Generating enhanced primordial GWs during inflation with intermittent violation of NEC and diminishment of GW propagating speed

We investigate both the null energy condition (NEC) violating scenario and the cT-diminishing scenario for generating enhanced power spectrum of primordial gravitational waves (GWs) during inflation, where cT is the propagating speed of primordial GWs. Both of these two scenarios can be realized stably with theories beyond Horndeski, hence can be uniformly implemented within the framework of the effective field theory. We calculate the power spectrum of primordial GWs by assuming that the inflationary Universe undergoes three phases, where the violation of NEC or the diminishment of cT occurs in the intermediate phase. A template of the spectrum is given for the NEC-violating scenario. We also discuss the underlying relation and discrepancy between these two scenarios with a disformal transformation.

A wide multi-frequency range of observations aimed at searching for GW signals will lead us to the era of GW astronomy.
The primordial GWs [20,21] predicted by inflation have yet to be confirmed by future observations.In the slow-roll inflation scenario, the power spectrum of primordial GWs is usually assumed as a power-law.As a result, the primordial GWs can hardly be detected by the pulsar timing arrays (PTA) or laser interferometers in the near future, since the tensor-to-scalar ratio r is severely constrained by the data of cosmic microwave background (CMB) B-mode polarization [22].However, the primordial GW background spans a broad frequency-band (10 −18 − 10 10 Hz), the structure of the power spectrum and the underlying physics might be far richer than expected [23][24][25].
Fully stable NEC violation can be realized in scalar-tensor theories beyond Horndeski, as discovered in the exploration of the effective field theory (EFT) of nonsingular cosmology [49][50][51][52].
Additionally, the propagating speed of GWs (i.e., c T ) can be time-dependent in some gravity theories beyong general relativity.Although the joint detection of GWs and the gamma rays from the binary neutron star merger GW170817 has strongly supported that the propagating speed of GWs is c T = 1 at present [53][54][55][56][57], with deviations smaller than O(10 −15 ), it still cannot place stringent bounds on c T in the very early Universe, see e.g., [58][59][60][61].The nontrivial variation of c T is able to imprint interesting features in the power spectrum of GWs [62,63], see also [64][65][66].An enhanced (or blue) power spectrum of primordial GWs may also be attributed to the diminishment of c T [67][68][69][70][71][72][73], which can be implemented with scalar-tensor theories beyond Horndeski.Intriguingly, it is found that a scenario in which c T gradually diminishes during inflation is actually a disformal dual to the superinflation [68].
In this paper, we investigate both the NEC-violating scenario and the c T -diminishing scenario for generating enhanced power spectrum of primordial GWs during inflation within the uniform framework of EFT.In Sec. 2, we briefly review the EFT method.In Sec. 3, we explicitly calculate the power spectrum of primordial GWs for a scenario in which the NEC-preserving slow-roll inflation is followed by an NEC-violating expanding phase and a later NEC-preserving slow-roll inflation with a larger Hubble parameter, while c T = 1 throughout.Templates of such spectra are provided.We also calculate the power spectrum of primordial GWs in Sec. 4 for a scenario in which c T decreases with time during an intermediate phase of the slow-roll inflation, while the NEC is preserving throughout.
The underlying relation and discrepancy between these two scenarios are discussed with a disformal transformation in Sec. 5.The speed of light is set as c = 1 throughout this paper.

