In Cosmology, the space curve represents the curve of the space sections of the Univers in a homogeneous model and isotropic of the type Friedmann-Lemaître-Robertson-Walker. Intuitively, it gives a scale length which delimits the distances in on this side which the universe can locally be described using metric Euclidean, i.e. the results of Solid geometry usual (as the Théorème of Pythagore) remain valid. In such a cosmological Model, the space curve is the only local geometrical parameter which characterizes the structure of space. Like habit in Geometry, the space curve corresponds (with the possible sign near) contrary to the square of the Radius of curvature of the Hypersurface S of density constants existing in these models.

Three possible cases

. Three cases are possible, according to the sign of the curve:

• a null space curve corresponds to space sections described by the Euclidean Géométrie. In particular the Théorème of Pythagore is valid there, and angles of a triangle is equal to 180 ° summons it.
• a positive space curve corresponds to the three-dimensional analog of the spherical Géométrie. The theorem of Pythagore is not valid any more, and angles of a triangle is higher than 180 ° summons it. Corollary, the angular Taille of an object of size given decrease less quickly with the distance than in the preceding case (and increases even with the distance for an object located nearer to the antipodean point than of the observer). One can easily visualize a space with two dimensions of constant positive curve: it is about the Sphère. Its three-dimensional analog is on the other hand more difficult to visualize.
• a negative space curve corresponds to a hyperbolic Géométrie. The theorem of Pythagore is not valid either, and angles of a triangle is lower than 180 ° summons it. Consequently, the angular size of the objects decrease more quickly with the distance than in the preceding cases. On scales larger than the radius of curvature, it decrease even Exponentielle lies and not linearly with the Distance. A simple example in two dimensions is given by the hyperboloidal to a tablecloth (visually it is a saddle of horse when it is plunged in space with three dimensions). Once again it is not easy to visualize a hyperbolic three-dimensional space simply.

Cosmological aspects

The equations of Friedmann connect the Paramètre of Hubble $H$ to the curve $K$ and the Densité of energy $\backslash \; rho$ of the matter according to the formula

$3\; \backslash \; left\; (\backslash \; frac\; \{H^2\}\; \{c^2\}\; +\; \backslash \; frac\; \{K\}\; \{a^2\}\; \backslash \; right)\; =\; 8\; \backslash \; pi\; \backslash \; frac\; \{G\}\; \{c^4\}\; \backslash \; rho$,

where $G$ is the Constante of Newton, $c$ the Speed of light and $a$ the Scale factor. Space curve (unit: the reverse of the square a length) corresponds here to $K/a^2$. By introducing the Densité criticizes $\backslash \; rho\_\; \{\backslash \; rm\; C\}$ and the Paramètre of density $\backslash \; Omega\; =\; \backslash \; rho/\backslash \; rho\_\; \{\backslash \; rm\; C\}$, it is possible to rewrite the preceding equality according to

$1\; +\; \backslash \; frac\; \{K\; c^2\}\; \{a^2\; H^2\}\; =\; \backslash \; Omega$.

The radius of curvature $R\_c$ of the space sections can thus be written in term of the variation with 1 of the parameter of density and the Rayon of Hubble, $R\_H\; =\; C/H$:

$R\_c\; =\; \backslash \; frac\; \{R\_H\}\; \{\backslash \; sqrt\}$

This last equality makes it possible to see which possible variation with 1 of the parameter of density one can hope to measure. So that the effect geometrical (related to the relation between angular size and distance) are measurable because of a nonnull curve, it is necessary that the radius of curvature is not too large compared to the ray of the observable universe. In the standard Model of cosmology, the latter is about three rays of Hubble. Thus, the geometrical effects due to a nonnull space curve are measurable as soon as the quantity

$3\; \backslash \; sqrt$

is not too small in front of 1. In a a little unexpected way, that proves that values from $\backslash \; Omega$ of 0,97 or 1,03 can be distinguished without too much from difficulty, when well even uncertainties on the critical density and the density of matter (of which the report/ratio is equal to $\backslash \; Omega$) are important.

Curve and to become expansion of the universe

It is sometimes known as that the sign of the space curve determines to become to it Expansion of the universe, this one knowing an eternal expansion if the curve is negative or null, or a stop of this expansion followed by a Big Crunch when the curve is positive. This assertion is erroneous because it depends on the material contents of the universe. If all matter shapes of universe are of the null or negligible pressure, then the preceding assertion is exact. If there is ordinary matter and a cosmological Constante the situation becomes very different. In particular a universe with positive and constant curve cosmological positive can either result from a Big Bang and finish by recontracter (when the cosmological constant is weak), or to have the same past, but an eternal expansion if the cosmological constant is sufficiently large, or to be static (it is the Univers of Einstein), or to have known in the past a phase of contraction, followed by a phase of rebound and an eternal expansion (one of the possible cases of the Univers of Sitter).

Importance for the cosmological models

The standard Modèle of cosmology is at present dominated by the idea that universe knew a phase of extremely violent expansion in its past, called inflation. This model predicts that the space sections of the universe are Euclidean, in any case on scales about the size of the observable universe. A proven variation of the space curve to the zero value would be regarded as a very strong argument in discredit of inflation, even if this one could put up with such a result, but by requiring natural parameters enough not very.

Current data

The most precise data on the space curve of the universe are those resulting from the analysis of the anisotropies of the cosmological diffuse Fond. The last data of the satellite WMAP give (see, p. 51)

$\backslash \; Omega\; =\; 1,\; \backslash !\; 003^\; \{+0,017\}\; \_\; \{-\; 0,013\}$,

in perfect agreement with the theory of inflation. With the commonly allowed value for the constant of Hubble, that gives a radius of curvature at least equal to 30  000 Mégaparsec S, is more of the double of the ray of the observable Univers.

See too

• Problème of flatness
• Courbure (in mathematics)
• Équations of Friedmann
• Paramètre of density
• Densité criticizes
• Topologie of the Universe

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