On the Stability of the Einstein Static Universe

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provided by CERN Document Server John D. Barrow1, George F.R. Ellis2, Roy Maartens2,3, Christos G. Tsagas2

1DAMTP, Centre for Mathematical Sciences, Cambridge University, Cambridge CB3 0WA, UK 2 Department of Mathematics & Applied Mathematics, University of Cape Town, Cape Town 7701, South Africa 3 Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 2EG, UK

We show using covariant techniques that the Einstein static universe containing a perfect fluid is always neutrally stable against small inhomogeneous vector and tensor perturbations and neutrally 2 1 stable against adiabatic scalar density inhomogeneities so long as cs > 5 , and unstable otherwise. We also show that the stability is not significantly changed by the presence of a self-interacting scalar field source, but we find that spatially homogeneous Bianchi type IX modes destabilise an Einstein static universe. The implications of these results for the initial state of the universe and its pre-inflationary evolution are also discussed.

I. INTRODUCTION absorbed into the potential V . The fluid has a barotropic equation of state p = p(ρ), with sound speed given by 2 The possibility that the universe might have started cs = dp/dρ. The total equation of state is wt = pt/ρt = out in an asymptotically Einstein static state has recently (p + pφ)/(ρ + ρφ). been considered in [1], thus reviving the Eddington- Assuming no interactions between fluid and field, Lemaˆıtre cosmology, but this time within the inflationary they separately obey the energy conservation and Klein- universe context. It is therefore useful to investigate sta- Gordon equations, bility of the family of Einstein static universes. In 1930, Eddington [2] showed instability against spatially homo- ρ˙ +3(1+w)Hρ =0, (1) geneous and isotropic perturbations, and since then the φ¨ +3Hφ˙ + V 0(φ)=0. (2) Einstein static model has been widely considered to be unstable to gravitational collapse or expansion. Never- The Raychaudhuri field equation theless, the later work of Harrison and Gibbons on the a¨ 8πG 1 entropy and the stability of this universe reveals that the = (1 + 3w)ρ + φ˙2 V (φ) , (3) a − 3 2 − issue is not as clear-cut as Newtonian intuition suggests. In 1967, Harrison [3] showed that all physical inhomo- has the Friedmann equation as a first integral, geneous modes are oscillatory in a radiation-filled Ein- stein static model. Later, Gibbons [4] showed stability of 8πG 1 K H2 = ρ + φ˙2 + V (φ) , (4) a fluid-filled Einstein static model against conformal met- 3 2 − a2 ric perturbations, provided that the sound speed satisfies 2 1 cs dp/dρ > 5 . where K =0, 1, and together they imply The≡ compactness of the Einstein static universe, with ± K an associated maximum wavelength, is at the root of H˙ = 4πG φ˙2 +(1+w)ρ + . (5) this “non-Newtonian” stability: the Jean’s length is a − a2 significant fraction of the maximum scale, and the maxi- h i mum scale itself is greater than the largest physical wavelength. Here we generalize Gibbons’ results to include II. THE EINSTEIN STATIC UNIVERSE general scalar, vector, and tensor perturbations and a self-interacting scalar field. We also show that the Ein- stein static model is unstable to spatially homogeneous The Einstein static universe is characterized by K =1, gravitational wave perturbations within the Bianchi type a˙ =ä = 0 . It is usually viewed as a fluid model with IX class of spatially homogeneous universes. a cosmological constant that is given a priori as a fixed Consider a Friedmann universe containing a scalar field universal constant, and this is the view taken in previous stability investigations [2–4]. However, in the context of φ with energy density and pressure given by inflationary cosmology, where the Einstein static model 1 1 may be seen as an initial state, we require a scalar field ρ = φ˙2 + V (φ),p = φ˙2 V (φ), φ 2 φ 2 − φ as well as a fluid, and Λ is not an a priori constant but determined by the vacuum energy of the scalar field. and a perfect fluid with energy density ρ and pressure p = The vacuum energy is determined in turn by the potential wρ,where 1

