Probing pre-inflationary anisotropy with directional variations in the gravitational wave background
PPrepared for submission to JCAP
Probing pre-inflationary anisotropywith directional variations in thegravitational wave background
Yu Furuya, Yuki Niiyama and Yuuiti Sendouda
Graduate School of Science and Technology, Hirosaki University,3 Bunkyocho, Hirosaki, Aomori 036-8561, JapanE-mail: [email protected], [email protected],[email protected]
Abstract.
We perform a detailed analysis on a primordial gravitational-wave backgroundamplified during a Kasner-like pre-inflationary phase allowing for general triaxial anisotropies.It is found that the predicted angular distribution map of gravitational-wave intensity on largescales exhibits topologically distinctive patterns according to the degree of the pre-inflationaryanisotropy, thereby serving as a potential probe for the pre-inflationary early universe withfuture all-sky observations of gravitational waves. We also derive an observational limit onthe amplitude of such anisotropic gravitational waves from the B -mode polarisation of thecosmic microwave background. Keywords: gravitational waves and CMBR polarization, cosmological perturbation theory,inflation, primordial gravitational waves (theory) a r X i v : . [ a s t r o - ph . C O ] S e p ontents t mid ( ψ i ) > t iso t mid ( ψ i ) < t ini t ini < t mid ( ψ i ) < t iso B -mode polarisation 205 Conclusions 22 There has been an accumulation of observational evidence for cosmic inflation [1, 2] in thepast decades, one of the most crucial being the precise measurements of the cosmic microwavebackground (CMB) [3]. Among others, a subject currently attracting a particular attentionin this field is the search for the primordial gravitational waves (PGWs) emerging from thequantum nature of space-time during an early accelerated expansion [4], which is thought tobe a direct evidence of the occurrence of inflation. Direct detections of such PGWs in low-frequency bands are indeed one of the primary aims of future planned laser-interferometerexperiments in space, such as eLISA [5], DECIGO [6], and Big-Bang Observer [7].There are, however, still many challenges to overcome in laser-interferometer experimentsin spite of the recent success of the first observation of gravitational waves from coalescingbinary black holes [8]. As an alternative approach, there have been attempts at measuringthe B -mode polarisations of the CMB as an imprint of the PGWs [9–11]. Past and ongoingprojects for CMB B -mode polarisation measurement at low multipoles include POLARBEAR[12], ACTpol [13], BICEP2/Keck array and Planck [14] (see also [15]), and SPTpol [16]. Manyfuture experiments are also planned, such as POLARBEAR-2 and Simons Array [17], andLiteBIRD [18]. Several constraints on tensor perturbations from the current bound on theCMB B -mode have been obtained in [14, 19].One of the goals beyond confirming the occurrence of cosmic inflation would be deter-mination of the initial condition of inflation. If there are earlier stages preceding the onset ofinflation, it is well anticipated that there were anisotropies and/or inhomogeneities of orderthe energy scale of unified theories such as Grand Unification Theories (GUTs) or superstrings.– 1 –nvestigations of early anisotropies may bring us useful information to construct the theoryof elementary particles beyond the Standard Model and even quantum gravity. This servesas another motivation for investigating the anisotropic stages before an isotropic inflation.Then, what is to be understood is how the universe has evolved into the currentlyobserved homogeneous and isotropic state. A key clue to this issue would be Wald’s cosmicno-hair theorem [20], stating that the Bianchi models for homogeneous anisotropic universe[21] inevitably evolve towards isotropic de Sitter space in the presence of a large enoughpositive cosmological constant Λ .It is generically expected that anisotropies of cosmological expansion would have greatimpacts on the evolution of cosmological perturbations. Indeed, several observational signa-tures of an anisotropic pre-inflation were discussed in [22–24]. Among others, a remarkablefinding is that amplification of gravitational waves occurs during the pre-inflationary Kasnerregime [23, 25], whose efficiency varies with the direction in the sky. In particular, in [23],Gümrükçüoğlu et al. investigated such gravitational waves from an inflationary backgrounddriven by a scalar field.The purpose of the present paper, therefore, is to give further insights into the connectionbetween direction-dependent gravitational waves and primordial pre-inflationary anisotropies.While Gümrükçüoğlu et al.’s analysis in [23] was restricted to a particular background withaxisymmetry, in contrast, we treat general triaxial backgrounds. To do so, we take the so-called Kasner-de Sitter metric as a simpler background in which initially anisotropic expansionof space is isotropised due to a cosmological constant rather than a scalar field. As analysedin [23, 25], the amplification of gravitational waves originates from an instability in the tensorsector and we expect even our simple model captures its essential features. This simple modelalso allows us to generate the all-sky map of gravitational-wave intensity from which we coulddecode the degree of the primordial anisotropy. Also we give some tentative constraints onthe model parameters from the current bounds on the CMB B -mode polarisation.The organisation of the paper is as follows. In section 2, we describe the basic equationsfor the background and perturbations. In particular, we employ a gauge-invariant formulationfor cosmological perturbations in the Bianchi type-I universe recently constructed by Pereiraet al. [26] (see also [27] for an earlier attempt). In section 3, we show how gravitationalwaves evolve depending on the direction of wavevectors as well as on the anisotropy of thebackground universe. In section 4, we demonstrate how limits can be given to the initialanisotropy and the initial power spectrum of the pre-inflationary gravitational waves fromfuture all-sky observations of the PGWs. Finally, in section 5, we conclude.Throughout the paper we use natural units with c = (cid:126) = k B = 1 . The Latin indices i, j are spatial and run through , , . We will use the arrow notation to denote spatialcontravariant vectors such as (cid:126)V ≡ ( V , V , V ) . Hereafter we consider the so-called Kasner-de Sitter (KdS) metric as a simple model foran inflationary universe with an initial anisotropy. It is an exact solution to the Einsteinequations in the presence of a positive cosmological constant Λ , whose line element is givenby g µν d x µ d x ν = − d t + a sinh / (3 Ht ) (cid:88) i =1 tanh q i (cid:18) Ht (cid:19) (d x i ) , (2.1)– 2 –here a iso is an arbitrary positive constant, H ≡ (cid:112) Λ / , and the three exponents q i ( i =1 , , ) satisfy the constraints (cid:88) i =1 q i = 0 , (cid:88) i =1 q i = 23 . (2.2)The KdS metric is classified into the Bianchi type-I cosmological model [21]. We define theaverage scale factor a ( t ) and the spatial metric γ ij ( t ) by a ( t ) ≡ a iso sinh / (3 Ht ) , γ ij ( t ) ≡ tanh q i (cid:18) Ht (cid:19) δ ij . (2.3)At earlier times with t (cid:28) H − , the metric behaves like a vacuum Kasner solution a ( t ) (cid:39) a iso (3 Ht ) / , γ ij ( t ) (cid:39) (cid:18) Ht (cid:19) q i δ ij , (2.4)while at later times with t (cid:29) H − , the metric asymptotes to a de Sitter solution, a ( t ) (cid:39) − / a iso e Ht , γ ij ( t ) (cid:39) δ ij . (2.5)Hence the KdS metric describes a universe evolving from an anisotropic initial phase to anisotropic inflation driven by Λ . The isotropisation is achieved around the time t iso ≡ H − and the constant a iso serves as the normalisation of the scale factor at the beginning of theisotropic de Sitter inflation. The duration of isotropic de Sitter expansion in this model isnecessarily bounded from above.This simple (pre-)inflationary model has some flaws as a realistic setup for the earlyuniverse such as lacking of mechanisms to realise a decline of the expansion rate and to exitfrom the inflationary phase, i.e., reheating. Precisely, what we assume is that the superhorizonfluctuations reentering the cosmological horizons at late times after reheating have exited thehorizon at the early era when the metric is well approximated by (2.1). Since the presence ofglobal anisotropies on cosmological scales is observationally not favoured, we should require a iso H (cid:46) a ( t ) H ( t ) , where t is the present age of the universe. Note that this condition isequivalent to the one for evading the horizon and flatness problems.It is useful to introduce an angular parameter Θ to express the exponents q i as q = 23 sin (cid:18) Θ − π (cid:19) , q = 23 sin (cid:18) Θ − π (cid:19) , q = 23 sin Θ . (2.6)The parameter Θ quantifies the degree of anisotropy of pre-inflationary cosmological expan-sion. With no loss of generality, we can restrict our attention to a range π ≤ Θ ≤ π , withinwhich the order of the exponents is q ≥ q ≥ q , see figure 1. Any KdS metric is equiv-alent, up to relabelling the axes and/or reversing their orientations, to one with the valueof Θ lying in the above range. The expansion of the universe is axisymmetric for Θ = π ( ( q , q , q ) = ( , − , − ) ) and for Θ = π ( ( q , q , q ) = ( , , − ) ). In the following anal-ysis we shall often take Θ = π as a fiducial value, for which the anisotropy exponents are ( q , q , q ) = ( √ , , − √ ) .The KdS metric has an initial Kasner singularity at t = 0 unless Θ = π , so wenecessarily introduce an initial time t ini > , which is supposed to be before t iso . We do– 3 – π � � π � � π � Θ - �� - ������ � � � � � Figure 1 . Values of q i for π ≤ Θ ≤ π . not discuss the initial singularity problem, but for a possible resolution, see [28]. If inflationoccurred at the GUT scale with H ∼ H GUT ∼ − m Pl , then t ini > t Pl ∼ − H − is required so as to avoid the trans-Planckian problems. In what follows, we take t ini =10 − t iso = 10 − H − as a fiducial value for the initial time.The conformal time η is defined by d η = d t/a ( t ) . A prime will denote differentiationwith respect to η . The cosmic expansion is characterised by the average Hubble parameter H and the shear tensor σ ij defined respectively as H ≡ a (cid:48) a = a iso H cosh(3 Ht )sinh / (3 Ht ) , σ ij ≡ γ (cid:48) ij = 3 q i a iso H tanh q i (cid:0) Ht (cid:1) sinh / (3 Ht ) δ ij . (2.7)The inverse of γ ij is defined by γ ik γ kj = δ ij . The indices of spatial tensors will be raisedand lowered by γ ij and γ ij such as σ ij ≡ γ ik σ kj . As usual, we will parameterise the harmonic modes of waves by a set of constants ( k , k , k ) ,which are regarded as the covariant components of a wavevector in the ( x , x , x ) comovingcoordinate frame. What is unusual is that, on an anisotropic KdS background, the contravari-ant components k i ≡ k j γ ij are not constant. The contravariant wavevector (cid:126)k ≡ ( k , k , k ) changes its direction and norm, (cid:112) k i k i , during the anisotropic Kasner regime, t (cid:46) t iso , andafter the universe is isotropised, t (cid:38) t iso , it comes to coincide with its covariant dual as lim t →∞ (cid:126)k = ( k , k , k ) .It is useful to introduce time-dependent polar angles ( β, γ ) to parameterise the nor-malised wavevector (cid:126) ˆ k ≡ (cid:126)k/ (cid:112) k i k i as [24] (cid:126) ˆ k ≡ e − β sin β cos γ e − β sin β sin γ e − β cos β , (2.8)where we have introduced the notation e − β i ≡ (cid:112) γ ii = tanh − q i (cid:18) Ht (cid:19) . (2.9)– 4 –he orthonormal polarisation vector basis perpendicular to (cid:126) ˆ k is introduced as (cid:126)e (1) ≡ e − β (cos β cos γ cos α − sin γ sin α ) e − β (cos β sin γ cos α + cos γ sin α ) − e − β sin β cos α ,(cid:126)e (2) ≡ − e − β (cos β cos γ sin α + sin γ cos α ) − e − β (cos β sin γ sin α − cos γ cos α ) e − β sin β sin α , (2.10)where the arbitrary angle α represents the rotation degree of freedom of ( (cid:126)e (1) , (cid:126)e (2) ) around (cid:126) ˆ k .As in [24], we determine α by imposing a condition α (cid:48) = − γ (cid:48) cos β . (2.11)Accordingly, the orthonormal tensor basis is defined in terms of the vector basis as ε + ij ≡ e (1) i e (1) j − e (2) i e (2) j √ , ε × ij ≡ e (1) i e (2) j + e (2) i e (1) j √ . (2.12)Using the polarisation basis, the shear tensor is decomposed into the scalar, vector, and tensorcomponents as σ (S) ≡ σ ij ˆ k i ˆ k j , σ (V)( a ) ≡ σ ij ˆ k i e j ( a ) , σ (T) λ ≡ σ ij ε ijλ , (2.13)where a = 1 , for the vector and λ = + , × for the tensor components, respectively. In order to analyse evolution of gravitational waves on a KdS spacetime, we apply the gauge-invariant formulation of cosmological perturbations in Bianchi type-I models developed byPereira et al. [26]. In this formalism, the general perturbed metric is given by ( g µν + δg µν ) d x µ d x ν = a ( η ) (cid:2) − (1 + 2 A ) d η + 2 B i d x i d η + ( γ ij + h ij ) d x i d x j (cid:3) , (2.14)where the (0 i ) and ( ij ) components are respectively decomposed into the scalar, vector, andtensor variables as B i = ∂ i B + ¯ B i , h ij = 2 ( γ ij + H − σ ij ) C + 2 ∂ ij E + 2 ∂ ( i E j ) + 2 E ij , (2.15)where the vector and tensor variables satisfy ∂ i ¯ B i = ∂ i E i = ∂ i E ij = E ii = 0 . In vacuum(plus a cosmological constant), the only dynamical degrees of freedom are represented by thetwo polarisation components of the gauge-invariant tensor variable E ij [26], defined using thetensor polarisation basis as E λ ≡ (cid:90) d x (2 π ) / e − i k l x l ε ijλ E ij ( λ = + , × ) . (2.16)Introducing µ λ ≡ a E λ for convenience, the equations of motion for gravitational waves aregiven by [26] µ (cid:48)(cid:48) + + ω µ + + ξ µ × = 0 ,µ (cid:48)(cid:48)× + ω × µ × + ξ µ + = 0 , (2.17)– 5 –here ω ≡ k i k i − a (cid:48)(cid:48) a − (cid:0) a σ (S) (cid:1) (cid:48) a − (cid:16) σ (T) × (cid:17) − a a (cid:16) σ (T)+ (cid:17) H − σ (S) (cid:48) ,ω × ≡ k i k i − a (cid:48)(cid:48) a − (cid:0) a σ (S) (cid:1) (cid:48) a − (cid:16) σ (T)+ (cid:17) − a a (cid:16) σ (T) × (cid:17) H − σ (S) (cid:48) , (2.18)and ξ ≡ σ (T)+ σ (T) × − a (cid:32) a σ (T)+ σ (T) × H − σ (S) (cid:33) (cid:48) . (2.19)One could guess from the expressions for “squared frequencies”, ω λ , that the shear cangive negative contributions and might lead to some instabilities. Indeed, it will be revealedthat the contributions from the shear control how gravitational waves grow during the pre-inflationary anisotropic expansion. In the sequel, we will pay particular attention to how muchand how long ω λ receives negative contributions from the shear, which crucially depends onthe values of ( k , k , k ) .The ξ term characterises the interaction between the two polarisation modes. Occurrenceof interactions in general makes the analyses more complicated, but eq. (2.19) implies that ξ vanishes if either of σ (T) λ is zero. We will take full advantage of this property in the followinganalysis. In this section we analyse time evolutions of gravitational waves labelling them with the final wavevector lim t →∞ (cid:126)k = ( k , k , k ) . The main object of interest in our current study is thedistribution of gravitational-wave intensity on a sphere of radius k ≡ (cid:112) k + k + k definedin the k -space, which, after being reflected through the origin and mapped to the celestialsphere, will provide an all-sky map of a gravitational wave background which we will be ableto observe at a comoving wavenumber k .Since there is a natural correspondence between the k -space and the position space, wesometimes mix up them with each other and even use common terminologies. For instance,a term “principal axes of expansion” may refer to both the x i -axes in the position space andthe k i -axes in the k -space depending on the context.Since we are interested in gravitational waves currently on large scales and since werequire a iso H (cid:46) a ( t ) H ( t ) , we only consider modes which had a shorter wavelength thanthe Hubble radius at the isotropisation time t iso = H − . We will take k = O (10 – ) × a iso H as reference values for the wavenumber. First we explain how a wavevector evolves with time. Figure 2 is a portrait of the timederivatives of (cid:126)k drawn on the unit sphere in the k -space (left) and the Mercator projection ofa portion of the sphere (right). As implied by eqs. (2.8) and (2.9), the general trend is that awavevector (cid:126)k changes its direction towards the principal axis of the slowest expansion of thethree, i.e., with the smallest exponent q i . Since q is always the smallest in our setup (exceptfor Θ = π ), wavevectors generically rotate towards the k -axis.– 6 – ′ � π � π � � ′ Figure 2 . Left: A phase portrait of the time derivative (cid:126)k (cid:48) in the k -space. Right: The Mercatorprojection of the half hemisphere specified by k ≥ and k ≥ . The background is characterisedby the anisotropy parameter Θ = π . In the exceptional cases when (cid:126)k is aligned to one of the principal axes, only the normchanges with time but the direction does not, as understood from (2.8). Also, those quartercircumferences of the spheres connecting two principal axes (e.g. equator) are special in thatif the endpoint of a wavevector lies on some of them, then it remains to do so, rotatingfrom the direction of the faster axis to the slower. This is understood as follows. In order toparameterise a wavevector pointing somewhere on one of such quarter circumferences, we shalldenote quantities associated with the axis of faster expansion by a superscript “(fast)” andthose with the slower axis by “(slow)”. For example, if one considers a circumference specifiedby k i = 0 , then k (fast) i and k (slow) i denote the covariant components of the wavevector alongthe faster and the slower axes, respectively. Then, we introduce an angle parameter ψ i and amodulus k as tan ψ i ≡ k (fast) i k (slow) i , k ≡ (cid:114)(cid:16) k (slow) i (cid:17) + (cid:16) k (fast) i (cid:17) . (3.1)Hence, the ratio of the contravariant components evolves during the anisotropic phase, t (cid:28) H − , as k i (fast) k i (slow) = tan ψ i (cid:20) tanh (cid:18) Ht (cid:19)(cid:21) q (slow) i − q (fast) i ) (cid:39) tan ψ i (cid:18) Ht (cid:19) q (slow) i − q (fast) i ) . (3.2)If k (fast) i < k (slow) i , the wavevector is finally aligned closer to the slower axis. Fromeq. (3.2), the time at which (cid:126)k comes to the midpoint of the two axes, i.e., at an angle of π ,is estimated as Ht mid (cid:39) (cid:32) k (fast) i k (slow) i (cid:33) / [2( q (fast) i − q (slow) i )] . (3.3)– 7 –n the other hand, if k (fast) i > k (slow) i , (cid:126)k is initially closer to the faster axis and remains sountil the end of the anisotropic Kasner regime.From the above analysis, we can deduce that the direction of a wavevector pointing ingeneral directions can only move within one of the quarter hemispheres bounded by suchspecial quarter circumferences. It follows that, thanks to the symmetry of the background,it is sufficient for our purpose to consider only one of those quarter hemispheres and we shallhereafter restrict on the one specified by k i ≥ ( i = 1 , , ).In the k -space, a normalised wavevector (cid:126) ˆ k = ( k / | (cid:126)k | , k / | (cid:126)k | , k / | (cid:126)k | ) with one vanishingcomponent k i = 0 ( i = 1 , , ) points on one of the quarter circumferences of the unit sphere,which we shall call C i . See figure 3 for the definitions of the circumferences C i and angles ψ i ( i = 1 , , ). Figure 3 . Illustration of the angles ψ i and circumferences C i on the unit sphere in the k -space. Thearrows are examples of normalised final wavevector lim t →∞ (cid:126) ˆ k = ( k /k , k /k , k /k ) lying on each C i . We first consider the evolution of modes whose wavevector is aligned to either of the principalaxes. Let k i be the only non-vanishing covariant component. Then the wavevector (cid:126)k is keptaligned with the k i -axis while its squared norm evolves with time as k i k i = k i tanh − q i (cid:18) Ht (cid:19) . (3.4)The projected components of the shear tensor are greatly simplified in this case as σ (S) = 3 q i a iso H ( a/a iso ) , (cid:12)(cid:12)(cid:12) σ (T)+ (cid:12)(cid:12)(cid:12) = 3∆ i √ a iso H ( a/a iso ) , σ (V)(1) = σ (V)(2) = σ (T) × = 0 , (3.5)where we have introduced ∆ i ≡ max j (cid:54) = i q j − min j (cid:54) = i q j , which quantifies the + -componentof the shear. See figure 4 for the dependence of ∆ i on Θ for each i . For σ (T) × = 0 , the The sign of σ (T)+ is not needed because only its square will appear. – 8 –nteraction term ξ in the gravitational-wave equations vanishes identically, see (2.19), so thetwo polarisation modes are decoupled. � π � � π � � π � Θ �� � �� Δ � � � � Figure 4 . Values of ∆ i = max j (cid:54) = i q j − min j (cid:54) = i q j . Substituting eqs. (3.4) and (3.5) into (2.18), we obtain the squared frequencies ω λ as ω × a H = k i a H tanh − q i (cid:18) Ht (cid:19) + 2 − cosh(6 Ht )( a/a iso ) − i ( a/a iso ) ,ω a H = k i a H tanh − q i (cid:18) Ht (cid:19) + 2 − cosh(6 Ht )( a/a iso ) + 6∆ i ( a/a iso ) [(2 /
