Inflation with stable anisotropic hair: is it cosmologically viable?
aa r X i v : . [ g r- q c ] D ec Prepared for submission to JHEP
Inflation with stable anisotropic hair:is it cosmologically viable?
Sigbjørn Hervik, a David F. Mota b and Mikjel Thorsrud b a Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger, Norway b Institute of Theoretical Astrophysics, University of Oslo, N-0315 Oslo, Norway
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Recently an inflationary model with a vector field coupled to the inflaton wasproposed and the phenomenology studied for the Bianchi type I spacetime. It was foundthat the model demonstrates a counter-example to the cosmic no-hair theorem since thereexists a stable anisotropically inflationary fix-point. One of the great triumphs of infla-tion, however, is that it explains the observed flatness and isotropy of the universe todaywithout requiring special initial conditions. Any acceptable model for inflation should thusexplain these observations in a satisfactory way. To check whether the model meets thisrequirement, we introduce curvature to the background geometry and consider axisymmet-ric spacetimes of Bianchi type II,III and the Kantowski-Sachs metric. We show that theanisotropic Bianchi type I fix-point is an attractor for the entire family of such spacetimes.The model is predictive in the sense that the universe gets close to this fix-point after a fewe-folds for a wide range of initial conditions. If inflation lasts for N e-folds, the curvatureat the end of inflation is typically of order ∼ e − N . The anisotropy in the expansion rate atthe end of inflation, on the other hand, while being small on the one-percent level, is highlysignificant. We show that after the end of inflation there will be a period of isotropiza-tion lasting for ∼ N e-folds. After that the shear scales as the curvature and becomesdominant around N e-folds after the end of inflation. For plausible bounds on the reheattemperature the minimum number of e-folds during inflation, required for consistency withthe isotropy of the supernova Ia data, lays in the interval (21 , Keywords:
Cosmology of Theories beyond the SM, Classical Theories of Gravity
ArXiv ePrint: ontents
It is plausible that inflation occurred around the energy scale of the grand unification.Since we do not have direct experimental access to such energy scales we need to beopen-minded regarding the physics possibly occurring there. Indeed, when theoreticalmodels of inflation are compared to observations they might provide the clue to understandfundamental physics at non-accessible energies. In this paper we are concerned with aspecific model that violates isotropy; i.e. three-dimensional rotational invariance. Althoughrotational invariance is a well established feature of low-energy physics, several puzzlingfeatures of the large scale CMB anisotropies hints that this symmetry might have beenbroken during inflation [1–6]. At the same time recent progress in theoretical models hasclarified that inflation with anisotropic hair is a theoretical possibility [7–16]. It has becomeclear, however, that specific realizations are often plagued by instabilities, either ghosts orunstable growth of the linearized perturbations [17–20].In a series of recent papers M. Watanabe, S. Kanno and J. Soda studied the cosmologyof a model with anisotropic hair which, apparently, is free of instabilities [21–25] (forrelated works by other authors also see [26–30]). Their model is inspired from supergravitywhich includes a massless vector field coupled to the scalar field(s). The vector part ofthe supergravity action has so far been neglected in cosmology, but for an appropratelychosen coupling, the authors demonstrated interesting cosmological implications when thebackreaction to geometry is properly accounted for. In particular, it is shown that theanisotropy in the expansion rate is stable and proportional to the slow roll parameter [21].Even in the case when the anisotropy is very small, say on the micro level, the statistical– 1 –nisotropy imprinted in primordial fluctuations can be significant [24]. Thus, it has becomeclear that, in high precision cosmology, one cannot always neglect the back-reaction ofvector-fields to geometry.One of the essential features of the conventional inflationary scenarios, however, is thatthey explain the observed flatness and isotropy of the universe today without requiringspecial initial conditions. For spatially homogeneous models of non-positive curvaturecontaining comoving fluids obeying the strong energy condition, this is a consequence of the cosmic no-hair theorem [31–33] which guarantees that the curvature and shear of the spatialgeometry decay rapidly during inflation. Given that inflation is stable for a sufficiently largetime, the universe will, in agreement with observations, remain almost flat and isotropicuntil today. The above mentioned papers, however, clearly demonstrate that the theoremmay be violated if a massless vector field is appropriately coupled to the scalar field (andnecessarily violating the assumptions given in the theorem). The phenomenology of themodel has so far only been studied in the context of the spatially flat Bianchi type I model.The implications of curvature are therefore, until now, unexplored. A crucial question iswhether the model, while violating the cosmic no-hair theorem, is still consistent with theobserved isotropy and flatness of the universe today. It is therefore important to understandthe behavior of the model with more general initial conditions. In this paper we shallintroduce curvature to the model by considering axisymmetric spacetimes of Bianchi typeII,III and the Kantowski-Sachs metric. To be specific we shall consider the well motivatedcase where both the potential of the scalar field and the coupling function between thevector and scalar fields are exponentials (this specific model is analyzed without spatialcurvature in [23]).The organization of the paper is as follows. In section 2 we give a brief introductionto the model, while in section 3 we motivate our class of considered spacetimes. In section4 we derive the field equations. Section 5 is devoted to dynamical system analyses. First,in 5.1, we characterize the phase-space by identifying and classifying all fix-points. Then,in 5.2, we study the phase flow with arbitrary initial conditions. As we shall see themodel is predictive in the sense that it provides unambiguous initial conditions for thepost-inflationary era. Finally, in section 6, we show that these initial conditions provide aviable cosmological scenario.
