Null energy condition violation during inflation and pulsar timing array observations

Recently, evidence of stochastic gravitational wave background (SGWB) signals observed by pulsar timing array (PTA) collaborations, has prompted investigations into their origins. We explore the compatibility of a proposed inflationary scenario, incorporating an intermediate null energy condition (NEC)-violating phase, with the PTA observations. The NEC violation potentially amplifies the primordial tensor power spectrum, offering a promising explanation for PTA observations. Numerical analyses, primarily focused on NANOGrav's 15-year results, reveal the model's compatibility with PTA data. Notably, the model predicts a nearly scale-invariant GW spectrum in the mHz frequency range, which sets our scenario apart from other interpretations predicting a red primordial GW spectrum on smaller scales.

Gravitational Waves (GWs) are promising tools for exploring the physics of the early universe.Observations of GWs from binary pulsar systems [1] and the merger of binary black holes [2] have ushered in the era of GW detection.Recently, pulsar timing array (PTA) collaborations, including NANOGrav [3,4], EPTA [5], PPTA [6], and CPTA [7], announced compelling evidence of a signal consistent with stochastic gravitational wave background (SGWB) at the reference frequency f = 1 yr −1 .Shortly after the release of the PTA results, various studies emerged regarding the possible origin of the observed signal, see, e.g., .
Primordial GWs [81][82][83], which are the tensor fluctuations generated quantum mechanically in the very early universe, serve as an important possible source of the SGWB.Detecting the primordial GW background would provide valuable physical insights into the origin and evolution of the universe.It is intriguing to explore whether recent observations by PTA can be interpreted as signals of the primordial GWs with a blue-tilted power spectrum.Recently, a lot of research has been conducted in this direction, see, e.g., [21,55,56,61,[84][85][86][87].
Inflation [88][89][90][91], as a leading paradigm for the very early universe, elegantly addresses the horizon and flatness problems of the Big Bang cosmology.It also predicts a nearly scale-invariant power spectrum of scalar perturbations, which is consistent with the cosmic microwave background (CMB) observations.Furthermore, the conventional slow-roll inflationary scenario also predicts a nearly scale-invariant power spectrum of primordial GWs.
The violation of the null energy condition (NEC) may play a crucial role in the very early universe (see e.g.[112] for a review).It has been demonstrated that a fully stable NEC violation can be realized in the "beyond Horndeski" theories [113][114][115][116][117][118][119][120], see also [121][122][123][124][125][126].Basically, the violation of NEC implies an increase in the Hubble parameter H (i.e., dH/dt > 0).Given that the power spectra of scalar and tensor perturbations are proportional to H 2 , an intermediate NEC violation during inflation leads to intriguing phenomena of observational interest at certain scales, including: 1) a significant enhancement in the power spectrum of primordial GWs [127,128], 2) a notable amplification of the parityviolation effect in primordial GWs [129] (see also [130]), 3) the generation of primordial black holes with masses and abundances of observational interest [131], along with the associated scalar-induced GWs.Remarkably, the scenario proposed by [127] naturally yields a broken power-law power spectrum, which may potentially be consistent with observations from both CMB and PTA, see also [55] for recent studies.
In this paper, we examine the predicted primordial GW background from the model proposed by [127] in light of observations from PTA, with a specific focus on the NANOGrav signal.Our paper is organized as follows.In Sec.II, we provide a brief overview of the background dynamics in our model.Sec.III introduces the power spectrum and its parameterization as predicted by our model.In Sec.IV, we perform numerical comparisons between the primordial GW background predicted by our model and the NANOGrav signal.
Sec. V is dedicated to our conclusion.

