Relic Gravitational Waves Research Articles

Constraint on the early Universe by relic gravitational waves: From pulsar timing observations

Recent pulsar timing observations by the Parkers Pulsar Timing Array and European Pulsar Timing Array teams obtained the constraint on the relic gravitational waves at the frequency $f_*=1/{\rm yr}$, which provides the opportunity to constrain $H_*$, the Hubble parameter when these waves crossed the horizon during inflation. In this paper, we investigate this constraint by considering the general scenario for the early Universe: we assume that the effective (average) equation-of-state $w$ before the big bang nucleosynthesis stage is a free parameter. In the standard hot big-bang scenario with $w=1/3$, we find that the current PPTA result follows a bound $H_*\leq 1.15\times10^{-1}\mpl$, and the EPTA result follows $H_*\leq 6.92\times10^{-2}\mpl$. We also find that these bounds become much tighter in the nonstandard scenarios with $w>1/3$. When $w=1$, the bounds become $H_*\leq5.89\times10^{-3}\mpl$ for the current PPTA and $H_*\leq3.39\times10^{-3}\mpl$ for the current EPTA. In contrast, in the nonstandard scenario with $w=0$, the bound becomes $H_*\leq7.76\mpl$ for the current PPTA.

Detection of relic gravitational waves in the CMB: prospects for CMBPol mission

Detection of relic gravitational waves, through their imprint in the cosmic microwave background radiation, is one of the most important tasks for the planned CMBPol mission. In the simplest viable theoretical models the gravitational wave background is characterized by two parameters, the tensor-to-scalar ratio r and the tensor spectral index nt. In this paper, we analyze the potential joint constraints on these two parameters, r and nt, using the potential observations of the CMBPol mission, which is expected to detect the relic gravitational waves if r≳0.001. The influence of the contaminations, including cosmic weak lensing, various foreground emissions, and systematical errors, is discussed.

A New Theoretical Technique for the Measurement of High-Frequency Relic Gravitational Waves

Under most models of the early universe evolution, high-frequency gravitational waves (HFGWs) were produced. They are referred to as “relic” high-frequency gravitational waves or HFRGWs and their detection and measurement could provide important information on the origin and development of our Universe – information that could not otherwise be obtained. So far three instruments have been built to detect and measure HFRGWs, but so far none of them has achieved the required sensitivity. This paper concerns another detector, originally proposed by Baker in 2000 and patented, which is based upon a recently discovered physical effect (the Li effect); this detector has accordingly been named the “Li-Baker detector.” The detector has been a joint development effort by the P. R. China and the United States HFGW research teams. A rigorous examination of the detector’s performance is important in the ongoing debate over the value of attempting to construct a Li-Baker detector and, in particular, an accurate prediction of its sensitivity in the presence of significant noise will decide whether the Li-Baker detector will be capable of detecting and measuring HFRGWs. The potential for useful HFRGW measurement is theoretically confirmed.

Estimate of diffraction from Gaussian Beam in Li-Baker HFGW detector

Recent experimentation and speculation about the design of a sensitive detector for high-frequency gravitational waves (HFGW) has centered around a number of principles. Those detectors that have been built so far have not yet realized sensitivity sufficient to investigate the cosmic high-frequency relic gravitational wave background, analogous to the cosmic microwave background. A proposal for a more sensitive HFGW detector due to Baker and based upon a principle first enunciated by Li and co-workers has become known as the Li-Baker detector. Its possible design details are currently the subject of scientific debate. One significant aspect concerns the diffraction of microwave power from the intrinsic transmitter producing a microwave beam. This beam is required for interaction with the gravitational waves to be detected and diffraction will not be distinguishable from photons produced by interaction with the HFGW. This means that diffraction is potentially a source of shot noise at the microwave receivers and, if extreme, may also swamp the receivers. In this paper some estimates of this diffraction are obtained. These estimates show that the Li-Baker detector must be designed in such a way that the diffraction reaching the microwave receivers is reduced as far as possible by employing a suitable geometry and highly absorbent walls for the interaction volume.

