Technical-note-on-implementation.md

+# Introduction
+
+The bilby implementation follows the implementation of the dispersion implemented in LAL simulation. The parametrization of LIV has the form:
+```math
+E^2=(pc)^2+m_g^2c^4+A_{\alpha}(pc)^{\alpha}
+```
+Source `https://arxiv.org/pdf/1110.2720.pdf` derives how it translates to the phase correction of the GW signal. The exact form of the correction implemented in LAL can be found for example here `https://dcc.ligo.org/public/0156/P1800316/010/o2_tgr.pdf` and looks like:
+```math
+\delta \Phi_{\alpha}(f)=sign(A_\alpha) \begin{cases}\frac{\pi D_L}{\alpha-1}\lambda_{A,eff}^{\alpha-2}(\frac f c)^{\alpha-1} & \alpha  \neq 1\\\frac{\pi D_L}{\lambda_{A,eff}}\ln(\frac{\pi G \mathcal{M}f}{c^3})& \alpha = 1\end{cases} 
+```
+where
+
+```math
+\lambda_{\alpha}=hc|A_{\alpha}|^{1/(\alpha-2)}
+```
+from `https://arxiv.org/pdf/1110.2720.pdf` (eq. 13), $`c`$ is suppressed in the reference
+
+```math
+A_{\alpha, eff} = \frac{D_{\alpha}}{D_L}(1+z)^{\alpha-1}A_{\alpha}
+```
+```math
+\frac{\lambda_{\alpha, eff}}{\lambda_{\alpha}} = \frac{hc|A_{\alpha, eff}|^{1/(\alpha-2)}}{hc|A_{\alpha}|^{1/(\alpha-2)}} = (\frac{D_{\alpha}}{D_L}(1+z)^{\alpha-1})^{1/(\alpha-2)}
+```
+from `https://dcc.ligo.org/public/0156/P1800316/010/o2_tgr.pdf` (eq. 4), though the authors made a typo and made proportionality factor be the inverse of the correct one written here.
+
+```math
+D_{\alpha} = \frac{c(1+z)^{(1-\alpha)}}{H_0}\int_0^z \frac{(1+\bar{z})^{(\alpha-2)}d\bar{z}}{\sqrt{\Omega_M(1+\bar{z})^3+\Omega_\Lambda}}
+```
+from `https://arxiv.org/pdf/1110.2720.pdf` (eq. 15). $`c`$ is suppressed in the reference, $`H_0`$ is the Hubble constant, $`\Omega_M`$ and $`\Omega_\Lambda`$ are matter and dark energy densities.
+
+
+The phase correction used here has the form:
+
+```math
+\delta \Phi_{\alpha}(f)= -\frac{\pi D_L h^{\alpha-2}}{c}A_{\alpha,eff}(f )^{\alpha-1}
+```
+
+The difference stems from the 2 points:
+1. The correction is parametrized in terms of $` A_{\alpha,eff}`$ instead of $` \lambda_{A,eff}`$
+2. The correction used in lal inference was derived using phase velocity, while this one uses group velocity, following `https://dcc.ligo.org/DocDB/0182/P2200154/001/GW_phase_degeneracies.pdf`.
+
+# Reparametrizing phase velocity correction in terms of A
+
+1. Route one - plug in the definition of $`\lambda_{\alpha, eff}`$.
+```math
+\begin{aligned}
+\delta \Phi_{\alpha}(f)
+=sign(A_\alpha) &\begin{cases}\frac{\pi D_L}{\alpha-1}\lambda_{A,eff}^{\alpha-2}(\frac f c)^{\alpha-1} & \alpha  \neq 1\\\frac{\pi D_L}{\lambda_{A,eff}}\ln(\frac{\pi G \mathcal{M}f}{c^3})& \alpha = 1\end{cases} \\
+= sign(A_\alpha) &\begin{cases}\frac{\pi D_L}{\alpha-1}(hc|A_{\alpha, eff}|^{1/(\alpha-2)})^{\alpha-2}(\frac f c)^{\alpha-1} & \alpha  \neq 1\\\frac{\pi D_L}{hc|A_{\alpha, eff}|^{-1}}\ln(\frac{\pi G \mathcal{M}f}{c^3})& \alpha = 1\end{cases} \\
+=&\begin{cases}\frac{\pi D_L h^{\alpha-2}}{(\alpha-1)c}A_{\alpha, eff}f^{\alpha-1} & \alpha  \neq 1\\\frac{\pi D_L}{hc}A_{\alpha, eff}\ln(\frac{\pi G \mathcal{M}f}{c^3})& \alpha = 1\end{cases} 
+\end{aligned}
+```
+2. Route two - use explicitly derivation form `https://arxiv.org/pdf/1110.2720.pdf`
+Combining equations 28-32 (remembering that they are in units $`G=c=1`$) together and neglecting graviton mass terms, as was done in the LAL implementation:
+```math
+\begin{aligned}
+\delta \Phi_{\alpha}(f)
+&=\begin{cases}-\frac{\mathrm{\pi}^{2-\alpha}D_{\alpha}(G\mathcal{M}/c^2)^{1-\alpha}}{(1-\alpha)\lambda_\alpha^{2-\alpha}(1+z)^{1-\alpha}}(\frac{\mathrm{\pi}G\mathcal{M}f}{c^3})^{\alpha-1} & \alpha  \neq 1\\\ \frac{\mathrm{\pi}D_{\alpha}}{\lambda_\alpha}\ln(\frac{\pi G \mathcal{M}f}{c^3})& \alpha = 1\end{cases} \\
+&=\begin{cases}\frac{\mathrm{\pi}D_{\alpha}(1+z)^{\alpha-1}}{(\alpha-1)}\lambda_\alpha^{\alpha-2}(\frac{f}{c})^{\alpha-1} & \alpha  \neq 1\\\ \frac{\mathrm{\pi}D_{\alpha}}{\lambda_\alpha}\ln(\frac{\pi G \mathcal{M}f}{c^3})& \alpha = 1\end{cases} \\
+&=\begin{cases}\frac{\mathrm{\pi} D_L}{\alpha-1}\lambda_{A,eff}^{\alpha-2}(\frac f c)^{\alpha-1} & \alpha  \neq 1\\\frac{\pi D_L}{\lambda_{A,eff}}\ln(\frac{\pi G \mathcal{M}f}{c^3})& \alpha = 1\end{cases} \, ,
+\end{aligned}
+```
+and then follow the steps of route 1.
+
+# Parametrization in terms of group velocity.
+
+Reference `https://dcc.ligo.org/DocDB/0182/P2200154/001/GW_phase_degeneracies.pdf` derives the LIV correction if one uses group velocity instead of phase velocity. The result is eq. 2.9:
+
+```math
+\delta \Phi_{\alpha}(f)= -\frac{D_\alpha [\omega(1+z)]^{\alpha-1}}{2}A_{\alpha}
+```
+The paper uses $`\hbar=1`$ and does not include $'c'$ in the definition of $`D_\alpha`$ (so it is in units of time). Accounting for these, the phase correction becomes:
+
+```math
+\delta \Phi_{\alpha}(f)= -\frac{\pi D_\alpha [f(1+z)]^{\alpha-1}h^{\alpha-2}}{c}A_{\alpha}= -\frac{\pi D_L h^{\alpha-2}}{c}A_{\alpha,eff}(f )^{\alpha-1}
+```
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