Energy Density Of Gravitational Waves Research Articles

Constraining the gravitational wave energy density of the Universe using Earth’s ring

The search for gravitational waves is one of today's major scientific endeavors. A gravitational wave can interact with matter by exciting vibrations of elastic bodies. Earth itself is a large elastic body whose so-called normal-mode oscillations ring up when a gravitational wave passes. Therefore, precise measurement of vibration amplitudes can be used to search for the elusive gravitational-wave signals. Earth's free oscillations that can be observed after high-magnitude earthquakes have been studied extensively with gravimeters and low-frequency seismometers over many decades leading to invaluable insight into Earth's structure. Making use of our detailed understanding of Earth's normal modes, numerical models are employed for the first time to accurately calculate Earth's gravitational-wave response, and thereby turn a network of sensors that so far has served to improve our understanding of Earth, into an astrophysical observatory exploring our Universe. In this paper, we constrain the energy density of gravitational waves to values in the range 0.035--0.15 normalized by the critical energy density of the Universe at frequencies between 0.3 and 5 mHz, using ten years of data from the gravimeter network of the Global Geodynamics Project that continuously monitors Earth's oscillations. This work is the first step towards a systematic investigation of the sensitivity of gravimeter networks to gravitational waves. Further advances in gravimeter technology could improve sensitivity of these networks and possibly lead to gravitational-wave detection.

Search for a stochastic gravitational-wave background using a pair of torsion-bar antennas

We have set a new upper limit on the stochastic gravitational wave background (SGWB) using two prototype Torsion-bar Antennas (TOBAs). TOBA is a low-frequency gravitational-wave detector with bar-shaped test masses rotated by the tidal force of gravitational waves. As a result of simultaneous 7-hour observations with TOBAs in Tokyo and Kyoto in Japan, our upper limit with a confidence level of 95% is $\Omega_{\rm gw}h_0^2 < 1.9 \times 10^{17}$ at 0.035 - 0.830 Hz, where $h_{0}$ is the Hubble constant in units of 100 km/s/Mpc and $\Omega_{\rm gw}$ is the gravitational wave energy density per logarithmic frequency interval in units of the closure density. We successfully updated the upper limit and extended the explored frequency band.

Stochastic gravitational wave background generated by cosmic string networks: Velocity-dependent one-scale model versus scale-invariant evolution

We compute the power spectrum of the stochastic gravitational wave background generated by cosmic string networks, described by the velocity-dependent one-scale (VOS) model for a wide range of macroscopic and microscopic parameters. The VOS model---which has been shown to provide an accurate macroscopic description of the evolution of cosmic string networks---is used to demonstrate that cosmic string networks are unable to rapidly attain scale-invariant evolution after the transition between the radiation and matter eras. However, in computations of the stochastic gravitational wave background, it is often assumed that the networks experience scale-invariant evolution throughout cosmological history. We demonstrate that this assumption leads to an underestimation of the amplitude and broadness of the peak of the spectrum. As a result, the total energy density of gravitational waves at the present day obtained using the VOS model may be up to 70% larger than that obtained under the linear scaling assumption.

Modified amplitude of the gravitational wave spectrum

The spectrum of thermal gravitational waves is obtained by including the high-frequency thermal gravitons created from extra-dimensional effects and is a new feature of the spectrum. The amplitude and spectral energy density of gravitational waves in a thermal vacuum state are found to be enhanced. The amplitude of the waves is modified in the frequency range (10−16–108 Hz) but the corresponding spectral energy density is less than the upper bound of various estimated results. With the addition of higher frequency thermal waves, the obtained spectral energy density of the wave in the thermal vacuum state does not exceed the upper bound put by the nucleosynthesis rate. The existence of cosmologically originated thermal gravitational waves due to extra dimension is not ruled out.

Upper Limit on Gravitational Wave Backgrounds at 0.2 Hz with a Torsion-Bar Antenna

We present the first upper limit on gravitational wave (GW) backgrounds at an unexplored frequency of 0.2 Hz using a torsion-bar antenna (TOBA). A TOBA was proposed to search for low-frequency GWs. We have developed a small-scaled TOBA and successfully found Ω(gw)(f)<4.3×10(17) at 0.2 Hz as demonstration of the TOBA's capabilities, where Ω(gw)(f) is the GW energy density per logarithmic frequency interval in units of the closure density. Our result is the first nonintegrated limit to bridge the gap between the LIGO band (around 100 Hz) and the Cassini band (10(-6)-10(-4) Hz).