The starting point of the model is the following action for the metric g µν , scalar field φ and vector field A µ : S = Z d x √− g " M p R −
12 ( ∇ φ ) − V ( φ ) − f ( φ ) F µν F µν , (2.1)where ( ∇ φ ) = g µν ∇ µ φ ∇ ν φ is the kinetic term of the scalar field, and F µν = ∇ µ A ν − ∇ ν A µ is the field strength of the vector field. In agreement with conventional notation g is the As a rule of thumb, 60 e-folds is sufficient. – 2 –eterminant of the metric, R is the Ricci scalar and M p is the reduced Planck mass.We shall use metric signature ( − , , , V ( φ ) = V e λ φMp , (2.2) f ( φ ) = f e Q φMp , (2.3)where λ and Q are constant parameters that characterize the model. We shall assumeslow-roll inflation which implies λ ≪
1. We shall treat the coupling constant Q as afree parameter, and study the implications on inflation for different values. We considera positive potential which implies V >
0. There are no restrictions on the constant f .Variation with respect to g µν , φ and A µ , respectively, gives the equations of motion: M p E µν = T µ ( φ ) ν + T µ ( A ) ν , (2.4) ∇ µ T µ ( φ ) ν = − Q L A M p ∇ ν φ, (2.5) ∇ µ T µ ( A ) ν = + Q L A M p ∇ ν φ, (2.6)where E µν is the Einstein tensor, and L A = − f ( φ ) F µν F µν is the Lagrangian for the vectorfield. Notice that the total energy-momentum tensor is conserved ∇ µ ( T µ ( φ ) ν + T µ ( A ) ν ) = 0.It is manifest from the equations of motions that the coupling leads to exchange of energyand momentum between the scalar field and the vector field. The rate is determined bythe coupling constant Q . It is the vector field that sources the shear degree of freedom ofthe expansion rate in this model. Without the coupling to the scalar field, however, thevector field would decay rapidly leading to isotropization. Finally we write down the components of the energy-momentum tensors for the scalarfield and the vector field: T µ ( φ ) ν = − δ µν (cid:18)
12 ( ∇ φ ) + V ( φ ) (cid:19) + ∇ µ φ ∇ ν φ (2.7) T µ ( A ) ν = − f ( φ ) F µα F αν − f ( φ ) δ µν F αβ F αβ , (2.8)where δ µν = g µα g αν . The usual approach for anisotropic models where the shear is sourced by a homogeneousvector field, is to assume axisymmetric geometries, often called local rotational symmetric Since we do not consider any shear in the matter source, the term shear will unambiguously refer tothe geometrical freedom in the expansion rate. As an interesting digression we mention that in a model where inflation is driven by non-Abelian gaugevector fields (without any scalar field), the anisotropic hair vanish exponentially fast although the field issufficiently stable to drive inflation, see [34] and references therein. – 3 –LRS) spacetimes. The vector field is assumed to be aligned parallel with the axis ofsymmetry, ie. orthogonal to the plane of rotational symmetry. Although this is a widelyused assumption in the literature [10, 21–24, 35], it seems like a proper discussion is lacking. Beside being a simplifying assumption we think that the approach is appealing since, onthe background level, one has the same symmetry in the spacetime geometry as in the(total) matter field. One can also loosely argue that if the symmetry was broken initially,spacetime would quickly become axisymmetric again, since there is no way to source apreferred direction in the plane orthogonal to the vector field. The purpose of this sectionis to show rigorously that this is actually what happens in the Bianchi type I spacetime.We find that the universe isotropizes in the plane orthogonal to the vector field and thatin the case of inflation this is a very rapid process.We consider the most general Bianchi type I geometry: ds = − dt + a x dx + a y dy + a z dz , (3.1)where a x ( t ), a y ( t ) and a z ( t ) are three independent scale factors. It is convenient to intro-duce three functions α ( t ), σ ( t ) and σ ( t ) defined by a x = e α e − σ − σ , a y = e α e σ , a z = e α e σ . (3.2)In the special case σ ( t ) = σ ( t ) we recover the axisymmetric geometry usually assumed.The three Hubble factors H i ≡ ˙ a i a i can then be written: H x = ˙ α − ˙ σ − ˙ σ , H y = ˙ α + ˙ σ , H z = ˙ α + ˙ σ , (3.3)where a dot denotes differentiation with respect to the cosmic time t . It is also useful tointroduce a mean Hubble rate H ≡
13 ( H x + H y + H z ) = ˙ α. (3.4)As in the Friedman-Lemaˆıtre-Robertson-Walker (FLRW) metric, acceleration can be quan-tified in terms of the deceleration parameter q defined by: q = − − ˙ HH . (3.5)Acceleration is then defined as q <
0. Note that q < d dt e α >
0. Thefunction e α can be interpreted as an isotropic scale factor. More precisely it is the geometricmean of the scale factors. Thus our definition of acceleration is equivalent to acceleration Ref. [24] comments that the shear variable (in the considered two dimensional hypersurface) is exponen-tially decaying in an expanding universe. But the physical interesting quantity is the dimensionless sheardegree of freedom which, roughly speaking, measures the fraction between the anisotropic and isotropicparts of the expansion rate. As we shall see, this one is not necessarily exponentially decaying in anexpanding universe. Note that in quadratic theories the curvature, in some sense, may source itself creating the possibilityfor non-axisymmetric Bianchi type I solutions [9]. – 4 –f the geometric mean of the scale factors. These definitions are standard in homogenouscosmologies, and generalize the definitions of the FLRW model in a natural way.To proceed we impose the metric (3.1) on the gravitational equation (2.4). We assumea single electric-type field. Since there are no spatial symmetries, apart from homogeneityof course, in the considered spacetime, we can without loss of generality align the fieldin the x -direction: F = F µν ( t ) dx µ ∧ dx ν = F ( t ) dx ∧ dt . In the gauge A = 0 this fieldconfiguration corresponds to the vector potential A = A µ dx µ = A x ( t ) dx (where A x arerelated to F by F = − ˙ A x ). By taking linear combinations of the components of (2.4)one can eliminate the matter sources leading to the pure geometrical equation:(¨ σ − ¨ σ ) = − α ( ˙ σ − ˙ σ ) . (3.6)We now introduce a shear degree of freedom: X ⊥ ≡ H z − H y H = ˙ σ − ˙ σ ˙ α . (3.7)The quantity | X ⊥ | is a measure of the anisotropy in the plane orthogonal to the vectorfield (as indicated by the subscript). For | X ⊥ | = 0 the hypersurface is isotropic and wehave the axisymmetric metric usually assumed. Equation (3.6) then implies: d | X ⊥ | dα = −| X ⊥ | (2 − q ) . (3.8)In this equation we have changed to the number of e-folds, α , as time parameter by theidentity dαdt = H . From the gravitational equations it is easy to verify that q < V ( φ ) >
