The Evidence for a
Non-Zero Cosmological Constant

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Let's start with the very basics. What do we know about the Universe, really:

• Its expanding at a current rate of somewhere between 40 and 100 km/s mpc^-1
• For uniform expansion, the expansion age is in the inverse of the rate: The expansion age of the Universe then is 10--25 billion years. Evidence for uniform expansion started with Hubble in 1920 and is confirmed today out to large distances by using Supernova as Standard Candles (at this point real physicists should protest)

• The universe is filled with an isotropic and homogenous background of photons whose energy distribution is just like that of a blackbody
• The ratio of baryons to photons S , is 5 -- 10 x 10^-10
• The unit of mass aggeregate is a galaxy. The local distribution of galaxies is very inhomogenous
• If the universe is closed, then luminous material (\eg stars in galaxies) contribute 0.5% of the closure density.

What large scale features we don't we know:

• The total matter density in the Universe. This is unknown by a factor of 10.
• The large scale geometry of the Universe (which is driven mostly by the distribution of its mass)
• The Nature and distribution of the Dark Matter (if there is any)
• How structure formed and the bias between the distributions of mass vs. light
• The behavior of gravity on very large scales
• What Cambridge University thinks we don;t know

We have to create a framework in which we can relate observationally determined parameters to the large scale charactistics of the Universe:

We assume the Universe to be isotropic and homogenous. In this situation the length of a Geodesic is described by the Robertson-Walker metric:

Where

• R(t) is the Universal Scale Factor that describes the expansion as a function of time
• k is a constant (-1,0,+1) that specifies the geometry (curvature) of the surface
• r is the co-moving coordinate
• Gravity enters the RW metric through R(t) and k

The gravitational acceleration of the Universe is given by Poisson's Equation:

In the early Universe, pressure from the photon field was very high and the universe acts as an isotropic fluid. This Poisson's equation is modified to

The effective gravitional mass density has been increase due to the pressure (making it hard for structure to form early on). Consider a spherical region with some net gravitational acceleration. The total mass of the sphere is:

As the acceleration on the surface of the sphere goes as GM/r_s² then

Within this sphere there is net energy E_n = rho*V

From conservation of Energy we have:

The volume and its derivative are:

Solving for p and inserting that into the previous equation yields a relation between only density and its derivatives and the acceleration:

This differential equation can be solved as a non-trivial exercise for the reader to yield

Now let's assume the universe is static: all derivates with respect to r_s dissappear and we have

Since mass is positive, then pressure must be negative to solve this equation. Normal matter doesn't have negative pressure so Einstein introduces the cosmological constant to compensate:

Thus we have:

and

This last term shows that the rate of change of R(t) depends on

• the density which determines the net gravitational acceleartion
• the curvature of the Universe
• the vacuum energy term which acts as a long range repulsive force

The quantity is called the Hubble constant or H_o If = 0 and H_o can be measured then K is known. Another way to look at this is the following:

Suppose K=0 (space is flat) and = 0 and pressure = 0 then we have:

Which is solved with:

This is also the solution which satisfies the condition that the mass within a sphere remains constant with time.

Which is only solved if R(t) goes as t^2/3. Hence we get

So we have expressions for the critical density and the expansion age of the Universe in terms of the observable H_o.

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