---
title: 'LIGO S5: Constraining Inflationary RGWs'
url: https://www.emergentmind.com/topics/s5
type: topic
---

# LIGO S5: Constraining Inflationary RGWs

S5, in the context of relic-gravitational-wave searches, denotes the **LIGO S5 run** whose achieved design sensitivity, published strain data, and H1–L1 cross-correlation stochastic-background result made direct constraints on inflationary **relic gravitational waves (RGWs)** realistic. In the framework of “Constraints upon the spectral indices of relic gravitational waves by LIGO S5” [1004.2944], S5 is used to compare analytic RGW spectra against LIGO sensitivity in the \(\sim 40\)–\(500\) Hz band and against the published stochastic-background upper limit \(\Omega_0<6.9\times10^{-6}\) around \(\sim 100\) Hz, with the aim of constraining the inflationary parameters \((\beta,\alpha_t,r)\).

## 1. Observational role of S5

S5 is relevant because it provided both **single-detector strain sensitivity** and, more importantly, a **cross-correlated H1–L1 stochastic-background result**. In this usage, “constraints by LIGO S5” means that one computes the predicted present-day RGW spectrum for given \((\beta,\alpha_t,r)\), compares it with the S5 sensitivity in the \(\sim 40\)–\(500\) Hz range, and computes the expected cross-correlation signal-to-noise ratio for the H1–L1 detector pair over the full S5 observing time [1004.2944].

The observing duration adopted is
\[
T = 59{,}961{,}600~\mathrm{s},
\]
corresponding to Nov. 5, 2005 to Sep. 30, 2007. The S5 upper limit quoted in the paper,
\[
\Omega_0 < 6.9\times 10^{-6},
\]
is a bound for a **flat stochastic background** near \(\sim 100\) Hz. This distinction matters because the RGW spectra considered in the paper are generally not flat.

## 2. RGW parameterization constrained by S5

The RGW spectrum is parameterized by three inflationary or initial-condition quantities: the spectral index parameter \(\beta\), the tensor running index \(\alpha_t\), and the tensor-to-scalar ratio \(r\). The parameter \(\beta\) is related to the inflationary scale factor through
\[
a(\tau)\propto |\tau|^{1+\beta},
\]
with \(\beta\simeq -2\) giving a nearly scale-invariant spectrum. Larger \(\beta\), meaning less negative values such as \(-1.88\) instead of \(-2.0\), strongly boosts the high-frequency amplitude in the LIGO band. The running \(\alpha_t\) introduces logarithmic bending of the primordial tensor spectrum, and even very small positive \(\alpha_t\) appreciably raises the amplitude at \(\sim 100\) Hz. The ratio
\[
r \equiv \frac{\Delta_h^2(k_0)}{\Delta_\mathcal{R}^2(k_0)}
\]
sets the overall normalization, and the computed SNR satisfies \(\mathrm{SNR}\propto r\) [1004.2944].

The initial spectrum at horizon crossing is taken to be
\[
h(k,\tau_i) = \Delta_\mathcal{R}(k_0)\, r^{1/2} \left(\frac{k}{k_0}\right)^{2+\beta+\frac{1}{4}\alpha_t \ln(k/k_0)} ,
\]
with pivot \(k_0\) corresponding to \(0.002\,\mathrm{Mpc}^{-1}\) and
\[
\Delta_\mathcal{R}^2(k_0) = (2.445\pm 0.096)\times 10^{-9}.
\]
This parameterization makes the S5 sensitivity especially dependent on \(\beta\) and \(\alpha_t\), because small changes in either can change the LIGO-band amplitude by orders of magnitude.

## 3. Present-day observables and detection formalism

The quantity directly constrained by S5 is the stochastic-background energy density
\[
\Omega_g(f) = \frac{2\pi^2}{3}\, h_c^2(f)\left(\frac{f}{H_0}\right)^2,
\]
with
\[
H_0 = 3.24\, h \times 10^{-18}\,\mathrm{Hz},
\]
and frequency–wavenumber relation
\[
f = \frac{k H_0}{2\pi \gamma},
\qquad \gamma \simeq 1.97
\]
for \(\Omega_\Lambda\simeq 0.73\). These formulas connect the inflationary RGW model to the quantity bounded by the S5 stochastic search [1004.2944].

