A plausible model of inflation driven by strong gravitational wave turbulence

It is widely accepted that the primordial universe experienced a brief period of accelerated expansion called inflation. This scenario provides a plausible solution to the horizon and flatness problems. However, the particle physics mechanism responsible for inflation remains speculative with, in particular , the assumption of a scalar field called inflaton. Furthermore, the comparison with the most recent data raises new questions that encourage the consideration of alternative hypotheses. Here, we propose a completely different scenario based on a mechanism whose origins lie in the nonlin-earities of the Einstein field equations. We use the analytical results of weak gravitational wave turbulence to develop a phenomenological theory of strong gravitational wave turbulence where the inverse cascade of wave action plays a key role. In this scenario, the space-time metric excitation triggers an explosive inverse cascade followed by the formation of a condensate in Fourier space whose growth is interpreted as an expansion of the universe. Contrary to the idea that gravitation can only produce a decelerating expansion, our study reveals that gravitational wave turbulence could be a source of inflation. The fossil spectrum that emerges from this scenario is shown to be in agreement with the cosmic microwave background radiation measured by the Planck mission.

It is widely accepted that the primordial universe experienced a brief period of accelerated expansion called inflation. This scenario provides a plausible solution to the horizon and flatness problems. However, the particle physics mechanism responsible for inflation remains speculative with, in particular, the assumption of a scalar field called inflaton. Furthermore, the comparison with the most recent data raises new questions that encourage the consideration of alternative hypotheses. Here, we propose a completely different scenario based on a mechanism whose origins lie in the nonlinearities of the Einstein field equations. We use the analytical results of weak gravitational wave turbulence to develop a phenomenological theory of strong gravitational wave turbulence where the inverse cascade of wave action plays a key role. In this scenario, the space-time metric excitation triggers an explosive inverse cascade followed by the formation of a condensate in Fourier space whose growth is interpreted as an expansion of the universe. Contrary to the idea that gravitation can only produce a decelerating expansion, our study reveals that gravitational wave turbulence could be a source of inflation. The fossil spectrum that emerges from this scenario is shown to be in agreement with the cosmic microwave background radiation measured by the Planck mission.

I. INTRODUCTION
Understanding the origin of the universe -before or around the Planck time τ P ∼ 10 −43 s -is currently out of reach because it would require using a quantum theory of gravity that remains to be built. For a time significantly greater than τ P , the situation is different because the general relativity theory [1] provides a theoretical framework for describing the evolution of the universe as a whole [2]. It is believed that around 10 −36 s the primordial universe experienced an accelerated expansion called inflation which led to an increase of the size of the universe by a factor of at least 10 28 [3,4]. This superluminal expansion is supposed to have eventually stopped around 10 −32 s. The inflation scenario has met a growing success (see, however, [5] for an alternative) because it can explain why the cosmic microwave background (CMB) radiation appears so uniform at large scales and why the universe is flat [6]. While the inflationary paradigm is well accepted, the detailed particle physics mechanism responsible for inflation -like the existence of a scalar field called inflaton -remains unknown [2]. Furthermore, current experiments (in LHC) can only provide limited * sebastien.galtier@lpp.polytechnique.fr † j.laurie@aston.ac.uk ‡ sergey.nazarenko@inphyni.cnrs.fr information with conditions corresponding to the age of ∼ 10 −12 s [7]. Inflation finds its energy from the vacuum and a phase transition associated with the grand-unified-theory symmetry breaking [3,8]. The inflationary scenario has, however, several tuning parameters leading to a wide spectrum of speculative models. The most recent data from the Planck satellite shows, with a high precision, that we live in a remarkably simple universe with e.g. a small spatial curvature and a nearly scale-invariant density fluctuation spectrum with nearly Gaussian statistics [6]. These observations have made it possible to exclude several models, while raising new questions that could weaken the inflationary paradigm and encourage consideration of alternatives [9][10][11][12].
Here, we propose a plausible alternative based on the nonlinearities of the general relativity equations which have been neglected so far when considering the primordial universe. As the fundamental hypothesis of our study we will neglect the role of inflaton in the mechanism of inflation and we will focus our attention only on gravitational wave (GW) turbulence. Since the problem is highly non-trivial, we will examine a simplified theoretical framework from which analytical results were recently derived for the regime of weak GW turbulence [13]. We will use these results to develop a theory of strong GW turbulence which is phenomenological by nature because, unlike for weak turbulence, the problem of strong turbulence is unsolvable perturbatively. In this way we will follow a very classical approach of turbulence based on the idea of critical balance [see e.g. [14][15][16][17][18][19].