Relic gravitational waves: Latest revisions and preparations for new data

The forthcoming release of data from the Planck mission, and possibly from the next round of Wilkinson Microwave Anisotropy Probe (WMAP) observations, make it necessary to revise the evaluations of relic gravitational waves in the existing data and, at the same time, to refine the assumptions and data analysis techniques in preparation for the arrival of new data. We reconsider with the help of the commonly used CosmoMC numerical package the previously found indications of relic gravitational waves in the 7-year (WMAP7) data. The CosmoMC approach reduces the confidence of these indications from approximately $2\ensuremath{\sigma}$ level to approximately $1\ensuremath{\sigma}$ level, but the indications do not disappear altogether. We critically analyze the assumptions that are currently used in the cosmic microwave background (CMB) data analyses, and outline the strategy that should help avoid the oversight of relic gravitational waves in the future CMB data. In particular, it is important to keep away from the unwarranted assumptions about density perturbations. The prospects of confident detection of relic gravitational waves by the Planck satellite have worsened, but they are still good. It appears that more effort will be required in order to mitigate the foreground contamination.

PHASE TRANSITION GENERATED COSMOLOGICAL MAGNETIC FIELD AT LARGE SCALES

We constrain a primordial magnetic field (PMF) generated during a phase transition (PT) using the big bang nucleosynthesis bound on the relativistic energy density. The amplitude of the PMF at large scales is determined by the shape of the PMF spectrum outside its maximal correlation length scale. Even if the amplitude of the PMF at 1 Mpc is small, PT-generated PMFs can leave observable signatures in the potentially detectable relic gravitational wave background if a large enough fraction ($1-10%$) of the thermal energy is converted into the PMF.

Relic gravitational wave spectrum, the trans-Planckian physics and Hořava–Lifshitz gravity

We calculate the spectrum of the relic gravitational wave due to the trans-Planckian effect in which the standard linear dispersion relations may be modified. Of the modified dispersion relations suggested in the literature which has investigated the trans-Planckian effect, we especially use the Corley–Jacobson dispersion relations. The Corley–Jacobson-type modified dispersion relations can be obtained from Hořava–Lifshitz gravity which is non-relativistic and UV complete. Although it is not clear how the transitions from Hořava–Lifshitz gravity in the UV regime to Einstein gravity in the IR limit occur, we assume that the Hořava–Lifshitz gravity regime is followed by the inflationary phase in Einstein gravity.

Testing metric-affine f(R)-gravity by relic scalar gravitational waves

We discuss the emergence of scalar gravitational waves in metric-affine f(R)-gravity. Such a component allows to discriminate between metric and metric-affine theories The intrinsic meaning of this result is that the geodesic structure of the theory can be discriminated. We extend the formalism of cross-correlation analysis, including the additional polarization mode, and calculate the detectable energy density of the spectrum for cosmological relic gravitons. The possible detection of the signal is discussed against the sensitivities of the VIRGO, LIGO and LISA interferometers.

Relic gravitational waves in light of the 7-year Wilkinson Microwave Anisotropy Probe data and improved prospects for the Planck mission

The new release of data from Wilkinson Microwave Anisotropy Probe improves the observational status of relic gravitational waves. The 7-year results enhance the indications of relic gravitational waves in the existing data and change to the better the prospects of confident detection of relic gravitational waves by the currently operating Planck satellite. We apply to WMAP7 data the same methods of analysis that we used earlier [W. Zhao, D. Baskaran, and L.P. Grishchuk, Phys. Rev. D 80, 083005 (2009)] with WMAP5 data. We also revised by the same methods our previous analysis of WMAP3 data. It follows from the examination of consecutive WMAP data releases that the maximum likelihood value of the quadrupole ratio $R$, which characterizes the amount of relic gravitational waves, increases up to $R=0.264$, and the interval separating this value from the point $R=0$ (the hypothesis of no gravitational waves) increases up to a $2\sigma$ level. The primordial spectra of density perturbations and gravitational waves remain blue in the relevant interval of wavelengths, but the spectral indices increase up to $n_s =1.111$ and $n_t=0.111$. Assuming that the maximum likelihood estimates of the perturbation parameters that we found from WMAP7 data are the true values of the parameters, we find that the signal-to-noise ratio $S/N$ for the detection of relic gravitational waves by the Planck experiment increases up to $S/N=4.04$, even under pessimistic assumptions with regard to residual foreground contamination and instrumental noises. We comment on theoretical frameworks that, in the case of success, will be accepted or decisively rejected by the Planck observations.