The effective field theory method
It is convenient to use the 3 + 1 decomposed metric where N is the lapse function, N i is the shift vector, h ij is the 3-dimensional metric.In the unitary gauge, the EFT of cosmological perturbations up to quadratic order can be written as where we have disregarded those higher-order spatial derivatives; δg 00 = g 00 + 1, δK µν = K µν −h µν H with H being the Hubble parameter, h µν = g µν +n µ n ν is the induced metric, n µ is the unit normal vector of the constant time hypersurfaces, K µν is the extrinsic curvature, R (3) is the induced 3-dimensional Ricci scalar, S m represents the action of matter sector, which is minimally coupled to the metric g µν .
The background evolution of the Universe is only determined by the first line of (2.2).
Particularly, for f (t) ≡ 1, the Friedmann equations can be given as c With a variety of designs of those coefficient functions, the EFT action (2.2) is versatile in applications to a variety of cosmological background, including inflation [81], dark energy [82][83][84] and nonsingular cosmology [49][50][51].Furthermore, the EFT method has great advantages in the study of cosmological perturbations.
The derivation of quadratic actions for scalar and tensor perturbations of the action (2.2) can be found in [49].In this paper, we will focus on the tensor perturbations γ ij (i.e., primordial GWs), which is traceless and divergence-free, i.e., γ ii = 0 = ∂ i γ ij , where we have set Generally, the quadratic actions of γ ij can be formulated as where ˙= d/dt, c T is the propagating speed of primordial GWs.In order to avoid the ghost and gradient instabilities, Q T > 0 and c 2 T > 0 are required, respectively.From (2.2), we have where f and m 2 4 are time-dependent coefficients defined in (2.2).When f = 1 and m 2 4 = 0, we have Q T = 1 and c T = 1, which are same as those in general relativity.
In the momentum space, we have where The power spectrum of primordial GWs is which is evaluated after the perturbation modes exited their horizons, i.e., aH/ (c T k) ≫ 1.
The energy density spectrum of primordial GWs (see e.g.[85], see also [86][87][88][89]) can be given by where k eq is the comoving wavenumber of the perturbation mode that entered the horizon at the matter-radiation equality, Ω m = ρ m /ρ c , ρ c = 3M 2 p H 2 0 is the critical energy density.The observation of a stochastic GW background would provide us with significant information or constraints on the primordial GWs as well as the evolution history of the very early Universe.

Enhanced power spectrum of primordial GWs from intermittent NEC violations during inflation
Primordial NEC violation would imprint a blue-tilted power spectrum of primordial GWs.
A series of intermittent NEC violations during inflation would generate a spectrum of primordial GWs with far richer structures, such as a Great Wall-like spectrum [23].An enhanced power spectrum of primordial GWs at the frequency ∼ 1/yr induced by NEC violation might be compatible with the recent result of NANOGrav, provided the reported common-spectrum process could be interpreted as GWs.
In this section, we analytically calculate the power spectrum of primordial GWs in a scenario proposed by [23] and provide a template of such an enhanced GW spectrum.
Follow [23], we assume that after an NEC-preserving slow-roll inflation with Hubble parameter H ≃ H inf 1 , the Universe experienced an NEC-violating expansion, and then enters a subsequent NEC-preserving slow-roll inflation again but with a higher Hubble scale H (= H inf 2 ≫ H inf 1 ).

Set up
Instead of explicitly implementing this scenario in scalar-tensor theories, we adopt the EFT action where we have disregarded the action of the matter sector (i.e., S m ), which should make a negligible contribution to the evolution of the very early Universe in our scenario.The action (3.1) corresponds to setting f (t) = 1 and m 2 4 (t) = 0 in action (2.2).As a result, we have in Eq. (2.4).Although the covariant correspondence of action (3.1) (see [52,90]) belongs to the "beyond Horndeski" theory, the propagation of GWs is exactly same as that in general relativity, at least at quadratic order.
The functions c(t) and Λ(t) can be determined in a reverse manner by the background evolution with relations c(t 4 (t) can be determined or constrained by the requirement that the scalar perturbations should be consistent with observations.Additionally, a nonzero m2 4 (t), i.e., a contribution from the operator R (3) δg 00 , is crucial for avoiding instabilities of scalar perturbations induced by a violation of the NEC [49][50][51].However, since we will focus on the primordial GWs in the following calculations, we will not be interested in the explicit formulations of these coefficient functions in the following.
The evolution history of the Universe that we will consider can be divided into three phases, which will be represented by phase 1, 2 and 3.During phase j (j = 1, 2 or 3), the scale factor could be parameterized with the conformal time τ as for τ < τ j , where τ j is the conformal time at the end of phase j, τ R,j = τ j −(ϵ j −1) The continuities of a(τ ) and da/dτ are required at τ 1 and τ 2 .Since phase 1 and 3 are assumed as slow-roll inflation while phase 2 is NEC-violating expansion, we will set ϵ 1 ≈ ϵ 3 ≈ 0 and ϵ 2 < 0. A specific design of such a model can be found in [23].The detailed variation of ϵ around the beginning or the end of the NEC-violating phase could be model-dependent.For our purpose, the simplification will not make a qualitative difference.

