Primordial Tensor Perturbations Research Articles

Hunting for statistical anisotropy in tensor modes with B -mode observations

We investigate a possibility of constraining statistical anisotropies of the primordial tensor perturbations by using future observations for the Cosmic Microwave Background (CMB) B-mode polarization. By parameterizing a statistically-anisotropic tensor power spectrum as $P_h ({\boldsymbol{k}}) = P_h (k) \sum_n g_n \cos^n \theta_{\boldsymbol{k}}$, where $\theta_{\boldsymbol{k}}$ is an angle of the direction of $\hat{k}={\boldsymbol{k}}/k$ from a preferred direction, we find that it would be possible for future B-mode observations such as CMB-S4 to detect the tensor statistical anisotropy at the level of $g_n \sim {\mathcal O} (0.1)$.

Viable super inflation scenario from F(T) modified teleparallel gravity

Viability of the super inflation scenario is discussed in F(T) modified teleparallel gravity. We calculate the power spectrum of the primordial curvature and tensor perturbations, and we derive two important parameters corresponding to these perturbations, i.e., spectral index parameter n_{s} and scalar-to-tensor ratio r. We observe that the super inflation scenario provides a nearly scale invariant power spectrum of scalar and tensor perturbations in the observational bounds of n_{s} and r, according to choosing the proper values interval of the power-term of the scale factor, asim t^{h}.

Numerical relativity investigation of the effects of gravitational waves on the inhomogeneity of the universe

We numerically integrate the Einstein’s equations for a spatially flat Friedmann–Lemaire–Robertson–Walker (FLRW) background spacetime with a spatial curvature perturbation and evolving primordial tensor perturbations using the Einstein Toolkit. We find that although the primordial tensor perturbation does not play an important role in the evolution of the overdensity produced by the scalar perturbation, there is an obvious imprint left by the primordial tensor perturbation on the distribution of the fractional density perturbation in the nonlinear region. This imprint may be a possible probe of a gravitational waves background in the future.

Berry phase of primordial scalar and tensor perturbations in single-field inflationary models

In the framework of the single-field slow-roll inflation, we derive the Hamiltonian of the linear primordial scalar and tensor perturbations in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes in terms of the Lewis–Riesenfeld phase. We conclude by discussing the discrepancy in the results of Pal et al. (2013) [21] for these Berry phases, which is resolved to yield agreement with our results.

Some topics on scale-invariant perturbations from noninflationary universe

In this paper, we review some topics on generations of scale-invariant primordial scalar and tensor perturbations in the early universe models different from inflation. The content includes generation of scale-invariant and Gaussian scalar perturbation in the ekpyrotic/cyclic universe, and production scale-invariant tensor perturbation in contracting universe. The main property of the models reviewed in this paper is the nonminimal couplings, include nonminimal couplings between the scalar fields and those to the gravity. By introducing these couplings, it is not difficult to achieve scale-invariances for the perturbations in the early universe models alternative to inflation.

Cosmology with a light ghost

We study the creation and evolution of cosmological perturbations in renormalizable quadratic gravity with a Weyl term. We adopt a prescription that implies the stability of the vacuum at the price of introducing a massive spin-two ghost state, leading to the loss of unitarity. The theory may still be predictive regardless the interpretation of non-unitary processes provided that their rate is negligible compared to the Universe expansion rate. This implies that the ghost is effectively stable. In such a setup, there are two scalar degrees of freedom excited during inflation. The first one is the usual curvature perturbation whose power spectrum appears to coincide with that of single-field inflation. The second one is a scalar component of the ghost encoded in the shift vector of the metric in the uniform inflaton gauge. The amplitudes of primordial tensor and vector perturbations are strongly suppressed. After inflation the ghost field starts to oscillate and its energy density shortly becomes dominant in the Universe. For all ghost masses allowed by laboratory constraints ghosts should have ``overclosed'' the Universe at temperatures higher than that of primordial nucleosynthesis. Thus, the model with the light Weyl ghost is ruled out.

