Gravity Waves Signatures from Anisotropic pre-Inflation
We show that expanding or contracting Kasner universes are unstable due to the amplification of gravitational waves (GW). As an application of this general relativity effect, we consider a pre-inflationary anisotropic geometry characterized by a Kasner-like expansion, which is driven dynamically towards inflation by a scalar field. We investigate the evolution of linear metric fluctuations around this background, and calculate the amplification of the long-wavelength GW of a certain polarization during the anisotropic expansion (this effect is absent for another GW polarization, and for scalar fluctuations). These GW are superimposed to the usual tensor modes of quantum origin from inflation, and are potentially observable if the total number of inflationary e-folds exceeds the minimum required to homogenize the observable universe only by a small margin. Their contribution to the temperature anisotropy angular power spectrum decreases with the multipole as , where depends on the slope of the initial GW power-spectrum. Constraints on the long-wavelength GW can be translated into limits on the total duration of inflation and the initial GW amplitude. The instability of classical GW (and zero-vacuum fluctuations of gravitons) during Kasner-like expansion (or contraction) may have other interesting applications. In particular, if GW become non-linear, they can significantly alter the geometry before the onset of inflation.
I Introduction: Anisotropic pre-Inflation
The inflationary stage of the very early universe explains the dynamical origin of the observed isotropic and homogeneous FRW geometry. The patch of the FRW geometry covers the cosmological horizon and beyond if inflation lasted
e-folds or longer. Here is the potential energy of the inflation, and is a correction from the (p)reheating stage after inflation, which is not essential for our discussion. Chaotic inflationary models, associated with a large energy ( GUT scale) of GeV, predict a very large number of inflationary e-folds, . Long-lasting inflation erases all classical anisotropies and inhomogeneities of the pre-inflationary stage. However, scalar and tensor vacuum fluctuations during inflation lead to almost scale free post-inflationary scalar and tensor metric inhomogeneities around our smooth observable FRW patch.
In particular, the amplitude of the gravitational waves generated from the vacuum fluctuations during inflation is proportional to
, (where is the reduced Planck mass). There are significant efforts to measure the -mode of polarizations, since this will provide a direct probe of the scale of inflation.
The current C.L. limits on (ratio of the tensor to scalar amplitudes of cosmological fluctuations) (WMAP-only) and (WMAP plus acoustic baryon oscillation, plus supernovae) WMAP () shall be improved to by the Planck mission planck (), to
by the over clover (), EBEX ebex (), and Spider spider () experiments (see epic () for the study of a mission that can improve over these limits). While these limits imply a detection in the case of high energy inflation, a number of other
inflationary models, including many of the string theory constructions have
lower energy, and therefore lead to GW of much smaller amplitude, which are virtually
unobservable through mode polarization
In anticipation of the null signal observation of the primordial GW from inflation, it is worth thinking about other implementations of this result for the theory of inflation, besides putting limits on the energy scale . There are models of inflation (including many string theory inflationary models) where the total number of e-folds, , does not exceed the minimum (1) by a large number. If the extra number of e-folds beyond (1) is relatively small then pre-inflationary inhomogeneities of the geometry are not erased completely, and their residuals can be subject to observational constraints. In the context of this idea, in this paper we suggest an additional mechanism to have observable gravitational waves associated with inflation. These gravitational waves are very different from the GW generated from the vacuum fluctuations during inflation. Firstly, they are the residual tensor inhomogeneities from the pre-inflationary stage. Secondly, they can be of a classical, rather than quantum, origin. Thirdly, while their initial amplitude and spectrum are given by the initial conditions, they are significantly affected by the number of “extra” e-folds . Therefore, observational limits on gravity waves result in constraints on a combination of and of the initial amplitude.
The choice of the initial geometry of the universe before inflation is wide open. In principle, one may assume an arbitrary geometry with significant tensor inhomogeneities component, and much smaller scalar inhomogeneities. This choice is, however, very artificial. A much more comfortable choice of the pre-inflationary stage will be a generic anisotropic Kasner-like geometry with small inhomogeneities around it. The origin of the anisotropic universe with the scalar field can be treated with quantum cosmology, or can be embedded in the modern context of the tunneling in the string theory landscape. In fact, a Kasner-like (Bianchi I) space was a rather typical choice in previous papers on pre-inflationary geometry, see e.g. pre (). Most of the works on an anisotropic pre-inflationary stage aimed to investigate how the initial anisotropy is diluted by the dynamics of the scalar field towards inflation starob83 ().