Power spectrum
With Eqs.(2.4) and (3.2), the equation of motion for γ λ (τ, k) can be given as where we have defined u k = γ λ (τ, k)aM p /2.The parameterization in Eq. (3.3) gives The solutions to Eq. (3.4) during the three phases can be given as where α j and β j are k-dependent coefficients for j = 1, 2, 3; H ν j and H (2) ν j are the first and second kind Hankel functions of the ν j th order, respectively.
For perturbations that exit the horizon in the first inflationary phase, we have k ≪ H1 ≪ H2 , i.e., x 1 , x 2 , y 1 , y 2 ≪ 1, provided 1/2 < ν 2 < 3/2.By straightforward series expansion, we find As can be inferred, the power spectrum is (nearly) scale-invariant in the limit of k ≪ H1 .
Note that, according to Eq. (3.3), the leading order term in the right hand side of Eq. (3.9) can also be written as ( H1 / H2 ) for ν 2 = 1/2, where there is a difference of a factor 4/9 at the leading order compared with (3.9).However, it should be pointed out that when ν 2 = 1/2, i.e., ϵ 2 → −∞, the parameterization given by Eq. (3.3) cannot guarantee the continuity of da/dτ at τ 1 or τ 2 any more.The above derivation will no longer be applicable.Therefore, we will disregard the case of ν 2 = 1/2 in this paper.
For perturbations that exit the horizon in the NEC-violating phase, we will focus on the perturbation modes which satisfy H1 ≪ k ≪ H2 , i.e., x 1 , x 2 ≫ 1 and y 1 , y 2 ≪ 1.We obtain for 1/2 < ν 2 < 3/2.Therefore, the GWs generated during the NEC-violating phase have a blue spectrum, where the spectrum index is As for the perturbation modes that exit the horizon in phase 3, namely, the second inflationary phase, we have k ≫ H2 , which indicates x 1 , x 2 ≫ 1 and y 1 , y 2 ≫ 1.The result is trivially given as where there is some oscillating corrections included in the subleading order terms.
Apparently, when ν 2 → 3/2, all of Eqs.(3.9) to (3.11) reduce to P T /P T,inf 2 → 1, which is just the result of a single inflationary phase.In a realistic inflation model which is constructed with a slowly rolling scalar field, the slow roll parameter ϵ should slightly deviate from 0. The resulting power spectrum would also deviate from scale-invariance.
We assume that such a deviation could be absorbed into P T,inf 2 , for simplicity.