New CMB constraints for Abelian Higgs cosmic strings

We present cosmic microwave background (CMB) power spectra from recent numerical simulations of cosmic strings in the Abelian Higgs model and compare them to CMB power spectra measured by Planck. We obtain revised constraints on the cosmic string tension parameter Gμ. For example, in the ΛCDM model with the addition of strings and no primordial tensor perturbations, we find Gμ < 2.0 × 10−7 at 95% confidence, about 20% lower than the value obtained from previous simulations, which had 1/64 of the spatial volume. The increased computational volume also makes it possible to simulate fully the physical equations of motion, in which the string cores shrink in comoving coordinates. We find however that this, and the larger dynamic range, changes the amplitude of the power spectra by only about 10%. The main cause of the stronger constraints on Gμ is instead an improved treatment of the string evolution across the radiation-matter transition.

Primordial black holes as a novel probe of primordial gravitational waves. II. Detailed analysis

Recently we have proposed a novel method to probe primordial gravitational waves from upper bounds on the abundance of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations generated in the early Universe is fairly large, they induce substantial scalar perturbations due to their second-order effects. If these induced scalar perturbations are too large when they reenter the horizon, then PBHs are overproduced, their abundance exceeding observational upper limits. That is, primordial tensor perturbations on superhorizon scales can be constrained from the absence of PBHs. In our recent paper we have only shown simple estimations of these new constraints, and hence in this paper, we present detailed derivations, solving the Einstein equations for scalar perturbations induced at second order in tensor perturbations. We also derive an approximate formula for the probability density function of induced density perturbations, necessary to relate the abundance of PBHs to the primordial tensor power spectrum, assuming primordial tensor perturbations follow Gaussian distributions. Our new upper bounds from PBHs are compared with other existing bounds obtained from big bang nucleosynthesis, cosmic microwave background, LIGO/Virgo and pulsar timing arrays.

Signal photon flux generated by high-frequency relic gravitational waves**Supported by National Natural Science Foundation of China (11305181,11322545,11335012) and Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (Y5KF181CJ1)

The power spectrum of primordial tensor perturbations increases rapidly in the high frequency region if the spectral index nt > 0. It is shown that the amplitude of relic gravitational waves ht(5 × 109 Hz) varies from 10−36 to 10−25 while nt varies from −6.25 × 10−3 to 0.87. A high frequency gravitational wave detector proposed by F.-Y. Li detects gravitational waves through observing the perturbed photon flux that is generated by interaction between relic gravitational waves and electromagnetic field. It is shown that the perturbative photon flux (5 × 109 Hz) varies from 1.40 × 10−4 s−1 to 2.85 × 107 s−1 while nt varies from −6.25 × 10−3 to 0.87. Correspondingly, the ratio of the transverse perturbative photon flux to the background photon flux varies from 10−28 to 10−16.

High-order primordial perturbations with quantum gravitational effects

In this paper, we provide a systematic investigation of high-order primordial perturbations with nonlinear dispersion relations due to quantum gravitational effects in the framework of {\em uniform asymptotic approximations}. Because of these effects, the equation of motion of the mode function in general has multiple-turning points. After obtaining analytically approximated solutions to any order in different regions, associated with different types of turning points, we match them to the third one. To this order the errors are less than $0.15\%$. General expressions of the power spectra of the primordial tensor and scalar perturbations are derived explicitly. We also investigate effects of back-reactions of the quantum gravitational corrections, and make sure that inflation lasts long enough in order to solve the underlying problems, such as flatness, horizon and monopole. Then, we study various features of the spectra that are observationally relevant. In particular, under a moderate assumption about the energy scale of the underlying theory of quantum gravity, we have shown that the quantum gravitational effects may alter significantly the ratio between the tensor and scalar power spectra, thereby providing a natural mechanism to alleviate the tension between observations and certain inflationary models, including the one with a quadratic potential.

Structure of Kähler potential for D-term inflationary attractor models

Minimal chaotic models of D-term inflation predicts too large primordial tensor perturbations. Although it can be made consistent with observations utilizing higher order terms in the K\"ahler potential, expansion is not controlled in the absence of symmetries. We comprehensively study the conditions of K\"ahler potential for D-term plateau-type potentials and discuss its symmetry. They include the alpha-attractor model with a massive vector supermultiplet and its generalization leading to pole inflation of arbitrary order. We extend the models so that it can describe Coulomb phase, gauge anomaly is cancelled, and fields other than inflaton are stabilized during inflation. We also point out a generic issue for large-field D-term inflation that the masses of the non-inflaton fields tend to exceed the Planck scale.