1 model. An initial Einstein static state of the universe involves a scalar field with nontrivial potential V (φ), so arises if the field starts out in an equilibrium position: that the Einstein static model is an initial state, corresponding to an equilibrium position (V00 = 0). This V 0(φ0)=0, Λ=8πG V (φ0) . (6) general case is a two-component model, since in addition to the scalar field, a fluid is necessary to provide kinetic We will first discuss the simple case where is trivial, V energy and keep the initial model static. i.e., a flat potential. A general Einstein static model has ¨ ρ˙ =0=V 0 = φ by Eqs. (1)–(5), and satisfies 1 III. INHOMOGENEOUS PERTURBATIONS OF (1 + 3w)ρ + φ˙2 = V , (7) 2 0 0 0 THE FLUID MODEL 1 (1 + w)ρ + φ˙2 = . (8) 0 0 4πGa2 We consider first the effect of inhomogeneous density 0 perturbations on the simple one-component fluid mod- ˙ els. Density perturbations of a general Friedmann uni- If the field kinetic energy vanishes, that is if φ0 =0,then verse are described in the 1+3-covariant gauge-invariant by Eq. (8), (1 + w)ρ0 > 0, so that there must be fluid in 2 2 2 order to keep the universe static; but if the static universe approach by ∆ = a D ρ/ρ,whereD is the covariant spatial Laplacian. The evolution of ∆ is given by [6] has only a scalar field, that is if ρ0 = 0, then the field must have nonzero (but constant) kinetic energy [5,1]. K Qualitatively, this means that there must be effective to- ∆+¨ 2 6w +3c2 H∆+˙ 12 w c2 − s − s a2 tal kinetic energy, i.e. ρt + pt > 0, in order to balance the curvature energy 2 in the absence of expansion or 2 2 2 a− +4πG 3w +6cs 8w 1 ρ +(3cs 5w)Λ ∆ contraction, as shown by Eq. (5). − − − K Equations (7) and (8) imply c2D2∆ w D2 +3 =0, (13) − s − a2 E 1 wt = , (9) where is the entropy perturbation, defined for a one- −3 componentE source by 1 V0 +(1+3w)ρ0/2 wφ = . (10) −3 V0 (1 + 3w)ρ0/6 p = a2D2p ρc2∆ . (14) − E − s It follows that w 1 since w> 1 and ρ φ ≤−3 − 3 0 ≥ For a simple (one-component perfect-fluid) Einstein 0. In other words, the presence of matter with (non- static background model, = 0 and Eq. (13) reduces inflationary) pressure drives the equation of state of the to E 1 scalar field below 3 . If the field has no kinetic energy, then by Eqs. (7) and− (8), its equation of state is that of ¨ 2 2 ∆k =4πG(1 + w) 1+(3 k )cs ρ0∆k , (15) a cosmological constant, − where we have decomposed into Fourier modes with co- ˙ φ0 =0 wφ = 1 . (11) 2 2 2 ⇒ − moving index k (so that D k /a0). It follows that ∆ is oscillating and non-growing,→− i.e. the Einstein static If there is no fluid (ρ = 0), then the field has maximal k 0 universe is stable against gravitational collapse, if and kinetic energy, only if 1 ρ0 =0 wφ = = wt . (12) (k2 3)c2 > 1 . (16) ⇒ −3 − s In this case, the field rolls at constant speed along a flat In spatially closed universes the spectrum of modes is dis- 2 potential, V (φ)=V0. Dynamically, this pure scalar field crete, k = n(n + 2), where the comoving wavenumber n case is equivalent to the case of w = 1 pure fluid with cos- takes the values n =1, 2, 3 ... in order that the harmon- mological constant. This can be seen as follows: equating ics be single-valued on the sphere [3,6]. Formally, n =0 the radii a0 of the pure-field and fluid-plus-Λ cases, using gives a spatially homogeneous mode (k = 0), correspond- Eq. (8), implies V0 =(1+w)ρ0, by Eq. (7). This then ing to a change in the background, and the violation of leads to wφ = (3 + 5w)/(5 + 3w), by Eq. (10). How- the stability condition in Eq. (16) is consistent with the ever, we know− that w = 1 for the fluid-plus-Λ case, known result [2] that the Einstein static models collapse φ − and equating the two forms of wφ leads to w =1. or expand under such perturbations. This follows from Thus there is a simple one-component type of Einstein the Raychaudhuri equation (3). static model, which may be realised by perfect fluid uni- For the first inhomogeneous mode (n = 1), the sta- verses with cosmological constant (the pure scalar field bility condition is also violated, so that the universe is case is equivalent to the w = 1 fluid model). This is unstable on the corresponding scale if there is initial the case considered by Eddington, Harrison and Gib- data satisfying the constraint equations. However this bons. The general case, of most relevance to cosmology, is a gauge mode which is ruled out by the constraint

