Stochastic Gravitational Wave Background

• Stochastic gravitational wave background is a persistent, broadband random field created by the superposition of unresolved GW signals from both compact astrophysical events and cosmological processes.
• The energy-density spectrum is modeled via cosmic rate and waveform synthesis, revealing characteristic dual peaks near 500–800 Hz with amplitudes around 10⁻¹⁴ to 10⁻¹³.
• This topic underlines its role as a verification baseline for core-collapse models while distinguishing subdominant astrophysical backgrounds from stronger, cosmological signals.

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 (Finkel et al., 2021, Chowdhury et al., 3 Sep 2024, Pacheco, 2020).

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 (Finkel et al., 2021, Chowdhury et al., 3 Sep 2024).

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$^{-1}$ (Finkel et al., 2021).

No analytic formula for the full spectrum is given in (Finkel et al., 2021), 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 $$\nu=0.178\ M_\odot\ \text{yr}^{-1}\ \text{Mpc}^{-3}$$, $z_m=2.0$, $p=2.37$, $q=1.80$ (Finkel et al., 2021). 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 (Finkel et al., 2021) 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 (Finkel et al., 2021, Pacheco, 2020, Christensen, 2018).

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 (Finkel et al., 2021, Chowdhury et al., 3 Sep 2024).

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 (Finkel et al., 2021):

• 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 (Finkel et al., 2021, Chowdhury et al., 3 Sep 2024).
• 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 (Finkel et al., 2021, Chowdhury et al., 3 Sep 2024, Pacheco, 2020).

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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 (Finkel et al., 2021, Chowdhury et al., 3 Sep 2024).