Removing Ostrogradski’s ghost from cosmological perturbations in f(R,Rμν2,Cμνρσ2) gravity

Recently, it was argued that gravity with the square of the Ricci tensor can be stabilized by adding constraints to the theory in a Lorentz violating way. This was so far demonstrated for fluctuations on the Minkowski/de Sitter background. We show that the same scheme works equally well for removing Ostrogradski’s ghost from fluctuations on a cosmological background in generic [Formula: see text]-type theories of gravity. As an application, we derive the general formula for the spectrum of primordial tensor perturbations from the stabilized theory. The evolution of matter density perturbations is also discussed.

Primordial black holes as a novel probe of primordial gravitational waves

We propose a novel method to probe primordial gravitational waves by means of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations on comoving scales much smaller than those relevant to Cosmic Microwave Background is very large, it induces scalar perturbations due to second-order effects substantially. If the amplitude of resultant scalar perturbations becomes too large, then PBHs are overproduced to a level that is inconsistent with a variety of existing observations constraining their abundance. This leads to upper bounds on the amplitude of initial tensor perturbations on super-horizon scales. These upper bounds from PBHs are compared with other existing bounds.

Resonant primordial gravitational waves amplification

We propose a mechanism to evade the Lyth bound in models of inflation. We minimally extend the conventional single-field inflation model in general relativity (GR) to a theory with non-vanishing graviton mass in the very early universe. The modification primarily affects the tensor perturbation, while the scalar and vector perturbations are the same as the ones in GR with a single scalar field at least at the level of linear perturbation theory. During the reheating stage, the graviton mass oscillates coherently and leads to resonant amplification of the primordial tensor perturbation. After reheating the graviton mass vanishes and we recover GR.

Primordial gravitational waves measurements and anisotropies of CMB polarization rotation

Searching for the signal of primordial gravitational waves in the B-modes (BB) power spectrum is one of the key scientific aims of the cosmic microwave background (CMB) polarization experiments. However, this could be easily contaminated by several foreground issues, such as the interstellar dust grains and the galactic cyclotron electrons. In this paper we study another mechanism, the cosmic birefringence, which can be introduced by a CPT-violating interaction between CMB photons and an external scalar field. Such kind of interaction could give rise to the rotation of the linear polarization state of CMB photons, and consequently induce the CMB BB power spectrum, which could mimic the signal of primordial gravitational waves at large scales. With the recently released polarization data of BICEP2 and the joint analysis data of BICEP2/Keck Array and Planck, we perform a global fitting analysis on constraining the tensor-to-scalar ratio r by considering the polarization rotation angle [α(nˆ)] which can be separated into a background isotropic part [α¯] and a small anisotropic part [Δα(nˆ)]. Since the data of BICEP2 and Keck Array experiments have already been corrected by using the “self-calibration” method, here we mainly focus on the effects from the anisotropies of CMB polarization rotation angle. We find that including Δα(nˆ) in the analysis could slightly weaken the constraints on the tensor-to-scalar ratio r, when using current CMB polarization measurements. We also simulate the mock CMB data with the BICEP3-like sensitivity. Very interestingly, we find that if the effects of the anisotropic polarization rotation angle could not be taken into account properly in the analysis, the constraints on r will be dramatically biased. This implies that we need to break the degeneracy between the anisotropies of the CMB polarization rotation angle and the CMB primordial tensor perturbations, in order to measure the signal of primordial gravitational waves accurately.

On the possibility of blue tensor spectrum within single field inflation

We present a series of theoretical constraints on the potentially viable inflation models that might yield a blue spectrum for primordial tensor perturbations. By performing a detailed dynamical analysis we show that, while there exists such possibility, the corresponding phase space is strongly bounded. Our result implies that, in order to achieve a blue tilt for inflationary tensor perturbations, one may either construct a non-canonical inflation model delicately, or study the generation of primordial tensor modes beyond the standard scenario of single slow-roll field.

Primordial perturbations from dilaton-induced gauge fields

We study the primordial scalar and tensor perturbations in inflation scenario involving a spectator dilaton field. In our setup, the rolling spectator dilaton causes a tachyonic instability of gauge fields, leading to a copious production of gauge fields in the superhorizon regime, which generates additional scalar and tensor perturbations through gravitational interactions. Our prime concern is the possibility to enhance the tensor-to-scalar ratio r relative to the standard result, while satisfying the observational constraints. To this end, we allow the dilaton field to be stabilized before the end of inflation, but after the CMB scales exit the horizon. We show that for the inflaton slow roll parameter ϵ ≳ 10−3, the tensor-to-scalar ratio in our setup can be enhanced only by a factor of \U0001d4aa(1) compared to the standard result. On the other hand, for smaller ϵ corresponding to a lower inflation energy scale, a much bigger enhancement can be achieved, so that our setup can give rise to an observably large r≳ 10−2 even when ϵ≪ 10−3. The tensor perturbation sourced by the spectator dilaton can have a strong scale dependence, and is generically red-tilted. We also discuss a specific model to realize our scenario, and identify the parameter region giving an observably large r for relatively low inflation energy scales.

