Bianchi type-I space and the stability of the inflationary Friedmann-Robertson-Walker solution
W. F. Kao*Institute of Physics, Chiao Tung University, Hsin Chu, Taiwan 共Received 19 April 2001; published 24 October 2001兲
A stability analysis of the Bianchi type-I universe in pure gravity theory is studied in detail. We first derive the nonredundant field equation of the system by introducing the generalized Bianchi type-I metric. This nonredundant equation reduces to the Friedmann equation in the isotropic limit. It is shown further that any unstable mode of the isotropic perturbation with respect to a de Sitter background is also unstable with respect to anisotropic perturbations. The implications of the choice of physical theories are discussed in detail in this paper.
DOI: 10.1103/PhysRevD.64.107301 PACS number共s兲: 98.80.Cq, 04.20.⫺q
I. INTRODUCTION
Inflationary theory provides an appealing resolution to the the flatness, monopole, and horizon problems of our present universe described by standard big-bang cosmology关1兴. It is known that our universe is homogeneous and isotropic to a very high degree of precision关2,3兴. Such a universe can be described by the well-known Friedmann-Robertson-Walker
共FRW兲 metric 关4兴.
It is also known that gravitational physics should be dif-ferent from standard Einstein models near the Planck scale
关5,6兴. For example, quantum gravity or string corrections
could lead to interesting cosmological consequences 关5兴. Moreover, some investigations have addressed the possibility of deriving inflation from higher-order gravitational correc-tions关7–10兴.
A general analysis of the stability condition for a variety of pure higher derivative gravity theories is very useful in many respects. In fact, it was shown that a stability condition should hold for any potential candidate of inflationary uni-verse in the flat Friedmann-Robertson-Walker 共FRW兲 space
关10兴.
In addition, the derivation of the Einstein equations in the presence of higher derivative couplings is known to be very complicated. The presence of a scalar field in induced gravity models and the dilaton-gravity model makes the derivation even more tedious. In order to simplify the complications in the derivation of the field equations, an easier way has been described关11,12兴. We will try to generalize the work of Ref.
关12兴 in order to obtain a general and model-independent
for-mula for the nonredundant field equations in the Bianchi type-I共BI兲 anisotropic space. This equation can be applied to provide an alternative and simplified method to obtain the stability conditions in pure gravity theories. In fact, this gen-eral and model-independent formula for the nonredundant field equations is very useful in many areas of interest. In particular, it will be applied to study a large class of pure gravity models with inflationary BI-FRW solutions in this paper. Any Bianchi type-I solution that leads itself to an asymptotic FRW metric at time infinity will be referred to as the BI-FRW solution in this paper for convenience.
Note that there is no particular reason why our universe is initially isotropic to such a high degree of precision. Even anisotropy can be smoothed by the proposed inflationary process, it is also interesting to study the stability of the FRW space during the post-inflationary epoch. Nonetheless, it is natural to expect that our universe starts out as an anisop-tropic universe. The universe is then expected to evolve from a certain anisotropic universe, e.g., a Bianchi type-I universe, to an isotropic universe, such as the flat FRW space. Indeed, it was shown that there exists such kind of BI-FRW solution for a NS-NS model with a metric field, a dilaton, and an axion field 关13兴. This inflationary solution is also shown to be stable against small field perturbations关14兴. Note also that stability analysis has been studied in various fields of interest
关15,16兴.
A large class of models with the BI-FRW solutions will be shown to be unstable against arbitrary anisotropic perturba-tions in this paper. We will first derive a stability equation which turns out to be identical to the stability equation for the existence of the inflationary de Sitter solution discussed in Refs. 关10,12兴. Note that an inflationary de Sitter solution in pure gravity models is expected to have one stable mode and one unstable mode for the system to undergo inflation with the help of the stable mode. Consequently, the inflation-ary era will come to an end once the unstable mode takes over after a brief period of inflationary expansion. The method developed in Refs. 关10,12兴 was shown to be helpful in choosing a physically acceptable model for our universe. Our result indicates, however, that the unstable mode will also tamper with the stability of the isotropic space. To be more specific, if the model has an unstable mode for the de Sitter background perturbation with respect to isotropic per-turbation, this unstable mode will also be unstable with re-spect to any anisotropic perturbations.
