Gravitational wave events sampling (Analytical formulation)

Initial setup

Note: I will interchangeably use terms like observable events and detectable events.

Define all the parameters involved.

  • Source parameters: \(\theta \in \{\) \(m_1\) (mass of the heavier one), \(m_2\) (mass of the lighter one), \(\iota\) (inclination-angle), \(\phi\) (phase-of-coalescence), \(\psi\) (polarization-angle), \(ra\) (right-ascension), \(dec\) (declination) \(\}\) and \(z_s\) : red-shift of the source.

Given \(d N^U_{obs}(z_s)\) is the number of detectable gravitational wave (GW) events from sources at red-shift \(z_s\), then, let rate of observing GW (number of such events happening per unit time) is given by,

\begin{equation} \begin{split} \mathcal{R}_U &= \int_{z_{min}}^{z_{max}} \frac{d N^U_{obs}(z_s)}{d \tau} \\ &= \int_{z_{min}}^{z_{max}} \frac{d N^U_{obs}(z_s)}{d \tau \;dV_c} \frac{dV_c}{dz_s} dz_s \end{split} \tag{1} \end{equation}

\(\frac{d N^U_{obs}(z_s)}{d \tau \;dV_c}\) is the observed merger rate density at source-frame, and \(\frac{dV_c}{dz_s}\) is the differential co-moving volume at red-shift \(z_s\). After taking care of time-dilation, the expression looks,

\begin{equation} \begin{split} \mathcal{R}_U &= \int_{z_{min}}^{z_{max}} \frac{d N^U_{obs}(z_s)}{d t\;dV_c} \frac{1}{1+z_s} \frac{dV_c}{dz_s} dz_s \\ &= \int_{z_{min}}^{z_{max}} R^U_{obs}(z_s) dz_s \end{split} \tag{2} \end{equation}

Observed merger rate density of GW events at source red-shift \(z_s\) (detector-frame) is given by \(R^U_{obs}(z_s) = \frac{d N^U_{obs}(z_s)}{d t\;dV_c}\). And, let \(R(z_s)\) be the rate of the merger of GW sources (source frame) at \(z_s\), regardless of whether it is detectable or not.

\begin{equation} \begin{split} \mathcal{R}_U &= \int_{z_{min}}^{z_{max}} R^U_{obs}(z_s) dz_s \\ &= \int_{z_{min}}^{z_{max}} R(z_s) P(obs|z_s) dz_s \\ &= N_1 \int_{z_{min}}^{z_{max}} P(z_s) P(obs|z_s) dz_s \\ \end{split} \tag{3} \end{equation}

\(P(obs|z_s)\) is the probability of observing a GW event at red-shift \(z_s\). Normalizing factor: \(N_1 = \int_{z_{min}}^{z_{max}} R(z_s) dz_s\).

Rate equation (statistical formulation)

Let’s now consider all the source parameters \(\theta \in \{m_1,m_2,z_s,\iota,\phi,\psi,ra,dec,\text{spins}\}\),

\begin{equation} \begin{split} \mathcal{R}_U &= N_1 \int_{z_{min}}^{z_{max}} P(z_s) P(obs|z_s) dz_s \\ & \text{consider} \int \rightarrow \int_{z_s}\int_{\theta} \rightarrow \int_{z_s}\int_{m_1}\int_{m_2}\int_{\iota}\int_{\phi}\int_{\psi}\int_{ra}\int_{dec}\int_{\text{spins}} \\ \mathcal{R}_U &= N_1 \int P(z_s) P(obs|z_s, \theta) P(\theta) d\theta dz_s \end{split} \tag{8} \end{equation}

Here, I have assumed \(P(\theta)\) is a normalized multidimensional pdf. All parameters in \(\theta\), other than \(m_1\) and \(m_2\), are sampled independently. The pdf \(P(\theta)\) is a product of individual pdfs for each parameter.

Final expression for the rate of observing GW events is given by,

\begin{equation} \begin{split} \mathcal{R}_U &= N_1 \bigg< P(obs|\theta, z_s) \bigg>_{z_s\in P(z_s), \theta\in P(\theta)} \\ P(\theta) &= P(m_1,m_2)P(\iota)P(\phi)P(\psi)P(ra)P(dec)P(\text{spins}) \end{split} \tag{4} \end{equation}

The order of sampling and rate calculation steps in LeR are listed below.

  1. Sample \(z_s\) from \(P(z_s) = \frac{R(z_s)}{N_1}\).

  2. Sample \(\theta\) from \(P(\theta)\).

  3. Calculate SNR with gwsnr

  4. Apply the SNR threshold and check whether the event is detectable or not.

  5. Calculate rate of GW events, \(\mathcal{R}_U\) using equation 4.

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