Fluid source

$$\frac{d \Omega_{\text{gw},0}}{d \ln(f)} = 0.68 F_{\text{gw},0} \Gamma^2 \overline{U}^4_\text{f} (H_n R_*) \tilde{\Omega}_\text{gw} C\left(\frac{f}{f_{\text{p},0}}\right),$$

where

0.68 is $2/(8\pi)^{1/3}$

ðŸ˜…

Fluid source

$$\frac{d \Omega_{\text{gw},0}}{d \ln(f)} = 0.68 F_{\text{gw},0} \Gamma^2 \overline{U}^4_\text{f} (H_n R_*) \tilde{\Omega}_\text{gw} C\left(\frac{f}{f_{\text{p},0}}\right),$$

where

$$F_{\text{gw},0} = (3.57 \pm 0.05)\times 10^{-5} \left(\frac{100}{g_*}\right)^{\frac{1}{3}}$$

(redshifting)

Fluid source

$$\frac{d \Omega_{\text{gw},0}}{d \ln(f)} = 0.68 F_{\text{gw},0} \Gamma^2 \overline{U}^4_\text{f} (H_n R_*) \tilde{\Omega}_\text{gw} C\left(\frac{f}{f_{\text{p},0}}\right),$$

where

$\Gamma = 1 + \frac{\large \overline{p}}{\large \overline{\varepsilon}} \approx \frac{4}{3}$

Fluid source

$$\frac{d \Omega_{\text{gw},0}}{d \ln(f)} = 0.68 F_{\text{gw},0} \Gamma^2 \overline{U}^4_\text{f} (H_n R_*) \tilde{\Omega}_\text{gw} C\left(\frac{f}{f_{\text{p},0}}\right),$$

where

$$\overline{U}^2_\text{f} = \frac{1}{(\overline{p} + \overline{\varepsilon}) V}\int_{V} d^3 x \, \tau^{\mathrm{(f)}}_{ii} = f(\alpha, v_\mathrm{w})$$

(fluid kinetic energy)

What is $f(\alpha, v_\mathrm{w})$?

$$\overline{U}_\mathrm{f}^2 = \frac{3}{4} \alpha\, \kappa_\mathrm{v}(v_\mathrm{w},\alpha)$$
[Espinosa et al.]

Fluid source

$$\frac{d \Omega_{\text{gw},0}}{d \ln(f)} = 0.68 F_{\text{gw},0} \Gamma^2 \overline{U}^4_\text{f} (H_n R_*) \tilde{\Omega}_\text{gw} C\left(\frac{f}{f_{\text{p},0}}\right),$$

where

$$\tilde{\Omega}_\text{gw} \approx 0.12$$

(universal amplitude fitted from simulations)

Fluid source

$$\frac{d \Omega_{\text{gw},0}}{d \ln(f)} = 0.68 F_{\text{gw},0} \Gamma^2 \overline{U}^4_\text{f} (H_n R_*) \tilde{\Omega}_\text{gw} C\left(\frac{f}{f_{\text{p},0}}\right),$$

where

$$C(s) = s^3\left(\frac{7}{4+3s^2}\right)^{7/2}$$

(spectral shape supported by latest simulations)

Fluid source

$$\frac{d \Omega_{\text{gw},0}}{d \ln(f)} = 0.68 F_{\text{gw},0} \Gamma^2 \overline{U}^4_\text{f} (H_n R_*) \tilde{\Omega}_\text{gw} C\left(\frac{f}{f_{\text{p},0}}\right),$$

where

$$f_{\text{p},0} \simeq 26 \left( \frac{1}{H_\mathrm{n}R_*} \right) \left( \frac{z_\mathrm{p}}{10} \right) \left( \frac{T_\mathrm{n}}{10^2 \, \text{GeV}} \right) \left( \frac{g_*}{100} \right)^{\frac{1}{6}} \; \mu\text{Hz}$$

(peak frequency)

Peak frequency

$$f_{\text{p},0} \simeq 26 \left( \frac{1}{H_\mathrm{n}R_*} \right) \left( \frac{z_\mathrm{p}}{10} \right) \left( \frac{T_\mathrm{n}}{10^2 \, \text{GeV}} \right) \left( \frac{g_*}{100} \right)^{\frac{1}{6}} \; \mu\text{Hz}$$

where

$z_\mathrm{p}$ is the peak $k R_* \approx 10$ (weak function of $v_\mathrm{w}$, ...)

Online tool?

• Web-based, open to all (mostly?)
• Power spectra and SNR plots
• Output vector graphics (SVG)
• ... but not slow to run
• ... securely containerised

Features

• Python (w/ Django framework)
• Matplotlib
• Git repo, collaborators welcome
• Source code available
• Can also be run standalone