---
title: Stochastic Gravitational Wave Background
url: https://www.emergentmind.com/topics/stochastic-gravitational-wave-background-fe23a5d1-7645-4656-91da-dd3bbee01572
type: topic
---

# Stochastic Gravitational Wave Background

The stochastic gravitational wave background (SGWB) is the superposition of myriad unresolved gravitational-wave signals from diverse sources throughout the universe, forming a persistent, broadband random field. This ensemble encodes both the collective astrophysical history of compact object mergers, stellar core-collapse, and other dynamical phenomena, as well as signals from cosmological processes such as inflation, particle production, and topological defects. Within the context of astrophysics, stellar core-collapse events—primarily core-collapse supernovae (CCSNe)—represent a well-characterized but subdominant component of the total extragalactic SGWB, with distinctive spectral and morphological features as established by waveform modeling and rate estimates [2110.01478; 2409.01542; 2001.09663].

## 1. Energy Density Spectrum Formalism

The SGWB is quantified through the dimensionless energy-density spectrum
\[
\Omega_{\mathrm{GW}}(f) = \frac{1}{\rho_c} \frac{d\rho_{\mathrm{GW}}}{d\ln f}
\]
where $\rho_c = 3 H_0^2 c^2 / (8 \pi G)$ is the critical density of the universe, $H_0$ the Hubble parameter, and $f$ the gravitational-wave frequency. In a cosmological context, $\Omega_{\mathrm{GW}}(f)$ gives the fraction of the cosmic energy density per logarithmic frequency interval stored in GWs. For a population of discrete, independent sources:
\[
\Omega_{\mathrm{GW}}(f) = \frac{f}{\rho_c c^3} \sum_i \int dz\ \frac{R_i(z)}{(1+z)H(z)}\ \frac{dE_{GW,i}}{df_s}\Big[(1+z)f\Big]
\]
where $R_i(z)$ is the comoving event rate density for class $i$, $H(z)$ is the Hubble expansion term, $f_s = (1+z) f$ the source-frame frequency, and $dE_{GW,i}/df_s$ the single-source energy spectrum [2110.01478; 2409.01542].

## 2. Core-Collapse Contribution: Waveforms and Population Synthesis

### Single-Event Spectral Characteristics

State-of-the-art numerical models categorize CCSN GW emission by progenitor rotation:

- **Non-rotating/slowly rotating progenitors**: $dE/df_s$ rises sharply from $\sim$100 Hz, peaks near 500–800 Hz (driven by proto-neutron star oscillations and fluid instability), and drops above 2 kHz. Peak single-event spectral densities reach $10^{46}–10^{47}$ erg Hz$^{-1}$.

- **Moderately rotating**: Additional spectral structure at 100–300 Hz driven by SASI (Standing Accretion Shock Instability) spiral and sloshing modes; peak $dE/df_s$ $\sim \text{few} \times 10^{46}$ erg Hz$^{-1}$, sometimes with secondary sub-peaks.

- **Rapidly rotating/extreme**: Strong coupling at 200–800 Hz from low-$T/|W|$ rotational instability and SASI; the most extreme models reach $dE/df_s \sim 10^{48}$ erg Hz$^{-1}$ [2110.01478].

No analytic formula for the full spectrum is given in [2110.01478], but the energy spectra display a robust two-peak morphology, with integrated emission per event ranging $\sim 10^{-9}$–$10^{-7}\ M_\odot c^2$.

### Cosmological Integration and Rate Modeling

The cosmic CCSN rate is modeled as $R_{CC}(z) = \lambda_{CC} R_*(z)$ where $R_*(z)$ is the cosmic star-formation rate (SFR) and $\lambda_{CC}$ the fraction of stellar mass in progenitors exceeding $8 M_\odot$. For a Salpeter initial mass function,
\[
\lambda_{CC} \approx 0.007\ M_\odot^{-1}
\]
The SFR parametrization follows [Vangioni et al. 2015]:
\[
R_*(z) = \nu p \frac{e^{q(z-z_m)}}{p-q+q e^{p(z-z_m)}}
\]
with $\nu=0.178\ M_\odot\ \text{yr}^{-1}\ \text{Mpc}^{-3}$, $z_m=2.0$, $p=2.37$, $q=1.80$ [2110.01478]. Event rates above $z \sim 2$ are negligible due to the decline in SFR.

## 3. SGWB Spectral Results and Scaling

### Peak Amplitudes and Frequency Scaling

Cosmological integration of the above yields:
- **Non-rotating/slowly rotating dominant case**: $\Omega_{\mathrm{GW}}(f) \sim 10^{-14}$–$10^{-13}$ between 100–1000 Hz, peaking broadly around 500 Hz.
- **Extreme all-rapid-rotator scenario**: $\Omega_{\mathrm{GW}}(f) \sim 10^{-11}$ near 400 Hz—achievable only if all CCSNe were maximally rapidly rotating, which is physically implausible.

