Constraints On Scalar-Induced Gravitational Waves Up To Third Order From Joint Analysis of BBN, CMB, And PTA Data

Recently, strong evidence for a gravitational wave background has been reported by collaborations of pulsar timing arrays (PTA). In the framework of scalar-induced gravitational waves (SIGWs), we concurrently investigate the second and third order gravitational waves by jointly analyzing PTA data, alongside big-bang nucleosynthesis (BBN), and cosmic microwave background (CMB) datasets. We determine the primordial curvature spectral amplitude as $0.021<A_\zeta<0.085$ and the spectral peak frequency as $10^{-7.3}\ \mathrm{Hz}<f_\ast<10^{-6.3}\ \mathrm{Hz}$ at a 95\% confidence interval, pointing towards a mass range for primordial black holes of $10^{-4.5}M_\odot<m_{\mathrm{PBH}}<10^{-2.5}M_\odot$. Our findings suggest that third order gravitational waves contribute more significantly to the integrated energy density than the second order ones when $A_\zeta\gtrsim0.06$. Furthermore, we expect future PTA projects to validate these findings and provide robust means to investigate the genesis and evolution of the universe, especially inflation.

In addition to the direct measurements of SIGWs from the PTA probe, early-universe probes offer indirect constraints on the integrated SIGW spectrum [25].Given that SIGWs behave like radiation in the universe, their energy can alter the growth of cosmological density perturbations and the universe's expansion rate at the time of decoupling.Consequently, the CMB probe is sensitive to the integrated SIGW spectrum [26,27].Simultaneously, the success of big-bang nucleosynthesis (BBN) theory can limit the number of relativistic species at the nucleosynthesis epoch.Therefore, the BBN probe is sensitive to the energy of SIGWs [28].Both probes independently measure cosmological GWBs of frequency bands above 10 −10 Hz, but are insensitive to other GWBs produced due to astrophysical processes in the late universe.
In this study, we simultaneously consider second and third order gravitational waves, and explore joint constraint on them from BBN, CMB, and PTA datasets.Previous research on third order gravitational waves is documented in Refs.[29,30].We will demonstrate that they dominate the SIGWs' energy density if the primordial curvature spectral amplitude exceeds O(0.06).We will also illustrate that they do not significantly alter the PTA bound but cause substantial changes in the BBN and CMB bounds.Consequently, the joint constraint will also experience significant alterations.Moreover, we will explore these possibilities and potential future improvements.
Tensor perturbations h ij induced by the linear scalar perturbations are referred to as SIGWs.Second order gravitational waves have been investigated in the literature [1][2][3][4][5][6], and we follow the conventions of Ref. [6].As shown in Appendix A, the equation of motion for third order gravitational waves is [30] h where H is conformal Hubble parameter, ∆ is Laplacian, Λ lm ij is transverse-traceless operator, and S (3) lm (η, x) denotes source terms, as expressed in Eq. (A2).Leveraging the two-point correlators of h h (k), we get the power spectrum for third order gravitational waves, i.e., where we introduce the quantities P * (k, p, q) and the kernel functions I (3) * (p 1 , p 2 , p 3 , p 4 , k, η) in Appendix A, the subscripts * and * * denoting different sources of third order gravitational waves, i.e., in terms of (ϕ (1) ) 3 , ϕ (1) ψ (2) , ϕ (1) Regarding the primordial black hole (PBH) formation, there should be a large-amplitude peak on the power spectrum of primordial curvature perturbations [32][33][34].Inflation models with sound speed resonance can generate a nearly monochromatic spectrum [35][36][37][38][39][40].For simplicity, we consider a delta-function spectrum where A ζ is the amplitude and k * is the pivot wavenumber.The energy-density fraction spectrum of SIGWs is defined as where ij , and ρ c is critical density at conformal time η.We determine it as (5) where the power spectrum of second order gravitational waves, P h (k, η), is calculated in pioneers' works [1][2][3][4][5][6], and the power spectrum of third order ones, P h (k, η), is shown in Eq. ( 3).Since gravitational waves behave like radiations, the energy-density fraction spectrum in the present universe is [41] where the physical energy-density fraction of radiations in the present universe is Ω r,0 h 2 ≃ 4.2 × 10 −5 with h = 0.6766 being the dimensionless Hubble constant [42], a subscript eq denotes cosmological quantities at the epoch of matter-radiation equality, and both g * ,ρ and g * ,s stand for the effective relativistic specifies in the universe [43].Moreover, cosmic temperature T is related with k, i.e., g * ,s (T ) 106.75 It should be emphasized that the contributions from the third order gravitational waves become significant on the small scales k ≫ k * .This is primarily due to the additional enhancement in their power spectrum originated from resonance of the higher order perturbations at late times.On the other hand, for the large scales k ≪ k * , the contributions from the third order gravitational waves can be neglected because their power spectrum is highly suppressed compared to that of the second order gravitational waves.
The theoretical predictions of the aforementioned spectrum are depicted in Fig. 1.In this case, we set the model parameters to A ζ = 0.1 (solid curves) and A ζ = 0.01 (dashed curves).The spectrum of the second order gravitational waves is represented in blue, while the combined spectrum of both the second and third order gravitational waves is shown in red.In comparison to the second order gravitational waves, the third order gravitational waves primarily contribute to the spectrum around the peak frequencies.
