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The structure of the Universe

   The starting point of a theoretical exploration of cosmology is the Einstein equation

Einstein equation

with a metric of the form

Friednmann-Robertson-Walker metric

The spacetime being modeled by this equation can be neatly separated into time and space, so we can talk of this spacetime as representing the evolution of space in time.
   The space part of this spacetime is homogeneous (looks the same at any point in a given direction) and isotropic (looks the same in any direction from a given point). This is an abstract ideal approximation to the Universe, but it's one that has worked extremely well from an observational point of view, as will be shown below.
   There are three options for the spatial geometry of a spacetime with the above metric, represented by three choices for the value of the parameter k: k=1, k=0 or k=-1. The condition of being spatially homogeneous and isotropic means that the surfaces of constant time t have constant curvature, which can be either positive, zero or negative.

A sphere A hyperboloid
A sphere has constant positive curvature. A hyperboloid has constant negative curvature.

   To solve the Einstein equation, we need to postulate some "stuff" in the spacetime, such as matter, radiation or vacuum energy, with energy momentum tensor Tmn whose components are the energy density r and pressure p of the "stuff" in question. The equations for the scale factor a(t) are

Equation for Hubble parameter

Equation for scale factor

Here we have truncated Newton's constant GN to plain G. The top equation contains the condition for the closure density of the Universe explained below.
   There are many different kinds of "stuff" that can be a part of the energy density r, with different equations of state relating r to the pressure p. For bookkeeping purposes, let's label each different r by an index i, so that ri refers to the energy density from the ith type of "stuff" in this spacetime. Let's also set something called the critical density rcrit and then make the above equation dimensionless by dividing everything by the critical density:

Critical density

Let's call W the density parameter. The equation that will tell us the curvature of space from the stuff content of the spacetime becomes

Density equation

The three possibilities for the value of the parameter k correspond the three different possibilities for the curvature of space in this spacetime. A value of k=1 corresponds to constant positive curvature, k=0 to zero curvature and k=-1 to constant negative curvature.
   The time evolution of space is more complicated because it depends on the equation of state of the stuff in spacetime. The equation of state is the relationship between pressure and density in the stuff. Energy conservation plus the equation of state determine how the energy density changes as space evolves in time.
   This is where vacuum energy becomes important. The energy densities for matter, radiation and vacuum energy change with the size of space (the scale factor a(t)) like

Densities for matter, radiation and vacuum energy

So as the Universe is getting bigger, the energy density from matter and radiation would be getting smaller, but vacuum energy density would remain the same. Another name for vacuum energy is the cosmological constant. A cosmological constant eventually controls the time evolution of an expanding universe, because its energy density stays the same while those of matter and radiation are getting smaller.
   In a spacetime with all three forms of energy present, the radiation part of the mix will dominate the dynamics when the scale factor a(t)<<1. In the Big Bang model this is called the radiation dominated era, and accounted for the first 10,000-100,000 years of the evolution of our Universe. Right now the dominant forms of energy in our Universe are matter and vacuum energy.
   That being said, we will avoid dealing with any vacuum energy right now and consider a spacetime with only matter, with no radiation or cosmological constant. In this case, the time evolution of space is related to the curvature of space as follows:

k W Topology Time Evolution
1 >1 Closed Space is positively curved and finite, expands from zero size to a maximum size and then shrinks back to zero again
0 =1 Open Space is flat and infinite, and expands forever
-1 <1 Open Space is negatively curved and infinite, and expands forever

   If the amount of energy density in the spacetime is over the critical density, so that W > 1, then the fate of the Universe is to expand in a Big Bang but then eventually contract back into a Big Crunch. Despite the fact that this would take place on a time scale of billions of years, humans today find this possibility philosophically undesirable. More importantly, the data do not support it.
   The visible matter in the Universe observed by humans today has barely a fraction of closure density. In fact, the Universe as observed today seems to have barely a fraction of the mass needed to keep galaxies from flying apart, based on the rotations of the stars in the galaxy about the galactic center.
   What keeps the galaxies from flying apart? It must be a lot of mass that we can't see. Which brings us to the subject of dark matter.

Next: Dark matter and the cosmological constant>>

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TASI Lectures: Cosmology for String Theorists by Sean M. Carroll

How old is the Universe? // Structure of the Universe // The Big Bang // Before the Big Bang? // What about string theory?

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