The mechanism of cascade in GW turbulence requires an initial excitation of the space-time metric denoted h i . We will assume that it happens at a wavenumber k i (or in a wavenumber window localized around k i ). Since the state of the universe before inflation is inherently unknown, the description of a source of excitation remains speculative and subject to criticism. However, it is likely that the primordial universe was in a tumultuous state -as a heritage of the quantum foam [20] -with for example the creation of primordial black holes (PBH) due to space-time fluctuations [21] (see [22] for the PBH formation by bubble collisions). They are expected to disappear quickly by radiation [23] but the merger of PBH is nevertheless possible and could be actually a potent source of GWs. Our scenario of GW turbulence may start around 10 −36 s or later, which is far enough from the Planck time to consider the general relativity model as applicable [13].
The case h i 1 is favorable to the development of weak GW turbulence for which a theory has been recently derived [13]. The theory describes the nonlinear evolution of weak ripples on the Poincaré-Minkowski flat space-time metric. It is limited to a 2.5 + 1 diagonal metric tensor which includes only one type (+) of GW (the × GW being excluded). Weak GW turbulence is, however, limited because i) it leads to strong turbulence at large scales (see below) and ii) the GW dilution will overpower the nonlinear GW interactions [24]. In this theory, like in the rest of the paper, the cosmological constant is neglected (Λ = 0) and we consider a vacuum space. Therefore, we will consider the vacuum Einstein model which in terms of the Ricci tensor reads R µν = 0.
The paper is structured as follows: section II is devoted to the Friedmann equations in order to recall the assumptions of the standard model and notations; in section III we briefly present the analytical results of weak GW turbulence published recently and then deduce the phenomenological theory of strong GW turbulence; the formation of a condensate and its interpretation in terms of inflation is discussed in section IV; in section V we show that our prediction is in good agreement with the CMB measured by the Planck mission; in the last section we conclude with a summary and a discussion. A consideration of the GW dilution in an expanding universe, where the expansion is caused by the GW themselves, is presented in the Appendix.

II. FRIEDMANN EQUATIONS
In this section, we recall the Friedmann equations in the simplified case where the cosmological constant is neglected (Λ = 0) and the metric is flat; they read with H ≡ȧ/a the Hubble parameter,˙the time derivative, a(t) the cosmic scale factor, c the speed of light, ρ(t) the density and P (t) the pressure. These equations are derived from the Friedmann-Lemaître-Robertson-Walker (FLRW) diagonal metric with interval where x are the co-moving coordinates. The basic assumption behind the FLRW metric is that the universe is homogeneous and isotropic (assumption only valid at large scale) and thus the density and pressure are uniform functions (in space) which depend only on t. The Friedmann equations describe, therefore, the large-scale evolution of the universe.
The assumption of isotropy leads, in fact, to the suppression of nonlinearities. It is a striking fact that in a vacuum space (ρ = P = 0) the solution of the Friedmann equations (1)-(2) is trivially a static universe, whereas the regime of weak GW turbulence (see below) provides a rich dynamics with a dual cascade (isotropy is also assumed but in the statistical sense). Note that inflation models can use the Friedmann equations where the density and pressure are substituted with new quantities [25]. It is known [26], and explicitly shown in the Appendix, that even in vacuum GW fields (metric disturbances) produce effective density and pressure in the Friedmann equations which have form typical for radiation. However, the difference with a usual (e.g. electromagnetic) radiation is that the GW field is nonlinear, which can affect in a substantial way the law of energy dilution due to the universe expansion.