2 equations (G0i =8πGT0i). This mode reflects a free- expansion, since the background is static. In the cos- dom to change the 4-velocity of fundamental observers. mological context, where an initial Einstein static state (For multiple fluids with different velocities, one can in begins to expand under a homogeneous perturbation, the principle have isocurvature modes with n = 1 that are expansion will damp the inhomogeneous perturbations. not forced to zero by the Einstein constraint equations.) However, the Einstein static will not be an attractor; in- The physical modes thus have n 2. Stability of all stead, the attractor will be de Sitter spacetime. these modes is guaranteed if ≥ Vector perturbations of a fluid are governed by the comoving dimensionless vorticity $ = aω , whose modes 1 a a c2 > . (17) satisfy the propagation equation s 5 $˙ = 1 3c2 H$ . (20) This condition was also found by Gibbons [4] in the re- k − − s k stricted case of conformal metric perturbations; we have generalized the result to arbitrary adiabatic density per- For a fluid Einstein static background, this reduces to turbations. Harrison [3] demonstrated stability for a $˙ =0, (21) radiation-filled model and instability of the dust-filled k model, which are included in the above result. so that any initial vector perturbations remain frozen. Thus an Einstein static universe with a fluid that satis- Thus there is neutral stability against vector perturbations fies Eq. (17) (and with no scalar field), is neutrally stable for all equations of state on all scales. against adiabatic density perturbations of the fluid for all Gravitational-wave perturbations of a perfect fluid allowed inhomogeneous modes.Thecasew =1coversthe may be described in the covariant approach [7] by the pure scalar field (no accompanying fluid) models. When 2 comoving dimensionless transverse-traceless shear Σab = cs > 0 the stability of the model is guaranteed by Eq. (16) aσ , whose modes satisfy for all but a finite number of modes. The universe be- ab comes increasingly unstable as the fluid pressure drops. 2 ¨ ˙ k K Clearly, a dust Einstein static universe ( 2 =0)isalways Σk +3HΣk + +2 cs a2 a2 unstable. In that case, Eq. (15) reads ∆¨ =4πGρ ∆ , k 0 k 8πG 2 implying exponential growth of ∆k (the Jeans instability; (1 + 3w)ρ + Λ Σ =0. (22) − 3 3 k see [3]). The physical explanation of this rather unexpected For the Einstein static background, this becomes stability lies in the Jeans length associated with the model [4]. Although there are always unstable modes Σ¨ +4πGρ (k2 + 2)(1 + w)Σ =0, (23) (i.e., with wavelength above the Jeans scale) in a flat k 0 k space, in a closed universe there is an upper limit on so that there is neutral stability against tensor perturba- wavelength. It turns out that, for sufficiently large speed tions for all equations of state on all scales. However this of sound, all physical wavelengths fall below the Jeans analysis does not cover spatially homogeneous modes. length. By Eq. (8), the maximum wavelength 2πa0 de- It turns out that there are various unstable spatially pends on the equation of state and is given by homogeneous anisotropic modes, for example a Bianchi π type IX mode which we discuss below. The associated λ = . (18) anisotropies can die away in the case where the insta- max Gρ (1 + w) r 0 bility results in expansion. If this is the case, despite the extra spatially homogeneous unstable modes, the ex- The Jeans length is 2πa /n ,where∆¨ =0andk2 = 0 j kj j panding inflationary universe will be an attractor. n (n + 2). Hence, by Eq. (15), we have j j In summary, for the simplest models (one-component perturbations), we find neutral stability on all physical cs λj = λmax . (19) inhomogeneous scales against adiabatic density pertur- 2 4cs +1 cs ! bations if 2 1 , and against vector and tensor per- − cs > 5 turbations for any c2 and w, thus generalizing previous Stability means λ<λp , which leads to Eq. (17). For dust, s j results. λj = 0 and all the modes are clearly unstable. For radiation, λj =0.61λmax, and for a stiff fluid λj =0.81λmax. In both of these cases the Jeans length comprises a con- IV. SCALAR-FIELD PERTURBATIONS siderable portion of the size of the universe, and is greater than all allowed wavelengths. This would also place important restrictions on the evolution of non-linear density We turn now to the case of a self-interacting scalar inhomogeneities by shock damping or black hole forma- field, i.e., where the scalar field is governed by a non- tion. flat potential V (φ), with initial Einstein static state at ˙ We also note that the stability condition entails neu- φ = φ0, i.e., V00 =0=φ0. This is the general dynami- tral stability; the oscillations in ∆ are not damped by cal problem: the stability analysis of the Einstein static