Templates
With the asymptotic behaviors of P T /P T,inf 2 given by Eqs.(3.9) to (3.11), we can provide a template of the GW power spectrum in our scenario as where there are only three model-dependent parameters, i.e., ν 2 ≡ 1 2 + 1 1−ϵ 2 , H1 and H2 .Here, H1 and H2 can also be replaced by H 1 and H 2 , since ( H1 / H2 ) 3−2ν 2 = (H 1 /H 2 ) 2 .The condition H1 ≪ H2 or H 1 ≪ H 2 is required for the template to be applicable.Obviously, in the limit k ≪ H1 , H1 ≪ k ≪ H2 and k ≫ H2 , Eq. (3.12) reduces to Eq. (3.9), (3.10) and (3.11) up to the leading order, respectively.A comparison between the template (3.12) and the numerical result of P T /P T,inf 2 is displayed in Fig. 1.The parameterization of the template (3.12) is quite simple and straightforward.However, a shortcoming of (3.12) is the loss of the oscillating features in P T /P T,inf 2 around k ∼ H1 and k ∼ H2 .These oscillations could be model-dependent to some extent.Additionally, since H1 ≪ H2 in our model, the oscillations around k ∼ H1 are suppressed greatly compared with those around k ∼ H2 , as can be seen from Eq. (3.8), see also Fig. 1.Therefore, only the oscillations around k ∼ H2 need to be included in the template.For this purpose, we can use the condition k ≫ H1 without considering the relation between k and H2 .The template can be given as where there are still three model-dependent parameters, i.e., ν 2 , H1 and H2 .It is obvious that Eq. (3.13) reduces to (3.9) in the limit k ≪ H1 .For k ≳ H1 , Eq. (3.13) is equivalent to (3.8) at the leading order, so that the oscillating features around k ≪ H2 is preserved, see Fig.  4 Enhanced power spectrum of primordial GWs from the diminishment of propagating speed of inflationary GWs With a disformal transformation of the metric, a time-dependent c T can be reset as unity [97].However, the background evolution may no longer be inflationary.For example, it has been found that slow-roll inflation in which c T decreases rapidly with time is a disformal dual to a superinflation (ϵ < 0) in which c T ≡ 1 [67].The power spectrum, which is observable in principle, has to be invariant under a nonsingular disformal transformation.Therefore, a time-dependent c T during inflation is able to imprint interesting features in the power spectra of primordial GWs [62,63,67,68]. 1 In this section, we calculate the enhanced power spectra of primordial GWs for a specific time-dependent c T while assuming the background evolution as the slow-roll inflation.

Set up
Using the EFT method, we can write the action as where ' ˆ's are used to mark the frame in which the propagating speed of GWs is timedependent.We assume that the background evolution is the slow-roll inflation throughout, which is determined by the first line of action (4.1).Again, we have neglected S m .However, it should be remembered that the matter sector is crucial for identifying the physical background evolution as measured relative to physical rulers, see e.g., [98].Additionally, operators (δĝ 00 ) 2 , δ Kδĝ 00 and R(3) δĝ 00 can also appear in the action (4.1), since they will not affect the tensor perturbations at quadratic order, as can be seen from Eq. (2.5).For simplicity, we will focus on the tensor perturbations and consider only the contribution of a nonzero m2 4 (t), which contributes nothing to the scalar perturbation at quadratic order.According to Eqs. (2.2) to (2.5), we can write the quadratic action of γ ij as where As in the last section, we consider three phases, during which the scale factor can be parameterised as for τ < τj , where τj is the conformal time at the end of phase j, τR,j = τj − (ε j − 1) −1 Ĥ−1 (τ j ), Ĥ(τ ) ≡ â−1 dâ/dτ .Since the background evolution is assumed as slow-roll inflation throughout these three phases, we will set ε1 = ε2 = ε3 = 0 in the following, for simplicity.