0. This implies d | X ⊥ | dα <
0. Thus we have shown that | X ⊥ | decays monotonically, although, in general, it will not necessarily decay rapidly. Inthe case of inflation, however, we have by definition q <
0. From (3.8) it then follows that d | X ⊥ | dα < − | X ⊥ | . Thus inflation guarantees that | X ⊥ | ( α ) decays faster than the function e − α . To summarize we have showed that the Bianchi type I spacetime isotropizes in theplane orthogonal to the vector field, and that in the case of inflation this is a very rapid(exponential) process.Let us finally use this result to prove that there will be no inflationary fix points withanisotropic expansion in the considered hypersurface. A dynamical system analysis forthe spacetime (3.1) would require two shear variables, say X = ˙ σ ˙ α and X = ˙ σ ˙ α . Notethat from the definition of X ⊥ we have X ⊥ = X − X . It follows that for a fix-point( dX dα = dX dα = 0), we must have dX ⊥ dα = 0. But, as shown above, this is not possible inour model. Thus there will be no additional fix-points if introducing a new shear degree offreedom.Although this analysis has been restricted to the Bianchi type I metric, we believe itsheds some light on the more general class of homogenous geometries, and we will use thisas a motivation for the class of spacetimes to be introduced in the next section.– 5 – Field equation for a class of homogenous and axisymmetric spacetimes
We shall now impose a class of homogenous spacetimes to the field equations. Motivatedby the discussion above we consider a class of axisymmetric versions of Bianchi types I,II, III and the Kantowski-Sachs metric. We shall refer to these as BI, BII, BIII and KS,respectively. Type BI is spatially flat while BII, BIII and KS have anisotropic curvaturein addition to the shear. For these spacetimes there is an intrinsic rotational symmetry inthe spatial curvature which is aligned with the rotational symmetry of the expansion rate.Our considered class of spacetimes can be written on the form: ds = − dt + e α ( t ) (cid:16) e − σ ( t ) w ⊗ w + e σ ( t ) w ⊗ w + e σ ( t ) w ⊗ w (cid:17) , (4.1)where w i are three time-independent and mutually orthogonal one-forms which are relatedto the coordinate basis in table 1. These spacetimes have rotational symmetry in the planespanned by w and w . The rotational symmetry in the expansion rate is manifest from theline-element on this form. The rotational symmetry of the anisotropic curvature, on theother hand, is manifest first after an appropriate coordinate transformation. To obtainthe same symmetry in the (total) matter as in the spacetime geometry, we shall align thevector field parallel to w .Spacetime R w w w BI 0 dx dy dz
BII − k e − α − σ dx + k ( ydz − zdy ) dy dz BIII & KS 2 ke − α − σ dx (1 − ky ) − / dy ydz Table 1 . Geometric variables in the various spacetimes. w i are the one-forms in the line element(4.1). R is the spatial Ricci scalar. k is a constant. For BIII k <
0, while k >
Since the one-forms are time-independent one can read the scale factors directly fromthe line-element (4.1): a k = e α − σ , a ⊥ = e α + σ , (4.2)where a k and a ⊥ are the scale factors in the direction parallel and perpendicular to thevector field, respectively. The corresponding Hubble factors are: H k = ˙ α − σ, H ⊥ = ˙ α + ˙ σ. (4.3)The mean Hubble rate becomes similar as in the previous section, H = ˙ α .We shall now introduce two spacetime dependent coefficients s and s with valuesspecified in table 2. The former is defined by ˙ R = −
2( ˙ α + s ˙ σ ) R , where R is the three-dimensional Ricci scalar of constant time hypersurfaces. The functions R for the variousspacetimes are specified in table 1. The latter ( s ) determines the strength of a coupling As an example, in the Bianchi type II one can use y = r cos( φ ) and z = r sin( φ ). – 6 –etween the curvature and the energy density of the vector field. We can then treat theentire class of spacetimes in a unified way by expressing the equations in terms of s , s and R .Next, let us consider the matter sources. The most general field strength compatiblewith the class of spacetimes is on the form F = ˙ v w ∧ w + ( b + 2 s kv ) w ∧ w , where v = v ( t ) is the dynamical degree of freedom and b is a constant. This field is homogeneous,axisymmetric and satisfy the identity d F = d A = 0. Note that the field includes bothan electric-type and a magnetic-type component which are parallel and pointing in the w direction. Previous studies (which are restricted to the BI metric) have consideredthe case corresponding to b = 0, ie. neglecting the possibility of a magnetic component. Since our intention is to generalize the geometry, and not the matter fields, we will put b = 0. In that case there is no magnetic-type component in the BI, BIII and KS spacetimes(since s = 0). In the BII spacetime, however, where s = 1, there is still a magnetic-typecomponent. This is the minimal magnetic component required to satisfy the source-freeMaxwell equations (d F = 0). Our considered field strength is therefore: F = ˙ v ( t ) w ∧ w + 2 s kv ( t ) w ∧ w . (4.4)The corresponding vector potential in the gauge A = 0 is A ≡ A µ w µ = vw .We shall continue to specify all tensors, and perform all calculations, relative to theone-forms ( dt, w , w , w ). In this basis the energy-momentum tensor is diagonal and wewrite: T µν = diag( − ρ, p k , p ⊥ , p ⊥ ) . (4.5)Since T µν is a diagonal mixed tensor of rank (1 , w , w , w ), ie. under a change to an orthonormal basis.Thus the components of T µν are physical quantities representing the energy density andpressure as measured in the fluid rest frame. We split the energy and pressure in thecontributions from the scalar field ( φ ) and vector field (A): ρ = ρ φ + ρ A ,p k = p φ + ( p A ) k ,p ⊥ = p φ + ( p A ) ⊥ . (4.6)The energy density and pressure of the scalar field takes the standard form in the entirefamily of spacetimes: ρ φ = 12 ˙ φ + V ( φ ) , p φ = 12 ˙ φ − V ( φ ) . (4.7)The energy density (and pressure) of the vector field, on the other hand, depends on thespacetime: ρ A = f e − α +4 σ (cid:18)
12 ˙ v − s ( R ) v (cid:19) . (4.8) As we shall see, it turns out that such a coupling exists in BII, but not in BIII or KS. Reference [22] treated magnetic fields perturbatively. – 7 –s s = 0 only for BII it turns out that the spatial curvature couples to the energy densityonly in this case. Clearly, this is due to the magnetic-type field unique for BII. Note that,since R < p A ) k = − ρ A , ( p A ) ⊥ = + ρ A . (4.9)The relation ( p A ) k = − ( p A ) ⊥ also hold in the case of a massive vector field [28] (at leastin the Bianchi type I metric).The field equations (2.4)-(2.6) for the considered family of spacetimes can be written:Spacetime s s BI 0 0BII 4 1BIII & KS 1 0
