An Upper Limit on the Stochastic Gravitational-Wave Background of Cosmological Origin

TL;DR

We address the challenge of directly measuring the stochastic gravitational-wave background (SGWB), which encodes information from the early universe, by applying a cross-correlation method to LIGO S5 data. The approach uses an optimally filtered cross-correlation estimator to combine strain data from interferometer pairs, yielding a stringent upper limit on the frequency-domain energy density $\Omega_{\rm GW}(f)$ in a targeted band. The main result is a 95% CL upper limit of $\Omega_0 < 6.9 \times 10^{-6}$ (for a flat spectrum, $\alpha=0$, in $41.5$--$169.25$ Hz), improving on previous LIGO limits by more than an order of magnitude and surpassing indirect BBN and CMB bounds in this band. This constrains models of early-universe evolution and cosmic strings, and demonstrates the feasibility and value of advanced detector networks for mapping isotropic and non-isotropic SGWB signals. Looking ahead, Advanced LIGO/Virgo will push bounds toward $\Omega_{\rm GW} \sim 10^{-9}$, enabling more stringent tests of cosmology and high-energy physics during the pre-BBN era and beyond.

Abstract

A stochastic background of gravitational waves is expected to arise from a superposition of a large number of unresolved gravitational-wave sources of astrophysical and cosmological origin. It is expected to carry unique signatures from the earliest epochs in the evolution of the universe, inaccessible to the standard astrophysical observations. Direct measurements of the amplitude of this background therefore are of fundamental importance for understanding the evolution of the universe when it was younger than one minute. Here we report direct limits on the amplitude of the stochastic gravitational-wave background using the data from a two-year science run of the Laser Interferometer Gravitational-wave Observatory (LIGO). Our result constrains the energy density of the stochastic gravitational-wave background normalized by the critical energy density of the universe, in the frequency band around 100 Hz, to be less than 6.9 x 10^{-6} at 95% confidence. The data rule out models of early universe evolution with relatively large equation-of-state parameter, as well as cosmic (super)string models with relatively small string tension that are favoured in some string theory models. This search for the stochastic background improves upon the indirect limits from the Big Bang Nucleosynthesis and cosmic microwave background at 100 Hz.

Figures (12)