3 8πG 1 2 universe as an initial equilibrium position within a phys- H2 = φ˙2 + V + a , (30) ically motivated (non-flat) potential. (Some realizations (1) 3 2 (1) (1) a3 (1) 0 of this scenario are discussed in [1].) Since we are inter- φ¨ = V 0 . (31) ested only in the behaviour close to the Einstein static (1) − (1) solution, we can treat H and V 0 as small in the pertur- Using these results, Eq. (13) gives the evolution equation bation equations. The lowest-order solution for density for the contribution from scalar field perturbations: (0) perturbations, ∆k , thus corresponds to H =0=V 0, ¨ (1) 2 (1) ˙ (0) (0) and is given by the fluid perturbation solution, Eq. (15), ∆k +˜ω0∆k = F ∆k + G∆k , (32) (0) where ∆k (t)=Ak cos ω0t + Bk sin ω0t, (24) 2 2 2 2 2 2 ω =4πG(1 + w)ρ0 (k 3)c 1 , (25) ω˜ =4πG(1 + w)ρ (k 3)(c +2) 1 (33) 0 − s − 0 0 − s − 2 F (t)= (2 6w +3cs )H(1) , (34) with Ak and Bk constants for each mode. The next or- − − der contains contributions from the scalar field perturba- 1 a(1) G(t)= 12[c2 w ] + 24[w c2] tions, a2 (1) − (1) − s a 0 0 2 (0) (1) (1) +6ww(1) +6c(1) 8w(1) ∆k =∆k +∆k + , ∆k =∆φk, (26) − ··· 2 2 ρ(1) +[3w 8w +6cs 1] during the time when the background is close to the Ein- − − ρ0 stein static equilibrium position. 2 2 2 a(1) The entropy term in Eq. (13) has no intrinsic fluid + k c(1) 2cs . (35) − a0 contribution since we assume the fluid is adiabatic. A scalar field generically has intrinsic entropy perturba- The change in the frequency, ω0 ω˜0, shows the stabi- tions, which follow from Eq. (14) as [8] lizing effect of scalar-field entropy→ perturbations, which effectively increase the adiabatic sound speed term in ω0: 2 2 2 1 cφ cs cs + 2. The solution of Eq. (32) is given by Green’s φ = − ∆φ . (27) method→ in the form E wφ cosω ˜0t (See [9] for the corresponding expression in the metric- ∆(1) = ∆(0) (F sinω ˜ t) G sinω ˜ t dt k ω˜ k 0 · − 0 based perturbation formalism.) These entropy perturba- 0 Z sinω ˜ t tions have a stabilizing effect on density perturbations of 0 ∆(0) (F cosω ˜ t) G cosω ˜ t dt . (36) − ω˜ k 0 · − 0 the scalar field: the entropy term in Eq. (13) cancels the 0 Z preceding Laplacian term c2 D2∆ , which would pro- φ φ Since ˜2 0 for 2 3, we see that stability is not 2 − ω0 > , k duce instability when cφ < 0; what remains is the term changed by the introduction≥ of scalar-field perturbations. 2 D ∆φ, which contributes to stability. We note that if general relativity is extended to include − For an initial Einstein static state, the entropy contri- higher-order curvature corrections to the gravitational bution from the scalar field to lowest order is, Lagrangian, then the existence and stability conditions for the Einstein static universe are changed in interest- (1) (1) [wφ φ] =2∆ , (28) ing ways [11] which are linked to the situation of general E k k relativity plus a pure scalar field through the conformal 2 (0) where we used [cφ] = 1. There is also a relative en- equivalence of the two problems [12]. tropy contribution, arising− from the relative velocity between the field and the fluid. We expect this contribution to be negligible. V. SPATIALLY HOMOGENEOUS TENSOR We now expand all the dynamical quantities in PERTURBATIONS Eq. (13) to incorporate the lowest order effect of the self- interacting scalar field. In particular, a = a0 + a(1) and We now investigate the stability of the Einstein 1 ˙2 ρ = ρ0 + ρ(1),whereρ(1) = 2 φ(1) + V(1), and similarly for static universe to spatially homogeneous gravitational- the pressure. Then w w + w(1),where wave perturbations of the Bianchi type IX kind. The → anisotropy in these spatially homogeneous perturbations, p wρ which is absent within the Friedmann family, is what al- w = (1) − (1) , (29) (1) ρ lowsfortensormodes. The Einstein static universe is a particular exact so- and c2 c2 + c2 ,wherec2 is determined by the form lution of the Bianchi type IX, or Mixmaster, universe s → s (1) (1) of the potential V (φ)nearφ0. It also follows from the containing a perfect fluid and a cosmological constant. background equations that The Mixmaster is a spatially homogeneous closed (com- pact space sections) universe of the most general type. It