II. NONREDUNDANT FIELD EQUATION AND BIANCHI IDENTITY
Note that the generalized Bianchi type-I共GBI兲 metric can be read off directly from the following equation:
ds2⬅g GBIdxdx ⫽⫺b2共t兲dt2⫹a 1 2共t兲dx2⫹a 2 2共t兲dy2⫹a 3 2共t兲dz2, 共1兲
*Email address: wfgore@cc.nctu.edu.tw
PHYSICAL REVIEW D, VOLUME 64, 107301
with b(t) the lapse function restored on purpose. Note also that the Bianchi type-I共BI兲 metric can be obtained from GBI metric by setting the lapse function b(t) equal to 1, i.e., b
⫽1, in Eq. 共1兲.
Note that one can list all nonvanishing components of the curvature tensor as Rt jti⫽1 2关HiB˙⫹2B共H˙i⫹Hi 2兲兴␦ j i , 共2兲 Rkli j⫽HiHjBCkli j. 共3兲 Here Ckli j⬅⑀i jm⑀mklwith⑀i jkthe three space Levi-Civita ten-sor关4兴. Here ‘‘•’’ denotes differentiation with respect to t and Hi⫽a˙i/ai is the directional Hubble constant. We have also written B⬅1/b2 for later convenience. Note also that the indices i, j in both sides of the above equations are open indices. Moreover, Rkli j⫽0 if i⫽ j due to the symmetric prop-erties of the curvature tensor.
Given a pure gravity model one can cast the action of the system as S⫽兰d4x
冑
gL⫽N兰dt(a1a2a3/冑
B)L(Hi,H˙i,B,B˙ ) in the GBI spaces. Here N is a time-independent integration constant. We will denote the volume factor as V⬅a1a2a3b. If we take VL as an effectiveLa-grangian, one can show that the variation with respect to b gives ⫺1 2L⫹ 1 2HiLi⫹H˙iL i⫺1 2共H˙i⫹3HHi⫹Hid/dt兲L i⫽0 共4兲
after setting B⫽1. Here the last term comes from the varia-tion with respect to␦(HiB˙ ) with the help of the identity
␦L ␦HiB˙ ␦共HiB˙兲⫽ ␦L ␦2BH˙i ␦共HiB˙兲→ ␦L ␦2H˙i Hi␦B˙ 共5兲
once we set B⫽1. One also needs an integration-by-part with respect to ␦B in order to obtain the result indicated in the last term of Eq.共4兲. Note that the first equality in Eq. 共5兲 comes from the fact that the factor HiB˙ always shows up with 2BH˙ias indicated by the curvature tensor R t j
ti
in Eq.
共2兲. Therefore one ends up with the b equation as
L⫺HiLi⫽共H˙i⫺3HHi⫺Hid/dt兲Li 共6兲 after setting B⫽1. Here H⬅兺iHi/3, Li⬅␦L/␦Hi, and Li
⬅␦L/␦H˙i for convenience. In addition, one can also derive
L⫹共3H⫹d/dt兲2Li⫽共3H⫹d/dt兲Li 共7兲
as the variational equation of ai. The derivation of this equa-tion is tedious but straightforward. In addiequa-tion, VL is nor-mally referred to as the effective Lagrangian. We will also call L the effective Lagrangian unless confusion occurs. Note that Eq. 共7兲 is in fact the spacelike i j component of the Einstein equation
G⫽t 共8兲
with t denoting the generalized energy momentum tensor associated with the system. It is known that one of these equations is in fact a redundant equation. Indeed, one can define H⬅G⫺t and write the field equation as H
⫽0.
Hence one has
DH⫽0 共9兲
from the energy conservation (Dt⫽0) and the Bianchi identity (DG⫽0). Now we have three independent scale factors aiand four equations. Therefore one of the four equa-tions has to be redundant. Indeed, the extended Bianchi iden-tity共9兲 can be shown to give
共t⫹3H兲Htt⫹
兺
iHiHii⫽0, 共10兲
as soon as the BI metric is substituted into equation 共9兲. Therefore Eq. 共10兲 indicates that: ‘‘Htt⫽0 implies 兺iHiHii
⫽0.’’ Hence two of the Hii equations vanish which imply that the third one also vanishes.