II. INTERMEDIATE NEC VIOLATION DURING INFLATION
A. A brief review of the scenario In our scenario [127], the universe begins with an initial phase of slow-roll inflation char- For the scalar (or tensor) perturbation modes that exit the horizon during the first stage of slow-roll inflation (i.e., k < k 1 ), their power spectrum is nearly scale-invariant, making it consistent with the observations of temperature anisotropy in the CMB.Similarly, the power spectrum of perturbation modes that exit the horizon during the second stage of slow-roll inflation (i.e., k > k 2 ) is also nearly scale-invariant, but it has a significantly larger amplitude.The scale-invariance of the tensor power spectrum at large scales ensures the absence of a highly suppressed tensor-to-scalar ratio r, and consequently a highly suppressed slow-roll parameter in canonical single-field slow-roll inflation.At small scales, the scaleinvariance of the scalar and tensor power spectra prevents them from growing to O(1), thus preserving the validity of perturbation theory at higher frequencies.
The power spectrum of perturbation modes that exit the horizon during the NECviolating phase (i.e., k 1 < k < k 2 ) is blue-tilted (n s > 0 or n T > 0).Namely, the violation of the NEC enhances both the scalar and tensor power spectrum by increasing the Hubble parameter H, see [127][128][129]131]. Significantly, at the intermediate scales of the scalar power spectrum, the blue tilt and oscillatory features, particularly around the scale corresponding to the beginning of the second inflationary phase, lead to intriguing phenomena of observational interest, including the generation of primordial black holes and the associated scalar-induced GWs [131].Simultaneously, the power spectrum of tensor perturbations (or the primordial GWs), also exhibits a blue tilt and possesses oscillatory features around the transition scales [127][128][129].These distinctive features set our scenario apart from other single-field primordial black hole formation scenarios.The combination of primordial black holes, scalar-induced GW signals, and primordial GWs provides a valuable avenue for studying the violation of the NEC during inflation, especially in the era of multi-messenger and multi-band observations.
Our scenario can be realized with the EFT action (see e.g., [113,115,128]) where δg 00 = g 00 + 1, R (3) is the Ricci scalar on the 3-dimensional spacelike hypersurface, δK = K − 3H, K is the extrinsic curvature.The time-dependent functions c(t) and Λ(t) determine the background evolution, with the relations c(t) = −M2 P Ḣ and Λ(t) = M 2 P ( Ḣ + 3H 2 ).The functions M 4 2 (t), m 3 3 (t) and m2 4 (t) can be determined or constrained based on the condition that the scalar perturbations are in agreement with observations.
Notably, the operator R (3) δg 00 plays a crucial role in preventing scalar perturbations from becoming unstable when the NEC is violated, as demonstrated in [113][114][115].The covariant form of action (1), as discussed in [116,117], falls under the category of "beyond Horndeski" theory.Nonetheless, the propagation of primordial GWs is exactly the same as that in general relativity at quadratic order.In this paper, we will not delve into the specific formulations of these coefficient functions or the intricacies of the model construction.
Instead, we will employ a simplified parameterization of the background evolution for our scenario.Consequently, we can establish a parameterization for the power spectrum of primordial GWs.

B. Parametrization of background
We begin with a flat Friedmann-Lemaitre-Robertson-Walker universe described by the metric where t is the cosmic time and τ is the conformal time, related by dt = adτ .Throughout this paper, we will use a dot to denote d/dt and a prime to denote d/dτ .We will also define An inflationary stage is commonly characterized by quasi-de Sitter expansion, where the scale factor approximately evolves as a ∝ |τ | −1 for τ < 0. Additionally, we will parameterize our NEC-violating stage with a power-law scale factor, i.e., a(τ ) ∝ |τ | n . 2 The full-scale factor can be then parametrized as a piecewise function of τ : where τ j is the conformal time at the end of phase j, j = 1, 2, 3 corresponds to the first inflation stage, the NEC-violating stage and the second inflation stage, respectively; τ R,j = is the integration constant and ϵ j = − Ḣ/H 2 is treated as constant during phase j.Since phases 1 and 3 are assumed as slow-roll inflation, we will set ϵ 1 ≈ ϵ 3 ≈ 0 for simplicity.As for the NEC-violating phase (i.e., phase 2), we have ϵ 2 < 0, which indicates A specific design of such a model can be found in [127].
The provided parametrization overlooks the details of transitions between different phases, including the specific variations of ϵ near the beginning or the end of the NECviolating phase.The dynamics of the transition might depend on the specific model.For simplicity, we will describe the transition physics using the matching condition, ensuring the continuity of the scale factor a and its first derivative a ′ at τ 1 and τ 2 .For our purpose, such a simplification will not make a qualitative difference.
Using Eq. ( 3) and denoting the beginning and the end time of the NEC-violating stage to be τ 1 and τ 2 , respectively, the continuity of a gives The continuity of a ′ or H enables us to define the following quantities With the help of (3), we can solve the integration constants as Obviously, the consistency of ( 6) requires In terms of the scale factor and Hubble parameter; where we have defined H 2 ≡ H2 /a 2 (τ 2 ) and H 1 ≡ H1 /a 2 (τ 1 ) to be the Hubble parameter at τ = τ 2 and τ = τ 1 .

