The Complete Protocol of Dark Matter Analysis: A Bayesian Approach

## Introduction

Dark matter, the elusive substance that makes up about 85% of the matter in our universe, has puzzled scientists for decades. As we strive to understand its nature, we employ sophisticated statistical methods to analyze data from various experiments and observations. Among these methods, Bayesian hypothesis testing has emerged as a powerful tool for evaluating competing dark matter models.

In this blog post, we'll walk through the complete protocol of dark matter analysis using Bayesian methods. We'll cover each step in detail, from formulating hypotheses to interpreting results, using a hypothetical direct detection experiment as our running example. By the end, you'll have a comprehensive understanding of how Bayesian analysis is applied in cutting-edge dark matter research.

## 1. Formulating Hypotheses

The first step in any scientific analysis is to clearly state the hypotheses we want to test. In dark matter research, we typically have several competing models that we want to evaluate.

For our example, let's consider the following hypotheses:

- H0: Background only (null hypothesis)

- H1: WIMPs (Weakly Interacting Massive Particles)

- H2: Axions

- H3: Self-Interacting Dark Matter (SIDM)

Each of these hypotheses makes different predictions about what we should observe in our experiment.

## 2. Specifying the Model

For each hypothesis, we need to specify a detailed physical model that makes quantitative predictions. This involves translating the qualitative hypothesis into mathematical form.

### Example: WIMP Model

For the WIMP hypothesis (H1), our model might include:

1. WIMP mass (m_χ)

2. WIMP-nucleon cross-section (σ_χn)

3. Local dark matter density (ρ_0)

4. Dark matter velocity distribution (usually assumed to be a Maxwell-Boltzmann distribution)

The differential event rate in our detector would then be given by:

dR/dE = (ρ_0 / (m_χ m_N)) σ_χn F^2(E) ∫ (f(v) / v) dv

Where:

- m_N is the mass of the target nucleus

- F^2(E) is the nuclear form factor

- f(v) is the dark matter velocity distribution

- The integral is taken from the minimum velocity required to produce a recoil of energy E up to the escape velocity of the galaxy

Similar models would be specified for the other hypotheses, each with their own parameters and predicted event rates.

## 3. Defining Priors

In Bayesian analysis, we need to specify prior probabilities for our hypotheses and prior distributions for the parameters within each model. These priors represent our state of knowledge before considering the new data.

### Hypothesis Priors

We might assign the following prior probabilities to our hypotheses:

- P(H0) = 0.1 (we think it's unlikely, but possible, that we'll see no dark matter)

- P(H1) = 0.5 (WIMPs have strong theoretical motivation)

- P(H2) = 0.3 (Axions are well-motivated but perhaps slightly less so than WIMPs)

- P(H3) = 0.1 (SIDM is a more recent proposal)

### Parameter Priors

For each model, we need to specify priors for its parameters. For the WIMP model:

1. WIMP mass (m_χ): We might use a log-uniform prior from 1 GeV to 1000 GeV.

2. WIMP-nucleon cross-section (σ_χn): Log-uniform prior from 10^-48 cm^2 to 10^-40 cm^2.

3. Local dark matter density (ρ_0): Gaussian prior with mean 0.3 GeV/cm^3 and standard deviation 0.1 GeV/cm^3.

The choice of priors should be justified based on theoretical considerations, previous experiments, and physicality constraints.

## 4. Collecting Data

Now we perform our experiment and collect data. For a direct detection experiment, this might involve:

1. Operating a large detector (e.g., a tank of liquid xenon) deep underground to shield from cosmic rays.

2. Cooling the detector to ultra-low temperatures to minimize thermal noise.

3. Recording any events that deposit energy in the detector, noting the energy and timing of each event.

4. Carefully calibrating the detector and characterizing its response to both signal and background events.

Let's say our hypothetical experiment runs for one year and observes 100 events with energies ranging from 5 keV to 50 keV.

## 5. Calculating Likelihoods

The likelihood is the probability of observing our data given a particular hypothesis and set of parameter values. For each hypothesis, we need to calculate this likelihood.

### Example: WIMP Likelihood

For the WIMP hypothesis, the likelihood would be a product of Poisson probabilities for each observed event, plus the probability of observing no events in the rest of the energy range:

L(data | H1, θ) = [Π_i P(E_i | H1, θ)] * exp(-N_expected)

Where:

- E_i are the observed event energies

- θ represents the model parameters (m_χ, σ_χn, ρ_0)

- P(E_i | H1, θ) is the probability of observing an event with energy E_i given the WIMP model with parameters θ

- N_expected is the total number of events expected given the model and parameters

This likelihood would be calculated for many different combinations of parameter values within the WIMP model.