Gravitational waves from Abelian gauge fields and cosmic strings at preheating

Primordial gravitational waves provide a very important stochastic background that could be detected soon with interferometric gravitational wave antennas or indirectly via the induced patterns in the polarization anisotropies of the cosmic microwave background. The detection of these waves will open a new window into the early Universe, and therefore it is important to characterize in detail all possible sources of primordial gravitational waves. In this paper we develop theoretical and numerical methods to study the production of gravitational waves from out-of-equilibrium gauge fields at preheating. We then consider models of preheating after hybrid inflation, where the symmetry breaking field is charged under a local $U(1)$ symmetry. We analyze in detail the dynamics of the system in both momentum and configuration space. We show that gauge fields leave specific imprints in the resulting gravitational wave spectra, mainly through the appearance of new peaks at characteristic frequencies that are related to the mass scales in the problem. We also show how these new features in the spectra correlate with stringlike spatial configurations in both the Higgs and gauge fields that arise due to the appearance of topological winding numbers of the Higgs around Nielsen-Olesen strings. We study in detail the time evolution of the spectrum of gauge fields and gravitational waves as these strings evolve and decay before entering a turbulent regime where the gravitational wave energy density saturates.

Gravitational waves from self-ordering scalar fields

Gravitational waves were copiously produced in the early Universe whenever the processes taking place were sufficiently violent. The spectra of several of these gravitational wave backgrounds on subhorizon scales have been extensively studied in the literature. In this paper we analyze the shape and amplitude of the gravitational wave spectrum on scales which are superhorizon at the time of production. Such gravitational waves are expected from the self ordering of randomly oriented scalar fields which can be present during a thermal phase transition or during preheating after hybrid inflation. We find that, if the gravitational wave source acts only during a small fraction of the Hubble time, the gravitational wave spectrum at frequencies lower than the expansion rate at the time of production behaves as ΩGW(f) ∝ f3 with an amplitude much too small to be observable by gravitational wave observatories like LIGO, LISA or BBO. On the other hand, if the source is active for a much longer time, until a given mode which is initially superhorizon (kη* << 1), enters the horizon, for kη ≳ 1, we find that the gravitational wave energy density is frequency independent, i.e. scale invariant. Moreover, its amplitude for a GUT scale scenario turns out to be within the range and sensitivity of BBO and marginally detectable by LIGO and LISA. This new gravitational wave background can compete with the one generated during inflation, and distinguishing both may require extra information.

Theory and numerics of gravitational waves from preheating after inflation

Preheating after inflation involves large, time-dependent field inhomogeneities, which act as a classical source of gravitational radiation. The resulting spectrum might be probed by direct detection experiments if inflation occurs at a low enough energy scale. In this paper, we develop a theory and algorithm to calculate, analytically and numerically, the spectrum of energy density in gravitational waves produced from an inhomogeneous background of stochastic scalar fields in an expanding universe. We derive some generic analytical results for the emission of gravity waves by stochastic media of random fields, which can test the validity/accuracy of numerical calculations. We contrast our method with other numerical methods in the literature, and then we apply it to preheating after chaotic inflation. In this case, we are able to check analytically our numerical results, which differ significantly from previous works. We discuss how the gravity-wave spectrum builds up with time and find that the amplitude and the frequency of its peak depend in a relatively simple way on the characteristic spatial scale amplified during preheating. We then estimate the peak frequency and amplitude of the spectrum produced in two models of preheating after hybrid inflation, which for some parameters may be relevant for gravity-wave interferometric experiments.

Three‐YearWilkinson Microwave Anisotropy Probe(WMAP) Observations: Implications for Cosmology

A simple cosmological model with only six parameters (matter density, Omega_m h^2, baryon density, Omega_b h^2, Hubble Constant, H_0, amplitude of fluctuations, sigma_8, optical depth, tau, and a slope for the scalar perturbation spectrum, n_s) fits not only the three year WMAP temperature and polarization data, but also small scale CMB data, light element abundances, large-scale structure observations, and the supernova luminosity/distance relationship. Using WMAP data only, the best fit values for cosmological parameters for the power-law flat LCDM model are (Omega_m h^2, Omega_b h^2, h, n_s, tau, sigma_8) = 0.1277+0.0080-0.0079, 0.02229+-0.00073, 0.732+0.031-0.032, 0.958+-0.016, 0.089+-0.030, 0.761+0.049-0.048). The three year data dramatically shrink the allowed volume in this six dimensional parameter space. Assuming that the primordial fluctuations are adiabatic with a power law spectrum, the WMAP data_alone_ require dark matter, and favor a spectral index that is significantly less than the Harrison-Zel'dovich-Peebles scale-invariant spectrum (n_s=1, r=0). Models that suppress large-scale power through a running spectral index or a large-scale cut-off in the power spectrum are a better fit to the WMAP and small scale CMB data than the power-law LCDM model: however, the improvement in the fit to the WMAP data is only Delta chi^2 = 3 for 1 extra degree of freedom. The combination of WMAP and other astronomical data yields significant constraints on the geometry of the universe, the equation of state of the dark energy, the gravitational wave energy density, and neutrino properties. Consistent with the predictions of simple inflationary theories, we detect no significant deviations from Gaussianity in the CMB maps.