3) cosh(3 Ht ) − q i ] . (3.6)The leading contributions from each term during the early anisotropic regime ( t (cid:28) t iso = H − )are found as ω × a H ⊃ k i a H (cid:18) Ht (cid:19) − q i + 1(3 Ht ) / − i (3 Ht ) / ,ω a H ⊃ k i a H (cid:18) Ht (cid:19) − q i + 1(3 Ht ) / + 6∆ i (3 Ht ) / (2 / − q i ) , (3.7)where we have assumed q i < . The shear tensor gives a negative contribution ∝ − ∆ i / ( Ht ) / to ω × , which is missing in ω in contrast.In figure 5, we show typical time evolutions of ω × (blue) and ω (red) for modes alignedwith either of the three principal axes. The three figures correspond to the cases when (cid:126)k (cid:107) k -axis (top-left), (cid:126)k (cid:107) k -axis (top-right), and (cid:126)k (cid:107) k -axis (bottom), where the finalwavenumber is k i = 100 a iso H ( i = 1 , , ) and the background anisotropy parameter is Θ = π , for which ( q , q , q ) = ( √ , , − √ ) and (∆ , ∆ , ∆ ) = ( √ , √ , √ ) . Since ∆ i > for all i in this case, the squared frequency ω × should turn negative once the shear termdominates as implied by (3.7). To illuminate this, the lines for ω × are dashed when takingnegative values. It is observed that there is a large negative contribution to ω × from the shearat the earliest stage of the anisotropic phase ( t (cid:28) t iso = H − ) except for the k -axis-alignedmode (top-left panel), for which ω × is dominated by the k i k i term throughout the time rangeconsidered, − < Ht (cid:46) , although the shear would eventually dominate if one were allowedto go back in time indefinitely. – 9 – � - � �� - � ����� ����� ����� � �� � � ����� � �� � ω λ � / � � � ∥ � � - ���� ⨯ + �� - � �� - � ����� ����� ����� � �� � � ����� � �� � ω λ � / � � � ∥ � � - ���� ⨯ + �� - � �� - � ����� ����� ����� � �� � � ����� � �� � ω λ � / � � � ∥ � � - ���� ⨯ + Figure 5 . Time evolutions of ω × (blue) and ω (red) for modes aligned with the k -axis (top-left), k -axis (top-right), and k -axis (bottom), for all of which the final wavenumber is k i = 100 a iso H ( i = 1 , , ) . The lines are dashed where ω × is negative. The background anisotropy parameter is Θ = π corresponding to ( q , q , q ) = ( √ , , − √ ) and (∆ , ∆ , ∆ ) = ( √ , √ , √ ) . Shown in figure 6 are the waveforms of the above three axis-aligned modes obtained bynumerically integrating (2.17). As expected from the behaviours of ω × , the × -mode growssubstantially while the shear term is giving a dominant contribution.These waveforms of gravitational waves can be analytically understood as follows. Firstlet us consider the × -mode. The asymptotic form of ω × for Ht (cid:28) is ω × ∼ k i (cid:0) / a iso Hη (cid:1) q i + 1 − i η , (3.8)where, in converting the time coordinates, we have used a relation Ht = 2 √
23 ( a iso Hη ) / (cid:34) a iso Hη ) O ( a iso Hη ) (cid:35) . (3.9)Since q i < for a triaxial KdS metric, ω × is dominated by the shear term ( ∝ η − ) at earliertimes and by the k i k i term ( ∝ η − q i ) at later times. The growth of × -mode is expected to– 10 – � - � �� - � ����� ����� ����� � �� � � �������� � λ ( � )/ � λ ( � � ) � ∥ � � - ���� ⨯ + �� - � �� - � ����� ����� ����� � �� � � �������� � λ ( � )/ � λ ( � � ) � ∥ � � - ���� ⨯ + �� - � �� - � ����� ����� ����� � �� � � �������� � λ ( � )/ � λ ( � � ) � ∥ � � - ���� ⨯ + Figure 6 . Waveforms of E × (blue) and E + (red) for the modes aligned with the k -axis (top-left), k -axis (top-right), and k -axis (bottom). The initial time is t ini = 10 − H − and the other parametersare the same as in figure 5. stall around the transition between these two regimes, whose time is estimated as a iso Hη stall = (cid:34) (cid:12)(cid:12) i − (cid:12)(cid:12) − q i (cid:18) a iso Hk i (cid:19) (cid:35) / (2 − q i ) (3.10)or equivalently Ht stall = 2 √ (cid:34) (cid:12)(cid:12) i − (cid:12)(cid:12) − q i (cid:18) a iso Hk i (cid:19) (cid:35) / [2 (2 / − q i )] . (3.11)For η (cid:28) η stall , the equation of motion for the × -mode is approximated as µ (cid:48)(cid:48)× + 1 − i η µ × ≈ (3.12)and its general solution is given by µ × ≈ C + ( Hη ) (1+3∆ i ) / + C − ( Hη ) (1 − i ) / . (3.13)Hence, up to the time η stall , the growing mode behaves as | E × | ∝ ( Hη ) i / ∝ ( Ht ) ∆ i .Once the shear term becomes subdominant after η stall , the × -mode starts to oscillateobeying µ (cid:48)(cid:48)× + k i (cid:0) / a iso Hη (cid:1) q i µ × ≈ . (3.14)– 11 –his cannot be integrated analytically for a general q i , but, since the exponents q i only mildlydepend on Θ , here we make a crude approximation that ( q , q , q ) ≈ ( , , − ) . Then, wehave a rough estimate | µ × | ∝ η / ( i = 1) η ( i = 2) η − / ( i = 3) (3.15)and, therefore, we may estimate the amplitude of E × between t stall and t iso = H − as | E × | ∝ ( Ht ) − p i with ( p , p , p ) ≡ (cid:18) , , (cid:19) . (3.16)As we will show numerically later, although quantitatively not quite precise, this approxima-tion captures certain qualitative features of the evolution of gravitational waves during theKasner epoch.The evolution of the + -mode can be deduced by setting ∆ i = 0 , see eq. (3.7). Hence E + oscillates constantly until t stall , and then decreasingly after t stall as | E + | ∝ ( Ht ) − p i . (3.17)The background is isotropised at ∼ t iso and enters the standard de Sitter inflation phase.During inflation, amplitudes of the both modes decay exponentially as E λ ∝ a − ∝ e − Ht untilthe modes exit the Hubble radius when a ( t ) = k i H − , and are then frozen. The amplitudeat the end of inflation is therefore suppressed by a factor a ( t iso ) H/k i relative to the value at t = t iso .Summing up all these effects, the linear growth factors for each polarisation mode alignedwith the principal axes are defined and estimated as D × ≡ lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12) E × ( t ) E × ( t ini ) (cid:12)(cid:12)(cid:12)(cid:12) ∼ a iso Hk i ( Ht stall ) p i (cid:18) Ht stall Ht ini (cid:19) ∆ i ,D + ≡ lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12) E + ( t ) E + ( t ini ) (cid:12)(cid:12)(cid:12)(cid:12) ∼ a iso Hk i ( Ht stall ) p i . (3.18)Equation (3.11) implies