The formalism of linear fluctuations about an anisotropic geometry driven by a scalar field toward inflation was constructed only recently Gumrukcuoglu:2006xj (); Pereira:2007yy (); Gumrukcuoglu:2007bx (); Pitrou:2008gk (). Besides the technical aspects of calculations of cosmological fluctuations, there is a substantial conceptual difference between computations in the standard inflationary setting and in the anisotropic case. For an isotropic space undergoing inflationary expansion, all the modes have an oscillator-like time-dependence at sufficiently early times, when their frequency coincides with their momentum. One can therefore use quantum initial conditions for these modes. This is no longer the case for an expansion starting from an initial Kasner singularity. In this case, a range of modes, which can potentially be observed today (if is not too large), are not oscillating initially and therefore cannot be quantized on the initial time hyper-surface; as a consequence, there is an issue in providing the initial conditions for such modes. For this reason we will adopt another perspective, namely, we will consider generic small classical inhomogeneities around the homogeneous background, as an approximation to the more generic anisotropic and inhomogeneous cosmological solution.
Equipped with this philosophy, we consider an anisotropic expanding universe filled up by the scalar field with a potential which is typical for the string theory inflation. We add generic linear metric fluctuations about this geometry. The evolution of these fluctuations is by itself an interesting academic subject. However, it acquires a special significance in the context of the GW signals from inflation, because of a new effect that we report here of amplification of long-wavelength GW modes during the Kasner expansion. This growth terminates when a mode enters the “average” Hubble radius (the average of that for all the three spatial directions), or, for larger wavelength modes, when the background geometry changes from anisotropic Kasner to isotropic inflationary expansion. We perform explicit computations in the case of an isotropy of two spatial directions. In this case the computation becomes much more transparent and explicitly dependent. Fluctuations for arbitrary were considered in the formalism of Pereira:2007yy (); Pitrou:2008gk (), where the dependence is not explicit. We verified that our results agree with Pereira:2007yy (); Pitrou:2008gk () in the axisymmetric limit. We find that only one of the two GW polarizations undergoes significant amplification. Therefore, even if we assume for simplicity equi-partition of the amplitudes of the three inhomogeneous physical modes of the system (the scalar and the two GW polarization) at the initial time, the final spectra that will be frozen at large scales in the inflationary regime will be very different from each other, in strong contrast to what is obtained in the standard inflationary computations.
This result can have different consequences, that we explore in the present work. Suppose that the growing GW mode is still linear (but significantly exceeds other modes) when the space becomes isotropic. Then, we can have significant yet linear classical GW fluctuations at the beginning of inflation, say of amplitude . If the modes which correspond to the largest scales that we can presently observe left the horizon after the first e-folds of inflation, their amplitude decreased by the factor in this period. If is relatively small, say the freeze out amplitude of these GW modes would be . The angular spectrum of these GW will rapidly decrease as the multipole number grows, since smaller angular scales are affected by modes which spend more time inside the horizon during the inflationary stage.
Suppose instead that the growing GW mode becomes non-linear before the onset of inflation. In this case the background geometry departs from the original onset.
Besides the phenomenological signatures, it is interesting to study the origin of the amplification of the GW mode. It turns out that the effect of GW amplification is related to the anisotropic Kasner stage of expansion. Therefore we will separately study GW in the pure expanding Kasner cosmology. For completeness, we also include the study of GW in a contracting Kasner universe, which is especially interesting due to the universality of anisotropic Kasner approach to singularity.
The plan of the paper is the following. In Section II we discuss the evolution of the anisotropic universe driven by the scalar field towards inflation. In Section III we briefly review the formalism of the linear fluctuations in the case of a scalar field in an anisotropic geometry, paying particular attention to the GW modes. In Section IV we compute the amplification of one of the two GW modes that takes place at large scales in the anisotropic era. In Section V we discuss instead the evolution of the other two physical modes of the system. In Section VI we study the evolution of the perturbations in a pure Kasner expanding or contracting Universe. In Section VII we return to the cosmological set-up, and we compute the contribution of the GW polarization amplified during the anisotropic stage to the CMB temperature anisotropies. In particular, by requiring that the power in the quadrupole does not exceed the observed one, we set some limits on the initial amplitude of the perturbations vs. the duration of the inflationary stage. In Section VIII we summarize the results and list some open questions following from the present study, which we plan to address in a future work.