III. PRIMORDIAL GRAVITATIONAL WAVES A. Tensor perturbations and mode functions
Since the gravity sector is minimally coupled to the matter sector, the quadratic action for tensor perturbation is simply where we have set the propagation speed of tensor perturbation to be unity (i.e., the speed of light), and M P = 1.In the momentum space, the dynamical equation for tensor perturbation is where u k ≡ γ λ k a/2 is the mode function and λ = +, × represent two different polarizations.The parameterization of scale factor, i.e., Eq. ( 3), gives Note that ϵ 2 < 0 would result in 1/2 < ν 2 < 3/2.As a result, in each phase, the general solution to Eq. ( 10) can be expressed in terms of the Hankel function as where H (1) ν j are the first and second kind Hankel functions of the ν j -th order, respectively; α j and β j are k-dependent coefficients.
For simplicity, we have assumed ϵ 1 ≈ ϵ 3 ≈ 0, which indicates ν 1 ≈ ν 3 ≈ 3/2 and the Hankel functions are simply where the asterisk denotes complex conjugation.We impose the Bunch-Davis vacuum initial condition, which gives |α 1 | = 1 and β 1 = 0.The other coefficients α j and β j for j = 2, 3 can be determined with the matching method, which requires the continuities of u k and u ′ k at the transition surface τ = τ 1 and τ = τ 2 .
More explicitly, we are interested in the final tensor spectrum, which is relevant to |α 3 − We present see [128], where we employ and an overall phase factor has been neglected.

B. Parametrization of the tensor spectrum
The tensor spectrum provided by ( 15) is complicated, necessitating further simplification.
To interpret the PTA data through primordial GWs from our model, the amplitude of primordial GWs must be enhanced from the order of O(10 −11 ), the upper bound on the amplitude of P T constrained by CMB observations, to 10 −3 within the PTA range.Given that in our scenario, n T < 2, the NEC violating stage must persist for scales spanning over four orders of magnitude, i.e., k 2 > 10 4 k 1 .Consequently, we can safely make the Next, we observe that the Hankel function behaves as a pure phase factor in the subhorizon region and as a power-law function in the super-horizon region.This asymptotic behavior allows us to derive approximate expressions for modes exiting the horizon in different stages.For example, for modes exiting the horizon during the second inflation stage, where k ≫ k 2 , leading to y 1 , y 2 , x 1 , x 2 ≫ 1, each Hankel function can be approximated as a pure phase function, resulting in Similarly, modes exiting the horizon during the first inflation stage satisfy k < k 1 , for which we have For perturbation modes that exit the horizon during the NEC-violating stage, the approximate tensor power spectrum is given by see [128] for more details.This spectrum shows a blue tilt (n T = 3 − 2ν 2 > 0) within the range k 1 < k < k 2 in our scenario.Since 1/2 < ν 2 < 3/2, we have 0 < n T < 2. Remarkably, when employing an exponential parameterization for a 2 (τ ), we reach n T = 2, as explicitly demonstrated in Appendix A.
The above treatment encounters challenges near the transition scales where a simple expansion of the Hankel function through its asymptotic behavior is not applicable.However, in the vicinity of the scale k ≃ k 1 , the corresponding tensor spectrum is expected to be too small for detection in the near future, allowing us to safely overlook the transition feature in this region.Conversely, near the scale k ≃ k 2 , the tensor spectrum is sufficiently large for potential detection.For k ≫ k 1 , all Hankel functions with arguments x 1 , x 2 ≫ 1 exhibit the asymptotic behavior H (1) This characteristic enables us to capture the features of P T around k ≃ k 2 while simplifying the formulation of In light of this, we can parametrize P T as where the auxiliary function is defined as It is important to note that we have replaced ν 2 by the corresponding tensor spectral index n T = 3 − 2ν 2 in the NEC violating stage.In the limit of k ≪ k 1 , P T ≈ P T,1 since the second term in Eq. ( 20) becomes negligible compared to P T,1 .Conversely, for k ≫ k 1 , the second term in Eq. ( 20) becomes dominant.Consequently, it ensures that the features around k ≃ k 2 are well captured.While this formulation may sacrifice accuracy around the first transition scale k ≃ k 1 , such a compromise has minimal impact on our interests.
The primordial tensor power spectrum P T can be converted to the observed GW energy spectrum by [132] where H 0 = 67.8km/s/Mpc, τ 0 = 1.41 × 10 4 Mpc, is the density fraction of matter today, and the wavenumber relates to the frequency as k = 2πf .