On the other hand, H1⫽H2⫽H3⫽0 implies instead (t
⫹3H)Htt⫽0. This implies that VHtt⫽const with V
⬅a1a2a3. Hence the Httequation is the nonredundant equa-tion while we are free to ignore one of the three Hii equa-tions. Hence any conclusion derived without the Httequation is known to be incomplete.
III. PERTURBATION AND STABILITY
One can then apply the perturbation, Hi⫽Hi0⫹␦Hi, to the field equation with Hi0 the background solution to the system. This perturbation will enable one to understand whether the background solution is stable or not. In particu-lar, one would like to learn whether a BI-FRW-type evolu-tionary solution is stable or not. It is known that our universe could start out anisotropic; even evidences indicate that our universe is isotropic to a very high degree of precision in the post inflationary era. Therefore one expects that any physical model should admit a stable BI-FRW solution.
Our result indicates that FRW inflationary solutions with a stable mode and an unstable mode is a negative result to our search for a physically acceptable model. Note that FRW inflationary solutions with a stable mode and an unstable mode will provide a natural way for the inflationary universe to leave the inflationary phase. Our result indicates, however, that such models will also be unstable against the anisotropic perturbations. Therefore such a solution will be harmful for the system to settle from BI space to FRW space once the graceful exit process is done.
First of all, one can show that the first-order perturbation equation from the nonredundant field equation 共6兲, with Hi
→H⫹␦Hi, gives
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L02H␦H¨i⫹关共3H2⫺H˙兲L02⫹3HH˙L12⫹3HH¨L03兴␦H˙i
⫹共6HL01⫹3H2L11⫺HL20⫹H¨L02⫹3HH¨L12
⫹3HH˙L21兲␦Hi⫽0, 共11兲
where all field variables are understood to be evaluated at the background FRW space where Hi⫽H⫽a˙/a for all direc-tions, with a the FRW scale factor. We will write H as the Hubble parameter for the FRW space for convenience from now on. Lab⬅␦a⫹bL/␦Hii␦Hi2•••Hia␦H˙j1␦H˙j2•••
␦H˙jb兩Hi→H. In deriving the above equations, we have used the following identities:
兺
i Hi ␦L ␦Hi→H ␦L ␦H, 共12兲 ␦L ␦Hi→ ␦L 3␦H, 共13兲 f冉
兺
i HiHi冊
→ f共3H2兲 共14兲 when we take the limit Hi→H carrying the system from the BI space to the flat FRW space limit. Here f (兺iHiHi) de-notes any functions of the variable 兺iHiHi. One can show that Eq.共11兲 reduces toL02␦H¨i⫹3H0L02␦H˙i⫹共6L01⫹3H0L11⫺L20兲␦Hi⫽0
共15兲
once we set H⫽H0⫽const which denotes the de Sitter
space. This equation is identical to the stability equation for the existence of an inflationary de Sitter solution discussed in Refs.关10,12兴. One notes that there are two other independent Hi jequations that remain to be checked. These equations can be shown to be redundant after the limiting case H0⫽const
is implemented. Indeed, the anisotropic perturbation on the Hi j equations are expected to reproduce the redundant iso-tropic perturbation equation shown in Ref.关12兴.
Note that an inflationary de Sitter solution is expected to have one stable mode and one unstable mode for the system to undergo inflation with the help of the stable mode. Indeed, one can show that Eq.共15兲 can be solved to give
␦Hi⫽ciexp关B⫹t兴diexp关B⫺t兴 共16兲 with B⫾⫽⫺3H0/2⫾
冑
⌬0/2L02and arbitrary constants ci,di to be determined by the initial perturbations. Here ⌬0⬅9H0 2L
02 2 ⫺4L
02(6L01⫹3H0L11⫺L20) is the discriminant
of the characteristic equation for Eq.共15兲,
L02x2⫹3H0L02x⫹共6L01⫹3H0L11⫺L20兲⫽0. 共17兲
There will be an unstable mode if B⫹⬎0. Therefore the inflationary era will come to an end once the unstable mode takes over. It was shown earlier to be a helpful way to select a physically acceptable model for our universe. Our result shown here indicates, however, that the unstable mode will
also tamper with the stability of the isotropic space. Indeed, if the model has an unstable mode for the de Sitter perturba-tion, this unstable mode will also be unstable against the anisotropic perturbation.