Ii Background Geometry
The anisotropic Bianchi-I geometry is described by
where are the scale factors for each of the three spatial directions.
We consider a scalar field in this geometry. Many string theory inflationary models (for examples see K2LM2T (); Conlon:2005jm ()) have a very flat inflationary potential which changes abruptly around its minima. Therefore, to mimic this situation, we will use a simple inflaton potential
which has quadratic form around the minimum, and is almost flat away from it. To obtain the correct amplitude of scalar metric perturbations from inflation, we set
The background dynamics is governed by the Einstein equations for the scale factors in the presence of the effective cosmological constant , plus a possible contribution from the kinetic energy of the scalar field. Quantum cosmology or tunneling models of the initial expansion favor a small scalar field velocity. Therefore we select the small velocity initial conditions . In this case the generic solutions of the Einstein equations with cosmological constant for are known analytically (see e.g. Pitrou:2008gk ()) and can be cast as
where are the Kasner indices, , , and is the characteristic time-scale of isotropization by the cosmological constant, , while are the normalizations of the three scale factors
For earlier times the anisotropic regime is described by the vacuum Kasner solution
which corresponds to an overall expansion of the universe (the average scale factor is increasing), although only two directions are expanding (two positive -s) while the third one is contracting (the remaining is negative).
For later time the universe is isotropic and expanding exponentially
where the constant normalizations are typically chosen to be equal.
It is instructive to follow the evolution of the curvature in the model. The Ricci tensor is (almost) constant throughout the evolution up to the end of inflation
At earlier times the Weyl tensor – i.e. the anisotropic component of the curvature tensor – gives
and, for , is much bigger than the isotropic components (8). This is why initially the contribution from the effective cosmological constant is negligible, and the vacuum Kasner solution (6) is a good approximation. In contrast, at later times
and the anisotropic part of the curvature becomes exponentially subdominant relative to its isotropic part driven by the cosmological constant. This is an illustration of the isotropization of the cosmological expansion produced by the scalar field potential. The timescale for the isotropization is .
In the following Sections we will study the equations for the linear fluctuations around the background (2), (5). These equations become significantly simpler and more transparent for the particular choice of an axi-symmetric geometry e.g. when , and the metric is
While the effect we will discuss is generic, for simplicity we will adopt the simpler geometry (11) rather than the general Bianchi-I space (2). In this case, the early time solution is a Kasner background with indices
At earlier times , while, at late times, . The equation for the homogeneous scalar field is
Since the value of is very large initially, , the Hubble friction keeps the field (practically) frozen at during the anisotropic stage. For , the universe becomes isotropic, and it enters a stage of slow roll inflation until rolls to the minimum of its potential.
We also will use another form of the metric (11), with the conformal time . There is ambiguity in the choice of , related to possible different choices of the scale factors in its relation with the physical time. We will use the average scale factor
and define through
which, at early times, gives . In this variable, the line element (11) reads
In the following, dot denotes derivative wrt. physical time , and prime denotes derivative wrt conformal time. Moreover, we always denote by and the Hubble parameters with respect to physical time.
Iii Linear fluctuations
In the FRW universe with a scalar field there are three physical modes of linear fluctuations. Two of them are related to the two polarizations and of the gravitational waves, and one to the scalar curvature fluctuations induced by the fluctuations of the scalar field . All three modes in the isotropic case are decoupled from each other. The formalism for the linear fluctuations on a FRW background has been extended to the Bianchi-I anisotropic geometry in Gumrukcuoglu:2006xj (); Pereira:2007yy (); Gumrukcuoglu:2007bx (); Pitrou:2008gk (). Again, there are three physical modes; however, in the general case of arbitrary the modes are mixed, i.e. their effective frequencies in the bi-linear action are not diagonal, as it is the case in the isotropic limit.
In a special case of the axi-symmetric Bianchi I geometry (11) one of the three linear modes of fluctuations, namely, one of the gravity waves modes, is decoupled from the other two. This makes the analysis of fluctuations much more transparent than the general case. While the effects we will discuss, we believe, is common for arbitrary , we will consider linear fluctuations around the geometry (11). The computation follows the formalism of Gumrukcuoglu:2007bx (), where the reader is referred for more details.
The most general metric perturbations around (11) can be written as
where the indices span the coordinates of the isotropic d subspace.
