Derivation of Marginal Likelihood with Stochastic Field Amplitude
2024-10-28 05:23:19 Author: hackernoon.com(查看原文) 阅读量:1 收藏

Authors:

(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;

(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;

(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;

(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University.

Abstract and 1 Introduction

2 Calculating the Stochastic Wave Vector Dark Matter Signal

2.1 The Dark Photon Field

2.2 The Detector Signal

3 Statistical Analysis and 3.1 Signal Likelihood

3.2 Projected Exclusions

4 Application to Accelerometer Studies

4.1 Recasting Generalised Limits onto B − L Dark Matter

5 Future Directions

6 Conclusions, Acknowledgments, and References

A Equipartition between Longitudinal and Transverse Modes

B Derivation of Marginal Likelihood with Stochastic Field Amplitude

C Covariance Matrix

D The Case of the Gradient of a Scalar

B Derivation of Marginal Likelihood with Stochastic Field Amplitude

The full signal in time space is given by

Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood:

where

This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent.


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