Table 2 . Spacetime dependent coefficients. H − ˙ σ = ρ M p − R , (4.10)˙ H + 3 H = 12 M p ( ρ − p k − p ⊥ ) − R , (4.11)¨ σ + 3 H ˙ σ = p ⊥ − p k M p − s R , (4.12)˙ R = −
2( ˙ α + s ˙ σ ) R, (4.13)˙ ρ φ + 3 H ( ρ φ + p φ ) = Q L A φM p , (4.14)˙ ρ A + 4( H + ˙ σ ) ρ A = − Q L A φM p , (4.15)where L A = f ( φ ) e − α +4 σ (cid:0) ˙ v + 2 s v ( R ) (cid:1) . From (4.12) it is manifest that the shear issourced by anisotropic pressure and anisotropic curvature.We will now introduce dimensionless variables and rewrite the field equations as anautonomous set of first order differential equations. First we introduce the shear variable X ≡ H ⊥ − HH = ˙ σ ˙ α , (4.16)and a variable for the curvature: Ω K = − R H . (4.17)Note that Ω K > Y = ˙ φM p H , Z = f e − α +2 σ ˙ vM p H , V = s vf e − α +2 σ M p . (4.18)– 8 –he variables X, Y and Z are similar to those used in [23]. The additional variables Ω K and V are required to study the more general family of spacetimes considered here. Weshall refer to the space spanned by the set of independent variables ( X, Y, Z, Ω K , V ) as state space . The space spanned by the constant parameters λ and Q , we shall refer to as parameter space . Note that V 6 = 0 only in BII due to the coefficient s in the definition of V . It is only in BII, where R couples to the energy density of the vector field, that thevariable V is needed.We can make use of the Hamiltonian constraint equation (4.10) to eliminate V ( φ ) fromthe equations of motions. Using the identity dαdt = H we change to the scale, α , as timeparameter, and the autonomous equations becomes: dXdα = 13 Z ( X + 1) + X (cid:20) X −
1) + 12 Y (cid:21) + Ω K (cid:0) s + X + 4 V (1 + X ) (cid:1) , (4.19) dYdα = ( Y + λ ) (cid:18) X −
1) + 12 Y (cid:19) + 13 Y Z + ( Q + λ Z (4.20)+ Ω K (cid:0) Y + 3 λ + 2 V (3 λ − Q + 2 Y ) (cid:1) ,dZdα = Z (cid:20) X −
1) + 12 Y − QY + 1 − X + 13 Z (cid:21) + Ω K (cid:2) Z + 4 Z V − V (cid:3) , (4.21) d Ω K dα = 2Ω K (cid:20) − − s X + 3 X + 12 Y + 13 Z + Ω K + 4Ω K V (cid:21) , (4.22) d V dα = ( QY + 2 X − V + s Z, (4.23)The dynamical variables are subject to the constraint: X + 16 Y + 16 Z + Ω K (cid:0) V (cid:1) < , (4.24)which follows from our considered case of a positive potential ( V ( φ ) > K ≥
0. Morespecifically X must be in the interval ( − , −√ , √ K (cid:0) V (cid:1) in (0 , K <
0. Due to the negative curvature, the KS spacetime might collapse.At the turning point, from expansion to contraction, we have H = 0, and all the variables(apart from V ) diverges ( → ∞ ) as seen from the definitions (4.16)-(4.18). For initialconditions leading to collapse, α is then usually not a suitable time-parameter. For ourpurposes, however, it is fine since we are just interested in whether , and eventually when ,the universe collapse for a given set of initial conditions.It is only in BII that both R = 0 and s = 0. For the other spacetimes the equationssimplifies somewhat: Ω K → , V → . Although we shall study the dynamics in terms of the independent variables ( X , Y , Z , Ω K , V ), it is useful to introduce some auxiliary variables in order to make a closer– 9 –onnection to the physics. The (energy) density parameters for the vector field and thescalar field can be expressed in terms of the independent variables in the following way:Ω A ≡ ρ A M p H = 16 Z + 2Ω K V (4.25)and Ω φ ≡ ρ φ M p H = 1 − X − Z − Ω K (1 + 2 V ) . (4.26)Furthermore, we can split the latter one in the contributions from the kinetic and thepotential energy: Ω φ = Ω kin +Ω V , where Ω kin = Y . The Hamiltonian constraint equation(4.10) can then be written on the generic form:1 = X + Ω kin + Ω V + Ω A + Ω K . (4.27)It is also useful to express the deceleration parameter defined in (3.5) in terms of theindependent variables: q = − X + 12 Y + 13 Z + Ω K (cid:0) V (cid:1) . (4.28)From the definition of Ω K it follows that d Ω K dα = − K ( s X − q ) . (4.29)We notice that, due to the shear , there is no guaranty for monotonically decaying curvatureduring inflation ( q < V is monotonicallyincreasing in a large region of state space. This leads the universe close to an anisotropicfix point with linear stability and vanishing curvature. Equipped with the field equations we shall now investigate the phase-space structure bydynamical system analysis and simulations. First we shall identify the fix points of thesystem and classify their linear stabilities. Although this is a powerful way to characterizephase space qualitatively, it gives unambiguous predictions only in the linear regime closeto the fix points. With arbitrary initial conditions it is therefore usually necessary to runsimulations. As we shall see, however, in our case the potential energy of the scalar field ismonotonically increasing in a large region of state space for BI, BII and BIII, leading thesystem close to a stable and unique anisotropic fix point of type BI. The KS spacetime isa bit more complicated. Firstly, in this case Ω V is not a monotone function. Secondly, theKS spacetime might collapse. We therefore investigate the phase flow in the KS universeby running simulations. Our results indicates that if the universe does not collapse, its fateis similar to that of the Bianchi type spacetimes. The model is therefore predictive and,as we shall demonstrate by simulations, the universe gets close to the stable anisotropicfix-point typically within a few e-folds. – 10 – .1 Fix-points and linear stability The fix points of the system are found by setting the left-hand side of the dynamicalequations (4.19)-(4.23) equal to zero and solving the algebraic equation. The stability isdetermined by linearizing the field equations around the fix-points, dδX i dα = M δX i , andevaluating the eigenvalues of the matrix M . If the real part of all eigenvalues are negative,the fix-point is stable and we call it an attractor . If not all values are negative, the fix-pointis unstable. Unstable fix-points are called saddles if there are both positive and negativeeigenvalues, and repellers if all are positive. In general the phase flow goes from repellers,possibly via saddles, towards attractors. Without loss of generality we shall assume λ > Under the assumption λ ≪ X Y Z Ω K V (a) ∼ λQ − Q ∼ − Q ∼ λQ − Q ∼ s Z (b) 0 − λ − λ Q − λ − Q + λ )2 Q − λ − − λ λ − λ λ − − λ λ ) ±√ − X Table 3 . Coordinates of fix-points. For fix-point (a) we have written down an approximation. Theexact position of (a) can be found in the text.
