Kink instability and stabilization of the Friedmann universe with scalar fields

The evolution of weak discontinuity is investigated in the flat FRW universe with a single scalar field and with multiple scalar fields. We consider both massless scalar fields and scalar fields with exponential potentials. Then we find that a new type of instability, i.e. kink instability develops in the flat FRW universe with massless scalar fields. The kink instability develops with scalar fields with considerably steep exponential potentials, while less steep exponential potentials do not suffer from kink instability. In particular, assisted inflation with multiple scalar fields does not suffer from kink instability. The stability of general spherically symmetric self-similar solutions is also discussed.


I. INTRODUCTION
Future precision observations of the cosmic microwave background radiation and/or gravitational lensing phenomena will provide a large number of new data to be analyzed and interpreted. The standard cosmology assumes that the observable part of the universe on scales of the order of the present Hubble distance is almost spatially homogeneous and isotropic. Actually, the data of WMAP for the cosmic microwave background radiation tell us that the universe at the last scattering surface is well approximated by the Friedmann-Robertson-Walker (FRW) universe so that understanding of the detailed properties of the FRW universe is an important step in cosmology [1].
One of the most important problems is how cosmological structures on various scales formed from the very homogeneous and isotropic universe at early times. It is known that the gravitational instability scenario with cold dark matter well describes structure formation from small primordial perturbations in our universe [2].
Recently, a new kind of instability has been found in the studies of self-similar solutions. Self-similar solutions are defined in terms of a homothetic Killing vector field in general relativity [3] and include a class of the flat FRW solutions. The instability concerns weak discontinuity. If the perturbations with weak discontinuity are inserted into a self-similar solution which is unstable against this mode, the discontinuity grows as time proceeds. This instability has been called kink instability. Kink instability of self-similar solutions was originally found by Ori and Piran in spherically symmetric isothermal gas systems in Newtonian gravity [4]. In general relativity, Harada investigated the kink instability of self-similar solutions for the spherical system of a perfect fluid with the equation * Electronic address:hideki@gravity.phys.waseda.ac.jp † Electronic address:T.Harada@qmul.ac.uk of state p = kµ [5]. The present authors investigated the kink instability of self-similar solutions for the spherical system of a stiff (k = 1) fluid and those of a massless scalar field [6]. These works have shown that the kink instability can occur in a large class of self-similar solutions. In particular, the work in general relativity showed that the flat FRW universe is unstable for a perfect fluid for 1/3 ≤ k ≤ 1 and for a massless scalar field.
In this letter we derive a stability criterion against kink mode perturbations for the power-law flat FRW universe with a single scalar field and with multiple scalar fields with and without exponential potentials, which arise naturally in supergravity [7] or theories obtained through dimensional reduction to effective four dimensional theories [8,9]. The stability of general spherically symmetric self-similar solutions with scalar fields is also discussed in Appendix A. We adopt the units such that c = G = 1 and the abstract index notation of [10].

