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### Scalar-induced stochastic gravitational waves

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*Oct 18,2022*Recently, IOP Publishing announced the "2022 Top Cited Paper Award from China", and three research articles on scalar-induced stochastic gravitational waves by Cai's group from the Institute of Theoretical Physics won the awards.

On September 14, 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO) in the United States detected the gravitational wave event GW150914 [1] originating from the merger of binary black holes. This is the first time that humans have directly detected gravitational waves, and it has been a century since A. Einstein predicted the existence of gravitational waves. Although in 1974, the binary pulsars discovered by R. A. Hulse and J. H. Taylor provided the first indirect evidence for the existence of gravitational waves [2], the direct detection in 2015 is still of great significance, it represents the coming of the era of gravitational-wave astronomy and gravitational-wave cosmology.

Before the detection of gravitational waves, our main tools of observing the universe was electromagnetic waves. Through the studies of the Cosmic Microwave Background (CMB), we know that there are primordial fluctuations, which have a nearly scale-invariant power spectrum and are Gaussian. The results of the CMB imply that the current observable universe originates from the accelerated expansion of a causally related region in the very early universe, i.e. , the so-called inflation [3-6]. During the period of inflation, the vacuum quantum fluctuations are stretched with the expansion of the universe and become the primordial fluctuations, which eventually lead to the anisotropy of the CMB and the large-scale structure. Therefore, we can study the primordial fluctuations and the inflation by observing the CMB. However, with the CMB we can only observe primordial fluctuations on the large scales, which correspond to the early stages of inflation. For smaller scales, corresponding to late stages of inflation, the CMB does not provide sufficient information. Since the photons interact with the passing matter during their propagation, the information carried by the photons is easy to be lost, it is difficult for us to obtain information about the early universe through electromagnetic wave observations. On the contrary, due to the weak interaction between gravitational waves and matter, the information carried by gravitational waves is not easy to be lost during their propagation, which means gravitational waves are very good information carriers. With gravitational wave observations, we can obtain valuable information about the early universe, which helps to study the origin and evolution of the universe. In addition, through gravitational waves, we can also obtain the source information of violent astronomical events, some of which are difficult to obtain by traditional electromagnetic wave observations, such as the merger of binary black holes. For other events, such as the merger of binary neutron stars, which generate both electromagnetic waves and gravitational waves, we can combine the information carried by gravitational waves and electromagnetic waves to conduct multi-messenger astronomy research.

In the early universe, many cosmological processes can produce gravitational waves, and these gravitational waves from various origins in various directions are superimposed and propagated up to now, forming a stochastic gravitational wave background. According to cosmological perturbation theory, perturbations can be decomposed into scalar perturbations, vector perturbations and tensor perturbations, and their evolution equations are independent of each other in linear order. However, in the nonlinear order, these three perturbations will be coupled together, so the linear order scalar perturbations can be the source to induce the nonlinear order tensor perturbations, that is the scalar-induced gravitational waves. The CMB tells us that on the scale larger than about 1 Mpc, the magnitude of the primordial curvature perturbations is in the order of $10^{-5}$, so the corresponding scalar-induced gravitational waves are difficult to be detected experimentally. However, the primordial curvature perturbations at small scales are still unclear. If the primordial curvature perturbations on small scales are relatively large, then the induced gravitational waves may be detected by current or future experiments, and even if these induced gravitational waves are not detected, they can still provide constraints on the small-scale primordial curvature perturbations. On the other hand, if the primordial curvature perturbations on small scales are large enough, they may lead to the formation of primordial black holes after entering the event horizon [7-9]. We know that baryon matter only accounts for 5% of the universe, the nature of dark matter accounting for 26% and dark energy accounting for 69% is still unclear. The primordial black hole is a good candidate for dark matter, and it may also be the source of the gravitational wave events detected by LIGO and could serve as the seeds for galaxy formation. The scalar-induced gravitational waves corresponding to the primordial black hole was generated in the early universe and propagated with the evolution of the universe, they carry very valuable information about the early universe, which provides us with an excellent opportunity for studying the physics of early universe. Such information about the very early universe is usually difficult to obtain through traditional electromagnetic wave observations. It can be seen that the scalar-induced gravitational waves are closely related to the inflation and the primordial black holes. The study of scalar-induced gravitational waves can promote the understanding of the early universe, cosmic evolution and dark matter, and promote the development of related fields.

We noticed that the shape of the spectrum of scalar-induced gravitational waves is sensitive to the shapes and positions of the peaks in the scalar perturbation spectrum. This motivates us to study the gravitational waves induced by scalar perturbations whose spectrum has multiple peaks, which can be easily generated by inflation model with multiple fields or multiple inflection points. In JCAP05(2019)013 [10], we studied the gravitational waves induced by scalar perturbations whose spectrum has multiple peaks. We found a nontrivial multiple-peak structure in the spectrum of induced gravitational waves due to the resonant amplification (there are at most $C_{n+1}^{2}$ and at least $n$ peaks in the gravitational wave spectrum when there are $n$ peaks in the spectrum of scalar perturbations), and identified analytically the positions of these peaks. Under the narrow-width approximation, we found a universal factor due to the momentum conservation. This work can help us to study the structure of the primordial scalar perturbation spectrum by using gravitational waves, and further constrain the inflation model.