III. METRIC CASCADES IN GW TURBULENCE
Weak GW turbulence is characterized by a direct cascade of energy E and an inverse cascade of wave action N [13]. The respective turbulent spectra are defined from the space-time fluctuations h µν which are introduced as perturbations of the Minkowski metric η µν , with g µν the metric and |h µν | 1. We shall give a phenomenological argument to explain such a double cascade [27]. Let us assume that at wavenumber k i ∼ 1/λ i we have an injection of wave action flux ζ i and energy flux ε i . For the demonstration, we will assume the presence of sinks at small and large wavenumbers, k 0 and k ∞ respectively. We define ζ 0 , ε 0 , ζ ∞ and ε ∞ as the flux values at the corresponding sinks. By virtue of conservation of wave action and energy fluxes, the equations should be satisfied in the steady state. The energy and the wave action spectral densities are linked through the relationÊ k = ωN k , which leads to ε = ωζ (with ω = ck). Then, for a sufficiently large inertial range (k 0 k i k ∞ ), we obtain in the limit which means that the energy and the wave action fluxes are opposite to each other in k-space. As explained by [13] and shown numerically in [28], the inverse cascade is explosive with in principle the possibility for the wave action spectrum, excited at k i , to reach the wavenumber k = 0 in a finite time. However, the description fails at scale k s (or λ s ∼ 1/k s ) where turbulence becomes strong. We may evaluate λ s by using the weak GW turbulence theory for which we know the exact power law solutions [29]. We have the following one-dimensional isotropic constant-flux stationary solution corresponding to the inverse cascade (see Fig. 1 where the metric spectrum is schematically reported), Let us denote by h the typical value of the metric disturbance at length scale and, by E and N the energy and the wave action contained in the scales greater than respectively. By using the scaling relation at scale [30], . GW turbulence becomes fully strong when h = h s ∼ 1; with an initial excitation h i ∼ 10 −1 it leads to = λ s ∼ 10 3 λ i . However and as explained above, it is better to see the range of scales between k s and k i as a region where GW turbulence is already partly strong. Therefore, the weak wave turbulence regime in Fig. 1 is illustrated mainly for pedagogical reasons to emphasize the overall wave-kinetic scenario.
Weak GW turbulence can be seen as a local description for which the assumption of a perturbed Minkowski space-time metric (3) applies well. For larger scales, however, the metric could be different with, for example, the Strong wave turbulence Weak wave turbulence One-dimensional metric spectrum produced by an injection of wave action and energy fluxes at wavenumber ki. The GW turbulence regime is localized between the Planck wavenumber kP and ks, which determines the wavenumber below which turbulence is strong. In this scenario, the inverse cascade leads to the formation of a condensate at k = 0. The growth of the condensate corresponds to an increase of the cosmic scale factor.
presence of a non-zero curvature. The phenomenological scenario described hereafter applies to this situation as well. The presence of an inverse cascade in the regime of weak GW turbulence is an indication that at wavenumber k < k s the inverse cascade continues in the regime of strong turbulence. Following the classical theory of strong wave turbulence [for different applications see e.g. 14-19], we assume that the dynamics is given by a critical balance between the wave period τ GW ∼ 1/ω and the nonlinear time τ N L ∼ /(h c). The scale-by-scale balance relation τ GW ∼ τ N L leads to statistical fluctuations h ∼ 1. This result means that strong turbulence may develop on the background of the Minkowski space-time keeping the fluctuations finite. Note that the presence of such fluctuations may be accompanied by the creation of structures like PBH, which does not modify our statistical prediction. The space-time spectrum of critical balance corresponds to the scaling |ĥ k | 2 ∼ k −1 (see Fig.  1), and thus a wave action spectrumN k ∼ k 0 . Then, the phenomenology of strong GW turbulence tells us that the spectrum can reach the mode k = 0 in finite time because it is a finite-capacity turbulence system (i.e. ki 0N k dk converges; see the numerical simulations in Galtier et al. [28]). It is important to note that the excitation of the metric from k i > 0 to k = 0 in a finite time does not violate the causality principle because it does not correspond to the propagation of information in physical space from a given position to infinity. Instead, the inverse cascade means a continuous increase of the wavelength of a fluctuation as a consequence of its interaction with other fluctuations of predominately similar wavelengths. Clearly, this can be done locally in the physical space. Then, the mode k = 0 corresponds to the level of the background over which there are fluctuations of different wavelengths (see e.g. [31][32][33] where the zero mode plays an important role). Note also that this description does not require to fix the size (finite or infinite) of the system.

IV. CONDENSATE AND INFLATION
When the spectral front reaches the mode k = 0 a condensate emerges in Fourier space (see Fig. 1). This situation is similar to the formation of a non-equilibrium Bose-Einstein condensation generated by an inverse cascade (see e.g. [31,[33][34][35][36][37]). In our case, the growth in time of the condensate is at the expense of the fluctuations which can be maintained as long as the wave action flux ζ is finite (possibly constant).
