# Horizon Distance

The **horizon distance**, denoted $D_{\rm hor}$, is a key metric for assessing the sensitivity of a gravitational-wave detector or network. It is defined as the maximum luminosity distance at which a specific GW source can be reliably detected, i.e., where its signal-to-noise ratio (SNR) meets or exceeds a given threshold, typically $\rho_{\rm th} = 8$.

This distance is computed under the optimal scenario: the source is directly overhead and face-on with respect to the detector(s), providing the strongest possible signal. The `gwsnr` package offers two approaches for calculating the horizon distance: an **analytical scaling method** and a **numerical root-finding method**.

## 1. Analytical Method

The analytical approach calculates the horizon distance by scaling a known SNR measurement. For any generic inspiral–merger–ringdown (IMR) waveform, the horizon distance can be determined from the optimal SNR ($\rho_{\rm opt}$) at a given [effective distance](interpolation.md#mathematical-formulation) ($D_{\rm eff}$):

$$
\begin{align}
D_{\rm hor} &= \frac{\rho_{\rm opt}}{\rho_{\rm th}} D_{\rm eff}\, \tag{1}
\end{align}
$$

Notably, this approach does not require explicit optimization over sky location, as the dependence cancels between the SNR and the effective distance.

In `gwsnr`, $\rho_{\rm opt}$ is computed using the core functionalities, such as the noise-weighted inner product or the Partial Scaling interpolation method. The package also includes a JIT-compiled function for efficient evaluation of $D_{\rm eff}$.

This general formula in Eq.(1) is rooted in the work of [Allen et al. (2012)](https://arxiv.org/pdf/gr-qc/0509116), which provides a specific analytical expression for simple inspiral waveforms:

$$
\begin{align}
D_{\rm hor} &= \left( \frac{1~\mathrm{Mpc}}{\rho_{\rm th}} \right) \mathcal{A}_{1~\mathrm{Mpc}}(\mathcal{M}) \sqrt{ 4 \int_{f_{\rm min}}^{f_{\rm max}} \frac{f^{-7/3}}{S_n(f)}\,df }, \tag{2}
\end{align}
$$

where $\mathcal{M}$ is the chirp mass, and $\mathcal{A}_{1~\mathrm{Mpc}}$ denotes the waveform's intrinsic amplitude at a distance of 1 Mpc. While Eq.(1) was originally derived for simple inspiral signals, it remains applicable to general IMR waveforms dominated by the quadrupolar (2,2)-mode, provided an appropriate waveform model is used to compute $\rho_{\rm opt}$.

## 2. Numerical Method

The numerical method directly determines the horizon distance by finding the specific distance at which the SNR matches the detection threshold. This involves two main steps:

* **Optimization:** For a given set of relevant GW parameters (masses, spins, phase, etc.), `gwsnr` first finds the optimal sky location (for each detector) that maximize the SNR, providing the best-case SNR for that source type at a reference distance.

* **Root-Finding:** With the optimal configuration fixed, the algorithm then numerically solves for the luminosity distance, $d_L$, that satisfies the equation:

  $$
  F(d_L) = \rho(d_L) - \rho_{\rm th} = 0
  $$

This approach is also applicable to a network of detectors, in which case the optimal sky location is found for the detector network, where the single-detector SNR, $\rho$, is replaced by the network SNR, $\rho_{\rm net}$, and then the root-finding is performed to find the distance at which $\rho_{\rm net} = \rho_{\rm th}$.


## Example Usage

```python
# loading GWSNR class from the gwsnr package
from gwsnr import GWSNR

# initializing the GWSNR class with default configuration and interpolation method
gwsnr = GWSNR(mtot_min=2., mtot_max=6., ifos=['H1', 'L1', 'V1'])

# 1. Analytical Method
d_hor_analytical = gwsnr.horizon_distance_analytical(mass_1=1.4, mass_2=1.4)

# 2. Numerical Method
d_hor_numerical, _, _ = gwsnr.horizon_distance_numerical(mass_1=1.4, mass_2=1.4)

# print the type of the SNRs
print(f"horizon distance (analytical): {d_hor_analytical} Mpc")
print(f"horizon distance (numerical): {d_hor_numerical} Mpc")
```

```
horizon distance (analytical): {'L1': array([416.49490771]), 'H1': array([416.49490771]), 'V1': array([317.93916075])} Mpc
horizon distance (numerical): {'L1': 416.49483499582857, 'H1': 416.4948722301051, 'V1': 228.76581222284585, 'snr_net': 557.9508489044383} Mpc
```

## Horizon Distance of a BNS System across various Detectors

<div align="center">
<figure>
  <img src="_static/horizon_distance_for_BNS.png" alt="Horizon distance for BNS systems across detectors" />
  <figcaption align="left">
    <b>Figure.</b> Horizon distance ($D_{\rm hor}$) for binary neutron star (BNS) systems with $m_1=m_2=1.4\,M_\odot$ across various detectors. The bar heights show the maximum detection range (in Mpc, log scale) for the LIGO detectors (L1, H1, V1), their network (L1+H1+V1), and future third-generation observatories (Einstein Telescope, ET; Cosmic Explorer, CE). The annotated values indicate the corresponding redshift at each horizon distance, highlighting the enhanced reach of next-generation detectors.
  </figcaption>
</figure>
</div>