• Figure 1: Sensitivities of LIGO interferometers. LIGO interferometers reached their design sensitivity in November 2005, resulting in the interferometer strain noise at the level of $3\times 10^{-22}$ rms in a 100 Hz band around 100 Hz. This figure shows typical strain sensitivities of LIGO interferometers during the subsequent science run S5. Also shown is the strain amplitude corresponding to the upper limit on the GW energy density presented in this paper (gray dashed line). Note that this upper limit is $\sim 100$ times lower than the individual interferometer sensitivities, which illustrates the advantage of using the cross-correlation technique in this analysis.
• Figure 2: Comparison of different SGWB measurements and models. The 95% upper limit presented here, $\Omega_0 < 6.9 \times 10^{-6}$ (LIGO S5), applies in the frequency band 41.5-169.25 Hz, and is compared to the previous LIGO S4 result S4paper and to the projected Advanced LIGO sensitivity aligo1. Note that the corresponding S5 95% upper bound on the total gravitational-wave energy density in this band, assuming frequency independent spectrum, is $9.7 \times 10^{-6}$. The indirect bound due to BBN maggioreBBN applies to $\Omega_{\rm BBN} = \int \Omega_{\rm GW} (f) d(\ln f)$ (and not to the density $\Omega_{\rm GW} (f)$) over the frequency band denoted by the corresponding horizontal line, as defined in Equation \ref{['BBNeq']}. A similar integral bound (over the range $10^{-15}$ - $10^{10}$ Hz) can be placed using CMB and matter power spectra smith. Projected sensitivities of the satellite-based Planck CMB experiment smith and LISA GW detector LISA are also shown. The pulsar bound pulsar is based on the fluctuations in the pulse arrival times of millisecond pulsars and applies at frequencies around $10^{-8}$ Hz. Measurements of the CMB at large angular scales constrain the possible redshift of CMB photons due to the SGWB, and therefore limit the amplitude of the SGWB at largest wavelengths (smallest frequencies) BBN. Examples of inflationary PA1PA2, cosmic strings kibble76cstringsCS3cspaper, and pre-big-bang pbbBMUMB models are also shown (the amplitude and the spectral shape in these models can vary significantly as a function of model parameters).
• Figure 3: Constraining early universe evolution. The GW spectrum $\Omega_{\rm GW}(f)$ is related to the parameters that govern the evolution of the universe boylebuonanno: $\Omega_{\rm GW}(f) = A \; f^{\hat{\alpha}(f)} \; f^{\hat{n}_t(f)} \; r$, where $\hat{\alpha}(f) = 2 \; \frac{3\hat{w}(f) - 1}{3\hat{w}(f)+1}$, $r$ is the ratio of tensor and scalar perturbation amplitudes (measured by the cosmic microwave background (CMB) experiments), $\hat{n}_t(f)$ and $\hat{w}(f)$ are effective (average) tensor tilt and equation of state parameters respectively, and $A$ is a constant depending on various cosmological parameters. Hence, the measurements of $\Omega_{\rm GW}$ and $r$ can be used to place constraints in the $\hat{w}-\hat{n}_t$ plane, independently of the cosmological model. The figure shows the $\hat{w}-\hat{n}_t$ plane for $r=0.1$. The regions excluded by the BBN cyburt, LIGO, and pulsar pulsar bounds are above the corresponding curves (the inset shows a zoom-in on the central part of the figure). The BBN curve was calculated in boylebuonanno. We note that the CMB bound smith almost exactly overlaps with the BBN bound. Also shown is the expected reach of Advanced LIGO aligo1. Note that these bounds apply to different frequency bands, so their direct comparison is meaningful only if $\hat{n}_t(f)$ and $\hat{w}(f)$ are frequency independent. We note that for the simplest single-field inflationary model that still agrees with the cosmological data, with potential $V(\phi) = m^2\phi^2/2$ (where $\phi$ is a scalar field of mass $m$), $r=0.14$ and $n_t(100 {\rm \; Hz}) = -0.035$wmap, implying a LIGO bound on the equation-of-state parameter of $\hat{w}(100 {\rm \; Hz}) < 0.59$.
• Figure 4: Cosmic strings models. The network of cosmic strings is usually parametrized by the string tension $\mu$ (multiplied by the Newton constant $G$), and reconnection probability $p$. The CMB observations limit $G\mu<10^{-6}$. If the size of the cosmic string loops is determined by the gravitational back-reaction gbrloops1, the size of the loop can be parametrized by a parameter $\epsilon$CS3 which is essentially unconstrained. The mechanism for production of GWs relies on cosmic string cusps: regions of string that move at speeds close to the speed of light. If the cusp motion points toward Earth, a detectable burst of gravitational radiation may be produced CS3SCMMCR. The superposition of GWs from all string cusps in the cosmic string network would produce a SGWB cspaper. This figure shows how different experiments probe the $\epsilon - G\mu$ plane for a typical value of $p=10^{-3}$cspaper ($p$ is expected to be in the range $10^{-4} - 1$). The excluded regions (always to the right of the corresponding curves) correspond to the S4 LIGO result S4paper, this result, BBN bound BBNcyburt, CMB bound smith, and the pulsar limit pulsar. In particular, the bound presented in this paper excludes a new region in this plane ($7\times 10^{-9} < G\mu < 1.5\times 10^{-7}$ and $\epsilon< 8\times 10^{-11}$), which is not accessible to any of the other measurements. Also shown is the expected sensitivity for the search for individual bursts from cosmic string cusps with LIGO S5 data SCMMCR. The region to the right of this curve is expected to produce at least one cosmic string burst event detectable by LIGO during the S5 run. Note that this search is complementary to the search for the SGWB as it probes a different part of the parameter space. Also shown is the region that will be probed by the Planck satellite measurements of the CMB smith. The entire plane shown here will be accessible to Advanced LIGO aligo1 SGWB search.
• Figure 5: Pre-Big-Bang models. In the pre-Big-Bang model, the GWs are produced via the mechanism of amplification of vacuum fluctuations, analogously to the standard inflationary model. The typical GW spectrum increases as $f^3$ up to a turn-over frequency $f_s$, above which $\Omega_{\rm GW}(f) \sim f^{3-2\mu}$ with $\mu<1.5$. The spectrum cuts off at a frequency $f_1$, which is theoretically expected to be within a factor of 10 from $4.3\times 10^{10}$ Hz (dashed horizontal line). This figure shows the $f_1-\mu$ plane for a representative value of $f_s = 30$ Hz. Excluded regions corresponding to the S4 result and to the result presented here are shaded. The regions excluded by the BBN BBNcyburt and the CMB smith bounds are above the corresponding curves. The expected reaches of the Advanced LIGO aligo1 and of the Planck satellite smith are also shown.