The Big Bang hyperbolic universe neither needs inflation nor dark matter and dark energy
Authors: Mabkhout Salah A. 
Published: July 2015

Although the perspective for nearby objects in hyperbolic space is very nearly identical to Euclidean space (i.e., the universe locally is approximately flat consistent with local observations), the apparent angular size of distant objects falls off much more rapidly, in fact exponentially.
The universe is globally hyperbolic as we did prove mathematically [S. A. Mabkhout, Phys. Essays 25, 112 (2012)]. Such a solution predicts the equation of state of cosmology <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi>
<mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>ρ</mml:mi></mml:mrow></mml:math> . The hyperbolic structure of the space causes the accelerated expansion of the universe equivalent to its negative pressure, without need for dark energy. The
dark matter halo is nothing but instead of it we have a cell of same hyperbolic negative curvature as the negative curvature of the whole hyperbolic universe. The Virial theorem <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mrow>
<mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi></mml:mrow> <mml:mrow> <mml:mn>2</mml:mn></mml:mrow></mml:msup> <mml:mi>R</mml:mi> <mml:mo>/</mml:mo>
<mml:mi>G</mml:mi> <mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> does no longer hold for non-Euclidean space. We developed the equation of motion in the hyperbolic space-time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>e</mml:mi></mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi></mml:mrow></mml:msup>
<mml:msqrt> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo>
<mml:mi>a</mml:mi> <mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msqrt></mml:mrow></mml:math> that describes the speed up motion in the hyperbolic space-time and predicts the flat curve. Galaxies farthest away from the center are moving
fastest until they have reached a large distance from the center, the space-time turns flat, and they possess hyperbolic trajectory: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo>
<mml:msqrt> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo>
<mml:mi>a</mml:mi> <mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msqrt></mml:mrow></mml:math> , according to the Vallado theorem, with constant speed called hyperbolic excess velocity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
display="inline"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi></mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:msub> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mrow>
<mml:mo>-</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>/</mml:mo> <mml:mi>a</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:math> that can explain the galaxy flat rotation curve problem, where a is the negative semi-major
axis of orbit's hyperbola. Instead of the unphysical inflation epoch, the hyperbolic universe grows exponentially, preserves a legitimate inflation, and covers the current observed large structure (1028 cm).