Power spectrum
In order to analytically calculate the power spectrum induced by a time-dependent ĉT (τ ), we define so that the quadratic action (4.2) can be rewritten as The equation of motion for γ λ (τ, k) is where Using Eq. (2.7), the power spectrum can be given as From Eq. (4.5) to (4.7), we can write the solutions to Eq. (4.10) in the three phases as where where PT,inf2 = , Ĥj is the Hubble parameter evaluated at τj .Since we have assumed that ε3 = 0, we have Ĥ3 = Ĥ2 = Ĥinf2 .For a specific model of slow-roll inflation construct by scalar-tensor theories, the slow-roll parameter εj can slightly deviate from 0.
Therefore, there could be a correction in PT,inf2 , which will be ignored in this paper for simplicity.
We can obtain α3 and β3 by requiring the continuities of v k and dv k /dσ at σ 1 and σ 2 .Note that the continuities of Ĥ at τ1 and τ2 actually requires that τR,1 = τR,2 = τR,3 .
For the perturbation modes exited horizon during phase 2 (i.e., σ 1 ≤ σ ≤ σ 2 ), we consider the situation in which z 1 ≫ 1 and z 2 ≪ 1.Then, we find It can be checked that PT ( Ĥ1 / cT,1 ≪ k ≪ Ĥ2 / cT,2 ) → PT,inf2 in the limit p → 0. Again, the discontinuity of dĉ T (τ )/dτ at τ1 or τ2 could contribute a p-dependent factor, which is not physical.What we shall be interested in is the spectrum index, i.e., nT,2 = 3 − 2ν 2 = p/(1 + p), which is consistent with that obtained in [67,68].Note that we have assumed p > 0 in the derivation of Eq. (4.17).For a general p, we actually have ν2 = 3+2p 2+2p , which is displayed in Fig. 3.It can be checked that nT,2 = 3 − 2ν 2 = p/(1 + p) is also valid for an increasing c T as long as −1 < p ≤ 0, which will predict a red-tilted (i.e., nT,2 < 0) spectrum of primordial GWs.It is interesting to see whether a blue spectrum can be obtained by an increasing c T with p < −1.
Unfortunately, the result of Eq. (4.17) is no longer applicable for the situation p < −1.
Actually, according to Eqs. (4.4), (4.6) and (4.10), the effective comoving sound horizon of tensor perturbation mode, which can be defined as will not decrease with time when p ≤ −1.In other words, we will have Ĥ1 / cT,1 ≫ Ĥ2 / cT,2 in Eq. (4.17) for p < −1.Therefore, the tensor perturbation modes are not able to exit their sound horizon when p ≤ −1.In fact, the tensor perturbation modes with comoving wavenumber Ĥ1 / cT,1 ≫ k ≫ Ĥ2 / cT,2 should reenter their sound horizon during the p < −1 phase and eventually exit their horizon during the third phase (τ 2 < τ ≤ τ3 ), see Fig. 4 for an illustration.With this in mind, we still obtain a red-tilted spectrum, i.e., PT / PT,inf2 ∼ k −(3+2ν 2 ) , for Ĥ1 / cT,1 ≫ k ≫ Ĥ2 / cT,2 .In a short summary, the case in which c T increases with time (i.e., p < 0) during inflation cannot generate GWs with a blue spectrum.
For the perturbation modes exited horizon during phase 3 (i.e., σ 2 < σ ≤ σ 3 ), we have z 1 ≫ 1 and z 2 ≫ 1.In this limit, the effect of the discontinuity of dĉ T (τ )/dτ can be negligible, as can be inferred from Eq. (4.10).Therefore, we can obtain that where the leading order term is trivial.
When τ > τ3 , ĉT may regain the value of the speed of light, so that the predictions will be consistent with observations of GWs at higher frequency [53].As a consequence, the power spectrum of GWs could become red-tilted at much higher frequencies.Additionally, the discontinuity of dĉ T (τ )/dτ at τ1 or τ2 is a defect of the parameterizations (4.5) to (4.7).
For a generic smooth ĉT (τ ), the power spectrum can be solved numerically from the action see [97] for details.Therefore, we find dĉ T /dτ ĉT (τ ) . (5.2) In Sec.4.1, we have assumed that ĉT (τ ) is constant during phase 1 and phase 3.As a result, we can obtain H1 / H2  (4.14) for the definitions of P T,inf 2 and PT,inf2 , respectively.Therefore, the power spectra of the two scenarios coincide with each other at the leading order for the perturbation modes exited their horizon in phase 1 and phase 3.
Regarding the NEC-violating phase and the ĉT -diminishing phase, we can find p = − 2ϵ 2 1+ϵ 2 by using (5.1), which indicates that ν2 = ν 2 , provided there exists such a disformal transformation.Therefore, for perturbation modes that exited their horizon during the intermediate phase, their spectra indexes are same in these two scenarios at the leading order.However, there is a discrepancy in the coefficients of the leading order terms in Eq. (3.10) and (4.17), which should be attributed to the discontinuity of dĉ T (τ )/dτ .As can be seen from Eq. (5.2), even if dĉ T (τ )/dτ is continuous everywhere, the oversimplified parameterizations of a (or H) and ĉT cannot result in the equivalence of these two scenarios around the matching time τ 1 (or τ1 ) and τ 2 (or τ2 ), unless dĉ T /dτ /ĉ T is negligible.For this reason, we should focus on the comparison of spectra indices of these two scenarios, which are evaluated for Ĥ1 / cT,1 ≪ k ≪ Ĥ2 / cT,2 so that they can hardly be affected by the details around the matching time.
In the NEC-violating scenario, ν 2 = 1/2 + 1/(1 − ϵ 2 ), which indicates that 1/2 < ν 2 < 3/2 and 0 < n T = 3 − 2ν 2 < 2 . (5.4) However, in the ĉT -diminishing scenario, ν2 = (3 + 2p)/(2 + 2p) with p > 0, which results in 1 < ν2 < 3/2 and where the background evolution has been assumed as inflation.During inflation, a bluetilted power spectrum of primordial GWs with 0 < n T < 1 could be generated by either the NEC violation or the diminishment of the propagating speed of primordial GWs, while n T > 1 can only be attributed to the violation of NEC.In fact, in the limit p → ∞, the relation (5.2) will be invalid due to the divergence of dĉ T /dτ /ĉ T , namely, the correspondence of the two scenarios will be broken.This is also due to the fact that we have required an inflationary background in the ĉT ̸ = 1 case.
The NEC-violating case with ϵ 2 < −1, which gives 1 < n T < 2, cannot be related to a varying c T with p > −1 on an inflationary background by a disformal transformation.To see this, we can assume that â2 (τ ) ∼ (τ R,j − τ ) where ŵ2 = p2 /ρ 2 is the equation of state parameter of phase 2. It can be checked that p > −1 is still required so that the tensor perturbation modes are able to exit their effective sound horizon during phase 2 (τ 2 < τ ≤ τ3 ).Under the conditions ϵ 2 < −1 and p > −1, the disformal transformation (or the relation given by Eq. (5.1)) actually gives ŵ2 > −1/3 or ŵ2 < −5/3, which indicates that the background evolution in the ĉT ̸ = 1 frame should be either a contraction2 or an NEC-violating expansion, namely non-inflationary, see Fig. 5 for an illustration.In other words, even if we give up the requirement of having an inflationary background in the ĉT ̸ = 1 frame, we would conclude that an observation of n T > 1 would support the existence of a primordial NEC violation.
Since we have focused on the ĉT -diminishing scenario on the inflationary background in Sec. 4, the resulted range of nT given by (5.5) cannot cover that of the NEC-violating scenario (i.e., (5.4)).Therefore, in the sense of the prediction of blue-tilted primordial GWs, these two scenarios may be identified by observations of the spectra index when n T > 1, provided the background is assumed as inflation in the ĉT ̸ = 1 frame.Note that physics should not depend on the chosen frame.Our result does not mean the breakdown of the equivalence between the two scenarios which give 1 ≤ n T < 2 in Fig. 5, but highlighted the significance of primordial NEC violation in generating GWs with n T > 1.
In the context of 0 < n T < 1, we can numerically demonstrate the correspondence of the NEC-violating scenario and ĉT -diminishing scenario.For simplicity, we consider two examples of the step-like ĉT and obtain the corresponding H with Eqs.(5.1) and (5.2), which are displayed in Fig. 6.Obviously, when dĉ T /dτ /ĉ T is significant, the corresponding H can no longer be mimicked by the parameterization (3.3), see Fig. 6(a).The resulted power spectra in two scenarios exactly coincide with each other for smooth ĉT regardless of the significance of dĉ T /dτ /ĉ T , as displayed in Fig. 7, see also [67,68].As a contrast, we also plotted the template (3.12) in Fig. 7(b).Note that the numerical method used here follows that in [67].
In order to be consistent with the current observations of LIGO and Virgo, the spectrum may decrease with respect to the frequency at much higher frequencies.We can assume that ĉT will regain the value of the speed of light or H will go down to a smaller reasonable value before the end of inflation.Figure 6.The diminishing step-like ĉT and the corresponding H, where the parameters N and N satisfy ∆N = ∆ ln a and ∆ N = ∆ ln â, respectively.See Ref. [67] for more details about the numerical method.