Joint constraints.It is known that the PTA probe is directly sensitive to the energy density of SIGWs.Following the methodology of Ref. [12], we examine the parameter space by conducting a Bayesian analysis over the NANOGrav 15-year (NG15) dataset [9].The priors of log 10 (f * /Hz) and log A ζ are uniformly set within the intervals of [-11,-5] and [-3,1], respectively.In fact, our results are robust with respect to priors.Here, we neglect the very likely presence of an astrophysical foreground, which has been considered in Refs.[44,45].
We consider two scenarios related to SIGWs.The first scenario (Scenario I) includes only the second order gravitational waves, while the second scenario (Scenario II) incorporates both the second and third order gravitational waves.For both scenarios, we derive the posteriors of f * and A ζ , which are illustrated in Fig. 2. Statistically, it is challenging to differentiate between the two scenarios since their posteriors nearly overlap.We conclude that the third order gravitational waves, when being compared with the second order ones, have negligible impact on the interpretation of the observed PTA signal in terms of SIGWs.
The BBN and CMB probes are indirectly sensitive to the energy density of SIGWs 1 .Specifically, they are 1 Dr. Carlo Tasillo tells us via email that they have studied phasetransition gravitational waves by jointly analyzing the BBN, CMB, and NANOGrav 12.5-year data [46].Deep green line denotes f PBH = 1, indicating that all of dark matter is composed of PBHs [12].
sensitive only to the integrated energy-density fraction, as described by where k min = 2πf min sets the lower bound of the integral, and N eff represents the number of relativistic species.As f min is dependent on the physical process that occurred during the epochs of BBN and CMB formation, we adopt f BBN ≃ 1.5×10 −11 Hz for BBN and f CMB ≃ 3×10 −17 Hz for CMB [24].According to the Planck 2018 CMB plus BAO dataset [42], the right hand side of Eq. ( 8) equals 2.9 × 10 −7 [27], resulting in an upper limit of A ζ ≤ 0.130 for Scenario I and A ζ ≤ 0.085 for Scenario II.In contrast, for BBN the right hand side equals 1.3 × 10 −6 [28], yielding an upper limit of A ζ = 0.275 for Scenario I and A ζ = 0.150 for Scenario II.These upper limits are illustrated in Fig. 2. Though the contours are almost the same as those in Ref. [12], the BBN and CMB upper bounds would significantly alter the posteriors via reducing a large portion of the posteriors.This indicates the importance of the data combination.
The results from the joint analysis are as follows.In both scenarios, the parameter region inferred from the NG15 data is notably refined by the inclusion of the BBN and CMB data.The permissible upper limit on A ζ is somewhat smaller in Scenario II than in Scenario I, highlighting the significance of third order gravitational waves.Starting from the peak of the posterior, we derive the combined constraints on A ζ and f * as 5.0 × 10 −8 Hz < f * < 5.0 × 10 −7 Hz , (10) at the 95% confidence level.To our knowledge, these findings represent the state-of-the-art and most stringent constraints on the model parameters.
It should be noted that when A ζ ≳ 0.06, the third order gravitational waves contribute more to the integrated energy density than the second order ones.This outcome suggests that the third order gravitational waves cannot be disregarded in the data analysis of BBN and CMB.
Taking into account both the second and third order gravitational waves, we also find that the CMB bound, denoted by the dashed red line in Fig. 2, is comparable to the deep green line which indicates all of dark matter to be composed of PBHs, i.e., f PBH = 1 [12].There may be a risk of overproducing PBHs.However, the allowed maximum peak amplitude of the power spectrum is A ζ ≃ 0.058, an amplitude making the third order contribution as nearly equal as the second order one.Therefore, it is important to take into account the third order gravitational waves in our analysis.
Anticipated Constraints.It is anticipated that the energy-density fraction spectrum of SIGWs, and subsequently the power spectrum of primordial curvature perturbations, will be potentially measured by the Square Kilometre Array (SKA) [47][48][49], µAres [50], Laser Interferometer Space Antenna [51,52], big bang observer [53,54], Deci-hertz Interferometer Gravitational wave Observatory [55,56], Einstein Telescope [57], and Advanced LIGO and Virgo [58][59][60].Complementing CMB, which is sensitive to the earliest stages of inflation, gravitational-wave probes offer the capability to investigate the physics of the early universe that occurred during later stages of inflation.Conducting multi-band gravitational-wave observations, we expect to explore the origin and evolution of the universe throughout the entire inflationary era.Following Ref. [61], we examine this subject.During a GWB search, if neglecting the very likely presence of an astrophysical foreground, the optimal signal-to-noise Projected constraint capabilities of future gravitational-wave detection projects.The allowable parameter region inferred from Fig. 2, as well as the upper limits from present BBN and CMB observations, are also displayed for comparison.Here, we set SNR = 1 and neglect the very likely presence of an astrophysical foreground.
ratio is defined as [62 where n det represents the number of detectors, T obs is the observation duration, the frequency band extends from f l to f u , and S eff n denotes the effective noise power spectral density of the detector network.Here, H 0 = 100h km/s/Mpc is the Hubble constant.For the aforementioned experiments, we employ the setups summarized in Table 2 of Ref. [44].
Requiring SNR = 1, we illustrate the anticipated constraint contours in the A ζ -f * plane for SKA in Fig. 3.The allowable parameter region (blue shaded contours) inferred from this study is presented for comparison.Notably, we find that our inferred contours can be further tested with SKA and µAres.We note superb performance in measuring the primordial curvature spectral amplitude, i.e., A ζ ∼ 10 −6 for PTA projects, A ζ ∼ 10 −8 for space-borne projects, and A ζ ∼ 10 −5 for groundbased projects.The high sensitivities should enable us to explore the early universe more comprehensively.

