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CritPt / Challenge_56_main / Consider a scalar field ϕ linearly coupled to the standard model…

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 $\Lambda_\gamma^{-1}$.

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

    Outputs
    ----------
    lambda_inv: list[float]
        Smallest coupling strength $\Lambda_\gamma^{-1}$ the Cosmic Explorer can probe at 200 Hz
        with an observation time of $\{1000\,\text{s}, 0.7\,\text{yrs}\}$ and a signal-to-noise of 1.
    """

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

    return lambda_inv

problem description

# Problem setup: Consider a scalar field

\phi

linearly coupled to the standard model electromagnetic field with coupling strength

\Lambda_\gamma^{-1}

. We can write the interaction term of the Lagrangian as

\mathcal{L}_\text{int}=\frac{\phi}{4 \Lambda_\gamma}F^{\mu\nu}F_{\mu\nu}

. Assuming the scalar field makes up all the dark matter, it may induce observable effects in laser interferometers. We consider signals of scalar field in the Cosmic Explorer, whose planned strain sensitivity is about

2\times 10^{-25}\, \text{Hz}^{-1/2}

in the frequency range of

50\,\text{Hz} \leq f \leq 500\,\text{Hz}

. The interferometer arm length of Cosmic Explorer is designed to be

40\, \text{km}

. We assume the Cosmic Explorer uses a similar beamsplitter as LIGO, with thickness

6\,\text{cm}

and index of refraction

3.5

. # Main problem: If there is an overdensity of dark matter at the device that is

178

times the local dark matter density (take the local dark matter density to be

0.4\,\text{GeV}/\text{cm}^3

and velocity to be

230\,\text{km}/\text{s}

), what is the smallest

\Lambda_\gamma^{-1}

the Cosmic Explorer can probe at

200\,\text{Hz}

with a signal-to-noise ratio of 1 and with an observation time of

1000\,\text{s}

0.7\,\text{yrs}

, respectively? Keep at least three significant digits in the final result.

Plain-text mathematical notation (without MathML)


# Problem setup:
Consider a scalar field ϕ linearly coupled to the standard model electromagnetic field with coupling strength Λ_(γ)^(−1). We can write the interaction term of the Lagrangian as L_(int)=(ϕ)/(4Λ_(γ))F^(μν)F_(μν). Assuming the scalar field makes up all the dark matter, it may induce observable effects in laser interferometers. We consider signals of scalar field in the Cosmic Explorer, whose planned strain sensitivity is about 2×10^(−25) Hz^(−1/2) in the frequency range of 50 Hz≤f≤500 Hz. The interferometer arm length of Cosmic Explorer is designed to be 40 km. We assume the Cosmic Explorer uses a similar beamsplitter as LIGO, with thickness 6 cm and index of refraction 3.5.

# Main problem:

If there is an overdensity of dark matter at the device that is 178 times the local dark matter density (take the local dark matter density to be 0.4 GeV/cm³ and velocity to be 230 km/s), what is the smallest Λ_(γ)^(−1) the Cosmic Explorer can probe at 200 Hz with a signal-to-noise ratio of 1 and with an observation time of 1000 s or 0.7 yrs, respectively? Keep at least three significant digits in the final result.

Original LaTeX notation


# Problem setup:
Consider a scalar field $\phi$ linearly coupled to the standard model electromagnetic field with coupling strength $\Lambda_\gamma^{-1}$. We can write the interaction term of the Lagrangian as $\mathcal{L}_\text{int}=\frac{\phi}{4 \Lambda_\gamma}F^{\mu\nu}F_{\mu\nu}$. Assuming the scalar field makes up all the dark matter, it may induce observable effects in laser interferometers. We consider signals of scalar field in the Cosmic Explorer, whose planned strain sensitivity is about $2\times 10^{-25}\, \text{Hz}^{-1/2}$ in the frequency range of $50\,\text{Hz} \leq f \leq 500\,\text{Hz}$. The interferometer arm length of Cosmic Explorer is designed to be $40\, \text{km}$. We assume the Cosmic Explorer uses a similar beamsplitter as LIGO, with thickness $6\,\text{cm}$ and index of refraction $3.5$.

# Main problem:

If there is an overdensity of dark matter at the device that is $178$ times the local dark matter density (take the local dark matter density to be $0.4\,\text{GeV}/\text{cm}^3$ and velocity to be $230\,\text{km}/\text{s}$), what is the smallest $\Lambda_\gamma^{-1}$ the Cosmic Explorer can probe at $200\,\text{Hz}$ with a signal-to-noise ratio of 1 and with an observation time of $1000\,\text{s}$ or $ 0.7\,\text{yrs}$, respectively? Keep at least three significant digits in the final result.

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Source and history

upstream_id / Original problem ID: Challenge_56_main
task_id: 017ec79f-1e88-584b-8d8b-a6418ec87fcb
task_key: train--Challenge~5f56~5fmain
task_revision_id: 1
config / split: / train

Official source

initial import