The above modes are divided into d scalars () and d vectors (, subject to the transversality conditions )
The gauge choice
corresponding to , completely fixes the freedom of coordinate reparametrizations. It is convenient to work with the Fourier decomposition of the linearized perturbations. We can therefore fix a comoving momentum , and study the evolutions of the modes having that momentum. Since modes with different momenta are not coupled at the linearized level, this computation is exhaustive as long as we can solve the problem for any arbitrary value of . More precisely, we denote by the component of the momentum along the anisotropic direction, and by the component in the orthogonal plane. We denote by and by the corresponding components of the physical momentum. Finally, we denote by and the magnitudes of the comoving and physical momenta, respectively. Therefore, we have
To identify the physical modes, one has to compute the action of the system up to the second order in these linear perturbations. One finds that the modes , and , corresponding to the metric perturbations, are nondynamical, and can be integrated out of the action. This amounts in expressing the nondynamical fields (through their equations of motion) in terms of the dynamical ones, and in inserting these expressions back into the quadratic action. For instance, for the nondynamical d vector mode one finds
The analogous expressions for the d nondynamical scalar modes can be found in Gumrukcuoglu:2007bx ().
In this way, one obtains an action in terms of the three remaining dynamical modes . Once canonically normalized, these modes correspond to the three physical perturbations of the system. The canonical variables are
where is anti-symmetric and (we stress that encodes only one degree of freedom, since, due to the transversality condition of the d vector modes, ).
The dynamical equations for the modes and are coupled to each other, while that of the mode is decoupled
The explicit expressions for the frequency matrix are rather tedious and given in Gumrukcuoglu:2007bx ().
In the limit of isotropic background, , also the d scalars decouple, and the frequencies become
Therefore, the mode becomes the standard scalar mode variable musa (), associated to the curvature perturbation, while the modes and are associated to the two polarizations of the gravitational waves. Also notice that without the scalar field there are two physical modes which, due to the residual d isotropy, are decoupled.
Iv Decoupled tensor polarization
Since there are no vector sources, the d vector system describes a polarization of a gravitational wave obeying the “free field equations” (which reproduce the equation (20) and the first of (23)). As we now show, this mode undergoes an amplification, which does not occur for the other two modes. This can be understood from considering the frequency of this mode
In the isotropic case, for which is given by (24), each mode is deeply inside the horizon, , at asymptotically early times. This is due to the fact that is nearly constant, while is exponentially large at early times. As a consequence, in the asymptotic past, and the mode oscillates with constant amplitude. In the present case instead
at early times (). Namely, as we go backwards in time towards the initial singularity, the anisotropic direction becomes large, and the corresponding component of the momentum of the mode is redshifted to negligible values. On the contrary, the two isotropic directions become small, and the corresponding component of the momentum is blueshifted. However, the magnitude of the two Hubble parameters increases even faster. Therefore, provided we can go sufficiently close to the singularity, the early behavior of each mode is controlled by the negative term proportional to the Hubble parameters in eq. (25).
To be more precise, if we denote by and the values of the two scale factors at some reference time close to the singularity, we have
where the first term in the expansion comes from the terms proportional to
in eq. (25), while the second term from the component of the momentum in the isotropic plane (cf. the early time dependences with those given in eq. (26)). We see that the frequency
squared is negative close to the singularity, so that the mode experiences a growth
This simple description is affected by two considerations: firstly, we do not set the initial conditions for the modes arbitrarily close to the singularity, but at some fixed initial time ; Secondly, the geometry changes from (nearly) Kasner to (nearly) de Sitter due to the inflaton potential energy. Consequently, there are three types of modes of cosmological size. I: Modes with large momenta start inside the two horizons at . They oscillate () all throughout the anisotropic regime, and they exit the horizon during the inflationary stage. II: Modes with intermediate momenta, for which (27) is a good approximation at . These modes enter the horizons, and start oscillating, at some time during the Kasner era; they exit the horizon later during inflation. III: Modes with small momenta, that are always outside the horizons, and never oscillate during the Kasner and inflationary regimes.