IV. NUMERICAL RESULTS
In this section, we confront the parameterized spectrum (20) to the recent PTA data.
As explained in the previous section, we are interested in the case k 1 ≪ k ≲ k 2 where the tensor spectrum is blue.To this end, the first term in (20) is much smaller than the second one, thus we can safely neglect P T,1 .With this simplification, the theoretical spectrum ( 20) is fully parameterized by three parameters {P T,2 , n T , f c }, where we have defined the transition frequency from the NEC-violating stage to the second inflationary stage as f c ≡ 2πk 2 .We performed Monte Carlo Markov Chain (MCMC) analysis varying {P T,2 , n T , f c } plus the pulsar and nuisance parameters against the most recent public NANOGrav 15yr dataset [4].Following NANOGrav, we assume two models on the spatial correlation of the signal, uncorrelated common-spectrum red noise (CURN) and Hellings-Down (HD). 3 .Theoretically, our model predicts n T < 2. The validity of the perturbation theory requires P T,2 < 1. See Table .I for our prior choice of the model parameters.Therefore, a prior upper bound P T,2 < 1 gets translated to an upper bound on f c .

Fig.3 further compares the best-fit theoretical spectrum with violin points measured by
NANOGrav.The left panel of Fig. 3 illustrates the physical energy spectrum of GW denoted as Ω GW h 2 .While the NEC-violating phase satisfactorily accounts for the PTA signal, the tensor spectrum originating from the second inflationary phase also falls comfortably within 3 CURN assumes spatially uncorrelated signal while HD assumes that the signal at different pulsars has a spatial correlation described by the Hellings-Down curve [133].The later is the expected spatial correlation of SGWB.See [4] for details.
the detection sensitivity of forthcoming space-based GW observatories like LISA [134], Taiji [135], and Tianqin [136].The right panel of Fig. 3 zooms into the PTA data-constrained frequency range and plots the PTA timing excess, see [4] for a detailed definition.Only the first few data points detect GW background signal and the theoretical spectrum (solid black line) fits them as well as a power law (dashed black line).Close to f c , the solid line drops to a smaller n T , which is not favored by data.Therefore, in the MCMC, f c automatically shifts to a higher frequency than the range constrained by the data, leading to the observed lower bound on f c as depicted in Fig. 2.
The numeric results confirmed that the NEC-violating model can well explain the GW background signal observed by PTA experiments, but the current data is not precise enough to distinguish it from other models, e.g., a simple power law.Our model also predicts a flat tensor spectrum in the mHz frequency range and is detectable to next-generation space GW detectors.
Parameter Prior

