmological parameter inference: prior-volume effects in Bayesian analyses of models with non-gravitational dark matter (DM) interactions.

• The Model: The study focuses on velocity-independent dark matter-proton scattering. This model introduces three new parameters: the momentum-transfer cross-section ($\sigma_0$), the DM particle mass ($m_\chi$), and the fraction of DM that interacts ($f_\chi$).
• The Issue: In Bayesian inference, the posterior distribution is the product of the likelihood and the prior. When the interaction parameters ($\sigma_0$ or $f_\chi$) approach zero, the model becomes indistinguishable from the standard $\Lambda$CDM model. In this limit, the likelihood becomes flat with respect to the other parameters (mass and the remaining interaction parameters).
• The Consequence: If the prior volume in this "flat likelihood" region is large (e.g., a broad prior extending to very small cross-sections), the Bayesian MCMC sampler spends a disproportionate amount of time exploring this region. This artificially shifts the posterior distribution toward the $\Lambda$CDM limit, resulting in biased, overly stringent upper limits on the interaction parameters. These constraints reflect the choice of prior volume rather than the actual data constraints.

2. Methodology

The authors employ a dual approach to compare Bayesian and Frequentist statistical frameworks using Planck 2018 CMB data.

• Data: Planck 2018 temperature, polarization, and lensing power spectra (TT, TE, EE, and lensing).
• Codes:
  • CLASS: A modified version including dark matter-proton scattering physics (solving modified Boltzmann equations for density and velocity fluctuations).
  • Cobaya: Used for Bayesian MCMC sampling (Metropolis-Hastings algorithm).
  • Procoli: A public Python package used to generate Frequentist profile likelihoods via simulated annealing optimization.
• Statistical Frameworks:
  • Bayesian: Marginalized posterior distributions are generated using MCMC. The authors test various flat priors on $\log_{10}(\sigma_0)$ and $\log_{10}(f_\chi)$ to demonstrate sensitivity.
  • Frequentist (Profile Likelihood): The profile likelihood $\Delta\chi^2(\mu)$ is constructed by maximizing the likelihood over all nuisance parameters ($\nu$) for fixed values of the parameters of interest ($\mu$). Confidence intervals are derived using the graphical method (Wilks' theorem), where the 95.45% confidence limit corresponds to $\Delta\chi^2 = 6.18$ for two parameters.
• Experimental Design:
  • Fixed Fraction: Constraints are derived for fixed values of $f_\chi$ (ranging from 1 to 0.01) while varying $\sigma_0$ and $m_\chi$.
  • Varying Priors: The authors systematically widen the lower bounds of the priors for $\sigma_0$ and $f_\chi$ to observe how the Bayesian constraints shift.
  • Simultaneous Variation: A full analysis is performed where $f_\chi$, $\sigma_0$, and $m_\chi$ are varied simultaneously to assess the compounding effects of prior-volume in a multi-dimensional space.

3. Key Contributions

1. First Profile Likelihood Constraints: This work presents the first profile-likelihood constraints on velocity-independent dark matter-proton scattering, including the first constraints on fractional dark matter scattering (where only a fraction $f_\chi$ of DM interacts).
2. Quantification of Prior-Volume Bias: The study provides concrete evidence that Bayesian constraints on this model are systematically biased by prior choices. It demonstrates that the "strength" of the upper limits is directly correlated with the breadth of the prior range in the $\Lambda$CDM limit.
3. Methodological Recommendation: The authors advocate for supplementing Bayesian analyses with frequentist profile likelihoods for models susceptible to prior-volume effects. This provides a prior-independent benchmark to assess the true constraining power of the data.

4. Key Results

• Discrepancy in Constraints: The Bayesian MCMC constraints are systematically stronger (by a factor of 2 or more) than the frequentist profile-likelihood constraints across the entire mass range and for all interaction fractions.
  • Example: For a fixed fraction, the Bayesian upper limit on $\sigma_0$ is significantly lower than the frequentist limit.
• Prior Sensitivity:
  • Cross-Section Priors: As the lower bound of the prior on $\log_{10}(\sigma_0)$ is extended to smaller values (e.g., from $-27$ to $-31$), the Bayesian upper limit on $\sigma_0$ becomes tighter. This occurs because the MCMC samples the vast volume of the $\Lambda$CDM limit where the likelihood is flat.
  • Fraction Priors: Similarly, widening the prior on $f_\chi$ tightens the constraints on $f_\chi$.
  • Cross-Parameter Degeneracy: Widening the prior on one parameter (e.g., $\sigma_0$) actually loosens the constraints on the other (e.g., $f_\chi$) when marginalizing, due to the complex interplay of the flat likelihood region.
• Profile Likelihood Stability: The frequentist profile-likelihood constraints remain invariant regardless of the prior range chosen. The $\Delta\chi^2$ cutoff is determined solely by the data likelihood, not the prior volume.
• Physical Limits: The profile likelihoods confirm the expected physical behavior:
  • For $m_\chi \gg m_p$, constraints scale as $\sigma_0 \propto m_\chi$ (degeneracy).
  • For $m_\chi \ll m_p$, the temperature evolution of DM dictates the constraint shape.

5. Significance and Conclusions

• Caution in Bayesian Interpretation: The paper serves as a cautionary tale for interpreting Bayesian upper limits in models where the data cannot distinguish between the new physics and the standard model (the $\Lambda$CDM limit). In such cases, the "constraints" may be artifacts of the prior volume rather than evidence from the data.
• Robustness of Frequentist Methods: Profile likelihoods are shown to be robust against these specific types of prior-volume effects, providing a more reliable measure of the data's actual constraining power.
• Broader Applicability: While focused on DM-proton scattering, the authors argue that these prior-volume effects are ubiquitous in cosmology (citing Early Dark Energy and Effective Field Theories). They recommend that for any model with parameters that can vanish to recover the standard model, researchers should always supplement Bayesian results with frequentist profile likelihoods to disentangle data-driven limits from prior-induced biases.

In summary, Straight et al. demonstrate that without frequentist cross-checks, Bayesian analyses of interacting dark matter may yield misleadingly strong exclusion limits that are driven by the arbitrary choice of prior ranges rather than the physical evidence provided by the CMB.