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CritPt / Challenge_57_main / Consider a vector field A^(μ) coupled to the standard model B−L current…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

code template

Code

def answer():
    r"""
    Return the values of the smallest $\epsilon_{B-L}$ for three scenarios.

    Inputs
    ----------
    None

    Outputs
    ----------
    eps_B_L_min: list[float],
          smallest $\epsilon_{B-L}$ it can probe at $250\,\text{Hz}$ with an observation time of $13$ years and SNR of $1$
          for scenarios where $\delta q=\{0.074, 6\times 10^{-3}, 5\times 10^{-4}\}$.
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    eps_B_L_min = ...
    # ---------------------------------------------------------------

    return eps_B_L_min

problem description

# Problem setup: Consider a vector field

A^\mu

coupled to the standard model

B-L

current

J^\mu_{B-L}

through

\mathcal{L}\supset -\epsilon_{B-L}eJ^\mu_{B-L} A_\mu

. If the vector field makes up all the dark matter, the local dark matter density indicates that the field could exert a measurable force on mirrors in laser interferometers, with sufficiently large coupling. Consider LIGO, with a strain sensitivity of

3\times 10^{-24}\, \text{Hz}^{-1/2}

at frequency

250\,\text{Hz}

. # Main problem: Take the dark charge

Q_D

to mass

M

radio to be

\frac{Q_D}{M}\sim \frac{0.5}{m_n}

for the inner mirrors, where

m_n

is the mass of a neutron. We introduce a doped outer mirror, resulting in a differential forced between the inner and outer mirror. We assume the outer mirror gains an additiona

\delta \left(\frac{Q_D}{M}\right)\sim \frac{\delta q}{m_n}

, where

m_n

is the mass of a neutron. With an observation time of

13

years and SNR of

1

, what is the smallest

\epsilon_{B-L}

it can probe at

250\,\text{Hz}

? Provide answers for scenarios where

\delta q=\{0.074, 6\times 10^{-3}, 5\times 10^{-4}\}

Plain-text mathematical notation (without MathML)


# Problem setup:
Consider a vector field A^(μ) coupled to the standard model B−L current J_(B−L)^(μ) through L⊃−ϵ_(B−L)eJ_(B−L)^(μ)A_(μ). If the vector field makes up all the dark matter, the local dark matter density indicates that the field could exert a measurable force on mirrors in laser interferometers, with sufficiently large coupling. Consider LIGO, with a strain sensitivity of 3×10^(−24) Hz^(−1/2) at frequency 250 Hz.

# Main problem:

Take the dark charge Q_(D) to mass M radio to be (Q_(D))/(M)∼(0.5)/(m_(n)) for the inner mirrors, where m_(n) is the mass of a neutron. We introduce a doped outer mirror, resulting in a differential forced between the inner and outer mirror. We assume the outer mirror gains an additiona δ((Q_(D))/(M))∼(δq)/(m_(n)), where m_(n) is the mass of a neutron. With an observation time of 13 years and SNR of 1, what is the smallest ϵ_(B−L) it can probe at 250 Hz? Provide answers for scenarios where δq={0.074,6×10^(−3),5×10^(−4)}.

Original LaTeX notation


# Problem setup:
Consider a vector field $A^\mu$ coupled to the standard model $B-L$ current $J^\mu_{B-L}$ through $\mathcal{L}\supset -\epsilon_{B-L}eJ^\mu_{B-L} A_\mu$. If the vector field makes up all the dark matter, the local dark matter density indicates that the field could exert a measurable force on mirrors in laser interferometers, with sufficiently large coupling. Consider LIGO, with a strain sensitivity of $3\times 10^{-24}\, \text{Hz}^{-1/2}$ at frequency $250\,\text{Hz}$.

# Main problem:

Take the dark charge $Q_D$ to mass $M$ radio to be $\frac{Q_D}{M}\sim \frac{0.5}{m_n}$ for the inner mirrors, where $m_n$ is the mass of a neutron. We introduce a doped outer mirror, resulting in a differential forced between the inner and outer mirror. We assume the outer mirror gains an additiona $\delta \left(\frac{Q_D}{M}\right)\sim \frac{\delta q}{m_n}$, where $m_n$ is the mass of a neutron. With an observation time of $13$ years and SNR of $1$, what is the smallest $\epsilon_{B-L}$ it can probe at $250\,\text{Hz}$? Provide answers for scenarios where $\delta q=\{0.074, 6\times 10^{-3}, 5\times 10^{-4}\}$.

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Source and history

upstream_id / Original problem ID: Challenge_57_main
task_id: b886e409-9498-52b8-afb5-011b0496a07a
task_key: train--Challenge~5f57~5fmain
task_revision_id: 2
config / split: / train

Official source

initial import