In previous works on the dynamics of the Bose-Einstein condensation, it was shown that the condensate growth accelerates [38] and an explosive evolution (in (t c − t) x with x < 0 and t < t c ) was not excluded; in the weak turbulence regime a power-law in time was predicted [39]. Analogous accelerated condensate growth scenario in GW turbulence would mean inflation. An inflation would not be in contradiction with the causality principle because it would be the result of an amplification of the background metric. This mechanism is limited in time because an expansion of the universe leads also to a dilution (see in the Appendix and also in [13]). We may expect, however, that it is only when the source of wave action flux dries up that the GW fluctuations start to decay. This leads eventually to a saturation of the condensate (because of the weakening of the nonlinear transfers) and the end of inflation.
We may estimate the time-scale necessary for the formation of a condensate by using the turbulence phenomenology, which gives several predictions. First, we can say that the development of weak GW turbulence happens in the typical time-scale of the four-wave kinetic equation [13] where the GW time τ GW ∼ 1/(k i c) and the small parameter ∼ τ GW /τ N L ∼ h i ∼ 10 −1 . We find τ W T ∼ 10 4 λ i /c. But, as we have already discussed, the GW turbulence quickly ceases to be weak, and the characteristic time becomes the one of the dynamical (rather than the weak turbulence) equation, namely which is shown in the Appendix to be the same time as for the dilution. Note that this expression is also valid for weak GWs with h 1 if their phases are not random. Non-random phases could appear e.g. due to presence of coherent structures such as solitons, PBH or wormholes. With the parameters given above, we find τ a dyn ∼ 100λ i /c. In the critical balance state the nonlinearity parameter is of order one, so It is important to realize that expressions (4)- (6) arise from the Einstein model, which is free of tuning parameters, and its form is given by nonlinear physics.

V. FOSSIL SPECTRUM AND CMB
According to our scenario (inflation followed by dilution), the fossil gravitational spectrum should be the one given by strong GW turbulence but with a much smaller amplitude. We may try to compare our prediction with the CMB radiation measured by the Planck mission [6].
After inflation we are left with a Minkowski metric plus very small fluctuations, therefore we may simply use the Newtonian law with φ the gravitational potential, to derive the one dimensional scalar spectrum P φ (k) ∼ φ 2 (k) ∼ k ns−2 [40]. Expression (7) leads to the scaling relation φ ∼ ρ 2 . Connexion with our prediction can be made through the relation ρ φ ∼ E ∼ −2 , which is compatible with the critical balance phenomenology. Then, we find which is the Harrison-Zeldovich spectrum (since n s = 1) [40]. Note that the Planck data are actually compatible with n s 0.967 which corresponds to a metric spectrum |ĥ k | 2 ∼ k ns−2 ∼ k −1.033 slightly different from our prediction. However, it is known that finite-capacity turbulence systems, which are characterized by an explosive cascade, exhibit anomalous scalings with power laws slightly different from the predictions [33,34,41] (for other examples see e.g. [29,32,42,43] and for weak GW turbulence see [28]). Therefore, the slight discrepancy between our prediction and the Planck data could be the signature of an anomalous scaling. Finally, we can show that the tensor-to-scalar ratio r (ratio between the metric and scalar spectra) is independent of k. Its amplitude is, however, difficult to estimate without numerical simulation because it implies the knowledge of the Kolmogorov constants.

VI. CONCLUSION
In summary, we propose an alternative scenario for inflation which is driven by GW turbulence. We show that a small-scale excitation of the metric leads to an explosive inverse cascade that could lead to the formation of a condensate whose growth is interpreted as an expansion of the universe. The condensate growth is expected to accelerate -possibly explosively -leading to a phase of inflation. It is shown that the scalar spectrum obtained with this scenario is compatible with the CMB measured by the Planck mission (without introducing tuning parameters [44]). This new scenario does not preclude the appearance of a reheating phase of the universe and particle creation after inflation [10]. The nonlinear mechanism described here does not require the introduction of the cosmological constant and has no tuning parameters. It also sustains the idea of inflation which was recently questioned [11,12].
