AbstmcL Extending our recent proof of the cosmic no-hair theorem for Bianchi models in power-law inflation, we prove a more general cosmic no-hair theorem for all 0 < X < fi, where X is the coupling constant of an exponential potential of an inflaton +, exp(-Xrcc$). For any initially expanding Bianchi-type model except type IX, we find that the isotropic idationary solution is the unique attractor and that anisotropies always enhance inflation. For Bianchi IX, this conclusion is also true, if the initial ratio of the vacuum energy A.@ to the maximum 3-cumature (3) R,, is larger than 1/[3(1-Xz/2)] and its time derivative is initially positive. It t u n s out that the sufficient condition for inflation in Bianchi twe-IX spacetims with cosmological constant A, which is a special case of the theorem (A = 0), becomes less restrictive than Wald's one. For type lX, we also show a recollapse theorem. PACS numbers: 9880,0450 0264-9381/93/040703+32SO7.50 @ 1993 IOP Publishing Lid 703 704inflationary models were systematically investigated in the conformal frame. They found that new inflation as well as chaotic inflation with a massive inflaton allow not only natural initial data but also natural coupling constants. Such approaches may hence give us a natural inllationary model.Apart from the naturalness of inflationary models, we have another question about naturalness of an inflationary solution, namely whether inflation is the unique attractor in the most general spacetime with cosmological constant or vacuum energy. In the discussion about inflation, we usually assume Friedmann-Robertson-Walkertype spacetimes. Although we have not so far known anything about the initial state of the Universe, the ansatz of isotropy and homogeneity of the Universe might be also justified by inflation. In order to show it, we have to study whether or not inflation really occurs even in anisotropic and/or inhomogeneous spacetimes and whether the spacetime is isotropized and homogenized during inflation. This problem is related to the so-called cosmic no-hair conjecture.In the most general inhomogeneous case, however, the analysis is very difficult, and hence a numerical approach may be the only way to show whether or not it is true [6], except for perturbation analysis [7].However, in an anisotropic but homogeneous case, if a positive cosmological constant exirts, we have the cosmic no-hair theorem proved by Wald [SI, which tells us that all initially-expanding Bianchi models except type IX approach the de Sitter spacetime in one expansion time. For type IX, some spacetimes recollapse, and we know just a sufficient condition for inflation [8-10]. In the old or new inflationary models, the false vacuum energy is almost a constant, and thus behaves as a positive cosmological constant. This theorem is, therefore, applied. This theorem was also extended to chaotic inflationary models 191.A natural question may arise, namely whether or not this theorem is extendible to power-law inflationary models [Ill. This question becomes more important in ...