Conclusion
The history of the inflationary era could be much more complicated than expected [99][100][101].
The primordial GWs generated during inflation span a frequency band from 10 −18 Hz to 10 10 Hz, hence could encode rich information about the history of the inflationary Universe.
Inflationary GWs with enhanced power spectra may be promising to explain the commonspectrum process reported recently by NANOGrav and PPTA.for more details about the numerical method.
Within the framework of EFT, we investigated an intermittent NEC violation as well as a diminishment of propagating speed of primordial GWs for generating an enhanced power spectrum of GW background during inflation.Both of the NEC violation and the time-dependent ĉT can be realized stably in the "beyond Horndeski" theory.We explicitly calculated the power spectra of primordial GWs for both the NEC-violating scenario and the c T -diminishing scenario.Interestingly, the range of the spectra index in the NEC-violating scenario is 0 < n T < 2, while it is 0 < n T < 1 in the c T -diminishing scenario.Therefore, these two scenarios can be identified by the spectra index when n T > 1, while they could be dual to each other by a disformal transformation of the metric in the sense of generating primordial GWs for the spectra index n T < 1.
Compared with previous works [23,97], our work discussed explicitly the relation and discrepancy between two scenarios for different ranges of the spectra index n T of the primordial GW spectrum.We pointed out that the GW modes are not able to exit their effective sound horizon during the phase in which the propagating speed of GWs increases with time too fast (i.e., p < −1 in Eq. (4.6)).As a result, a blue-tilted spectrum of GWs with n T > 1 cannot be attributed to a diminishing or increasing c T during inflation, unless the background evolution is replaced by non-inflationary ones, i.e., a contraction or NEC-violating expansion.Therefore, our work highlighted the significance of primordial NEC violation in generating a blue spectrum of GWs with 1 < n T < 2.
Our work reinforces the possibility that the primordial GWs could be non-negligible and essential for surveys of a stochastic GW background.Future observational surveys of GWs at the frequency band of the CMB, PTA and interferometers could be combined to put constraints on the NEC-violating scenario as well as the c T -varying scenario.The result of this paper can be generalized to the case where there are multiple violations of the NEC or nontrivial variations of ĉT .Additionally, taking account of the enhancement of chirality of primordial GWs during inflation would also be interesting from both theoretical and observational standpoints [102][103][104][105][106][107][108][109][110].

