### The Cosmological Redshift Manifests the Curvature and Interpreted as a Degree of Hyperbolicity of the Spacetime

#### Abstract

Hubble’s law describes a uniformly expanding flat universe. Hubble’s law doesn`t explain why distant objects were receding fastest. There is an approximately linear relationship between redshift and distance at small scales for all the FLRW models, and departures from linearity at larger scales can be used to measure *spatial curvature*. Locally the spacetime is flat. For distant objects, the imprint of the curvature is significant, where the spacetime does no longer remain flat. The redshifts from such distant objects increase according to the increase in the curvature of the hyperbolic spacetime. The cosmological (gravitational) redshift can be interpreted as a degree of the hyperbolicity of the curved spacetime. 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 *P = - *𝝆. The hyperbolic structure of the spacetime–not dark energy- causes the accelerated expansion of the universe. Thus, in our non-existing dark energy hyperbolic universe, the increase in the cosmological redshift can only account for the increase in curvature that causes such an accelerated expansion relative to the observer. We developed [S. A. Mabkhout, Phys. Essays, **26**,422 (2013)] the equation of motion in the hyperbolic spacetime, that describes the speed up motion in the hyperbolic spacetime and predicts the flat rotation curve. In the hyperbolic spacetime, the free fall due to the curvature, causes the non-decreasing speed of the galaxies for large *r*. Thus, the Doppler redshift manifests such curvature. As an object is far distant apart, as much the spacetime appears relatively hyperbolic curved with a high redshift. Its velocity relatively appears to exceed the speed of light "*c*" due to the assumption of flat spacetime.

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