Similar likelihoods would be calculated for the other hypotheses.

## 6. Computing Evidences

The evidence for each hypothesis is the total probability of the data under that hypothesis, integrating over all possible parameter values:

P(data | H) = ∫ P(data | H, θ) P(θ | H) dθ

This integral is often computationally challenging and requires numerical techniques such as nested sampling or Markov Chain Monte Carlo methods.

Let's say we calculate the following evidences:

- P(data | H0) = 1e-5

- P(data | H1) = 2e-3

- P(data | H2) = 5e-4

- P(data | H3) = 1e-4

## 7. Calculating Posterior Probabilities

Now we can use Bayes' theorem to calculate the posterior probability for each hypothesis:

P(H | data) = P(data | H) * P(H) / P(data)

Where P(data) = Σ P(data | Hi) * P(Hi)

Plugging in our values:

P(data) = (1e-5 0.1) + (2e-3 0.5) + (5e-4 0.3) + (1e-4 0.1) = 1.16e-3

Then:

- P(H0 | data) = (1e-5 * 0.1) / 1.16e-3 = 0.0009

- P(H1 | data) = (2e-3 * 0.5) / 1.16e-3 = 0.8621

- P(H2 | data) = (5e-4 * 0.3) / 1.16e-3 = 0.1293

- P(H3 | data) = (1e-4 * 0.1) / 1.16e-3 = 0.0077

## 8. Parameter Estimation

For the most probable hypothesis (in this case, WIMPs), we can also estimate the posterior distributions of its parameters. This is typically done by sampling from the posterior distribution using techniques like Markov Chain Monte Carlo.

We might find, for example:

- WIMP mass: 50 GeV with 68% credible interval [30 GeV, 80 GeV]

- WIMP-nucleon cross-section: 5e-46 cm^2 with 68% credible interval [1e-46 cm^2, 2e-45 cm^2]

## 9. Model Comparison

We can formally compare models using the Bayes factor, which is the ratio of evidences:

B12 = P(data | H1) / P(data | H2) = 2e-3 / 5e-4 = 4

This indicates that the data favor the WIMP hypothesis over the axion hypothesis by a factor of 4.

## 10. Sensitivity Analysis

It's crucial to test how sensitive our results are to our choice of priors and other assumptions. We might repeat the analysis with different priors, or with variations in our background model or detector response function.

## 11. Interpretation of Results

Based on our analysis, we can draw several conclusions:

1. The data strongly favor the existence of dark matter over the background-only hypothesis.

2. Among the dark matter models considered, WIMPs are most strongly favored, with a posterior probability of about 86%.

3. Axions are the second most favored model, but with a much lower probability of about 13%.

4. The SIDM and background-only hypotheses are strongly disfavored by this data.

5. If the WIMP hypothesis is correct, our analysis suggests a WIMP mass around 50 GeV and a WIMP-nucleon cross-section around 5e-46 cm^2.

However, we should be cautious in our interpretation:

1. These results depend on our choice of priors and models considered. Other well-motivated dark matter candidates not included in this analysis could potentially explain the data.

2. The preference for WIMPs over axions, while noticeable, is not overwhelming. More data would be needed to strongly distinguish between these hypotheses.

3. Systematic uncertainties, such as in our background model or detector response, could significantly affect these results if not properly accounted for.

## 12. Reporting and Open Science

The final step in our protocol is to report our results transparently and comprehensively. This includes:

1. Clearly stating all assumptions and priors used in the analysis.

2. Providing details of the experimental setup and data collection process.

3. Sharing code and data (where possible) to allow others to reproduce the analysis.

4. Discussing the sensitiv

ity of the results to various assumptions and potential systematic uncertainties.

5. Contextualizing the results within the broader field of dark matter research.

## Conclusion

Bayesian hypothesis testing provides a powerful and flexible framework for analyzing dark matter experiments. By explicitly incorporating prior knowledge, dealing naturally with nuisance parameters, and offering a consistent approach to model comparison, it helps us navigate the complex landscape of dark matter theories and experimental results.

However, it's important to remember that this is an iterative process. As new data come in from this and other experiments, we update our priors and repeat the analysis, gradually refining our understanding of dark matter.

The quest to understand dark matter is one of the great scientific adventures of our time. While we haven't yet pinned down its nature, methodologies like the one outlined here are bringing us ever closer to unraveling this cosmic mystery. As we continue to push the boundaries of experimental sensitivity and theoretical modeling, we can look forward to exciting discoveries in the years to come.