Geometry of the Universe

By: Kyle MacKay

Local Geometry vs. Global Geometry of the Universe

The local geometry of the universe describes the curvature of the universe, In particular, it describes the observable universe which can be described as the space around us bounded by the event horizon, or in other words, as the distance to which light has travelled since the origin of the universe.

The global geometry of the universe describes the geometry and topology of the universe as a whole.

r = 4.3x1026 meters or 46 billion light years

Albert Einstein’s Theory of General Relativity (1916)

Einstein explains what we perceive as gravitational force, actually results from the curvature of space and time.

The local spacetime can evolve and warp in the presence of energy and matter, therefore causing smaller bodies such as planets moving through it to curve.

According to Einstein, gravity is spacetime curvature.

Not only does general relativity explain the movement of planets, it also explains the physics of black holes, the expansion of the universe and the bending of light from distant stars and galaxies.

Albert Einstein’s Theory of General Relativity (1916)

He derived what are known as the Einstein Field Equations, which are represented by:

where is the Ricci curvature tensor, is the scalar curvature, is the metric tensor, is the cosmological constant, is the Newtonian gravitational constant, is the speed of light and is the stress-energy tensor.

Friedmann-Robertson-Walker (FRW) Model of the Universe

Developed independently by Alexander Friedmann, Howard P. Robertson and Arthur Geoffrey Walker during the 1920s and 1930s.

Is an exact solution of Albert Einstein’s theory of general relativity. The speed of light limits us to viewing the universe’s volume since the Big Bang,

therefore the FLRW model is . The FLRW model describes the observable universe as approximately isotropic

and homogenous. It is supported by many astronomical observations, such as supernovae and cosmic microwave background radiation. The metrics which meets the criteria of homogeneity and isotropy is:

where c is the speed of light, t is time a(t) is the rate relative expansion of the earth and where ∑ ranges over a three dimensional space of uniform curvature

(Euclidean space, hyperbolic space or elliptical space).

Friedmann-Robertson-Walker (FRW) Model of the Universe

Given the metric mentioned previously (which is under the assumption of homogenous and isotropic conditions) and Einstein’s field equation, Friedmann derived what is known as the Friedmann equation:

From which we can derive the density of the universe if it were flat, which is known as the critical density, by substituting k=0:

and is estimated to be approximately 5 atoms per cubic meter.

Friedmann-Robertson-Walker (FRW) Model of the Universe

Since, according to Einstein’s theory of general relativity and the FRW model, the universe can be curved by mass then the density parameter (Ω) of the universe determines its shape.

Ω is the actual density of the universe divided by the critical density of the universe.

• If Ω > 1, the critical density is less than the actual density. In this case the universe is closed and finite, therefore the it contains enough mass to eventually stop its expansion. Once the expansion halts, the universe will begin to contract (the “Big Crunch”). The universe is a 3-dimensional spherical space.

• If Ω < 1, the critical density exceeds the actual density. In this case the universe is open and infinite, therefore it will continue to expand forever. The universe is a 3-dimensional hyperbolic space.

• If Ω = 1, the critical density equals the actual density. In this case the universe is flat and infinite or finite. The universe is a 3-dimensional Euclidean space.

Global Geometry: Zero Curvature

All three of the possible geometries of the local structure are classes of Riemannian geometry where parallel lines never meet (Euclidean geometry), must cross (spherical geometry) or always diverge (hyperbolic geometry).

In global structure is composed of the local geometry plus a topology.

If the curvature of the universe is zero, it may be infinite or finite. Euclidean space, simply connected and infinite while a structure such as the torus or the Klein bottle is flat, multiply connected and compact. According to relativity, they are all topologically equivalent.

Global Geometry: Zero Curvature

However, in the multiply connected universe there are multiple paths light could travel to an observer compared to one path in a simply connected universe. This would cause an observer to see multiple images of each galaxy, potentially misinterpreting them as distinct.

Global Geometry: Positive and Negative Curvature

A universe with positive curvature can be described by spherical geometry, which could be anything that is topologically equivalent to the hypersphere (specifically, the Poincaré dodecahedron).

With positive curvature, the universe has to be finite and compact, but can either be simply connected or multiply connected.

A universe with a negative curvature can be described by hyperbolic geometry. There are a great variety of potential of hyperbolic 3-manifolds, although their classification is not yet completely understood. Specifically, the Picard horn has been proposed as a possible model of the shape of the universe.

National Aeronautics and Space Administration’s (NASA) Discoveries

The Wilkinson Microwave Anisotropy Probe (WMAP) is a spacecraft that the NASA launched in 2001 to measure the cosmic microwave background (CMB) radiation – the radiant heat left over from the Big Bang.

Prior to WMAP, the universe was known to be flat within 15% accuracy. According to WMAP readings in 2013, we now know the universe is flat within 0.4% margin of error.

How can it predict this?

National Aeronautics and Space Administration’s (NASA) Discoveries

http://map.gsfc.nasa.gov/resources/featured_images.html

Exam Question

Q: Explain what the density parameter (Ω) represents, and what 3D-space the FRW model suggests for the three different cases.

A: The density parameter is the quotient of the actual density of the universe and the critical density of the universe. If Ω > 1, the universe is a 3D spherical space. If Ω < 1, the universe is a 3D hyperbolic space. If Ω = 1, the universe is a 3D Euclidean space.

References

Einstein, A. & Infeld, L. (1938). The evolution of physics: the growth of ideas from early concepts to relativity and quanta. New York, NY. Simon and Schuster.

Ferreira, P. (2010). Instant expert: general relativity. New Scientist. 2767. pp. 44-64.

Griswold, B. (2014). Wilkinson microwave anisotropy probe. National Aeronautics and Space Administration. Retrieved from: http://map.gsfc.nasa.gov/

Kurki-Suonio, H. (2012). Friedmann-Robertson-Walker universe. University of Helsinki. Retrieved from: http://www.helsinki.fi/~hkurkisu/cosmology/Cosmo4.pdf

Redd, N. (2014). What shape is the universe? Space. Retrieved from: http://www.space.com/24309-shape-of-the-universe.html

Schombert, J. (2013). Geometry of the universe. University of Oregon. Retrieved from: http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html