II. STABILITY CRITERION FOR THE FLAT FRW UNIVERSE
A. power-law flat FRW solution with and without exponential potentials We begin with the action which describes the selfgravitating system of scalar fields with and without exponential potentials.
, (2.1) where {V i |i = 1, · · · , n} and {λ i |i = 1, · · · , n} are n real non-negative and positive constants, respectively. In this model we currently have n real scalar fields {φ j |j = 1, · · · , n}, for which the total potential has the If V j = 0, then φ j is massless and λ j is meaningless. We assume 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n without loss of generality. This action has been considered in the context of assisted inflation [11,12,13]. The action involving a perfect fluid has been studied by several authors [13,14]. Then the total energy-momentum tensor is given by 3) while equations of motion for {φ j } is given by We consider the power-law flat FRW spacetime with the metric; where α and t 0 are constants and dΩ 2 ≡ dθ 2 + sin 2 θdϕ 2 . We first show that there are no power-law flat FRW solutions which include both nontrivial massless scalar fields and scalar fields with exponential potentials. When there exist m scalar fields {φ j |1 ≤ j ≤ m} with a total potential m i=1 V i exp(− √ 8πλ i φ i ) and (n − m) massless scalar fields {φ j |m + 1 ≤ j ≤ n}, the Einstein equations G ab = 8πT ab and the equations of motion for scalar fields are written as where the comma denotes the partial derivative. From Eqs. (2.6) and (2.7), we obtain so that {φ j } must have the form where {κ j } are constants satisfying and {C j } are constants. We can set C j = 0 for all j by redefining the scalar field. From Eq. (2.7) and the assumption of V j > 0 for 1 ≤ j ≤ m, κ j = −2/( √ 8πλ j ) must be satisfied for 1 ≤ j ≤ m. Then Eqs. (2.7) and (2.8) (2.12) On the other hand, we obtain from Eq. (2.11) that Eqs. (2.12) and (2.13) give a contradiction κ j = 0 for m+ 1 ≤ j ≤ n. Therefore, we conclude that if the power-law flat FRW solution includes both massless scalar fields and scalar fields with exponential potentials, those massless scalar fields must be constant, i.e., trivial.
When there exist only massless scalar fields, the Einstein equations and the equations of motion for scalar fields give α = 1/3 and 4π n i=1 κ 2 i = 1/3, while when there are no massless scalar fields, they give (2.14) The latter equation gives a condition n i=1 λ −2 i > 1/6 for V j > 0. Setting n = 1 in both cases, we can obtain the results for a single scalar-field case.
For the later convenience, we transform the metric (2.5) to the self-similar form (B2). As shown in Appendix B, the flat FRW solution is self-similar if and only if the scale factor obeys a power-law. Therefore it is concluded that there are no self-similar flat FRW solutions if there exist both nontrivial massless scalar fields and those with exponential potentials. Because the case with α = 1 is exceptional, in which the homothetic Killing vector is not tilted [15], we consider only the case with α = 1. With a coordinate transformation r = |1 − α| −1 t α 0r 1−α , the metric (2.5) is transformed to the self-similar form (B2) as where z ≡ t/r is the self-similarity coordinate. The particle horizon z = 1 plays an important role in the following analysis. This is because it is a similarity horizon, which is given by a radial null ray on which z is constant.

B. kink instability
We consider general full-order spherically symmetric perturbations with scalar fields such as Hereafter we adopt the isotropic coordinates, in which ω =η. We define the following coordinates: We consider perturbations σ, ω and {ψ j } which satisfy the following conditions: (1) The initial perturbations vanish outside the similarity horizon.
(2) σ, ω, {ψ j } and {ψ ′ j } are continuous, in particular at the similarity horizon Z = 0, where a prime denotes the derivative with respect to Z.
(3) {ψ ′′ j } and {ψ ′′ j } are discontinuous at the similarity horizon, although they have finite one-sided limit values as Z → −0 and Z → +0, where a dot denotes the derivative with respect to T . (4) The quasi-local mass is continuous, i.e., there are no singular hypersurfaces in the spacetime. Now we consider the behavior of the perturbations at the similarity horizon. The perturbations satisfy ψ j = ψ ′ j = 0 for all j and ψ ′′ j = 0 for some j at the similarity horizon at the initial moment T = T 0 due to condition (2). The evolution of the initially unperturbed region is completely described by the background flat FRW solution because information from the perturbed side cannot penetrate the unperturbed side due to condition (1). Then, we find σ = ω = 0, ψ j = ψ ′ j = 0 for all j and ψ ′′ j = 0 for some j at the similarity horizon for T ≥ T 0 due to conditions (2) and (3). The Misner-Sharp quasi-local mass m is defined by where R is the circumferential radius. Because of in the present case, ω ′ = 0 is satisfied at the similarity horizon due to condition (4). Then we find σ ′ = 0 from the equation of motion for scalar fields. From the (00)+(01) and (11)+(01) components of the Einstein equations , ω ′′ = 0 and σ ′′ = 0 are obtained, respectively. Differentiating the equation of motion for scalar fields with respect to Z and estimating both sides at the similarity horizon, we obtaiṅ for each j. We can show that the right-hand side vanishes [6]. Finally, the full-order perturbation equation for ψ ′′ j at the similarity horizon is obtained aṡ It should be noted that the perturbations are those of full-order although this equation is linear. This equation can be integrated to obtain Therefore, it is found that the perturbation decays exponentially for α > 1/2, it is constant for α = 1/2 and it grows exponentially for α < 1/2. It is noted that these perturbations are gaugeindependent as shown in Appendix C. Here we define instability by the exponential growth of discontinuity. Then we find the following criterion: the flat FRW universe with α > 1/2 are stable against the kink mode, while those with α < 1/2 are unstable. Solutions with α = 1/2 are marginally stable against this mode.
Although we concentrate on the flat FRW solutions in this letter, we can obtain the stability criterion for general spherically symmetric self-similar solutions with regular similarity horizons adopting the similar analysis as that in [6], which is summarized in Appendix A.