These considerations are crucial for the quantization of these modes. We can perform the quantization only as long as , and the mode is in the oscillatory regime. As we mentioned, during inflation, this is always the case in the past. Moreover, the frequency is adiabatically evolving (), and one can set quantum initial conditions for the mode in the adiabatic vacuum. This procedure is at the base of the theory of cosmological perturbations, and results in a nearly scale invariant spectrum at late times, once the modes have exited the horizon and become classical. In the case at hand, we cannot perform this procedure for modes of small momenta / large wavelength (modes III above). If inflation lasts sufficiently long, such modes are inflated to scales beyond the ones we can presently observe, so that the inability of providing initial quantum conditions is irrelevant for phenomenology. However, if inflation had a minimal duration, this problem potentially concerns the modes at the largest observable scales.
Irrespectively of the value of the frequency, it is natural to expect that the modes possess some “classical” initial value at . In the following we discuss the evolution of the perturbations starting with these initial conditions. Although we do not have a predictive way to set these initial values, we can at least attempt to constrain them from observations. In addition, we should worry whether the growth of can result in a departure from the Kasner regime beyond the perturbative level, in which case the background solution described in the previous Section may become invalid (we discuss this in Section VI).
As long as the frequency is accurately approximated by (27) in the Kasner regime, the evolution eq. for the mode is approximately solved by
where are two integration constants, while and are the Bessel and Neumann functions. The first mode increases at early times (small argument in the Bessel function) , while the second one decreases as . We disregard the decreasing mode in the following computations, (moreover, this mode diverges at the singularity).
Rather than the time evolution of , we show in Figure 1 that of the corresponding power. We note the very different behavior obtained for large (III), intermediate (II), and small (I) scale modes. We choose to define the power spectrum as
This definition coincides with the standard one (see for instance Riotto:2002yw ()) as the universe becomes isotropic. In particular, the power spectrum is frozen at large scales in the isotropic inflationary regime. Clearly, there is an ambiguity in this definition at early times, when the two scale factors differ (there is no a-priori reason for the choice of the average scale factor in this definition). This arbitrariness affects the time behavior shown in the Figure. However, it does not affect the relative behavior of the large vs. intermediate vs. small scale modes. Moreover, if we analogously define the power spectra for the d scalar modes, the relative behavior of these two types of mode (Figure 1 vs. Figure 4 ) is also unaffected by this arbitrary normalization.
The reality of this instability is demonstrated in Section VI, where we compute the squared Weyl invariant due to these fluctuations (more precisely, we do so in an exact Kasner background, which, as we have remarked, coincides with the cosmological background at asymptotically early times).
In Figure 2 we show the power spectrum (normalized to the initial value for each mode) for the same background evolution as in the previous Figure, at some late time during inflation, when all the modes shown are frozen outside the horizon. As the approximate solution (28) indicates, the growth of occurs as long as the transverse momentum is smaller than the Hubble rates . Therefore, modes with smaller experience a larger growth. We denote by the angle between the comoving momentum, and the anisotropic direction,
Therefore, in general, we expect a greater growth at smaller values of (for any fixed ) and at smaller values of (for any fixed ).
The large growth at small is more manifest in Figure 3. The smallest angles shown in the Figure correspond to , but to at the initial time (this is due to the different behavior of the two scale factors in the anisotropic era). In this region, the spectrum exhibits a milder dependence than for intermediate values of . Finally, one may also consider smaller angles than those shown in the Figure, for which initially. We have found that final spectrum becomes independent in this region.
We conclude this section by discussing how the growth scales with the initial time. As long as , the power (29) of a mode grows as (as can be easily seen by combining the time dependences ). We use the initial value of (the difference between the two expansion rates, defined in eq. (12)) as a measure on the initial time, since, contrary to the conformal time, this quantity is not affected by the normalization of the scale factors. Starting with a greater value of corresponds to starting closer to the initial singularity, and, therefore, to a longer phase in which grows. Since in the Kasner regime, the ratio in the region in which the growth takes place. Although we do not show this here, we have verified that this scaling is very well reproduced by the numerical results.
V Coupled tensor polarization-scalar mode pair
As discussed in Section III the two other physical modes of the system are coupled to each other in the anisotropic era. The evolution equations for the coupled system are formally given in (23). At early times, we find
Therefore the coupling between the two modes can be neglected also at asymptotically early times. The main difference with the analogous expression for the decoupled tensor polarization, eq. (27), is that the squared eigenfrequencies of the two modes are now positive close to the singularity; therefore the two modes do not experience the same growth as . Indeed, as long as the expressions (31) are good approximations, we find the solution
and an identical one for with the replacement of the integration constants. Close to the singularity, the two modes grow as and , respectively. Analogously to what we did for the polarization, we disregard the mode which grows less at early times, .