that smaller q i leads to larger t stall as long as k i (cid:29) a iso H .Thus, the mode aligned with the k -axis enjoys the longest period of growth, see figure 7 forcomparison of the values of t stall for the axis-aligned modes with the anisotropy parameter Θ varied. Note that Ht stall given by (3.11) for i = 1 formally saturates to near Θ = π (then q ≈ ), but, since the usefulness of (3.11) is limited in such a situation, we have not shownthe corresponding values for i = 1 in figure 7.Substituting eq. (3.11) into (3.18), we can make the parameter dependence of the growthfactors more explicit as D × ∝ (cid:18) k i a iso H (cid:19) P × ,i ( Ht ini ) − ∆ i , D + ∝ (cid:18) k i a iso H (cid:19) P + ,i (3.19)with P × ,i = − − p i + ∆ i / − q i , P + ,i = − − p i / − q i . (3.20)The validity of this expression for i = 1 is also degraded when Θ ≈ π ( q i ≈ ) for the samereason as above. In order to clarify the k i dependences, we plot the indices P × ,i and P + ,i in– 12 – π � � π � � π � Θ �� - � �� - � �� - � �� - � � � ����� � � � Figure 7 . Θ -dependences of t stall for modes aligned with each principal axis with the final wavenumber k i = 100 a iso H ( i = 1 , , ). � π � � π � � π � Θ - � - � - � - � - � - � � ⨯ � � � � � � π � � π � � π � Θ - � - � - � - � - � - � � + � � � � � Figure 8 . Θ -dependences of the indices P × ,i (left) and P + ,i (right). figure 8. As for the × -mode, since P × , is the largest of the three, the k -axis-aligned modesdominate at sufficiently large wavenumbers.In the above analysis, we have implicitly assumed that the time scales are ordered as t ini < t stall < t iso , but this does not necessarily hold, as in the case of the k -axis-alignedmode exemplified in this section. If t stall < t ini , both polarisation modes oscillate from thebeginning with decaying amplitudes estimated by (3.16) and (3.17), and the growth factor isestimated as D × ∼ D + ∼ a iso Hk i ( Ht ini ) p i . (3.21)On the other hand, although this is not likely to occur, if t iso < t stall , the growth of gravi-tational waves does not stall until the time of isotropisation and the growth factor may beestimated as D × ∼ D + ∼ a iso Hk i ( Ht ini ) − ∆ i . (3.22)Figure 9 shows the Θ -dependences of the growth factors for the axis-aligned modes withfinal wavenumber k i = 100 a iso H ( i = 1 , , ) for the initial time t ini = 10 − H − . The solidand dashed lines are the numerical and analytic estimate (3.18), respectively, which are in– 13 –airly good agreement with each other in the almost entire range of Θ . The k -axis-alignedmode is almost independent of Θ since it does not grow but constantly oscillates as impliedby eqs. (3.16) and (3.17). The k -axis-aligned and k -axis-aligned modes are comparable forthe chosen set of parameters, although, as could be deduced from (3.19), at sufficiently largewavenumbers the k -axis-aligned × -mode should dominate over the others. � π � � π � � π � Θ ������������������ � ⨯ � � � � π � � π � � π � Θ ������������������ � + � � � Figure 9 . Θ -dependences of the linear growth factors D × (left) and D + (right). The final wavenumberis k i = 100 a iso H ( i = 1 , , ) and the initial time is t ini = 10 − H − . The dashed lines indicate theanalytic estimate given by (3.18), in fairly good agreement with the numerical calculations (solid). Then, we extend our analysis to modes whose wavevector (cid:126)k is not aligned with either ofthe principal axes but between two of them. For such modes, as discussed in section 3.1,the direction of a wavevector (cid:126)k changes with time from the axis of faster expansion to theslower along one of the quarter circumferences C i defined as in figure 3. As in section 3.1,we introduce notations “(fast)” and “(slow)” for quantities associated with the faster and theslower axes, respectively. Wavevectors to be considered should have one vanishing component, k i = 0 ( i = 1 , , ), while the other two, to be denoted as k (fast) i and k (slow) i , are non-zero.To parameterise such a wavevector, we introduce an angle parameter ψ i and a modulus k asgiven by eq. (3.1).It is notable that the × -component of the shear tensor σ (T) × still vanishes in this case,so, since ξ = 0 , the two polarisation modes of gravitational waves are decoupled. Thereforethe evolution of each mode can still be analysed separately.As we will discuss shortly, the distribution of the intensity of gravitational waves on thecircumferences C i will serve as a key clue to establish the connection between the directionalvariation of gravitational-wave intensity and the pre-inflationary parameters. The situationon a circumference may be classified into the following three cases according to the finaldirection of the wavevector, i.e., the value of tan ψ i .First, if the final direction of a mode satisfies tan ψ i > , the wavevector (cid:126)k remains closeto the faster axis throughout the anisotropic Kasner regime. In terms of t mid given by (3.3),the condition for this to realise may be written as t mid ( ψ i ) > t iso ⇔ tan ψ i > . (3.23)We will approximately evaluate the evolution of the modes of this kind by regarding theirwavevectors as exactly aligned with the faster axis throughout the anisotropic phase.– 14 –ext, for modes with tan ψ i < , a turnover of the direction could happen at the time t = t mid given by eq. (3.3). However, if t mid ( ψ i ) < t ini ⇔ tan ψ i < (cid:18) Ht ini (cid:19) q (fast) i − q (slow) i ) , (3.24)the wavevector (cid:126)k is already close to the slower axis at the initial time t = t ini , hence noturnover occurs during the Kasner epoch. In this case the wavevector is regarded as alignedexactly with the slower axis throughout.Finally, a turnover actually happens during the anisotropic phase if t ini < t mid ( ψ i ) < t iso ⇔ (cid:18) Ht ini (cid:19) q (fast) i − q (slow) i ) < tan ψ i < . (3.25)We regard such modes as aligned with the faster axis before t mid and with the slower axisafter t mid . The analyses in the first and second cases above still apply to the time ranges t < t mid and t > t mid , respectively. t mid ( ψ i ) > t iso For such a mode, we approximate the growth rate by the one for a mode exactly aligned withthe faster axis and having a wavenumber k i = k (fast) i = k sin ψ i . Applying (3.18) or (3.21)with q i = q (fast) i , ∆ i = ∆ (fast) i , and p i = p (fast) i , we may estimate the growth factor for the × -mode as D × ( ψ i ) ∼ a iso Hk × (cid:16) Ht (fast)stall (cid:17) p (fast) i (cid:18) Ht (fast)stall Ht ini (cid:19) ∆ (fast) i ( t ini < t (fast)stall )( Ht ini ) p (fast) i ( t (fast)stall < t ini ) , (3.26)where Ht (fast)stall ( ψ i ) = 2 √ (cid:12)(cid:12)(cid:12) (fast) i ) − (cid:12)(cid:12)(cid:12) − q (fast) i (cid:18) a iso Hk sin ψ i (cid:19) / [2 (2 / − q (fast) i )] . (3.27)It is noted that, for both the circumferences