Cosmic gravitational-wave background in a cyclic universe

Inflation predicts a primordial gravitational wave spectrum that is slightly ``red,'' i.e., nearly scale-invariant with slowly increasing power at longer wavelengths. In this paper, we compute both the amplitude and spectral form of the primordial tensor spectrum predicted by cyclic/ekpyrotic models. The spectrum is blue and exponentially suppressed compared to inflation on long wavelengths. The strongest observational constraint emerges from the requirement that the energy density in gravitational waves should not exceed around 10 per cent of the energy density at the time of nucleosynthesis.

Analysis of first LIGO science data for stochastic gravitational waves

We present the analysis of between 50 and 100 h of coincident interferometric strain data used to search for and establish an upper limit on a stochastic background of gravitational radiation. These data come from the first LIGO science run, during which all three LIGO interferometers were operated over a 2-week period spanning August and September of 2002. The method of cross correlating the outputs of two interferometers is used for analysis. We describe in detail practical signal processing issues that arise when working with real data, and we establish an observational upper limit on a f^-3 power spectrum of gravitational waves. Our 90% confidence limit is Ω0h100(^2)<~23±4.6 in the frequency band 40–314 Hz, where h100 is the Hubble constant in units of 100 km/sec/Mpc and Ω0 is the gravitational wave energy density per logarithmic frequency interval in units of the closure density. This limit is approximately 104 times better than the previous, broadband direct limit using interferometric detectors, and nearly 3 times better than the best narrow-band bar detector limit. As LIGO and other worldwide detectors improve in sensitivity and attain their design goals, the analysis procedures described here should lead to stochastic background sensitivity levels of astrophysical interest.

Detection methods for non-Gaussian gravitational wave stochastic backgrounds

A gravitational wave stochastic background can be produced by a collection of independent gravitational wave events. There are two classes of such backgrounds, one for which the ratio of the average time between events to the average duration of an event is small (i.e., many events are on at once), and one for which the ratio is large. In the first case the signal is continuous, sounds something like a constant $hiss,$ and has a Gaussian probability distribution. In the second case, the discontinuous or intermittent signal sounds something like popcorn popping, and is described by a non-Gaussian probability distribution. In this paper we address the issue of finding an optimal detection method for such a non-Gaussian background. As a first step, we examine the idealized situation in which the event durations are short compared to the detector sampling time, so that the time structure of the events cannot be resolved, and we assume white, Gaussian noise in two collocated, aligned detectors. For this situation we derive an appropriate version of the maximum likelihood detection statistic. We compare the performance of this statistic to that of the standard cross-correlation statistic both analytically and with Monte Carlo simulations. In general the maximum likelihood statistic performs better than the cross-correlation statistic when the stochastic background is sufficiently non-Gaussian, resulting in a gain factor in the minimum gravitational-wave energy density necessary for detection. This gain factor ranges roughly between 1 and 3, depending on the duty cycle of the background, for realistic observing times and signal strengths for both ground and space based detectors. The computational cost of the statistic, although significantly greater than that of the cross-correlation statistic, is not unreasonable. Before the statistic can be used in practice with real detector data, further work is required to generalize our analysis to accommodate separated, misaligned detectors with realistic, colored, non-Gaussian noise.

EVOLUTION OF GRAVITATIONAL WAVES THROUGH THE COSMOLOGICAL QCD TRANSITION

The spectrum of gravitational waves that have been produced in inflation is modified during cosmological transitions. Significant decrease in the number of relativistic particles, like during the QCD transition or at e+ e- annihilation, leads to steps in the spectrum of gravitational waves. We calculate the transfer function for the differential energy density of gravitational waves for a first-order and for a crossover QCD transition.