Forecasting sensitivity on tilt of power spectrum of primordial gravitational waves after Planck satellite

By taking into account the contamination of foreground radiations, we employ the Fisher matrix to forecast the future sensitivity on the tilt of power spectrum of primordial tensor perturbations for several ground-based (AdvACT, CLASS, Keck/BICEP3, Simons Array, SPT-3G), balloon-borne (EBEX, Spider) and satellite (CMBPol, COrE, LiteBIRD) experiments of B-mode polarizations. For the fiducial model nt=0, our results show that the satellite experiments give good sensitivity on the tensor tilt nt to the level σnt≲0.1 for r≳2×10−3, while the ground-based and balloon-borne experiments give worse sensitivity. By considering the BICEP2/Keck Array and Planck (BKP) constraint on the tensor-to-scalar ratio r, we see that it is impossible for these experiments to test the consistency relation nt=−r/8 in the canonical single-field slow-roll inflation models.

Tensors, BICEP2 results, prior dependence, and dust

We investigate the prior dependence on the inferred spectrum of primordial tensor perturbations, in light of recent results from BICEP2 and taking into account a possible dust contribution to polarized anisotropies. We highlight an optimized parameterization of the tensor power spectrum, and adoption of a logarithmic prior on its amplitude $A_T$, leading to results that transform more evenly under change of pivot scale. In the absence of foregrounds the tension between the results of BICEP2 and Planck drives the tensor spectral index $n_T$ to be blue-tilted in a joint analysis, which would be in contradiction to the standard inflation prediction ($n_T<0$). When foregrounds are accounted for, the BICEP2 results no longer require non-standard inflationary parameter regions. We present limits on primordial $A_T$ and $n_T$, adopting foreground scenarios put forward by Mortonson & Seljak and motivated by Planck 353 GHz observations, and assess what dust contribution leaves a detectable cosmological signal. We find that if there is sufficient dust for the signal to be compatible with standard inflation, then the primordial signal is too weak to be robustly detected by BICEP2 if Planck+WMAP upper limits from temperature and E-mode polarization are correct.

Examining the consistency relations describing the three-pointfunctions involving tensors

It is well known that the non-Gaussianity parameter fNL characterizing the scalar bi-spectrum can be expressed in terms of the scalar spectral index in the squeezed limit, a property that is referred to as the consistency relation. In contrast to the scalar bi-spectrum, the three-point cross-correlations involving scalars and tensors and the tensor bi-spectrum have not received adequate attention, which can be largely attributed to the fact that the tensors had remained undetected at the level of the power spectrum until very recently. The detection of the imprints of the primordial tensor perturbations by BICEP2 and its indication of a rather high tensor-to-scalar ratio, if confirmed, can open up a new window for understanding the tensor perturbations, not only at the level of the power spectrum, but also in the realm of non-Gaussianities. In this work, we consider the consistency relations associated with the three-point cross-correlations involving scalars and tensors as well as the tensor bi-spectrum in inflationary models driven by a single, canonical, scalar field. Characterizing the cross-correlations in terms of the dimensionless non-Gaussianity parameters CNLℛ and CNLγ that we had introduced earlier, we express the consistency relations governing the cross-correlations as relations between these non-Gaussianity parameters and the scalar or tensor spectral indices, in a fashion similar to that of the purely scalar case. We also discuss the corresponding relation for the non-Gaussianity parameter hNL used to describe the tensor bi-spectrum. We analytically establish these consistency relations explicitly in the following two situations: a simple example involving a specific case of power law inflation and a non-trivial scenario in the so-called Starobinsky model that is governed by a linear potential with a sharp change in its slope. We also numerically verify the consistency relations in three types of inflationary models that permit deviations from slow roll and lead to scalar power spectra with features which typically result in an improved fit to the data than the more conventional, nearly scale invariant, spectra. We close with a summary of the results we have obtained.