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Astron. Astrophys. 321, 19-23 (1997)

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2. Effect of mixing of geodesics flow: maps

The idea of the effect of instability of trajectories of freely moving particles can be most clearly demonstrated via the Jacobi equation written for spaces with constant curvature k:

[EQUATION]

describing the behavior of the vector of deviation, n, of close geodesics. Solutions of this equation are determined by the sign of the curvature: when [FORMULA] one has exponentially deviating geodesics.

A more rigorous treatment (Gurzadyan & Kocharyan 1991, 1992) includes the study of the projection of geodesics from (3+1)-dimensional Lorentzian space to a 3D Riemannian one, and the behavior of time correlation functions for geodesic flows on homogeneous isotropic spaces with negative curvature. Geodesic flows, being Anosov systems (locally if the space is not compact), are exponentially unstable systems possessing: the strongest statistical properties (mixing), non-zero Lyapunov characteristic exponents, and positive Kolmogorov-Sinai (KS) entropy h (Arnold 1989).

For Anosov systems two geodesics in 3-space deviate exponentially according to the law

[EQUATION]

where [FORMULA] is the Lyapunov exponent.

For a homogeneous isotropic Friedmannian Universe with [FORMULA] the Lyapunov exponent is determined by the only parameter a, the diameter of the Universe:

[EQUATION]

while Lyapunov exponents vanish when [FORMULA].

Time correlation functions, describing the decrease of perturbations also decay exponentially as determined by KS-entropy: [FORMULA].

For the Universe expanding as

[EQUATION]

the relation between the quantitative measurement of the distortion of patterns, [FORMULA], and [FORMULA] is given by (Gurzadyan & Kocharyan 1993):

[EQUATION]

where [FORMULA] is its present value ([FORMULA]), [FORMULA] corresponds to the time when matter becomes non relativistic and [FORMULA] to the decoupling time. We deliberately used the power law representation for the expansion law, (instead of trivial use of the corresponding exact Einstein equation) to demonstrate the role of the expansion rate in the described effect. It is not excluded that in the future the geodesic mixing effect can be useful to gain information also on the expansion rate of the Universe after the decoupling epoch.

The parameter of elongation [FORMULA] defined via the divergence of geodesics in (3+1)-space:

[EQUATION]

approaches 1 when [FORMULA] tends to 1 as shown in Fig. 1.

[FIGURE] Fig. 1. Elongation parameter [FORMULA] as a function of [FORMULA] for two exponents for the law of the expansion rate of the Universe, [FORMULA] and [FORMULA].

The typical pattern of a hot spot as seen today in a [FORMULA] universe would exhibit a very complex shape. The elongation, [FORMULA], measures the smallest-to-largest ratio of diameters of one-connected regions and is related to the "degree of complexity" of anisotropies.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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