C and C , the k -axis serves as the axisof faster expansion, so q (fast) i = q ≥ for i = 2 , . Then, since k > a iso H by assumption, arough but strict upper limit on the time scale of the growth stall is given as Ht (fast)stall ( ψ i ) (cid:46) (cid:18) k a iso H (cid:19) − ( i = 2 , . (3.28)This implies that growth of gravitational waves does not take place near the k -axis if thewavelength is sufficiently short such that k (cid:29) a iso H . t mid ( ψ i ) < t ini Again, no turnover takes place. Similarly to Case 1 but with the use of k i = k (slow) i = k cos ψ i instead of k sin ψ i , the growth factor is estimated as D × ( ψ i ) ∼ a iso Hk × (cid:16) Ht (slow)stall (cid:17) p (slow) i (cid:18) Ht (slow)stall Ht ini (cid:19) ∆ (slow) i ( t ini < t (slow)stall )( Ht ini ) p (slow) i ( t (slow)stall < t ini ) (3.29)– 15 –ith Ht (slow)stall ( ψ i ) ≈ √ (cid:12)(cid:12)(cid:12) (slow) i ) − (cid:12)(cid:12)(cid:12) − q (slow) i (cid:18) a iso Hk cos ψ i (cid:19) / [2 (2 / − q (slow) i )] . (3.30)In contrast to the previous case, the exponent for this expression does not become large excepton C for a nearly axisymmetric background with Θ ≈ π . t ini < t mid ( ψ i ) < t iso Finally, we consider the intermediate case in which a turnover occurs during the Kasnerregime. We regard the wavevector (cid:126)k as aligned with the faster axis before t mid and with theslower axis after t mid .It is useful here to compare the three time scales t (fast)stall , t (slow)stall , and t mid . Shown infigure 10 are their values evaluated on the circumferences C (top-left), C (top-right), and C (bottom) for the parameters k = 100 a iso H and Θ = π . As we will discuss shortly, the × -mode can grow during the time range indicated as grey-shaded regions. �� - � �� - � ���� � ��� ��� ψ � �� - � �� - � ����� � � � � ( � � � ) � ����� ( ���� ) � ����� ( ���� ) � ��� �� - � �� - � ���� � ��� ��� ψ � �� - � �� - � ����� � � � � ( � � � ) � ����� ( ���� ) � ����� ( ���� ) � ��� �� - � �� - � ���� � ��� ��� ψ � �� - � �� - � ����� � � � � ( � � � ) � ����� ( ���� ) � ����� ( ���� ) � ��� Figure 10 . Time scales on the circumferences C (top-left), C (top-right), and C (bottom) for k = 100 a iso H and Θ = π . The horizontal dashed lines correspond to Ht iso = 1 and Ht ini = 10 − .For each value of tan ψ i , the mode can grow within the time range indicated in cray, see text. Before the time of turnover t mid is reached, i.e., below the green lines in figure 10, thewavevector (cid:126)k is considered to be aligned with the faster axis and the analysis in section 3.3.1– 16 –ay be applied. The growth stalls at t (fast)stall if it comes before t mid , but otherwise continuesup to t mid . Indeed, in figure 10, there are small intervals of tan ψ i in which t mid < t (fast)stall .After t mid , i.e., above the green lines in figure 10, the wavevector (cid:126)k is considered to bealigned with the slower axis and the analysis in section 3.3.2 may be applied. If t mid < t (slow)stall ,then the mode continues (or resumes) to grow beyond t mid up to t (slow)stall .Near the k -axis on the circumferences C and C , it may happen that the time t (fast)stall comes to earlier than t ini and then growth is substantially suppressed. Indeed, for our choice ofparameters, the values of t (fast)stall evaluated on C and C are much earlier than t ini = 10 − H − as in the top-right and the bottom panels of figure 10. If this is the case, a growth can onlyhappen near the slower axes on these circumferences. Therefore, a criterion for growth interms of the angle from the slower axis may be given by t (fast)stall (cid:38) t ini ⇔ ψ i (cid:46) (cid:18) k a iso H (cid:19) − ( Ht ini ) − (2 / − q (fast) i ) ( i = 2 , . (3.31)In contrast, for the current choice of parameters, such suppression is not working on C as understood from the top-left panel of figure 10, where t (fast)stall (tan ψ (cid:38) is comparable to t (slow)stall (tan ψ (cid:46) and never comes below t ini . However, for some parameters and wavelengthsit may happen that t (fast)stall on C becomes earlier than t ini , leading to suppression of growthnear the k -axis (the faster axis on C ). In the next sections, we will pay particular attentionto this possibility. In figure 11, we show the linear growth factors for the × -mode on the circumferences C (top-left), C (top-right), and C (bottom) for the parameters Θ = π , t ini = 10 − H − , and k = 100 a iso H . In each figure, ψ i measures the angle from the axis of slower expansion. Theanalytic estimates based on the arguments in the previous sections are also plotted as dashedlines.As understood from the figure, there is only weak contrast in growths on C , whereasrather sharp declines at some angle of O (0 . from the slower axes on C and C are indicated.This implies that the region of high-intensity gravitational waves on the celestial sphere wouldform a belt-like pattern around a great circle containing C , which we will numerically confirmin the next section.In figure 12, we show waveforms of E × (blue) and E + (red) for several modes on C ,i.e., with k = 0 . The modes have a common final wavenumber k = (cid:112) k + k = 100 a iso H but different final directions parameterised by tan ψ = k /k : ψ = 10 − (top-left), − (top-right) and − (bottom). The other parameters are the same as in figure 11. It isobserved that growths occur more efficiently as ψ decreases, i.e., as the final direction of thewavevector gets closer to the k -axis. Indeed, there appears a threshold value ψ ∼ − forgrowth as implied by eq. (3.31). Note also that these figures are to interpolate the top panelsof figure 6.It should be noted here that the situation on C can change dramatically according tothe wavenumber and the background parameters. As implied by (3.27), the time scale t (fast)stall near the k -axis on C ( tan ψ (cid:38) ) is a decreasing function of both k and Θ , the reasonfor the latter dependence being that the exponent q (fast)1 = q is an increasing function of Θ . If t (fast)stall declines to a value as small as t ini , then amplification of gravitational waves– 17 – ��� ��� � ��� ψ � ��������������������� � ⨯ � � ( � � � ) ��������� �������� ���� ��� � ��� ψ � ��������������������� � ⨯ � � ( � � � ) ��������� ������������ ��� � ��� ψ � ��������������������� � ⨯ � � ( � � � ) ��������� �������� Figure 11 . Linear growth factors for the × -modes on the circumferences C (top-left), C (top-right),and C (bottom). The background anisotropy parameter is Θ = π . The initial time and the finalwavenumber are t ini = 10 − H − and k = 100 a iso H , respectively. is prohibited near the k -axis. Therefore, a simple criterion for the growth on C to occuruniformly may be given by t ini ≤ t (fast)stall ( ψ = π/