Name Spacetime Existence Stability q Comment(a) BI Q ≫ ∼ ( − λQ )(b) FLRW - - − λ (c) KS 2 Q > λ saddle − − λ λ (e) BI - unstable 2 boundary(f) BIII - saddle 1 / Table 4 . Properties of fix-points. The stability of (b) depends on λ and Q (it is a saddle when (a)exists, and stable if not). See text for more details. In table 4 we give an overview of certain properties of the fix-points. Notice that thedeceleration parameter q ∼ − q > V ( φ ) = 0, while (a)-(d) satisfy the constraint (4.24). Notice that there are The situation λ → − λ is equivalent to φ → − φ . – 11 –o inflationary fix-points of type BII or BIII. For KS there are two inflationary fix-points,(c) and (d), but they are both saddles. A fix-point of special significance is the anisotropicattractor (a). Notice that it exists only when Q ≫
1. Thus, interestingly, the shear X issmall if it exists. The only fine tuning in the model is therefore the usual λ ≪ Q ≪ − Fixpoint (a)
This is the anisotropically inflationary fix-point of Bianchi type I firstidentified in [23]. In table 3 the coordinates of (a) are given only to lowest order in thesmall quantities λQ and Q − . The exact position is: X = 2( λ + 2 Qλ − λ + 8 Qλ + 12 Q + 8 , (5.1) Y = − λ + 2 Q ) λ + 8 Qλ + 12 Q + 8 , (5.2) Z = 18( λ + 2 Qλ − − λ + 4 Qλ + 12 Q + 8)( λ + 8 Qλ + 12 Q + 8) , (5.3)Ω K = 0 , (5.4) V = s Z − X − QY . (5.5)Since the energy of the vector field is positive, ie. Ω A >
0, we get the condition Z >
0. Thefix-point (a) therefore only exist in the parameter region where 2 Qλ + λ >
4. Since λ ≪ Q ≫
1. As seen most directly from the approximations in table 3, it followsthat (a) is near the origin of the state space ( X , Y , Z ,Ω K , V ). Consequently the fix-point isstrongly dominated by the potential energy of the scalar field. For clarity we expand the– 12 –igenvalues in λQ and Q − , and truncate at zero order. For BI the three eigenvalues are: − , − − i r λQ + λ − − , −
32 + i r λQ + λ − − ! . (5.6)For BII one has the additional eigenvalues − −
2, while for BIII and KS one has theadditional eigenvalue −
2. The real part of all eigenvalues are negative for all spacetimes.Thus we have showed that the flat and anisotropic fix point identified in [23], is an attractoralso for the more general class of spacetimes considered here. Moreover, as we will verifybelow, it turns out to be the unique attractor (when it exists) for our considered class. Fi-nally we mention that, as shown in [23], there is an exact power-law solution correspondingto this fix-point, where the line element takes the form: ds = − dt + t k − k dx + t k +2 k ( dy + dz ) , (5.7)where, to lowest order, k ∼ Qλ and k ∼ λQ − λQ . Fixpoint (b)
This is a flat inflationary fix-point of type FLRW containing a scalar fielddominated by its potential energy. The three eigenvalues for BI are: (cid:18)
12 (2 λQ + λ − , − λ , − λ (cid:19) . For BII one has the additional eigenvalues − − λQ and − λ , while for BIII and KS onehas the additional eigenvalue − λ . This means that for the entire class of spacetimes,(b) is a saddle when (a) exists. If (a) does not exist, ie. 2 λQ + λ − <
0, then (b) is anattractor for BI, BIII and KS. For BII it is also stable in a large parameter region if (a)does not exist, but not if Q ≪ −
1, in which case it is unstable.
Fixpoint (c)
This is an inflationary fix-point of type KS containing an electric-type fieldand a cosmological constant (since the kinetic part of φ is vanishing). Essentially it is ageneralization of the solutions with a pure cosmological constant found in [36] and [37].From the condition Z > Q > λ . The four eigenvalues are: − , , −
32 + 12 s − λQ Q + λ Q − λ , − − s − λQ Q + λ Q − λ ! . Notice that the real part of the two latter eigenvalues are always negative when 2
Q > λ ,ie. when it exists. We note that (c) is a saddle.