III. DISCUSSIONS
We have derived the stability criterion for spherically symmetric self-similar solutions with scalar fields with independent exponential potentials to suffer from kink instability. It can be applied to any spherically symmetric self-similar solutions with regular similarity horizons. The kink instability, which we have considered here, was studied in general relativity for a perfect fluid with an equation of state p = kµ [5,6] and for a massless scalar field [6]. We focus on the flat FRW universe in this letter, for which result is summarized in Tables I and II. This is a new kind of instability of the FRW universe. We first discuss the relations of our result to other studies in the literature on the system with scalar fields with or without exponential potentials. After that, we discuss astrophysical or cosmological implications of the kink instability of the flat FRW universe.
Let us see the flat FRW universe filled with a single matter field. The FRW universe is unstable for a perfect fluid with 1/3 ≤ k ≤ 1, for a massless scalar field and for a scalar field with an exponential potential for 4 < λ 2 < 6. These results suggest that if there exists a phase in the flat FRW universe in which a scalar field dominates other matter fields, the scalar field with an exponential potential for 0 < λ 2 < 4 is preferred rather than a massless scalar field. Such a potential naturally arises in supergravity [7] or theories obtained through dimensional reduction to effective four dimensional theories [8,9].
Kitada and Maeda [16,17] showed the cosmic nohair theorem, which is a generalization of the result by Wald [18] and states that all ever-expanding homogeneous models which contain a scalar field with an exponential potential together with a matter field satisfying the dominant energy condition asymptote to the inflationary flat FRW solution for 0 < λ 2 < 2. According to the criterion obtained in this letter, the flat FRW solution with 4 < λ 2 < 6 suffers from kink mode perturbations, which are kinds of inhomogeneous perturbations, so that our result does not affect the cosmic no-hair theorem.
Next let us move on to the case where the flat FRW universe is filled with multiple scalar fields. In such a situation, Liddle, Mazumdar and Schunck proposed assisted inflation, where an arbitrary number of scalar fields with independent exponential potentials evolve to the inflationary flat FRW solution even if each individual potential is too steep to support inflation by its own [11]. Coley and van den Hoogen showed that the assisted inflationary solution is a global attractor in the spatially homogeneous and isotropic universe if [13]. Together with the result on the spatially homogeneous scalar-field cosmological models by Billyard et al. [19], they concluded that the assisted inflationary solution is a global attractor for all everexpanding spatially homogeneous cosmological models with multiple scalar fields with exponential potentials provided n i=1 λ −2 i > 1/2 [13]. According to the criterion obtained in this letter, the expanding flat FRW solution with n i=1 λ −2 i > 1/4 does not suffer from kink instability, so that our result does not affect the genericity of assisted inflation even if we allow initial data with weak discontinuity in the second-order derivative of the scalar field.
For any case of the flat FRW universes we have seen in this letter, we find that the stability of the flat FRW universe against kink mode perturbation changes from stability to instability as the power index α of the scale factor increases through 1/2, which is clear from Tables I and II. This suggests that the cosmological kink instability is related to the expansion-law of the universe rather than the characteristics of the matter fields.
The kink instability is particularly important for the flat FRW universe in the radiation (k = 1/3) dominated era. It is critical for some cosmological models which assume a stage when the effective equation of state is "harder" than the radiation fluid. Because the flat FRW solution is homogeneous, we can choose any point as a symmetric centre in the present analysis. This implies that a bubble-like structure will develop and the bubble walls collide with each other. This consideration suggests that density perturbations can arise in different scales as a result of multiple bubble wall collisions. Although the product of kink instability has not been clear yet, it is plausible that it will bring shock-wave formation, which could be important for the cosmological structure formation. Actually, the shock-wave formation has been observed in the numerical simulations of the gravitational collapse of the density fluctuations in the radiation-dominated flat FRW universe [20]. The outcome of the instability and the effect on the structure formation should be comprehensively investigated.

Acknowledgments
HM is grateful to T. Hanawa and S. Inutsuka for helpful discussions. TH is grateful to A. Ishibashi for helpful discussions. This work was partially supported by a Grant for The 21st Century COE Program (Holistic Research and Education Center for Physics Self-Organization Systems) at Waseda University. TH was supported by JSPS Postdoctoral Fellowship for Research Abroad.