SeparatingEandBtypes of polarization on an incomplete sky

Detection of magnetic-type ($B$-type) polarization in CMB radiation plays a crucial role in probing the relic gravitational wave background. In this paper, we propose a new method to deconstruct a polarization map on an incomplete sky in real space into purely electric and magnetic polarization-type maps, $\mathcal{E}(\stackrel{^}{\ensuremath{\gamma}})$ and $\mathcal{B}(\stackrel{^}{\ensuremath{\gamma}})$, respectively. The main properties of our approach are as follows: First, the fields $\mathcal{E}(\stackrel{^}{\ensuremath{\gamma}})$ and $\mathcal{B}(\stackrel{^}{\ensuremath{\gamma}})$ are constructed in real space with minimal loss of information. This loss of information arises due to the removal of a narrow edge of the constructed map in order to remove various numerical errors, including those arising from finite pixel size. Second, this method is fast and can be efficiently applied to high resolution maps due to the use of the fast spherical harmonics transformation. Third, the constructed fields, $\mathcal{E}(\stackrel{^}{\ensuremath{\gamma}})$ and $\mathcal{B}(\stackrel{^}{\ensuremath{\gamma}})$, are scalar fields. For this reason various techniques developed to deal with temperature anisotropy maps can be directly applied to analyze these fields. As a concrete example, we construct and analyze an unbiased estimator for the power spectrum of the $B$ mode of polarization ${C}_{\ensuremath{\ell}}^{BB}$. Basing our results on the performance of this estimator, we discuss the relic gravitational wave detection ability of two future ground-based CMB experiments, the Q/U Imaging Experiment and Polarization of Background Microwave Radiation.

Constraints upon the spectral indices of relic gravitational waves by LIGO S5

With LIGO having achieved its design sensitivity and the LIGO S5 strain data being available, constraints on the relic gravitational waves (RGWs) become realistic. The analytical spectrum of RGWs generated during inflation depends sensitively on the initial condition, which is generically described by the index $\ensuremath{\beta}$, the running index ${\ensuremath{\alpha}}_{t}$, and the tensor-to-scalar ratio $r$. By the LIGO S5 data of the cross-correlated two detectors, we obtain constraints on the parameters $(\ensuremath{\beta},{\ensuremath{\alpha}}_{t},r)$. As a main result, we have computed the theoretical signal-to-noise ratio of RGWs for various values of $(\ensuremath{\beta},{\ensuremath{\alpha}}_{t},r)$, using the cross-correlation for the given pair of LIGO detectors. The constraints by the indirect bound on the energy density of RGWs by big bang nucleosynthesis and cosmic microwave background have been obtained, which turn out to be still more stringent than LIGO S5.

Inflation, renormalization, and CMB anisotropies

In single-field, slow-roll inflationary models, scalar and tensorial (Gaussian) perturbations are both characterized by a zero mean and a non-zero variance. In position space, the corresponding variance of those fields diverges in the ultraviolet. The requirement of a finite variance in position space forces its regularization via quantum field renormalization in an expanding universe. This has an important impact on the predicted scalar and tensorial power spectra for wavelengths that today are at observable scales. In particular, we find a non-trivial change in the consistency condition that relates the tensor-to-scalar ratio r to the spectral indices. For instance, an exact scale-invariant tensorial power spectrum, nt = 0, is now compatible with a non-zero ratio r ≈ 0.12 ± 0.06, which is forbidden by the standard prediction (r = −8nt). Forthcoming observations of the influence of relic gravitational waves on the CMB will offer a non-trivial test of the new predictions.