An intermediate stage of NEC violation during inflation has the potential to amplify
the primordial tensor power spectrum on certain scales, offering a potential explanation for the PTA observations.In this paper, we explore the primordial GW spectrum within an inflationary scenario featuring an intermediate NEC-violating phase.We present parame-  Additionally, our model predicts a nearly scale-invariant GW spectrum in the mHz frequency range, potentially detectable by upcoming space-based GW detectors like LISA, Taiji, and Tianqin.This distinctive characteristic distinguishes our scenario from other interpretations predicting a red primordial GW spectrum on smaller scales.

VI. DISCUSSIONS
Intriguingly, Ref. [129] found that, for the scenario discussed in this paper, the violation of the NEC naturally amplifies the parity-violating effect as well as its observability in primordial GWs, provided the scalar field determining the background evolution is coupled to a parity-violating term.The wavenumber k corresponding to the maximum of the parityviolating effect is approximately the same as the wavenumber corresponding to the maximum of the power spectrum.This intriguing feature also sets our model apart observationally from other models.
The scale corresponding to the significantly amplified parity-violating effects depends on the scale at which the NEC violation takes place.In this paper, the maximum of the primordial GW power spectrum appears in the vicinity of the PTA band, as shown by the black curve in Fig. 3. Therefore, the parity violation is still notably amplified in the PTA band.However, considering the power spectrum presented in Fig. 3, the parity violation is suppressed by the slow-roll condition in the LISA band, as the GW modes in the LISA band are generated during the second slow-roll inflation stage in our scenario.To achieve a notable amplification of the parity-violating effect in the LISA band, it would be necessary for the NEC violation to occur later than what is assumed in the present paper.
In Ref. [131], it has been demonstrated that the NEC violation can naturally enhance the primordial scalar power spectrum at certain scales, leading to the production of PBHs and scalar-induced GWs of observational interest.The primordial GWs (i.e., tensor perturbations) primarily depend on the Hubble parameter H.The scalar-induced GWs are induced by scalar perturbations, depending not only on H but also on ϵ ≡ − Ḣ/H 2 (or its generalized formulation).Therefore, the relative contribution to the GW background of each depends on the specific model's characteristics, particularly the relative magnitudes of the power spectra of scalar and tensor perturbations.
In this paper, we primarily focus on primordial GWs and have not explicitly calculated the primordial scalar perturbations.We assume that the contribution to the GW background in the PTA band mainly comes from primordial GWs rather than scalar-induced GWs.
In fact, for the same background evolution, scalar perturbations exhibit stronger model dependence compared to tensor perturbations.In principle, for models constructed with specific covariant action, we can quantitatively compare the contributions of both to the GW background.This remains a subject for future work.
parameterized by Eq. ( 3), with the assumption that ϵ 1 ≈ ϵ 3 ≈ 0. Therefore, we have and If the NEC-violating phase is very brief, we can approximate the Hubble parameter as a linear function.As a result, a 2 (τ ) can be parameterized by an exponential formulation as where The parameterization of the background can be related to the characteristic scales: where k 1 is the mode that crosses the horizon at the end of the first inflationary phase, τ = τ 1 , and k 2 is the mode that crosses the horizon at the beginning of the second inflationary phase, The continuity of a and H at τ 1 is already incorporated into the parametrization.
Therefore, we focus on ensuring continuity at τ 2 , which requires that a 3 (τ 2 ) = a 2 (τ 2 ) and In terms of k 1 and k 2 , and considering the condition k 1 ≪ k 2 , we can express leading to For the first inflationary phase (i.e., τ < τ 1 ), the mode function can be given as where we have taken the vacuum initial condition.It can be further simplified to In the NEC-violating phase, we have ϵ ≡ 1−H ′ /H 2 ≪ −1, indicating H ′ ≫ H 2 for nearly the entire phase.As a result, a ′′ /a = H 2 + H ′ ≃ H ′ = γ 2 .Consequently, the dynamical equation simplifies to The parameter ω k turns imaginary for k > γ and real for k < γ.Yet, we will primarily focus on analyzing the dynamics of perturbations for k < γ as it should be adequate for our purposes.In this scenario, ω k is real, and the general solution takes the form The mode function for the second inflationary phase can be expressed as where J and Y are Bessel functions.In a more explicit form, this can be written as where z ≡ k(τ R,3 − τ ).On super-horizon scales, the dominating term is c 4 , making it sufficient to evaluate c 4 .The tensor spectrum on these scales, expressed in terms of c 4 , is given by We match the perturbation at τ 1 and τ 2 for fluctuations with k < γ, resulting in A quick check: for modes crossing the horizon in the first inflation phase, we have Therefore, for k ≪ k 1 , the power spectrum is which is scale-invariant, as expected.The equation (A17) indicates that the super-horizon modes generally experience a modification factor during the NEC-violating phase, aligning with results from bouncing cosmology.
Modes with k > k 2 and k > γ remain sub-horizon at τ = τ 2 and cross the horizon in the second inflationary phase.Therefore, we can directly write down their corresponding tensor spectrum: To explain the PTA result, we aim for P T,inf2 ≃ O(10  In conclusion, we could adopt the following parametrization for P T : where