Fixpoint (d)
This is an inflationary fix-point of type KS containing only our consideredscalar field. First note that there are no condition on Q for the existence of this fix-point.To lowest order in λ the eigenvalues of (d) are: (cid:16) − O ( λ ) , − O ( λ ) , O ( λ ) , λQ + O ( λ ) (cid:17) . – 13 –ince λ ≪ Q . In any case (d) is a saddle. Although our analysisfocus on slow-roll inflation λ ≪
1, we mention that there exist a region in parameter space(where λ is larger than unity) where all the eigenvalues are negative and the fix point isa stable attractor. In this region, however, the deceleration parameter is positive and thefix-point is not inflationary. Fixpoint (e)
This is a decelerating fix-point of type BI containing only a pure kineticscalar field (thus the fluid is stiff ρ = p ). The solution is part of a broader solution commonlyreferred to as Jacobs disc [38]. See also [39] for a discussion of such solutions. Note that(e) is a curve of fix-points, satisfying 6 X + Y = 6. Since V ( φ ) = 0, on the curve, it layson the boundary of our considered state space. We have two sets of eigenvalues dependingon the sign of Y = ±√ − X . The eigenvalues for BI: (cid:16) , − X ∓ Q p − X , ± λ p − X (cid:17) The sign of the third eigenvalue is always positive since λ ≪
1, while the second eigenvaluedepends on Q and the sign of Y . For BII one has the additional eigenvalues 4 − X and − X ± Q √ − X , while for BIII and KS one has the additional eigenvalue 4 − s X .In any case (e) is unstable. Fixpoint (f )
This is a decelerating fix-point of type BIII containing no fluids. Essentiallyit is a Bianchi type III generalization of the Milne universe (but with a trivial flat direction).Like (e), it lays on the boundary V ( φ ) = 0. The four eigenvalues are:(3 , − / , − / , − / . Thus (f) is a saddle.
The stability analysis above determines the phase flow close to the fix-points. With ar-bitrary initial conditions, however, we need something more to determine the fate of thedynamical system. For the spacetimes of type BI, BII and BIII we shall now see that faraway from (a) the phase flow can be characterized by the potential energy of scalar field.In section 4 we defined an auxiliary variable Ω V , representing the potential energy ofthe scalar field. The equation of motion is: d Ω V dα = Ω V [ λY + 2 q + 2] = Ω V F, (5.8)where F = 6 X +( Y + λ ) − λ + Z +2Ω K (1+4 V ). Note that if Ω K ≥ and | X i | ≫ λ forat least one of the variables, then F >
0, and consequently, Ω V is monotonically increasing.Note that, as implied by (4.27), Ω V ∼ ≪ V ∼ ≪ V will therefore lead the system close to the origin of the state space ( X , Y , Z ,Ω K , V ). In this– 14 – IIBIIIKS BI X - - W _K0.000.020.040.06 Z Figure 1 . The phase flow of X , Z and Ω K with λ = 0 . Q = 50. The black, green, red andblue curves, respectively, represents simulations with BI, BII, BIII and KS initial conditions closeto fixpoint (a). All trajectories converge towards a common point which is the type BI anisotropicfix point (a). region we have the stable anisotropic fix-point (a). The system will therefore eventuallyend up at the anisotropic fix-point (a) for arbitrary, but non-special , initial conditions.In principle, it could take arbitrary long time for the system to get close to (a), since itmight spend long time at the some of the saddles (for example at (b) which is also locatedclose to the origin). Simulations, however, demonstrates that the typical time spent to getclose to (a), starting with arbitrary initial conditions, only represents a minor fraction ofthe total number of e-folds during inflation. To demonstrate this, we define the distanceto the fix-point (a) by D = p ( X − X a ) + ( Y − Y a ) + ( Z − Z a ) + (Ω K − (Ω K ) a ) + ( V − V a ) , (5.9)where X a is the value of X at (a) and similar for the other coordinates. Note that thedistance between (a) and (b) is of order ∼ λ ∼ Q − . We say that the system is close to (a)when D < λQ . With this definition we ensure that the system is effectively unaffected by(b) and attracted by (a). In table 5 we have summarized some simulations with λ = 0 . Q . For Q = 50 we note that it typically takes 4-14 e-folds forthe system to get close to fix-point (a). For Q = 1000 it typically takes 5-7 e-folds. One can construct special initial-conditions not leading to (a). With initially vanishing vector field forinstance, Z = 0, the universe will converge towards the isotropic fix-point (b) instead of (a). – 15 – c LH d L H a LH b L - - X - Y Z Figure 2 . The phase flow of X , Y and Z for KS initial conditions with λ = 0 . Q = 50.Initial conditions and direction of flow is indicated by the arrow. The fixpoint (a), (b), (c) and (d)are indicated by the colored points. Initial conditions are carefully chosen such that the solution”rides” on both KS-saddles (d) and (c) before ending up in the BI anisotropic attractor (a). The above analysis does not apply for KS since Ω V is not a monotone function whenΩ K <
0. Instead we study the KS spacetime by numerical simulations. Some results forrandomly chosen initial conditions are summarized in table 6. As mentioned above, the KSuniverse might collapse, in which case H →
0. Simulations shows that non-special initialconditions, either lead to collapse or the system will converge towards the fix-point (a). Forinitial conditions leading to collapse, the table shows how long time it takes before H = 0.We note that H → D < λQ from (a). The results are very similar as those for BI, BII and BIII if the universedoes not collapse. The system is close to (a) typically after 4-14 e-folds for Q = 50 and 5-7e-folds for Q = 1000.Supported by simulations we have showed that for initial conditions leading to in-teresting cosmologies (no immediate recollapse), the universe gets close to (a) relativelyquickly. When the system is close to (a) we see from (4.22) that the curvature decays ex- We should mention that for several of the (randomly chosen) initial conditions in tables 5 and 6, theuniverse is not accelerating, ie. q might be positive at the initial time α = 0. The time taken to get closeto (a) after the start of inflation, is therefore somewhat smaller than the times in the table for the cases – 16 –onentially, Ω K ∼ e − α . If inflation lasted for N e-folds, a rough estimate for the curvatureat the end of