Revising the observable consequences of slow-roll inflation

We study the generation of primordial perturbations in a (single-field) slow-roll inflationary universe. In momentum space, these (Gaussian) perturbations are characterized by a zero mean and a non-zero variance $\Delta^2(k, t)$. However, in position space the variance diverges in the ultraviolet. The requirement of a finite variance in position space forces one to regularize $\Delta^2(k, t)$. This can (and should) be achieved by proper renormalization in an expanding universe in a unique way. This affects the predicted scalar and tensorial power spectra (evaluated when the modes acquire classical properties) for wavelengths that today are at observable scales. As a consequence, the imprint of slow-roll inflation on the CMB anisotropies is significantly altered. We find a non-trivial change in the consistency condition that relates the tensor-to-scalar ratio $r$ to the spectral indices. For instance, an exact scale-invariant tensorial power spectrum, $n_t=0$, is now compatible with a non-zero ratio $r\approx 0.12\pm0.06$, which is forbidden by the standard prediction ($r=-8n_t$). The influence of relic gravitational waves on the CMB may soon come within the range of planned measurements, offering a non-trivial test of the new predictions.

New primordial-magnetic-field limit from the latest LIGO S5 data

Since the energy momentum tensor of a magnetic field always contains a spin-2 component in its anisotropic stress, stochastic primordial magnetic field (PMF) in the early universe must generate stochastic gravitational wave (GW) background. This process will greatly affect the relic gravitational wave (RGW), which is one of major scientific goals of the laser interferometer GW detections. Recently, the fifth science (S5) run of laser interferometer gravitational-wave observatory (LIGO) gave a latest upper limit $\Omega_{GW}<6.9\times10^{-6}$ on the RGW background. Utilizing this upper limit, we derive new PMF Limits: for a scale of galactic cluster $\lambda=1$ Mpc, the amplitude of PMF, that produced by the electroweak phase transition (EPT), has to be weaker than $B_{\lambda} \leq 4\times 10^{-7}$ Gauss; for a scale of supercluster $\lambda=100$ Mpc, the amplitude of PMF has to be weaker than $B_{\lambda} \leq 9\times 10^{-11}$ Gauss. In this manner, GW observation has potential to make interesting contributions to the study of primordial magnetic field.

Erratum to: Information on the inflaton field from the spectrum of relic gravitational waves

After a review of a traditional analysis, it is shown a variation of a more recent treatment on the spectrum of relic gravitational waves (GWs). Then, a connection between the two different treatments will be analysed. Such a connection permits to obtain an interesting equation for the inflaton field. This equation gives a value that agrees with the slow roll condition on inflation.

Massive relic gravitational waves from f(R) theories of gravity: production and potential detection

The production of a stochastic background of relic gravitational waves is well known in various works in the literature, where, by using the so called adiabatically-amplified zero-point fluctuations process, it has been shown how the standard inflationary scenario for the early universe can in principle provide a distinctive spectrum of relic gravitational waves. In this paper, it is shown that, in general, f(R) theories of gravity produce a third massive polarization of gravitational waves and the primordial production of this polarization is analysed adapting the adiabatically-amplified zero-point fluctuations process at this case. In this way, previous results, where only particular cases of f(R) theories have been analysed, will be generalized. The presence of the mass could also have important applications in cosmology, because the fact that gravitational waves can have mass could give a contribution to the dark matter of the Universe. An upper bound for these relic gravitational waves, which arises from the WMAP constrains, is also released. At the end of the paper, the potential detection of such massive gravitational waves using interferometers like Virgo and LIGO is discussed.

Information on the inflaton field from the spectrum of relic gravitational waves

After a review of a traditional analysis, it is shown a variation of a more recent treatment on the spectrum of relic gravitational waves (GWs). Then, a connection between the two different treatments will be analysed. Such a connection permits to obtain an interesting equation for the inflaton field. This equation gives a value that agrees with the slow roll condition on inflation.