Relic gravitational waves produced after preheating

We show that gravitational radiation is produced quite efficiently in interactions of classical waves created by resonant decay of a coherently oscillating field. As an important example we consider simple models of chaotic inflation, where we find that today's ratio of energy density in gravitational waves per octave to the critical density of the Universe can be as large as ${10}^{\ensuremath{-}12}$ at the maximal wavelength of order ${10}^{5}$ cm. In the pure $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}/4$ model with inflaton self-coupling $\ensuremath{\lambda}{=10}^{\ensuremath{-}13}$, the maximal today's wavelength of gravitational waves produced by this mechanism is of order ${10}^{6}$ cm, close to the upper bound of operational LIGO and TIGA frequencies. The energy density of waves in this model, though, is likely to be well below the sensitivity of LIGO or TIGA at such frequencies. We discuss the possibility that in other models the interaction of classical waves can lead to an even stronger gravitational radiation background.

Resonant detectors for the search for gravitational waves

An overview of the experiments for the search for gravitational waves by means of resonant detectors is given. Since 1990 cryogenic resonant antennas have been in operation and data have been recorded by Explorer, Allegro, Niobe and Nautilus. The sensitivity for pulse detection with SNR = 1 is now (corresponding to a total energy of less than 0.001 solar masses for a source in the Galactic Center). The sensitivity for monochromatic waves is for one year of integration and about for stochastic background detection. Large resonant detectors operating at 1 kHz might reach, in the near future, a spectral amplitude sensitivity of the order of and, for pulse and monochromatic wave sensitivity, respectively, and . Cross-correlating two such large antennas for one year can give a sensitivity for stochastic background detection of the order of , corresponding to a ratio between the gravitational wave energy density to that needed for a closed Universe of .

Inflation in an exponential-potential scalar field model.

A spatially flat cosmological scalar field $\ensuremath{\Phi}$ model with the scalar field potential $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\Phi}}{\sqrt{p}})$, $pg1$, provides a simple class of inflationary cosmologies (which includes the usual exponential expansion inflation) that may be used as an analytical testing ground to help understand the predictions of the inflation model of the very early Universe. We divide the evolution of this model into three distinct epochs: scalar-field dominance and conventional radiation and baryon dominance; in each epoch we only account for irregularities in the dominant form of matter. We present closed-form solutions of the (synchronous gauge) relativistic linear perturbation equations that govern the evolution of inhomogeneities. These classical solutions, augmented with quantum-mechanically motivated initial conditions and joining conditions to match the expressions for the irregularities at the scalar-field-radiation and radiation-baryon transitions, are used to estimate the large-time form of the spectrum of energy-density irregularities, of the local departure velocity from homogeneous expansion, of large-scale fluctuations in the microwave background temperature, and of the gravitational-wave energy density. The inflation epoch results agree with those found from a purely quantum-mechanical analysis. Depending on the value of $p$ this model can have more large-scale power than the usual scale-invariant spectrum (at the expense of less small-scale power) and would seem to be marginally better at forming large-scale structure than the canonical model; however, the decrease in small-scale power serves to exacerbate the problem of late galaxy formation. As the model approaches the exponential expansion inflation limit, the power spectrum tends towards the scale-invariant form, although, in this limit the numerical prefactor diverges. We find that transverse peculiar velocity perturbations are not generated. Normalizing by fitting to the observed large-scale departure velocity, we find that models which stop inflating around ${10}^{7}$-${10}^{16}$ GeV are not obviously observationally inconsistent.

Clustering correlations and limits on cosmological gravitational waves

A cosmological background of gravitational waves induces angular deviations in the propagation of light traversing it. All observed astrophysical sources might therefore have varying apparent positions, with the time dependence set by the wave period. Wavelengths greater than a kiloparsec are examined, so the positions are frozen but in general correlated. Comparison with observed galaxy-galaxy n-point correlation functions provide a spectrum-independent limit on the energy density of gravitational waves with wavelengths between a few tens of kiloparsecs and a few hundred megaparsecs of Omega(GW) less than 0.001.

Gravitational radiation from cosmic strings and the microwave background

Gravitational radiation sets the most stringent constraints on the cosmic string theory of galaxy formation. We calculate the temperature fluctuations in the microwave background radiation induced by gravitational radiation. The predicted anisotropies are consistent with the best current observational bounds. We also show that the total energy density in gravitational waves does not conflict with the standard nucleosynthesis arguments, and that the predicted timing residual for the millisecond pulsar does not (yet) exceed the measured value.