For a single interferometer, the paper uses a qualitative detectability criterion based on comparing
\[
\frac{h_c(f)\sqrt{F}}{\sqrt{2f}}
\]
with the detector strain sensitivity, where \(F=2/5\) is the angular factor for one interferometer. For two-detector cross-correlation, the sensitivity improves substantially because the narrow-band detectability condition acquires the factor \((2T\Delta f)^{1/4}\gg 1\) for long integration.

The main quantitative tool is the Allen–Romano stochastic-background SNR:
\[
\mathrm{SNR} = \frac{3H_0^2}{10\pi^2}\sqrt{T} \left[ \int_{-\infty}^{\infty} df\, \frac{\gamma^2(f)\,\Omega_g^2(f)} {f^6 P_1(f)P_2(f)} \right]^{1/2}.
\]
Here \(P_1(f)\) and \(P_2(f)\) are the one-sided noise power spectra of H1 and L1, \(\gamma(f)\) is the overlap reduction function, and \(\Omega_g(f)\) is the predicted RGW spectrum. Methodologically, S5 enters through the published H1/L1 noise spectra, the overlap reduction function, and the full observing time.

## 4. Direct constraints from S5

Using the single-interferometer H1/L1 design sensitivity achieved during S5, the paper finds for the benchmark model \(r=0.55\), \(\alpha_t=0\),
\[
\beta \le -1.85.
\]
Using the stronger **cross-correlation** bound from S5, the constraint becomes
\[
\beta \le -1.88
\qquad (r=0.55,\ \alpha_t=0).
\]
For the benchmark model \(r=0.55\), \(\beta=-2.0\), the corresponding limits are
\[
\alpha_t \le 0.018
\]
from single interferometers and
\[
\alpha_t \le 0.01
\]
from cross-correlated S5. The principal direct S5 limits are therefore
\[
\beta \le -1.88,
\qquad
\alpha_t \le 0.01
\]
for the benchmark choice \(r=0.55\), holding the other parameter fixed as specified [1004.2944].

The spectral response in the S5 band is extremely sensitive to these parameters. Changing \(\beta\) from \(-2.0\) to \(-1.85\) increases \(h_c(f)\) by about **4 orders of magnitude** around \(100\) Hz, and changing \(\alpha_t\) from \(0\) to \(0.018\) also increases \(h_c(f)\) by about **4 orders of magnitude** around \(100\) Hz. Representative energy-density slopes are
\[
\Omega_g(f)\propto f^{0.24}
\quad \text{for } \beta=-1.88,\ \alpha_t=0,
\]
and
\[
\Omega_g(f)\propto f^{0.45}
\quad \text{for } \beta=-2.0,\ \alpha_t=0.01.
\]
This is why the flat-spectrum S5 bound \(\Omega_0<6.9\times10^{-6}\) cannot be identified directly with the tilted RGW spectra treated in the paper.

The paper also emphasizes a \(\beta\)–\(\alpha_t\) degeneracy in the S5 band. The models
\[
\beta=-2.0,\ \alpha_t=0.011
\]
and
\[
\beta=-1.88,\ \alpha_t=0
\]
give essentially the same amplitude around \(100\) Hz and very similar \(\Omega_g(f)\) slopes. Because S5 probes a relatively narrow frequency interval,
\[
(41.5,\ 169.25)\ \mathrm{Hz},
\]
it is unlikely to distinguish such models cleanly.

