LIGO S5: Constraining Inflationary RGWs

• S5 is the LIGO run that delivered key strain sensitivity and cross-correlation data to constrain relic gravitational wave (RGW) spectra in the ~40–500 Hz band.
• It compares predicted analytic RGW spectra with observed stochastic background limits to constrain inflationary parameters (β, αt, r), setting bounds such as Ω0 < 6.9×10⁻⁶.
• The analysis reveals that minor changes in β and αt significantly boost high-frequency amplitudes, underscoring the sensitivity and methodological advancements in stochastic background detection.

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” (Zhang et al., 2010), 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 (Zhang et al., 2010).

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 $500$0 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 $500$1, the tensor running index $500$2, and the tensor-to-scalar ratio $500$3. The parameter $500$4 is related to the inflationary scale factor through

$500$5

with $500$6 giving a nearly scale-invariant spectrum. Larger $500$7, meaning less negative values such as $500$8 instead of $500$9, strongly boosts the high-frequency amplitude in the LIGO band. The running $\Omega_0<6.9\times10^{-6}$0 introduces logarithmic bending of the primordial tensor spectrum, and even very small positive $\Omega_0<6.9\times10^{-6}$1 appreciably raises the amplitude at $\Omega_0<6.9\times10^{-6}$2 Hz. The ratio

$\Omega_0<6.9\times10^{-6}$3

sets the overall normalization, and the computed SNR satisfies $\Omega_0<6.9\times10^{-6}$4 (Zhang et al., 2010).

The initial spectrum at horizon crossing is taken to be

$\Omega_0<6.9\times10^{-6}$5

with pivot $\Omega_0<6.9\times10^{-6}$6 corresponding to $\Omega_0<6.9\times10^{-6}$7 and

$\Omega_0<6.9\times10^{-6}$8

This parameterization makes the S5 sensitivity especially dependent on $\Omega_0<6.9\times10^{-6}$9 and $\sim 100$0, 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

$\sim 100$1

with

$\sim 100$2

and frequency–wavenumber relation

$\sim 100$3

for $\sim 100$4. These formulas connect the inflationary RGW model to the quantity bounded by the S5 stochastic search (Zhang et al., 2010).

For a single interferometer, the paper uses a qualitative detectability criterion based on comparing

$\sim 100$5

with the detector strain sensitivity, where $\sim 100$6 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 $\sim 100$7 for long integration.

The main quantitative tool is the Allen–Romano stochastic-background SNR: $\sim 100$8 Here $\sim 100$9 and $(\beta,\alpha_t,r)$0 are the one-sided noise power spectra of H1 and L1, $(\beta,\alpha_t,r)$1 is the overlap reduction function, and $(\beta,\alpha_t,r)$2 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 $(\beta,\alpha_t,r)$3, $(\beta,\alpha_t,r)$4,

$(\beta,\alpha_t,r)$5

Using the stronger cross-correlation bound from S5, the constraint becomes

$(\beta,\alpha_t,r)$6

For the benchmark model $(\beta,\alpha_t,r)$7, $(\beta,\alpha_t,r)$8, the corresponding limits are

$(\beta,\alpha_t,r)$9

from single interferometers and

$(\beta,\alpha_t,r)$0

from cross-correlated S5. The principal direct S5 limits are therefore

$(\beta,\alpha_t,r)$1

for the benchmark choice $(\beta,\alpha_t,r)$2, holding the other parameter fixed as specified (Zhang et al., 2010).

The spectral response in the S5 band is extremely sensitive to these parameters. Changing $(\beta,\alpha_t,r)$3 from $(\beta,\alpha_t,r)$4 to $(\beta,\alpha_t,r)$5 increases $(\beta,\alpha_t,r)$6 by about 4 orders of magnitude around $(\beta,\alpha_t,r)$7 Hz, and changing $(\beta,\alpha_t,r)$8 from $(\beta,\alpha_t,r)$9 to $\sim 40$0 also increases $\sim 40$1 by about 4 orders of magnitude around $\sim 40$2 Hz. Representative energy-density slopes are

$\sim 40$3

and

$\sim 40$4

This is why the flat-spectrum S5 bound $\sim 40$5 cannot be identified directly with the tilted RGW spectra treated in the paper.

The paper also emphasizes a $\sim 40$6–$\sim 40$7 degeneracy in the S5 band. The models

$\sim 40$8

and

$\sim 40$9

give essentially the same amplitude around $500$0 Hz and very similar $500$1 slopes. Because S5 probes a relatively narrow frequency interval,

$500$2

it is unlikely to distinguish such models cleanly.

5. SNR calculations and detectability

For $500$3, the paper computes H1–L1 cross-correlation SNRs for several $500$4 choices. For $500$5, the SNRs are $500$6, $500$7, $500$8, and $500$9 for $T = 59{,}961{,}600~\mathrm{s},$0, $T = 59{,}961{,}600~\mathrm{s},$1, $T = 59{,}961{,}600~\mathrm{s},$2, and $T = 59{,}961{,}600~\mathrm{s},$3, respectively. For $T = 59{,}961{,}600~\mathrm{s},$4, the corresponding values are $T = 59{,}961{,}600~\mathrm{s},$5, $T = 59{,}961{,}600~\mathrm{s},$6, $T = 59{,}961{,}600~\mathrm{s},$7, and $T = 59{,}961{,}600~\mathrm{s},$8. For $T = 59{,}961{,}600~\mathrm{s},$9, they are $\Omega_0 < 6.9\times 10^{-6},$0, $\Omega_0 < 6.9\times 10^{-6},$1, $\Omega_0 < 6.9\times 10^{-6},$2, and $\Omega_0 < 6.9\times 10^{-6},$3. For $\Omega_0 < 6.9\times 10^{-6},$4, they are $\Omega_0 < 6.9\times 10^{-6},$5, $\Omega_0 < 6.9\times 10^{-6},$6, $\Omega_0 < 6.9\times 10^{-6},$7, and $\Omega_0 < 6.9\times 10^{-6},$8 (Zhang et al., 2010).

These values scale linearly with $\Omega_0 < 6.9\times 10^{-6},$9, since $500$00. The numerical pattern is the paper’s clearest demonstration that detectability rises steeply as $500$01 increases, as $500$02 becomes positive, or as $500$03 increases. Conversely, for nearly scale-invariant and weakly running spectra such as $500$04, $500$05, 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 $500$06, 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

$500$07

It explicitly stresses that $500$08 and $500$09 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 $500$10, which is not generally true for the RGW models considered (Zhang et al., 2010).

The indirect bounds used are

$500$11

for BBN, and

$500$12

for CMB, with

$500$13

These yield tighter constraints than S5. In particular,

$500$14

$500$15

$500$16

and

$500$17

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 $500$18 it implies

$500$19

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 $500$20–$500$21 degeneracy. The paper notes that a broader-band detector such as LISA would have a better chance of distinguishing models with different $500$22 and $500$23. 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.