*Xiaoqi Yu*

The shape of universe have been discussed by scientists, philosophers, writers and almost everyone. There were used to be lots of controversy over this problem; the universe seems like a enigma for us. Thanks to modern knowledge and technology, universe is no longer a scare or absolute enigma for people. By finding the shape of universe, we can predict the future of universe. Based on mathematical analysis and astronomic observation, now it is wildly accepted that the universe is flat.

### Density of the universe

According to General relativity, the local geometry (the curvature of the observable universe at any arbitrary point) of the Universe is described by what we call the Friedmann-Lemaître-Robertson-Walker (FLRW) model. This model provides a curvature of the universe. Here, the matter within the Universe is modeled as a perfect fluid. Mass causes space to curve. The shape of the universe is then determined by the fight between the pull of gravity and the momentum of expansion.

In the FLRW model, the density parameter Ω is related to the curvature of space. This parameter is defined as the ratio of the actual density of the universe and the critical density (the required density for the universe to be flat).

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* *According to scientists' research, c is the critical density of universe.

include both regular and dark matter.

when Ω<1 hyperbolic geometry

when Ω=1 flat

when Ω>1 spherical geometry

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### The geometry of universe

According to three solutions of the equation, here are three possible scenarios:

In the first case, there is exactly enough mass to stop the expansion, the universe will expand forever and has no boundaries.

In the second case, there is enough mass to slow down and stop the present expansion. The universe is not infinite, but has no boundaries. Eventually, the universe will collapse producing what we call Big Crunch.

In the third case, there is not enough mass to stop the expansion. The universe will expand forever and has no boundaries.

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Resources:

http://skyserver.sdss.org/dr1/en/astro/universe/universe.asp#properties

http://atramateria.com/the-shape-of-the-universe/

http://snap.lbl.gov/multimedia/presentations/DUII/img1.html

http://www.einsteins-theory-of-relativity-4engineers.com/friedmann-equation.html

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