Specific numerical values for representative models (from [2110.01478] Fig. 2):

| Model   | Peak $\Omega_{\mathrm{GW}}$ | Peak Frequency (Hz) |
|---------|----------------------------|---------------------|
| s15nr   | $2 \times 10^{-15}$        | $\sim$500           |
| Rad25   | $5 \times 10^{-14}$        | $\sim$600           |
| Shib2   | $3 \times 10^{-11}$        | $\sim$400           |

For realistic core-collapse populations (slow-rotators dominate), the SGWB peak is consistently $>100$ times below third-generation detector reach.

## 4. Astrophysical and Cosmological Comparison

Relative to other astrophysical and cosmological sources:
- **Compact binary coalescences (CBC; BBH/BNS)**: $\Omega_{\mathrm{GW}} \sim 10^{-9}–10^{-8}$ at $\sim$25 Hz, dominating the 10–100 Hz band.
- **Cosmic strings**: Wide, flat plateau $\Omega_{\mathrm{GW}} \sim 10^{-11}–10^{-8}$, highly model-dependent but generally above the CCSN background.
- **Inflationary or first-order phase transition GWs**: Model-dependent, $\Omega_{\mathrm{GW}} \sim 10^{-16}–10^{-11}$ across 1–$10^4$ Hz [2110.01478; 2001.09663; 1811.08797].

The CCSN SGWB is always subdominant except (in extreme models) near $400$–$600$ Hz, and even there, it would only approach detectability with an all-rapid-rotator assumption.

## 5. Detector Sensitivity, Detectability, and Masking

Current and upcoming GW detectors have the following sensitivity at relevant frequencies:
- **Advanced LIGO/Virgo**: $\Omega_{\text{GW,~sens}} \sim 10^{-9}$–$10^{-10}$ at 100–1000 Hz.
- **Einstein Telescope, Cosmic Explorer (third generation)**: Power-law-integrated sensitivity at $\sim 10^{-13}$ (one-year, two-detector, cross-correlation).
- **Other projected SGWB search limits**: Down to $10^{-12}$–$10^{-13}$ with multiyear, networked operation [2110.01478; 2409.01542].

Even optimistically, the core-collapse background is $\gtrsim 100\times$ below third-generation experimental sensitivity for the most probable scenario. It therefore cannot mask or bias searches for other, stronger backgrounds such as from CBC or cosmological phenomena.

## 6. Uncertainties, Parameter Dependence, and Model Limitations

Uncertainties in the CCSN-SGWB predictions stem from several sources [2110.01478]:
- **Rate normalization ($\lambda_{CC}$, IMF)**: $\pm$ a factor of 2; translates linearly to $\Omega_{\mathrm{GW}}$.
- **Star-formation history**: $<$30% effect on amplitude for reasonable SFR models.
- **Single-event spectral diversity**: Factor of $\sim 5$ uncertainty from progenitor mass structure.
- **Rotation fraction**: Only $\lesssim10\%$ of CCSNe are expected to be rapidly rotating, strongly modulating the possible high-end tail of $\Omega_{\mathrm{GW}}$.
- **Neglected and subdominant effects**: Anisotropic neutrino memory (important only $<$1 Hz); strong magnetohydrodynamic effects at kHz; both yield subdominant or uncertain additions.

The high-redshift ($z>2$) contribution to CCSN-SGWB is negligible, and changes in the choice of SFR prescription alter the results only moderately.

## 7. Prospects, Significance, and Theoretical Implications

- **Practical detectability**: Realistic detection of the CCSN SGWB is outside the scope of current and planned third-generation ground-based detectors. Only under implausibly high GW emission per event or an unusually high fraction of rapidly rotating progenitors could $\Omega_{\mathrm{GW}}$ approach detectability thresholds [2110.01478; 2409.01542].

- **Scientific role**: The CCSN contribution is an astrophysical background of interest for theoretical completeness and as a verification baseline for population synthesis and 3D explosion modeling. Its "subdominant" nature ensures that it does not constitute a limiting foreground for the detection of primordial or CBC-generated backgrounds.

- **Distinctive spectral signature**: The central frequency ($\sim$500–800 Hz), two-peak structure, and broad plateau are unique identifiers. Detection (or upper limits) could probe aspects of progenitor rotation statistics, explosion mechanism variability, and nuclear equation-of-state under extreme conditions.

- **Impact on SGWB searches**: The CCSN background does not mask cosmological or binary merger signals nor introduce significant bias below the $\Omega_{\mathrm{GW}} \sim 10^{-13}$ sensitivity regime. Thus, it poses little to no hindrance to the interpretation or extraction of stronger stochastic components [2110.01478; 2409.01542; 2001.09663].

---

In summary, the SGWB from core-collapse events, as rigorously modeled with current simulations and cosmic rate estimates, is a well-characterized but subdominant feature of the gravitational-wave sky, peaking at $\Omega_{\mathrm{GW}} \sim 10^{-14}$–$10^{-13}$ at 500–800 Hz for realistic progenitor populations. Its amplitude sits $2$–$5$ orders of magnitude below both astrophysical (CBC) and many cosmological backgrounds, remaining unobservable with envisioned detector capabilities for the foreseeable future [2110.01478; 2409.01542].

Source: https://www.emergentmind.com/topics/stochastic-gravitational-wave-background-fe23a5d1-7645-4656-91da-dd3bbee01572