We define the power spectra for the tensor polarization, and for the comoving curvature perturbation with the same prefactor as , cf. eq. (29),
We see that, contrary to what happened for , the coupled perturbations, and the corresponding power spectra, do not grow while outside the horizons during the anisotropic era.
Vi Instability of Kasner solution against Gravitational Waves
The main result of the previous section was a significant amplification of the mode , compared to the milder amplification of the mode , in the anisotropic background which is undergoing isotropization due to the effect of a scalar field. This growth can be ascribed to the instability of the Kasner geometry, either contracting or expanding, against gravitational waves which we are going to report in this Section. Therefore in this Section we consider linearized gravity waves around an expanding and a contracting Kasner solution, without the presence of the scalar field, nor its fluctuation. In this case there are only the two decoupled modes and .
The claim of instability of the Kasner solution against the growth of the GW sounds at first glance heretic, at least for the contracting branch, in the light of the universality of the Belinskii-Khalatnikov-Lifshitz (BKL) oscillatory regime of the Kasner epochs approaching the singularity. In fact, it is not, and, on the contrary, it is compatible with the BKL analysis. Moreover, our finding of the GW instability suggests a new interpretation of the phenomena connected to the instability, discussed by BKL and others for the contracting Kasner geometry in very different formalism and language BKL82 (); uggla (); wain (); Damour:2002et (); Damour:2007nb ().
In this Section we first perform linearized calculations for the classical gravitational waves around expanding and contracting Kasner solutions, and demonstrate their instabilities in terms of the evolution of the Weyl tensor invariant , which is independent of the gauge choice. We then connect the result with the BKL analysis.
The background line element is given by equation (16), with the scale factors
This compact notation describes two disconnected geometries, at negative and positive conformal times, respectively. The algebraic expressions below simplify if we introduce the time in which the normalization of the two scale factors coincide. Therefore, rather than (34), we can also use
The two Hubble rates are
while the average scale factor is . The physical and conformal times are related by
For future use, we also define to be the physical time corresponding to .
The solution with negative conformal time describes an overall contracting geometry, , which crunches into the singularity at . The solution with positive conformal time describes instead an overall expanding space, , originating at the singularity at . As in the previous Sections, we restrict the computation to the simpler case of a residual d isotropy between two spatial directions. We expect that the instability occurs for general Kasner indices .
Now we turn to the linearized perturbations satisfying the vacuum equations . There are two gravitational waves polarization perturbations. We consider a single mode with a given comoving momentum with components and . We denote by the vector in the plane along the direction of ,
The two GW polarizations obey the equations
where the effective frequencies can be written in compact form
both for the contracting and the expanding backgrounds. We recall that the physical momenta are related to the comoving one by the relations (19).
After solving the two equations (39), we can compute the metric perturbations (17) and the Weyl tensor of the background plus perturbations. The square of the Weyl tensor, once expanded perturbatively, has the following schematic structure
where is the Weyl invariant for the non-perturbed background solution, is the term linear in and , and is the term quadratic in the perturbations. We do the computation for the two polarizations separately. For instance, for the d vector modes, we compute the Weyl tensor in terms of the background and of the metric perturbations and . We then relate the two perturbations to through Eqs. (20) and (22), expressing spatial derivatives in terms of the comoving momenta, see Eq. (38). In this way, we can write the expression (41) in terms of , and their time derivatives. Finally, we insert in this expression the solutions of Eq. (39). The procedure for the mode is analogous.
We compare the second and third term in (41) with the background term. A growth of the ratios (denoted as ), or , signals an instability of the Kasner geometry. The background (zeroth order) Weyl invariant is
The first order term vanishes identically for the polarization. It is nonzero for , and it oscillates in space as . The second order term for the Fourier modes or has a part which is constant in space, plus two parts that oscillate in space as . In the following, we disregard the oscillatory parts in .
The solutions and are either monotonically evolving in time, or oscillating, with an envelope that is monotonically evolving in time. The oscillatory regime takes place when the mode has a wavelength shorter than the Hubble radii, , and are absent in the opposite regime. Eqs. (26) show the time dependence of the momentum and of the Hubble rates for the expanding Kasner solution. We see that, if we consider a complete background evolution, any mode starts in the large wavelength regime, and then goes in the short wavelength regime. Therefore, we expect that a mode is in the non-oscillatory regime sufficiently close to the singularity, and in the oscillatory regime sufficiently far from it (this behavior is manifest in the time evolutions shown). The same is true for the contracting background solution (with the obvious difference that early and late times interchange).
