Physics
Scientific paper
Dec 2002
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2002nmgm.meet.1026c&link_type=abstract
"THE NINTH MARCEL GROSSMANN MEETING On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and R
Physics
Scientific paper
The different ways in which an homogeneous and isotropic Universe in the form of a barotropic fluid expands in the cosmological theory of Brans-Dicke [1] can be completly rendered, even if the space is not flat, almost only for the special stress-energy tensor that represents incoherent radiation or ultrarelativistic matter because the original, and well known, nonlinear field equations comprise two unknowns which makes their integration difficult, and more so when the space is non-flat, in contrast with the General Relativity case which only has a single unknown function to determine -the scale factor-. Therefore, a fruitful avenue that can be used to obtain cosmological solutions for this, and other scalar-tensor theories, originally developed in Chauvet [2], and extended elsewhere [3] has been to procure equations for a single variable by combining the two aforementioned functions into a single one. So far this, and other methods to obtain perfect fluid, analytic solutions for a non-flat space, have given the sought after, and complete results, mostly for the vacuum, incoherent radiation, and stiff "matter" cases in this, and in similar but more general scalar-tensor theories [4]. A salient fact for a non-flat space is that radiation, and the remaining fluids as well, can expand linearly in time which is the limit for accelerating universes that, nowadays, turn out to be significant [5]. This expansion comes about as the end product of the special form that the composite function assumes: a second degree polynomial whose discriminant is equal to zero, which then permits a time inversion onto "cosmic time" which translates into a common behavior for the non-flat FRW models, and is moreover the general cosmic solution to the flat space [6]. For the polynomial function different fluids, and different spaces as well, are distinguished essentially by the three constant factors some of which depend on the equation of state through n, and the coupling parameter ω. Space curvature not withstanding, all the solutions can be obtained by solving a third order "dynamic equation" which naturally imposes itself by the desire to work under every circumstance with one dependent variable only. For k ± 1, the common for all the fluids, a second degree polynomial seems to be the only existing solution which can be given in terms of polynomic, transcendental or other fairly well known functions, and is furthermore time invertible also. The type of expansion that this functional form gives, is found to be a linear expansion in the physical, standard metric "cosmic time". This common type of outspread, alike for all the perfect fluids means that in this theory a curved space which expands in linear form ignores the particular nature of the "matter" that drives it. For the open space, k = -1, a linear expansion is known to give a Milne Universe where, space-time is actually flat Minkowski space. The analysis of this solution also brings out other, perhaps unexpected, consequences some of which may also apply to the general non-flat spaces behavior of this or other similar scalar-tensor theories. These special solutions imply that given k, and n the numerical value of ω is controlled by the amount of "matter" present while its sign, likewise, also depends on n. In other words, if the space is not flat one is not free to choose randomly the sign or the coupling between the tensor and the scalar modes freely. On the other hand, a linear expansion for k = 1 is not a Milne solution but the numerical values forω still depend on the quantity of "matter" present, and its sign on n...
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