Conclusion and Discussion
. In this study, we have delved into the gravitational waves induced by scalar perturbations, up to the third order, by scrutinizing recent PTA datasets in conjunction with BBN and CMB data.We have calculated the energy-density spectrum of SIGWs up to third order.Through the analysis of the joint datasets of BBN, CMB, and PTA, we inferred the allowable parameter region, which was depicted in Fig. 2. The inferred constraints on A ζ and f * were presented in Eq. ( 9) and Eq.(10).
Interestingly, we found that the third order gravitational waves could contribute more to the integrated energy density than the second order ones when the primordial curvature spectral amplitude A ζ exceeds approximately 0.06.This finding underscores the importance of third order gravitational waves in our joint data analysis.
As illustrated in Fig. 3, we anticipated that the energydensity fraction spectrum of scalar-induced gravitational waves, and subsequently the power spectrum of primordial curvature perturbations, will be potentially measured by future gravitational-wave experiments, which should enable us to explore the early universe more comprehensively, and further test the predictions of our study.Our findings represent a significant step forward in our understanding of the universe, particularly in relation to cosmic inflation.
The next-generation CMB experiments, e.g., CMB-S4 [63], Simons Observatory [64], and LiteBIRD [65], are expected to reach better sensitivity that would lead to improvements of the present CMB upper limits on A ζ , also indicating potential improvements of the best bound inferred in our current work.
(A2) Third order gravitational waves are produced by both the linear scalar perturbations and second order metric perturbations that are produced by the linear scalar perturbations.The latter were studied in the literature [30,83].
In Fourier space, the evolution of the first and second order metric perturbations is given by ϕ where ] is that for tensor perturbations [31], Φ k is a stochastic variable characterizing the primordial scalar perturbations.During radiation domination, the initial conditions lead to Φ k = 2ζ k /3, where ζ k denotes primordial curvature perturbations with wavevector k.The transfer function T ϕ is obtained by solving the master equation for the linear scalar perturbations [84].As shown in Ref. [30], the kernel functions I ϕ , I ψ , I V , and I h are obtained by solving the equations of motion for ϕ (2) , ψ (2) , V (2) , and h (2) , respectively.

FIG. 1 .
FIG. 1.This figure contrasts the energy density spectra of second order (blue) with those of both second and third orders (red).The model parameters provided are A ζ = 0.1 (solid) and A ζ = 0.01 (dashed).

− 7 . 1 FIG. 2 .
FIG.2.The posteriors for the model parameters A ζ and f * derived from the NG15 PTA dataset (contours), in addition to the BBN (thick dotted lines) and CMB (thick dashed lines) upper limits on A ζ .For the one-dimensional posteriors, the 68% confidence intervals are indicated by thin dashed lines.Deep green line denotes f PBH = 1, indicating that all of dark matter is composed of PBHs[12].
FIG. 3.Projected constraint capabilities of future gravitational-wave detection projects.The allowable parameter region inferred from Fig.2, as well as the upper limits from present BBN and CMB observations, are also displayed for comparison.Here, we set SNR = 1 and neglect the very likely presence of an astrophysical foreground.