The time dependence of the Weyl invariant (or of its amplitude, when the mode is oscillating) is summarized in Table 1. More specifically, we show the ratio between the terms proportional to the perturbations and the background one, both for the expanding and the contracting Kasner. In the expanding case, a mode evolves from the large scale to the short scale regime. The opposite happens in the contracting case. Notice the the short scale behaviors in the expanding and contracting cases coincide. The same is not true for the two large scales behaviors. The reason is that, in the expanding case, we set to zero one of the two solutions of Eq. (39) that is decreasing at early times, and that would diverge at the initial singularity. The different behaviors are discussed in details in the next Subsection and in Appendix A.
vi.1 Contribution of the mode to the Weyl invariant
In this Subsection, we compute the contribution of the to the square of the Weyl tensor, both for an expanding and a contracting Kasner geometry. The analogous computation for the mode can instead be found in Appendix A.
mode during expansion
Consequently, we have the asymptotic solutions
where are three integration constants. In the early time solution we have disregarded a decaying mode that would diverge at , and where the expression in the second line is the large argument asymptotic expansion of the parabolic cylinder functions
which are solutions of the evolution equation with the short scales expanded frequency (43).
An extended computation of the square of the Weyl tensor (performed as outlined after Eq. (41)) leads to ; for the non-oscillatory part of the quadratic term in the fluctuations we find instead
Since the background square Weyl , this computation indicates that the ratio increases as in the large scales regime, while it oscillates in the short scales regime, with an amplitude that increases as .
In the left panel of Figure 6 we present the full numerical evolution of for three modes with different momenta. The behavior of the modes that we find here (pure Kasner geometry) should be compared with that discussed in the previous Sections, where the initial Kasner evolution was followed by an isotropic inflationary stage. In the evolutions shown in that case (for instance, Figs. 1 and 4 ) we had isotropization at the time , and we started with an initial time of about . In the present case, the geometry does not undergo isotropization. However, the two scale factors are normalized in such a way that they are equal to each other at the time , cf. eqs. (35). Therefore, we also choose in the present case. Also in analogy to the modes shown in Figs. 1 and 4, we define to be the comoving momentum of the modes which have parametrically the same size as the average horizon at the time , namely (cf. the expressions (36)). Moreover, we choose as in those two Figures.
Figure 6 confirms that each mode evolves from the non oscillatory large scales regime to the oscillatory short scales regime (the transition occurs at later times for modes of smaller momenta / larger scales). The time dependence of shown in the Figure agrees with the one obtained analytically, and summarized in Table 1. For comparison we also plot in the Figure 6 the evolutions of the mode during expansion considered in the Appendix A(a).
The results shown in the Figure confirm the instability of the background Kasner solution against the GW polarization . The growth in the large scales regime (early times) agrees with the amplification of the power spectrum shown in Figure 1. However, contrary to what one would deduce from Figure 1, we see that the growth continues also in the short scales (late times) regime. We recall that the definition of the power spectrum (29) contains an arbitrariness in the overall time dependence (since one may have used a different combination of the two scale factors as overall normalization). We nonetheless adopted it to show the strong scale dependence of the evolution of the power spectrum (which is not affected by the overall normalization), and the very different evolution experienced by the two GW polarizations (which is also independent of the arbitrary normalization, since and are normalized in the same way). To properly study the instability, one must study invariant and unambiguous quantities, such as the (scalar) square of the Weyl tensor which is investigated in this Section.
mode during contraction
We consider now the contribution to the square of the Weyl tensor from the mode in a contracting Kasner geometry. Plugging (35) into (40), the frequency of the mode on the contracting background has the short and large scales expansions
Once expressed in terms of absolute values of the time, the short and late time asymptotic expressions coincide with those of the expanding case, cf. eqs. (43). As in the expanding case, the short scales regime occurs asymptotically far from the singularity, while the large scales regime occurs asymptotically close to the singularity (notice, however, that a mode evolves from the large scales to the short scales regime in the expanding Kasner, while from the short scales to the large scales regime in the contracting Kasner background).
Consequently, also the short and large scales asymptotical solutions are identical, once expressed in terms of and .