APPENDIX A: STABILITY CRITERION FOR GENERAL SELF-SIMILAR SOLUTIONS
Here we show that kink instability may occur in a very large class of spherically symmetric self-similar spacetimes. The analysis for the case of a single massless scalar field [6] is straightforwardly extended to the case of multiple scalar fields.
We adopt the Bondi coordinates for spherically symmetric spacetimes as where g = g(u, R) andḡ =ḡ(u, R). For later convenience we define new functions {h j (u, R)} as where {κ j |j = 1, · · · , n} are constants. We define the following self-similar coordinates: We refer to u < 0 and u > 0 as early time and late time, respectively. As u is increased, U increases in early times, while U decreases in late times. As u is fixed and R is increased, X increases. Then the Einstein equation and the equations of motion for scalar fields reduce to the following partial differential equations: where the dot and prime denote the partial derivatives with respect to U and X, respectively.
These equations are singular atḡ = 2x, which is called a similarity horizon. We denote the value of x at the similarity horizon as x (s) and also the value of X as X (s) ≡ ln |x (s) |. We consider self-similar solutions with finite values of functions g,ḡ, {h j } and {h ′ j } and their gradients with respect to X at the similarity horizon. Then we find at the similarity horizon X = X (s) for κ j = 0, where the subscript (s) denotes the value at the similarity horizon. When κ j = 0 for all j, which can be satisfied in the case that all scalar fields are massless, i.e. V j = 0 for all j, we find at the similarity horizon. A similarity horizon corresponds to a radial null curve because is satisfied along a radial null curve. No information propagate inwardly beyond the similarity horizon in early-time (late-time) solutions for g (s)ḡ(s) > (<)0, while no information propagate outwardly beyond the similarity horizon in late-time (early-time) solutions for g (s)ḡ(s) > (<)0. A class of the expanding flat FRW solutions belongs to the late-time self-similar solutions with an analytic similarity horizon. The flat FRW solutions are characterized by the parameters {κ j }, which are gaugeindependent. Actually, the FRW solution in the same coordinates with those used in this appendix was obtained by Christodoulou for the case of a massless scalar field [23].

kink instability of self-similar solutions
We consider perturbations which satisfy the following conditions in the background self-similar solution: (1) The initial perturbations vanish inside the similarity horizon for early-time (late-time) solutions with g (s)ḡ(s) > (<)0. (Conversely, the initial perturbations vanish outside the similarity horizon for late-time (earlytime) solutions with g (s)ḡ(s) > (<)0.) (2) g,ḡ, {h j } and {h ′ j } are continuous everywhere, in particular at the similarity horizon.
(3) {h ′′ j } and {ḣ ′′ j } are discontinuous at the similarity horizon, although they have finite one-sided limit values as X → X (s) − 0 and X → X (s) + 0. We denote the full-order perturbations as where g (b) ,ḡ (b) and {h j(b) } denote the background selfsimilar solution. Now we consider the behavior of the perturbations at the similarity horizon, of which we mean X → X (s) −0 for g (s)ḡ(s) > 0, while X → X (s) +0 for g (s)ḡ(s) < 0. Applying the similar analysis as that in [6], we finally obtain the full-order perturbation equation for δh ′′ j at the similarity horizon is obtained as This equation can be integrated to obtain where Therefore, for early-time solutions, it is found that the perturbation decays exponentially for 8π n i=1 κ 2 i < 1, it is constant for 8π n i=1 κ 2 i = 1 and it grows exponentially for 8π n i=1 κ 2 i > 1. The situation is reversed for latetime solutions.
It is noted that these perturbations are gaugeindependent as shown in Appendix C. Here we define instability by the exponential growth of discontinuity. Then we find the following criterion: for early-time solutions, the solutions with regular similarity horizon and 8π n i=1 κ 2 i < 1 are stable against the kink mode, while those with 8π n i=1 κ 2 i > 1 are unstable. Solutions with 8π n i=1 κ 2 i = 1 are marginally stable against this mode. The situation is reversed for late-time solutions. Setting n = 1, we reproduce the result obtained in [6] for the case of a massless scalar field. I: Stability of kink mode perturbation in the flat FRW universe with a perfect fluid with the equation of state p = kµ. The power index α of the scale factor is given by α = 2/[3(1 + k)]. See [5] for details.