4 contains the closed isotropic Friedmann universes as par- term on the right-hand side arises from the matter per- ticular subcases when a fluid is present. Physically, the turbations. Thus we see that the perturbation grows and Mixmaster universe arises from the addition of expansion the Einstein static solution is unstable to spatially homo- anisotropy and 3-curvature anisotropy to the Friedmann geneous pure gravitational-wave perturbations of Bianchi universe. It displays chaotic behaviour on approach to type IX, the initial and final singularities if w<1. This is closely linked to the fact that, despite being a closed universe, 3 δA exp 2t . (44) its spatial 3-curvature is negative except when it is close ∝ a0 to isotropy. r The diagonal type IX universe has three expansion When the matter perturbations are included the insta- scale factors ai(t), determined by the Einstein equations, bility remains unless 1 + 3w<0, which we have ruled which are [10]: out for a fluid. In this case, the perturbations oscillate. This condition corresponds to a violation of the strong (˙a a a ) 4[(a2 a2)2 a4] energy condition and this ensures that a Mixmaster uni- 1 2 3 · = 2 3 1 − −2 a1a2a3 (a1a2a3) verse containing perfect fluid matter will expand forever +Λ+4πG(1 w)ρ, (37) and approach isotropy. Notice that the matter effects − disappear at this order when w = 1. This is a famil- and the two equations obtained by the cyclic interchanges iar situation in anisotropic cosmologies where a w =1 a1 a2 a3 a1, together with the constraint fluid behaves on average like a simple form of anisotropy → → → “energy”. a¨ a¨ a¨ 1 + 2 + 3 =Λ 4πG(1 + 3w)ρ, (38) The instability to spatially homogeneous gravitational a1 a2 a3 − wave modes is not surprising. Mixmaster perturbations allow small distortions of the Einstein static solu- and the perfect fluid conservation equation (with w con- tion to occur which conserve the volume but distort the stant) shape. Some directions expand whilst others contract. Note that these (3)-invariant homogeneous anisotropy ρ(a a a )w+1 =const. (39) SO 1 2 3 modes are different to the inhomogeneous modes consid- The Einstein static model is the particular solution with ered in the perturbation analysis. Mixmaster oscillatory behaviour is not picked out by the eigenfunction expan- all ai = a0 = const. We consider the stability of this static isotropic solution by linearizing the type IX equa- sions of perturbations of Friedmann models. tions about it: ai(t)=a0 + δai(t),ρ(t)=ρ0 + δρ(t), and δp = wδρ. The linearized field equations lead to VI. CONCLUSIONS