15 A
NEC violation during inflation A. A brief review of the scenario B. Parametrization of background III.Primordial gravitational waves A. Tensor perturbations and mode functions B. Parametrization of the tensor spectrum IV.Numerical results 11 .Power spectrum from an exponential parameterization of the scale factor during the NEC-violating phase 15 References 20 I. INTRODUCTION

FIG. 1 :
FIG. 1: An illustration of the evolution of the conformal Hubble horizon (represented by the solid black lines) and the perturbation modes (depicted as dashed blue lines with arrows) in our scenario.

Fig. 2 − 8 . 1 <
Fig.2shows the posterior distributions of the tensor spectrum parameters.Data puts lower bounds on P T,2 > 10 −2 (95% C.L.) and n T > 1.0 (95% C.L.).This is to be expected because it is known that the NANOGrav 15yr data includes a ∼ 3σ detection of GW background near 10 −8 Hz with n T ∼ 1.8.The distribution of f c is a peak −8.1 < log 10 (f c /Hz) < −6.6, but this should not be confused with a detection.Near f c , the spectrum transits from n T ∼ 2 to n T = 0 with oscillations.Having f c > 10 −8 Hz means that data does not favor the shape of transition so it is moved to the right of the data constraining region, see also Fig.3.The upper bound on f c is due to the prior upper bound on P T,2 .Because data fixes the amplitude of the spectrum near 10 −8 Hz, f c and P T,2 are strongly positively correlated, as can be seen from the log 10 f c -log 10 P T,2 contour of Fig.2.

)≃ k 3k 2 2 (2k 2 +
How does this relate to PTA observations?As we can see from Fig.4, the tensor spectrum is blue-tilted in the regionk 1 ≪ k ≪ k 2 .Since γ is comparable to k 2 , we also have γ ≫ k such that ω k ≃ γ.The expression (A24) simplifies to |f (k)| blue spectrum with n T = 2 in the range k 1 ≪ k ≪ k 2 .This result is in agreement with that in the main text in the limit n → ∞, as the power-law parametrization a 2 ∝ |τ | n approaches an exponential function.

TABLE I :
Prior choice for spectrum parameters in the MCMC analysis.
V. CONCLUSIONSThe recent SGWB signals reported by PTA collaborations have unveiled a new frontier in the exploration of gravitational wave physics.Notably, if these PTA signals originate from the primordial universe, it necessitates the presence of new physics beyond the standard slowroll inflation scenario.This necessity arises because PTA observations indicate a highly blue tensor spectrum with a spectral index of n T = 1.8 ± 0.3, whereas the conventional slow-roll inflation scenario predicts a nearly scale-invariant tensor spectrum.