inflation is therefore (Ω K ) end ∼ e − N . To check if the model gives rise to an acceptable cosmology we will now investigate the late-time behavior. We showed above that at the end of inflation the universe will be extremelyclose to the anisotropic fix-point (a), with a curvature of order (Ω K ) end ∼ e − N if inflationlasted for N e-folds. At the end of inflation we assume a period of reheating where theenergy of the scalar and vector fields are dumped primarily into radiation. Reheating is apoorly understood process, but we shall assume that it is sufficiently fast that we can takethe initial conditions for the shear ( X ) and the curvature (Ω K ) to be similar as at the endof inflation. In our analysis we shall, for simplicity, introduce a single perfect fluid withequation of state p = ωρ . The radiation dominated era after inflation can then be studiedby setting ω = , while the subsequent period dominated by non-relativistic fluids can bestudied by setting ω = 0. We shall neglect the cosmological constant, since it would notchange the estimates we seek significantly.The conservation equation in the class of spacetimes yields˙ ρ + 3 H (1 + ω ) ρ = 0 . (6.1)The associated density parameter is definedΩ M = ρ M p H . (6.2)The Hamiltonian constraint equation can then be written:1 = X + Ω M + Ω K , (6.3)where X and Ω K is defined in the same way as above. Due to the constraint (6.3) thesystem is effectively two dimensional, and we choose Ω M as the auxiliary variable. Theautonomous equations for the post-inflationary era with a single perfect fluid can then bewritten: dXdα = − X (cid:2) (1 + 3 ω )Ω K + 3(1 − ω )(1 − X ) (cid:3) + s Ω K , (6.4) d Ω K dα = Ω K (cid:2) (1 − Ω K )(1 + 3 ω ) − s X + 3(1 − ω ) X (cid:3) . (6.5)This system has four fix-points (P1)-(P4) summarized in table 7. The real parts of theeigenvalues can be found in table 8. Note that there are no fix-points of type KS. Withoutintroducing a dark energy in the model, the KS spacetime will eventually collapse [40].The Bianchi type I fix-point (P4) represents the two special points on the Kasner circlewhich are LRS [41]. Also note that it is a special case of (e) in table 3. Setting ω = 0, the where q > – 17 –ttractors (P2) and (P3) represents the late time solutions for BII and BIII, respectively,for a universe without dark energy. Fix-point (P2) corresponds to the Collins-StewartBianchi type II exact solution [42], while (P3) is similar as fix-point (f) in table 3. For ω = 0, (P2) and (P3) are the global attractors for the most general perfect fluids of typeII and III, respectively, even including tilt [43], [44].Note that the shear in the attractors (P2) and (P3), with X around unity, is quiteextreme. Supernova Ia data gives the bound − . < X today < .
012 to one sigmaconfidence level for a Bianchi type I spacetime with rotational symmetry [45]. The goal ofthis section is to find the minimum number of e-folds during inflation, N min , required fora shear today in agreement with supernova observations. For this estimate we shall use | X today | < .
01. More stringent bounds can probably be found from the CMB which isvery sensitive to shear [46–49]. For simplicity we shall use the supernovae since the finalresult is not very sensitive to the bound on X today . Thus, observational bounds certainlyimplies that the universe must have been very close to the flat FLRW saddle (P1) justbefore dark energy became significant. To determine N min we must therefore investigatethe dynamics close to the fix-point (P1). Linearizing (6.4)-(6.5) around (P1) yields: dXdα = −
32 (1 − ω ) X + s Ω K , (6.6) d Ω K dα = (1 + 3 ω )Ω K . (6.7)These equations can be solved exactly: X ( α ) = e − (1 − ω ) α (cid:20) ± s ω e (5+3 ω ) α − N (1 − e − (5+3 ω ) α ) + λQ − Q (cid:21) , (6.8)Ω K ( α ) = ± e (1+3 ω ) α − N , (6.9)where we have used the initial conditions X (0) = λQ − Q and Ω K (0) = ± e − N . Here N isthe number of e-folds during inflation. The initial condition for the shear corresponds tothe shear at the anisotropic fix-point (a), while for the curvature we have used the estimateat the end of inflation derived above. To get some intuition for the solution we shall nowset ω = 1 / < α < N wehave the approximation X ∼ λQ − Q e − α . Thus there is an era of isotropization startingright after reheating, where the shear decays as ∼ e − α , lasting until α ∼ N . Also noticethat X ∼ Ω K around α = N . This is also true for KS since the sign of X will changefrom positive to negative around the same time. In the period N < α < N we have theapproximation X ∼ ± s e α − N . In this period we have ln | X | ∼ ln | Ω K | ∼ α − N .We therefore say that the curvature ”tracks” the shear in this period. Around α = N the parameters X and Ω K will approach unity and the universe is no longer close to thematter dominated fix-point (P1). The solutions (6.8)-(6.9) are then no longer valid. Inthe regime α > N the universe will in general converge towards the attractors (P2) and(P3), respectively, for BII and BIII, while KS will eventually collapse. Simulations of theautonomous equations (6.4)-(6.5) verifies these approximations. For an example with BIIIspacetime, see figure (3). Note how the shear decays until it catches up with the curvature– 18 –
20 40 60 80 Α - - - - W _KX W _M Figure 3 . Simulation of BIII after inflation with equation of state ω = 1 / λ = 0 . Q = 50 and number of e-folds N = 60. slightly before α = N . After that the shear is tracked by the curvature, approaching unityaround α ∼ N . This behavior is in close agreement with the above approximations. It iseasily seen from the solutions (6.8)-(6.9) that the same tracking behavior occurs also with ω = 1 /
3. Thus the shear will be tracked by the curvature also in the matter dominatedera. Reheating is complete when practically all of the energy is in radiation at thermalequilibrium. We shall denote the time and temperature at this stage by α reh and T reh ,the reheat time and temperature, respectively. Furthermore we shall denote the time atradiation-matter equivalence by α eq , and today by α . Setting the initial time α reh = 0 at T = T reh we get α = ln T reh T , where T = 2 . K is the temperature of the CMB today.Here we have used that the temperature of a photon gas scales as ∝ e − α . Matter-radiationequivalence occurred around redshift 3300 corresponding to α eq ∼ α −