2) = t ( i =2)stall . (3.32)In figure 13, we show the boundaries in the (Θ , Ht ini ) -plane saturating the above inequalityfor several wavenumbers, which can be used to discriminate whether the intensity on C isuniform or not at a corresponding wavenumber. Finally we discuss time evolutions of modes with a wavevector pointing in a general direction.As indicated in figure 2, a wavevector (cid:126)k rotates from the k - to the k -axis (except on thecircumference C ). Since neither of the tensor components of the shear vanishes, the twopolarisation modes no longer evolve independently but affect each other. Hence we shallnumerically solve the coupled equations of motion (2.17).In figure 14, we show all-sky maps of the growth factor for intensity I ≡ (cid:113) E + E × ,which is invariant under rotations of polarisation basis, for the anisotropy parameters Θ = π (top) and . × π (bottom). The former represents a highly triaxial configuration whilethe latter is nearly axisymmetric around the k -axis, resembling the situation investigated in[23]. – 18 – � - � �� - � ����� ����� ����� � �� � � �������� � λ ( � )/ � λ ( � ��� ) ψ � ��� ⨯ + �� - � �� - � ����� ����� ����� � �� � � �������� � λ ( � )/ � λ ( � ��� ) ψ � ���� ⨯ + �� - � �� - � ����� ����� ����� � �� � � �������� � λ ( � )/ � λ ( � ��� ) ψ � ����� ⨯ + Figure 12 . Waveforms of several modes on the circumference C ( k = 0 ) with a common finalwavenumber k = 100 a iso H but with different final directions k /k = tan ψ for ψ = 10 − (top-left), − (top-right), and − (bottom). The background anisotropy parameter and the initial timeare Θ = π and t ini = 10 − H − , respectively. An obvious feature seen in the top panels is significantly higher intensities along the k = 0 great circle. This is a manifestation of the result of the analysis in the previoussections that the growth of gravitational waves only occurs in a narrow range near the sloweraxis on the circumferences C and C . Indeed, the width of the belt-shaped region of higherintensities is well estimated by ψ i =2 , given by eq. (3.31). On the other hand, in the bottompanels ( Θ = 0 . × π ) the growth is localised to near the k -axis.As we shall explain now, this topological property is crucial in determining not onlythe directions of the principal axes of the early anisotropic expansion but also the degree ofanisotropy Θ and the initial time of the anisotropic pre-inflationary stage t ini with the aid offigure 13.Let us imagine all-sky observations of gravitational waves at multiple wavelengths be-come put into practice in the future. If the gravitational waves from the anisotropic regime aredetected at some (long) wavelength and the intensity map is determined to have a topologylike the top panels of figure 14, then the principal axis of the fastest expansion (in our nota-tion, the x -axis) is determined as the normal direction to the plane containing the great circleof higher intensities. As for the pre-inflationary parameters, the combination of (Θ , Ht ini ) infigure 13 is restricted to lie below the boundary corresponding to the observation wavelength.Our analyses further predict that, as one goes from longer wavelengths to shorter, the– 19 – π � � π � � π � �� - �� �� - � �� - � �� - � �� - � � Θ � � � � � �� � ��� � �� � � ��� � �� � � ��� � �� � � ��� � Figure 13 . Boundaries in the (Θ , Ht ini ) -plane satisfying t ini = t stall evaluated for the k -axis-alignedmodes with several wavenumbers k . If a combination of the parameters (Θ , Ht ini ) lies below aboundary for a given wavenumber k , then the corresponding k -axis-aligned mode is supposed togrow and the intensity on C would look rather uniform. intensity maps will exhibit a transition from a circle-like one (as depicted in the top panels offigure 14) to a localised one (bottom panels) at some critical wavelength. From the map afterthe transition, the axis of the slowest expansion (the x -axis) is determined to be the directionpointing the highest intensity region on the celestial sphere, after which the remaining axis(the x -axis) is also determined to be the orthogonal direction to the other two axes.If such a transition is observed, the values of the parameters (Θ , Ht ini ) in figure 13should lie on a curve corresponding to the critical wavelength. Then, once either of the twoparameters is given observationally or theoretically, the remainder can be determined. B -mode polarisation Our final project is to constrain the gravitational wave background from the anisotropic pre-inflationary phase using the CMB observations. In the inflationary scenario considered in thepresent analysis, the total power of primordial tensor perturbations after inflation, denotedas P T , may be regarded as a sum of the pre-inflationary contribution P preT and the ordinaryinflationary contribution P infT . Their crucial difference is that, while the inflationary part isexpected to be a function of the modulus k ≡ (cid:112) k + k + k only, the pre-inflationary partshould depend on the components ( k , k , k ) differently [24]. Another complexity arises ingeneral due to different evolutions of the two polarisation modes and their interactions duringthe anisotropic phase [24].In this study, we do not perform a full analysis of the directional variations of the powerspectra but rather focus on the axis-aligned modes. By doing so, we can take advantage of thefact that the two polarisations are decoupled, and hence, the power spectrum after inflationcan be represented as a sum of those of the + -mode P preT+ and the × -mode P preT × [23, 24]. Theyare not identical in general because the two polarisations evolve differently on an anisotropicbackground and even their initial values are not necessarily the same. We here assume thatclassical gravitational waves already exist at the initial time t ini . Denoting the values at t ini – 20 – igure 14 . Qualitative difference in the intensity ratio log [ I ( ∞ ) /I ( t ini )] between the cases of Θ = π (top) and . × π (bottom). as P pre , iniT λ , the power spectra after inflation can be written in terms of the growth factor D λ as P preT λ ( k i ) = D λ P pre , iniT λ ( k i ) , (4.1)where k i is the only non-zero component. If P pre , iniT+ ( k i ) and P pre , iniT × ( k i ) are of the same order,since D × (cid:38) D + in general, we may ignore the + -mode in comparison with the × -mode atlater times. Then, we can regard the total primordial power spectrum of the pre-inflationarygravitational waves as P preT ( k i ) ≈ P preT × ( k i ) = D × P pre , iniT × ( k i ) , (4.2)whose explicit k i dependence, apart from the unknown initial part, is P preT ( k i ) ∝ (cid:18) k i a iso H (cid:19) − − i / − qi ) ( Ht ini ) − i P pre , iniT × ( k i ) . (4.3)– 21 –he upper limit on the primordial tensor power spectrum is given in terms of the tensor-to-scalar ratio P T / P S ≡ r (cid:46) . at around k = 0 .