For example, one can show that the model 关10兴
L⫽⫺R⫺␣R␥R␥R 共18兲 admits an inflationary solution when ␣⬍0. Note that this model is the minimal consistent effective low-energy two-loop renormalizable Lagrangian for pure gravity theory关17兴. Indeed, one can show that
L⫽
兺
i 兵 2H˙i⫹4Hi2⫺4␣关2共H˙i⫹Hi2兲3⫺H i 6兴其 ⫺4␣冉
兺
i Hi3冊
2 →6共H˙⫹2H2兲⫺24␣关共H˙⫹H2兲3⫹H6兴 共19兲when we set Hi→H. Hence one can show that the general-ized Friedmann equation 共6兲 gives
H04⫽⫺1/4␣. 共20兲
In addition, the stability equation共11兲 for␦Hican be shown to be 12␣H02␦H¨i⫹36␣H0 3␦H˙ i⫺共1⫹12␣H0 4兲␦H i⫽0. 共21兲 This equation can be solved to give
␦Hi⫽ciexp关共
冑
35/3⫺3兲H0t/2兴⫹diexp关⫺共冑
35/3⫹3兲H0t/2兴 共22兲
with arbitrary constants ci,dito be determined by the initial perturbations. This indicates that this model admits one stable mode and one unstable mode following the stability equation 共15兲 for the inflationary de Sitter solution. It is shown to be a positive sign for an inflationary model that is capable of resolving the graceful exit problem in a natural manner. Our result indicates, however, that this model also admits an unstable mode against anisotropic perturbation. Hence this model will have a problem with remaining iso-tropic for a long period of time. Therefore a pure gravity model of this sort will not solve the graceful exit problem. One will need, for example, the help of a certain scalar field to end the inflation in a consistent way.
One expects any unstable mode for a model to be of the form ␦Hi⬃exp关lH0t兴, to the lowest order in H0t, in a de
Sitter background with l some constant characterizing the stability property of the model. In such models, the inflation-ary phase will only remain stable for a period of the order
⌬t⬃1/lH0. The inflationary phase will start to collapse after
this period of time. This means that the de Sitter background fails to be a good approximation when tⰇ⌬t. Hence the anisotropy will also grow according to ␦Hi
→␦Hi0exp关lH0⌬t兴 with␦Hi
0denoting the initial perturbation.
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The measure of anisotropy can be estimated by computing the anisotropy parameter A⬅
冑
兺i␦Hi2/3H2⬃
冑
兺i(␦Hi 0)2/3H02exp关lH0t兴. Hence A
⬃
冑
兺i(␦Hi 0)2/3H02exp关60l兴 for a typical inflationary model which requires a 60 e-fold expansion. This gives us hope that the small anisotropy observed today can be generated by the initial inflationary instability for models with appropriate factor l.
On the other hand, it is known that the Einstein-Hilbert model
L⫽⫺R⫺2⌳ 共23兲
admits only stable modes, which requires ␦Hi⫽0, which is bad for the natural graceful exit. This model is, however, stable against anisotropic perturbations, which tends to keep the universe isotropic as long as the model is in charge.
IV. CONCLUSION
In short, the result of this paper shows that graceful exit and stability of any de Sitter model cannot work along in a naive way. The physics behind the inflationary de Sitter
mod-els appears to be much more complicated than one may ex-pect. In another words, the phase transition during and after the inflationary phase deserves more attention and requires extraordinary care in order to resolve the problem lying ahead.
Note that the nonredundant field equation for field theo-ries with many different sorts of fields coupled to the system can also be derived similar to the pure gravity models 关12兴. Similar arguments also apply to these theories. Therefore theories with one unstable mode under anisotropic metric perturbations with respect to the de Sitter background will not be able to hold the de Sitter space stable for a long period of time even if there exist stable modes. Note that, in gen-eral, one needs to consider two different fourth derivative curvature terms when higher derivative theories are consid-ered in four dimensions.
ACKNOWLEDGMENTS
This work was supported in part by the National Science Council of Taiwan under contract number NSC89-2112-M009-043.
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