Figure 4 .
Figure 4. Horizon-crossing of primordial GWs with time-dependent propagating speed c T parameterized by Eqs.(4.5) to (4.7).The green lines represent the evolution of the effective comoving sound horizon of GWs, which is defined as

Figure 5 .
Figure 5. Relation and discrepancy between the NEC-violating scenario and the c T -diminishing scenario.In the sense of the prediction of blue-tilted spectrum of primordial GWs with 0 < n T < 1, these two scenarios cannot be distinguished by observations of n T when there exists a disformal dual.However, the NEC-violating scenario with c T = 1 which predicts 1 ≤ n T < 2 cannot be related to the c T ̸ = 1 case by a disformal transformation, unless the background evolution is contracting or NEC-violating (namely, non-inflationary).

Figure 7 .
Figure 7. Comparisons between the power spectra of primordial GWs in the NEC-violating scenario (red dashed curves) and the ĉT -diminishing scenario (black solid curves), where P T,CMB is assumed to be the power spectrum of primordial GWs at the CMB frequency band.The green dotted curve is plotted by multiplying a factor 10 with the right hand side of Eq. (3.12), where ν 2 = 0.88, H 1 = 0.18 Mpc −1 and H 2 = 1.15 Mpc −1 .Left: The power spectra induced by ĉT and H displayed in Fig.6(a).Right: The power spectra induced by ĉT and H displayed in Fig.6(b).See Ref.[67] 3−2ν 2 = (H 1 /H 2 ) 2 , where H 1 and H 2 are evaluated at τ 1