The turbulent inflation introduced here is mainly a phenomenological theory, inspired by the analytical results obtained in weak GW turbulence. At present, an essential part of it remains in conjecture, specifically the view that the inverse cascade will continue through the strongly turbulent stage. Indeed, strictly speaking the dual cascade behavior relies on the conservation of the wave action, which is a property of the four-wave kinetic equation and therefore breaks down when this equation is no longer applicable. The situation here is similar to the behavior described by the Gross-Pitaevskii model: when the inverse cascade becomes strong, the energy invariant ceases to be quadratic, and the dual cascade argument becomes, technically, invalid. However, it is known from numerical simulations of the Gross-Pitaevskii model [45,46] that the condensation process started at the weakly turbulent regime as an inverse cascade, continues at the strongly turbulent stage with the appearance of strongly nonlinear defects which move like hydrodynamic vortices. These tend to continuously annihilate, so that no defects remain after a finite time, with the correlation length becoming infinite. By this analogy, we conjecture that in the vacuum Einstein model, the condensation process will also continue through the strongly turbulent stage, possibly with some singular coherent objects, such as wormholes, PBH or solitons, appearing in the system at a transient stage (similar to the appearance of the vortices in the Gross-Pitaevskii model). Obviously some work remains to be performed for such a conjecture to be confirmed by direct numerical simulations (this issue is left for future work) and if possible by analytical calculations.
Finally, it is interesting to note that the effects of small scale fluctuations on the large-scale dynamics has been studied by [47]: it was shown analytically that the back reaction is much stronger for GWs than for matter density fluctuations. While, in another study [48], it was suggested that solitonic GWs of cosmological origin can contribute to the expansion of the universe. Clearly, these few examples underline the need to better understand the role of nonlinearities in cosmology that have been underestimated so far.

ACKNOWLEDGMENTS
We thank C. Clough for useful discussions. Sergey Nazarenko is supported by the Chaire D'Excellence IDEX (Initiative of Excellence) awarded by Université de la Cte d'Azur, France and by the Simons Foundation Collaboration grant 'Wave Turbulence' (Award ID 651471).

Appendix A: Universe expansion driven by nonlinear gravitational wave interactions
In this appendix we systematically derive a set of mean and fluctuation equations (A10)-(A12) for nonlinear gravitational wave evolution in a flat empty universe based on the Einstein vacuum equations. We show how, for non-weak gravitational wave amplitudes or nonrandom phases, the leading nonlinear wave contribution is of the same order as the term corresponding to universe dilation, implying that consideration of gravitational wave-wave interactions is important for describing the expansion of the universe. This motivates the use of the critical balance argument for strong gravitational wave turbulence of the main text.
Consider the Einstein vacuum equations restricted to small perturbations h µκ of metric g µκ about the Friedmann-Lemaître-Robertson-Walker (FLRW) metricḡ µκ : where a(t) is the scale factor for the spatial expansion of the universe and c is the speed of light.
It is worth remarking that in [49] and subsequently in [50], the authors considered weak gravitational waves in FLRW space without assuming slowness of a(t). This makes sense if matter or (non-gravitational) radiation produces a fast expansion withȧ/a = O(1). However, in the vacuum case considered in this article, the temporal evolution of the scale factor is slow. We will show a posteriori thatȧ is of the same order as the weak gravitational wave perturbation h µκ , i.e.ȧ ∼ √ä ∼ h ∼ 1 and thus nonlinear gravitational wave interactions cannot be neglected when describing the expansion of the universe. In our analysis, we will utilize these scalings and perform a formal expansion on Eq. (A1) in . The Ricci tensor R µκ can be expressed in terms of Christoffel symbols Γ λ µκ via where the Christoffel symbol is defined in terms of the metric g µκ and co-metric g µκ as By substituting (A2) into the expressions (A4) and (A3) we produce a formal -expansion of the vacuum equations, with each order in denoted as follows, µκ + · · · = 0. As the Einstein vacuum equations lead to an overdetermined system of equations for the perturbation h µκ , we can fix the coordinate system without loss of generality. We apply the four harmonic gauge conditions defined up to the linear perturbation of h µκ . The harmonic gauge conditions (A5) still leaves coordinate freedom with respect to any additional harmonic perturbation. Therefore, we can additionally choose the spacetime coordinates in such a way that h 00 = 0, and h l0 = 0, for l = 1, 2, 3.