8. We can thenfind the minimum number of e-folds during inflation, N min , required for a shear todayconsistent with supernova data, ie. | X | < − . The initial condition for the curvature isΩ K ( α reh ) = e − N . Assuming that the universe is close to the saddle (P1) all the way up totoday, the curvature grows approximately as Ω K ∝ e α in the radiation dominated era, andas Ω K ∝ e α in the matter dominated era. The curvature today is then (Ω K ) ∼ e α eq + α − N .Since the shear is tracked by the curvature both in the radiation and matter dominatederas, we have X ∼ (Ω K ) . Let | X max | denote the maximum shear today consistent withobservations. The condition | X | ≤ | X max | , then leads to: N min = ln T reh T −
12 ln | X max | − . (6.10)As discussed above we shall use | X max | = 10 − , consistent with the supernova Ia data. Alower bound on the reheat temperature T reh & N min = 21. There is also an upper limit T reh < GeV if supersymmetry exists [50],corresponding to N min = 48. – 19 –o summarize, we have showed that plausible bounds on T reh implies N min in theinterval (21 , In this paper we have studied an inflationary scenario with a stable anisotropic hair. Sincethe model provides a counter-example to the cosmic no-hair theorem, it is important tostudy the model with more general initial conditions. Ultimately, as suggested in [21, 23],this might lead to a modified cosmic no-hair theorem. As a first step we took a dynamicalsystem approach and showed that the stable anistropic fix-point of type BI, identified in[23], is the unique attractor for a wider class of spacetimes exhibiting spatial curvature.Moreover, for the considered Bianchi type spacetimes, we showed that the potential energyof the scalar field (Ω V ) is monotonically growing in a large variable region leading theuniverse close to the origin of the state space. In this region we have the stable anisotropicfix-point. This observation explains the rapid convergence in numerical simulations witharbitrary initial conditions, and, in fact, it is a reminiscent of the cosmic no-hair theoremitself. The KS spacetime, however, which (depending on initial conditions) might col-lapse, is more complicated and our analysis relies on numerical simulations. For initialconditions not leading to collapse, our simulations shows that the behavior is quite simi-lar as for the considered Bianchi type spacetimes. For arbitrary, but non-special , initialconditions, the universe will typically get close to the attractor after a few e-folds for anyof the considered spacetimes. Thus, for the major fraction of e-folds required for inflation,the inflationary universe is well described by this fix-point.Thanks to these features the model provides unambiguous initial conditions for the eraafter inflation. After reheating, when the energy of the scalar and vector fields are dumpedprimarily into radiation, the only hair left is the anisotropic expansion. Interestingly, theseinitial conditions yields an acceptable late-time cosmology. After reheating there will be aperiod of isotropization lasting for ∼ N e-folds, where N is the number of e-folds duringinflation. After that the shear scales as the curvature and becomes dominant around N e-folds after the end of inflation. For plausible bounds on the reheat temperature, theminimum number of e-folds during inflation, required for consistency with the isotropy ofthe supernova Ia data, lays in the interval (21 , yes , the results obtained for our restricted class of spacetimes indicates thatinflation with anisotropic hair is cosmologically viable. The only difference is that, while the no-hair theorem relies only on the fact that Ω V is a monotonefunction, Ω V is only monotone until it is almost 1 in our considered model. Our argument therefore alsorelies on the dynamical system analysis. For example, if the energy density of the vector field is initially vanishing, it will always be vanishing,and the universe goes to the isotropic FLRW saddle point. – 20 – cknowledgments
DFM thanks the Research Council of Norway FRINAT grant 197251/V30. DFM is alsopartially supported by project PTDC/FIS/111725/2009 and CERN/FP/116398/2010. Wealso appreciate useful comments from an anonymous referee.
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D < λQ at time ( α )Spacetime X Y Z Ω K V Q = 50 Q = 1000BI 0 . − . . . . − . − . . . . − . . − . . . . . − . . . . − . − . . . − . − . . . − . . . − . − . − . . − . . . − . . − . . − . . . − . . − . . − . . . − . . − . . − . . . − . − . . . . . . . − . . . . − . − . − . . . . − . − . − . . . . . − . . . . . Table 5 . Simulations for Bianchi type spacetimes with initial conditions generated by a randomnumber generator. D is the distance to fix-point (a) defined in (5.9). When D < λQ the universe isin the linear regime close to fix-point (a), and the distance to (a) is much smaller than the distanceto (b). Two runs with different values for Q , while λ = 0 . Initial conditions
D < λQ at time ( α ) H = 0 at time ( α ) X Y Z Ω K Q = 50 Q = 1000 Q = 50 Q = 10001 0 . − . . − . . . − − . . − . − . . . − − . − . . − . . . − − − . . . − . − − . .
35 0 . − . . − . . . − − . . . − . − − .
03 0 .
037 1 . − . − . − . − − .
13 0 .
138 1 . − . − . − . − − .
03 0 . − . − . . − . − − .
17 0 . − . . − . − . − − .
04 0 . . − . . −
23 3 . . − −
12 0 . − . − . −
22 4 . . − −
13 0 . . . −
28 5 . . − −
14 0 . . − . −
21 5 . . − − − . − . − . −
24 4 . . − − Table 6 . Simulations for KS with various initial conditions. D is the distance to fix-point (a) definedin (5.9). Two runs with different values for Q , while λ = 0 . K wherepicked uniformly by a random number generator from the intervals ( − . , − . − . , − . − , − V ∼ | Ω K | . – 24 –ame Spacetime X Ω K Ω M Stability(P1) FLRW 0 0 1 saddle(P2) BII (1+3 ω ) (1+2 ω − ω ) − ω attractor(P3) BIII
12 34 ± Table 7 . Post-inflationary fix-points. The unstable fix-point (P4) is either a saddle or a repellerdepending on the spacetime and the sign of X (see eigenvalues). Name Spacetime Eigenvalues(P1) FLRW (cid:0) ω, − (1 − ω ) (cid:1) (P2) BII (cid:0) − (1 − ω ) , − (1 − ω ) (cid:1) (P3) BIII (cid:0) − , − ω (cid:1) (P4) BI (4 ∓ s , − ω )) Table 8 . Real part of eigenvalues for post-inflationary fix-points.. Real part of eigenvalues for post-inflationary fix-points.