Relic gravitational waves with a running spectral index and its constraints at high frequencies

We study the impact of a running index ${\ensuremath{\alpha}}_{t}$ on the spectrum of relic gravitational waves (RGWs) over the whole range of frequency $({10}^{\ensuremath{-}18}\ensuremath{\sim}{10}^{10})\text{ }\text{ }\mathrm{Hz}$ and reveal its implications in RGWs detections and in cosmology. Analytical calculations show that, although the spectrum of RGWs on low frequencies is less affected by ${\ensuremath{\alpha}}_{t}\ensuremath{\ne}0$, on high frequencies, the spectrum is modified substantially. Investigations are made toward potential detections of the ${\ensuremath{\alpha}}_{t}$-modified RGWs for several kinds of current and planned detectors. The Advanced LIGO will likely be able to detect RGWs with ${\ensuremath{\alpha}}_{t}\ensuremath{\ge}0$ for inflationary models with the inflation index $\ensuremath{\beta}=\ensuremath{-}1.956$ and the tensor-scalar ratio $r=0.55$. The future LISA can detect RGWs for a much broader range of $({\ensuremath{\alpha}}_{t},\ensuremath{\beta},r)$, and will have a better chance to break a degeneracy between them. Constraints on ${\ensuremath{\alpha}}_{t}$ are estimated from several detections and cosmological observations. Among them, the most stringent one is from the bound of the big bang nucleosynthesis, and requires ${\ensuremath{\alpha}}_{t}&lt;0.008$ rather conservatively for any reasonable $(\ensuremath{\beta},r)$, preferring a nearly power-law spectrum of RGWs. In light of this result, one would expect the scalar running index ${\ensuremath{\alpha}}_{s}$ to be of the same magnitude as ${\ensuremath{\alpha}}_{t}$, if both RGWs and scalar perturbations are generated by the same scalar inflation.

Stable indications of relic gravitational waves in Wilkinson Microwave Anisotropy Probe data and forecasts for the Planck mission

The relic gravitational waves are the cleanest probe of the violent times in the very early history of the Universe. They are expected to leave signatures in the observed cosmic microwave background anisotropies. We significantly improved our previous analysis [W. Zhao, D. Baskaran, and L. P. Grishchuk, Phys. Rev. D 79, 023002 (2009)] of the 5-year WMAP $TT$ and $TE$ data at lower multipoles $\ensuremath{\ell}$. This more general analysis returned essentially the same maximum likelihood result (unfortunately, surrounded by large remaining uncertainties): The relic gravitational waves are present and they are responsible for approximately 20% of the temperature quadrupole. We identify and discuss the reasons by which the contribution of gravitational waves can be overlooked in a data analysis. One of the reasons is a misleading reliance on data from very high multipoles $\ensuremath{\ell}$ and another a too narrow understanding of the problem as the search for $B$ modes of polarization, rather than the detection of relic gravitational waves with the help of all correlation functions. Our analysis of WMAP5 data has led to the identification of a whole family of models characterized by relatively high values of the likelihood function. Using the Fisher matrix formalism we formulated forecasts for Planck mission in the context of this family of models. We explore in detail various ``optimistic,'' ``pessimistic,'' and ``dream case'' scenarios. We show that in some circumstances the $B$-mode detection may be very inconclusive, at the level of signal-to-noise ratio $S/N=1.75$, whereas a smarter data analysis can reveal the same gravitational wave signal at $S/N=6.48$. The final result is encouraging. Even under unfavorable conditions in terms of instrumental noises and foregrounds, the relic gravitational waves, if they are characterized by the maximum likelihood parameters that we found from WMAP5 data, will be detected by Planck at the level $S/N=3.65$.

Stochastic Background of Relic Scalar Gravitational Waves tuned by Extended Gravity

A stochastic background of relic gravitational waves is achieved by the so called adiabatically-amplified zero-point fluctuations process derived from early inflation. It provides a distinctive spectrum of relic gravitational waves. In the framework of scalar-tensor gravity, we discuss the scalar modes of gravitational waves and the primordial production of this scalar component which is generated beside tensorial one. Then we analyze different viable $f(R)$-gravities towards the Solar System tests and stochastic gravitational waves background. The aim is to achive experimental bounds for the theory at local and cosmological scales in order to select models capable of addressing the accelerating cosmological expansion without cosmological constant but evading the weak field constraints. It is demonstrated that viable $f(R)$-gravities under consideration not only satisfy the local tests, but additionally, pass the PPN-and stochastic gravitational waves bounds for large classes of parameters.