## 5. SNR calculations and detectability

For \(r=0.1\), the paper computes H1–L1 cross-correlation SNRs for several \((\beta,\alpha_t)\) choices. For \(\beta=-2.0\), the SNRs are \(5.4\times10^{-6}\), \(8.0\times10^{-4}\), \(6.0\times10^{-3}\), and \(1.2\times10^{-1}\) for \(\alpha_t=0\), \(0.005\), \(0.007\), and \(0.01\), respectively. For \(\beta=-1.96\), the corresponding values are \(2.0\times10^{-4}\), \(3.0\times10^{-2}\), \(2.2\times10^{-1}\), and \(4.5\). For \(\beta=-1.90\), they are \(4.5\times10^{-2}\), \(6.7\), \(5.0\times10^{1}\), and \(1.0\times10^{3}\). For \(\beta=-1.88\), they are \(2.8\times10^{-1}\), \(4.1\times10^{1}\), \(3.0\times10^{2}\), and \(6.2\times10^{3}\) [1004.2944].

These values scale linearly with \(r\), since \(\mathrm{SNR}\propto r\). The numerical pattern is the paper’s clearest demonstration that detectability rises steeply as \(\beta\) increases, as \(\alpha_t\) becomes positive, or as \(r\) increases. Conversely, for nearly scale-invariant and weakly running spectra such as \(\beta=-2.0\), \(\alpha_t=0\), the expected SNR is extremely small.

A common misconception addressed by the paper is that the S5 stochastic limit alone determines detectability for all inflationary RGW models. The SNR calculations show instead that S5 constrains **parameter combinations** through the detailed shape and amplitude of \(\Omega_g(f)\), not through a single universal bound.

## 6. Comparison with indirect bounds and overall significance

The paper compares S5 with indirect bounds on the **integrated** RGW energy density
\[
\Omega_{gw} = \int_{f_{\rm low}}^{f_{\rm upper}} \Omega_g(f)\,\frac{df}{f}.
\]
It explicitly stresses that \(\Omega_{gw}\) and \(\Omega_g(f)\) should not be confused: the former is an integral over frequency and only approximates the latter when the spectrum is flat over a logarithmic interval of width \(\Delta\ln f\sim 1\), which is not generally true for the RGW models considered [1004.2944].

The indirect bounds used are
\[
\Omega_{gw} < \Omega_{\rm BBN} = 1.5\times 10^{-5},
\qquad f_{\rm low}\sim 10^{-10}\ \mathrm{Hz},
\]
for BBN, and
\[
\Omega_{gw} < \Omega_{\rm CMB}=1.62\times 10^{-5},
\qquad f_{\rm low}\sim 10^{-15}\ \mathrm{Hz},
\]
for CMB, with
\[
f_{\rm upper}=10^{10}\ \mathrm{Hz}.
\]
These yield tighter constraints than S5. In particular,
\[
\beta \lesssim -1.96
\quad \text{for } r=0.55,\ \alpha_t=0,
\]
\[
\beta \lesssim -1.98
\quad \text{for } r=0.1,\ \alpha_t=0,
\]
\[
\alpha_t \lesssim 0.004
\quad \text{for } r=0.55,\ \beta=-2.0,
\]
and
\[
\alpha_t \lesssim 0.005
\quad \text{for } r=0.1,\ \beta=-2.0.
\]

The significance of S5 is therefore specific and limited. S5 provides a realistic and meaningful **direct** constraint on the present-day RGW spectral amplitude in the LIGO band, and for the benchmark \(r=0.55\) it implies
\[
\beta \le -1.88,\qquad \alpha_t \le 0.01.
\]
At the same time, it does **not** beat indirect BBN/CMB constraints on the total RGW energy density, and its relatively narrow frequency window does not effectively break the \(\beta\)–\(\alpha_t\) degeneracy. The paper notes that a broader-band detector such as **LISA** would have a better chance of distinguishing models with different \(\beta\) and \(\alpha_t\). A plausible implication is that S5’s principal legacy in this framework is methodological: it established that achieved detector sensitivity and published cross-correlation data were already sufficient to translate stochastic-background measurements into direct constraints on inflationary RGW parameter space.

Source: https://www.emergentmind.com/topics/s5