Using the harmonic gauge conditions, we can simplify Einstein vacuum equations at O( ) to wave equations of the form: for l, m = 1, 2, 3. It is important to note that Eq. (A6) depends on a time dependent scale factor a(t) whose evolution can only be described by considering the subsequent order in . The term R (2) µκ involves a mix of contributions arising from terms quadratic in h (denote them hh R (2) µκ ) and linear in h contributions ∼ȧh (denote them ah R (2) µκ ), and finally contributions of type (ȧ/a) 2 andä/a (denote them a R (2) ). Then Using the harmonic gauge, the order O( 2 ) contribution to the Einstein vacuum equation simplifies to where l = 1, 2, 3 with no summation, with the rest of the components of a R (2) µκ being zero. We also determine that and ah R (2) 0m = ah R (2) l0 = 0. The form of (A7) is precisely what results from the second term of Eq. (8) in Ref. [50]. The complete expression for hh R (2) µκ is lengthy and will not be reproduced here. It can be determined through the use of computational algebra software such as Mathematica. Note that by considering the Einstein vacuum equations at order O( 2 ), it gives rise to two equations: one equation for the mean and one equation for the fluctuations. Let us introduce the spatial average · := lim V →∞ [(1/V ) V · dx], then up to order O( 2 ) we have where we have taken into account that R (0) µκ is linear in terms of the perturbation h µκ and vanishes upon the spatial average. (Actually, this implies that the averaged metric remains of FLRW form which will be seen from the structure of the final averaged equations.) Now note that ah R (2) µκ has zero mean (being linear in h µκ ) and that a R µκ . We evaluate terms hh R (2) µκ assuming a distribution of gravitational waves with amplitudes h µκ = (2π) −3 ĥ µκ (k, t) exp(ik · x) dk, whereĥ µκ (k, t) are time-dependent wave amplitudes for a wave with wave vector k = (k 1 , k 2 , k 3 ).
For the calculation below, it suffices to take the leading order time dependence ofĥ k (t) which follows from the wave equation (A6), namelyĥ µκ (k, t) ∼ exp(−i t 0 ω k dt ) with ω k = c|k|/a(t ). In terms of waves with + and × polarizations, we have that We consider the monochromatic wave amplitudesλ k andμ k of the form which we substitute into the expression for hh R (2) µκ and apply space averaging, with the aid of the Mathematica software, to get a rather simple result: , l, m = 1, 2, 3.
One can further consider the case of an homogeneous and isotropic distribution of gravitational waves by angleaveraging the above formulas. This gives where spectra n λ k and n µ k are defined via λ kλk =(2π) 3 n λ k δ(k − k ), μ kμk =(2π) 3 n µ k δ(k − k ), where averaging now is over many periods of wave oscillations or, equivalently, wave phases. By defining the gravitational wave energy density ρ [51] by that acts on the 00 component, the pressure P = c 2 ρ/3 as the contribution acting on the jj components, and G as the Einstein constant. Eqs. (A8) for the 00 and the jj components (j = 1, 2 or 3) become respectively 3ä a + 8πGρ =0, By rearranging and using the fact that P = c 2 ρ/3 one get the usual expressions of the Friedmann equations that can be rearranged further into the Friedmann equations of the form (1) and (2). These equations show that our initial scalingȧ ∼ h ∼ √ä ∼ is justified. Also, Eqs. (A8) for the off-diagonal components become 0 = 0 identities which means that the averaged metric remains of FLRW type.
Keeping in mind the relation P = c 2 ρ/3, which is typical for radiation or ultra-relativistic matter, we see that gravitational waves make the universe expand in a way that usual radiation would. However, it remains to be seen how the expansion itself affects the gravitational waves. The crucial difference with usual (e.g. electromagnetic) radiation is that the waves are nonlinear and, therefore, may interact with each other and thus modify the usual energy dilution process. To describe the evolution of the gravitational waves we need an equation for the fluctuating field up to order O( 2 ), i.e.
Recall that R for l, m = 1, 2, 3. If we neglect the right-hand side of (A12), we would arrive at the familiar equation for the gravitational wave dilution found in [49] and [50], and by neglecting the sub-leading order in , its solution is which leads to the typical dilution law of radiation ρ ∼ 1/a 4 .
The right-hand side of Eq. (A12) describes nonlinear wave-wave interactions. Neglecting this term would be justified if the universe would be filled with a substance causing it to expand fast, so thatȧ = O(1). The key point is that in vacuum space, considered here,ȧ and h are of the same order of magnitude and, therefore, in general the right-hand side of Eq. (A12) is of the same order as the dilution term ∼ȧḣ lm , i.e. the timescales of the dilution and the wave-wave interaction are comparable. Note that randomness of phases may weaken the wavewave interactions, as is the case for weak gravitational turbulence [13]. However, when the wave amplitudes are not weak and/or the phases are not random, the wavewave interaction term becomes important and one has to seek a reasonable model closure for its description. This is precisely the critical balance approach suggested in the main text.