LIGO has detected gravitational waves produced by the merger of binary black holes, but the origin of these black holes is still unclear. It is shown that these black holes may be primordial black holes that formed in the early universe. The peak frequency in the induced gravitational wave spectrum is determined by the peak frequency in the curvature perturbation spectrum, and the mass of the primordial black hole caused by the curvature perturbations also depends on the peak frequency of the curvature perturbation spectrum. If the black holes detected by LIGO are primordial black holes originating from curvature perturbations, there must be corresponding induced gravitational waves with a peak frequency of about nHz, which are within the observation range of the pulsar timing array (PTA). So far, the PTA has not detected the stochastic gravitational wave background, so the upper limit of gravitational waves in certain frequency intervals has been given. In JCAP10(2019)059 [11], we analyzed in detail the possibility that the black holes observed by LIGO are primordial black holes by using the data of the first and second runs of LIGO. We found that if these black holes are all primordial black holes originating from Gaussian curvature perturbations, the corresponding induced gravitational waves are in tension with the PTA constraints on the of stochastic gravitational wave background, and this tension can be avoided by considering non-Gaussian curvature perturbations. With more gravitational wave observations put into operation in the future, we can obtain a large amount of data on merger events, which can help us determine the origin of these black holes and their mass functions. If more data in the future show that the above tension still exists, our work provides a possible solution to it.

In order to form primordial black holes, and to generate induced gravitational waves with sufficient amplitude, so that they can be detected by the current or near future gravitational wave observations, the primordial curvature perturbations on small scales need to be large enough. On the CMB scale, the spectrum of curvature perturbations is on the order of $10^{-9}$. However, on relatively small scales the constraints on the curvature perturbations are very loose, the spectrum of curvature perturbations on small scales can be very large. Primordial curvature perturbations arise from quantum fluctuations during inflation, and inflation is an important part of the modern standard cosmological model, so it is crucial to determine a successful inflationary model through experimental constraints on primordial curvature perturbations. The observations of CMB give strict constraints on the primordial curvature perturbations on large scales, while the observations of induced gravitational waves and primordial black holes can help us study the primordial curvature perturbations on small scales, and further provide constraints on the inflation model. In JCAP06(2020)013 [12], we proposed a new model, a single-field inflation model with a small periodic oscillatory structure on the inflation potential, which is called oscillating-potential inflation. In this model, due to the existence of these small periodic oscillatory structures on the inflation potential, the primordial curvature perturbations can be amplified by resonance. In such a model, depending on the parameters of the model, the primordial black hole can be all or part of the dark matter, and the corresponding induced gravitational waves can be detected by the future PTA or space-based gravitational wave detectors.

The universe is so mysterious and fascinating, gravitational waves give us a pair of eyes to appreciate the beauty of the universe. In the era of electromagnetic waves, cosmology has developed from an ancient imagination to a modern science. In the era of gravitational waves, cosmology will surely enter a new stage of development.

[1] LIGO Scientific, Virgo collaboration, B. P. Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102, [1602.03837].

[2] R. A. Hulse and J. H. Taylor, Discovery of a pulsar in a binary system, Astrophys. J. Lett. 195 (1975) L51–L53.

[3] A. A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682–685.

[4] A. H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D 23 (1981) 347–356.

[5] A. D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B 108 (1982) 389–393. 2

[6] A. Albrecht and P. J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys. Rev. Lett. 48 (1982) 1220–1223.

[7] Y. B. Zel’dovich and I. D. Novikov, The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model, Soviet Astron. AJ (Engl. Transl. ), 10 (1967) 602.

[8] S. Hawking, Gravitationally collapsed objects of very low mass, Mon. Not. Roy. Astron. Soc. 152 (1971) 75.

[9] B. J. Carr and S. W. Hawking, Black holes in the early Universe, Mon. Not. Roy. Astron. Soc. 168 (1974) 399–415.

[10] R.-G. Cai, S. Pi, S.-J. Wang and X.-Y. Yang, Resonant multiple peaks in the induced gravitational waves, JCAP 05 (2019) 013, [1901.10152].

[11] R.-G. Cai, S. Pi, S.-J. Wang and X.-Y. Yang, Pulsar Timing Array Constraints on the Induced Gravitational Waves, JCAP 10 (2019) 059, [1907.06372].

[12] R.-G. Cai, Z.-K. Guo, J. Liu, L. Liu and X.-Y. Yang, Primordial black holes and gravitational waves from parametric amplification of curvature perturbations, JCAP 06 (2020) 013, [1912.10437].

**Author:**Xing-Yu Yang, graduated from the Institute of Theoretical Physics with a PhD degree in 2022 (PhD advisor: Prof. Rong-Gen Cai), and is currently doing postdoctoral research at the Korea Institute for Advanced Study, focusing on gravity and cosmology.

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