05 Mpc − , where the scalar powerspectrum is P S ( k pivot ) ∼ − [19]. Since it is expected that the k -axis-aligned mode growsthe most at shorter wavelengths, we require the primordial tensor power spectrum of themode travelling along the x -axis to be lower than r P S as a conservative limit. An additionalassumption to be made here is that the modes exiting the horizon at t = t iso re-enter thecosmological horizon at sufficiently late times, namely a iso H ∼ a ( t ) H ( t ) .In the left panel of figure 15, we show upper limits on the × -mode initial power spectrumof the pre-inflationary gravitational waves P pre , iniT , × plotted against the anisotropy parameter Θ . The initial time is fixed to be t ini = 10 − H − . The constraint becomes gradually strongeras Θ approaches to π , reflecting the tendency observed in the left panel of figure 9. In theright panel, we show lower limits on the initial time t ini . In this figure, we set P pre , iniT × = 10 − . � π � � π � � π � �� - �� �� - � �� - � �� - � �� - � ��� � �� � Θ � ⨯ � � � � � � � �� � ��� � �� � � ��� � �� � � ��� � �� � � ��� � � π � � π � � π � �� - �� �� - � �� - � �� - � �� - � � Θ � � � � � �� � ��� � �� � � ��� � �� � � ��� � �� � � ��� � Figure 15 . Left: Upper limits on the initial spectrum of the × -mode P pre , iniT , × for wavenumbers k = 10 a iso H (red), a iso H (yellow), a iso H (green), and a iso H (blue). The initial time isfixed as t i = 10 − H − . Right: Lower limits on the initial time H t i for the same set of wavenumbers.The initial amplitude of the × -mode spectrum is fixed as P pre , iniT , × = 10 − . In this paper, we performed a detailed analysis on the directional variations of a gravitationalwave background in a (pre-)inflation model described by the general triaxial Kasner-de Sittermetric, in which the degree of anisotropy is parameterised by an angle parameter Θ . Thepurpose of the study was to give some insights into the connection between such gravitationalwaves and the primordial anisotropies.We divided the whole analysis into the following three steps. First, in section 3.2, weinvestigated the evolution of gravitational waves whose wavevector is aligned with either of theprincipal axes of anisotropic expansion. We clarified that, with our choice of the polarisationbasis, the × -polarisation mode grows substantially while the shear term is giving a dominant– 22 –ontribution to the squared frequency ω × as given by eq. (2.18). The amount of growthreflects two factors: (i) the slower the expansion along the axis is, the longer ω × remainsnegative; (ii) the larger the projected shear component σ (T)+ is, the faster the mode grows.There was a competition between the k - and k -axis-aligned modes since the former growsfor longest while the latter “sees” the largest value of σ (T)+ . In an analytical manner, we haveobtained the explicit parameter dependencies of the growth factors as in eq. (3.19), clarifyingthat the k -axis-aligned × -mode should dominate over the other modes at sufficiently largewavenumbers except for an axisymmetric background with Θ ≈ π .Second, in section 3.3, we extended the analysis to the modes whose wavevector is alignedbetween two of the principal axes. The situations are different on each circumference C i introduced as in figure 3. We revealed that the distribution of gravitational-wave intensities on C can change dramatically according to the background parameters Θ and t ini as explained byfigure 13 : when Θ is sufficiently close to π , a value corresponding to one of the axisymmetriclimits, and H t ini is sufficiently large, the growth of gravitational waves is localised to nearthe k -axis. In the opposite case, the growth takes place rather uniformly on C . On theother circumferences C and C , we showed that the growth is localised to near the k - and k -axis, respectively, for wide ranges of parameters.Third, in section 3.4, we discussed time evolutions of modes with a wavevector pointingin a general direction. Specifically, we demonstrated all-sky maps of the growth factor of thepre-inflationary gravitational waves for the two anisotropy parameters Θ = π (triaxial) and . × π (nearly axisymmetric) as in figure 14. For Θ = π , the growth of gravitationalwaves occurs in a narrow range near the k = 0 great circle on the sphere whose width canbe well estimated by eq. (3.31), while for Θ = 0 . × π , the growth is localised to the k -axis. Using these results, we argued that the topological properties of the pre-inflationarygravitational waves in future all-sky, multiwavelength observations will provide us a crucialprobe for determination of the configuration of the primordial anisotropy, its degree Θ , andthe initial time t ini .Finally, in section 4, we gave some tentative constraints on the initial amplitude of thepre-inflationary gravitational waves, the anisotropy parameter Θ , and the initial time t ini from the B -mode polarisation of CMB observed by the Planck satellite.One of possible directions of extending the present work might be inclusion of a scalardegree of freedom which drives isotropisation and inflation instead of a cosmological constant.Also, it will be meaningful if this sort of analysis can be extended to other Bianchi-typecosmological models.Perturbation analyses in anisotropic cosmologies will be of particular importance inconstraining some kinds of modified theories of gravity. Among others, the Einstein-Weylgravity, whose Lagrangian is L = R − γ C αβγσ C αβγσ , has a special property that it admitsgeneral Einstein spaces, including KdS, as exact solutions. We will come back to this issuein future publications. Acknowledgments
Y.F. and Y.S. are grateful to Nathalie Deruelle for helpful comments and stimulating dis-cussions. They also acknowledge a financial support and hospitality received from YukawaInstitute for Theoretical Physics, Kyoto University, at an early stage of this study. This workwas in part supported by JSPS KAKENHI Grants No. 26800115 and No. JP16K17675 (Y.S.).– 23 – eferences [1] A. H. Guth,
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