¨ 12 8 There is considerable interest in the existence of pre- δai + 2 δai = 2 (δa1 + δa2 + δa3) a0 a0 ferred initial states for the universe and in the existence +4πG(1 w)δρ . (40) of stationary cosmological models. So far this interest − has focussed almost entirely upon the de Sitter universe The conservation equation to linear order gives as a possible initial state, future attractor, or global stationary state for an eternal inﬂationary universe. Of the Λ δa + δa + δa δρ = 4 + ρ 1 2 3 . (41) other two homogeneous spacetimes, the Einstein static − 8πG 0 a 0 provides an interesting candidate to explore whether it could play any role in the past evolution of our Universe. If we deﬁne an arithmetic-mean perturbed scale factor It is important to know whether it can provide a natural by initial state for a past eternal universe, whether it allows

δa1 + δa2 + δa3 the universe to evolve away from this state, and whether δA(t) , (42) under any circumstances it can act as an attractor for ≡ a0 the very early evolution of the universe. We might also then it obeys ask whether it is not possible for it to provide the glob- ally static background state for an inhomogeneous eternal 4 δA¨ =3δA (1 w)(Λ + 8πGρ ) universe in which local regions undergo expansion or con- a2 − − 0 0 traction, manifesting an instability of the Einstein static 3δA universe. With these questions in mind we have investi- = (1 + 3 ) (43) 2 w . gated in detail the situations under which Einstein static a0 universe is stable and unstable. The ﬁrst term on the right-hand side arises from the We have shown that the Einstein static universe is neu- pure spatially homogeneous gravitational wave modes of trally stable against inhomogeneous vector and tensor Bianchi type IX (i.e., with δρ =0=δp). The second

5 linear perturbations, and against scalar density pertur- 2 1 bations if cs > 5 , extending earlier results of Gibbons for purely conformal density perturbations. However, we find that spatially homogeneous gravitational-wave perturbations of the most general type destabilise a static universe. Our results show that if the universe is in a neighbourhood of the Einstein static solution, it stays in that neighbourhood, but the Einstein static is not an attractor (because the stability is neutral, with non- damped oscillations). Expansion away from the static state can be triggered by a fall in the pressure of the matter. Typically, expansion away from the static solution will lead to inflation. If inflation occurs, then perturbations about a Friedmann geometry will rapidly be driven to zero. The nonlinear effects (which will certainly be important in these models because of the initial infinite time scale envisaged) will be discussed in a further pa- per, as will other aspects of the spatially homogeneous anisotropic modes.

Acknowledgements We thank Marco Bruni, Anthony Challinor and Peter Dunsby for useful discussions. GE and CT are supported by the NRF. RM is supported by PPARC, and thanks the Cosmology Group at Cape Town for hospitality while part of this work was done.

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