8

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8.1 Gaussian Curvature and the Friedmann-Lematre-Robertson-Walker Interval

In this section we will construct the Friedmann-Lematre-Robertson-Walker interval for general relativity in various forms from an expanding universe (big bang) model, in particular

(8.1.1)

And discuss how the cosmological constant yields accelerated expansion.

Recall the discussion of Gaussian curvature for general relativity in section 6.1. Though it is popular to associate curvature with the Riemann tensor for the reason discussed there, Gaussian curvature is really what cosmologists mean when they talk about the universe being curved. Lets say we describe the motion of the bug on the sphere with the following coordinates.

R = the radius of the sphere.

= the polar angle coordinate.

= the azimuthal angle

Its easy to show that when the bug traces out a small circle around the pole it will find that "the curvature" of its space as defined by the equation in that section is

K = +1/R^{2}

(Hint: )

We next let the size of the sphere vary in time and note that the rigid ruler length for the path the bug travels is given by

(8.1.2)

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90 Chapter 8 The Big Bang

Since this type of space-like hyper-surface is totally uniform(it looks like the same sphere from any place on it [Sandage, A., G. A. Tamman, and E. Hardy, 1972] , isotropic [Collins, C. B., and S. W. Hawking, 1973] ) so we know that modeling a 3d+1 space-time by

(8.1.3)

where we have

(8.1.4)

will result in a solution of Einstein's field equations for general relativity for a totally uniform matter distribution for the universe.

Doing a coordinate transformation

(8.1.5)

results in

(8.1.6)

Now we see that for general relativity R(t) is now the meaning of the radius of the universe as a function of time. This is called the Robertson-Walker interval for general relativity. Though one can imagine positively curved and yet open universes with non-uniform matter distributions, according to this model a positively curved universe implies a closed curved universe. Any frame in which the interval is expressed in this form is called a comoving frame. The time according to a comoving frame given the initial condition of t = 0 at R(t) = 0 is called cosmological time. It is this time that is being referred to when a physicist is talking about the age of the universe. The r coordinate of a comoving frame is related to the Radius of the universe and the coordinate By the relation

(8.1.7)

(Note the subtle difference between this r and the bug's ruler distance based coordinate r)

We could also use the relation to the curvature

K = + 1/R_{0}^{2}

(8.1.8)

8.1 Interval Gaussian Curvature and the Robertson-Walker 91

to express the interval as:

(8.1.9)

Doing this its obvious that should the universe be negatively curved

K = -1/R_{0}^{2}

(8.1.10)

the interval would become

(8.1.11)

In which case a coordinate transformation also expresses it:

.

(8.1.12)

For general relativity a positively curved conceptual model can be either open or closed curved, though a Freidmann-Robertson-Walker positively curved model is always also closed. However, for general relativity there are no conceptual closed curved universe models that are also everywhere negatively curved. Mathematically the big bang creating an expanding universe can be modeled whether positively or negatively curved just fine. Conceptually in general relativity one can imagine a negatively curved universe using 2d cross sections just as we have done for the positively curved universe. Conceptually in general relativity one can imagine a positively and closed curved universe collapsing all the way back to the moment of the bang. However, there is no conceptualization for general relativity for an negatively curved universe extrapolating back to a big bang. As a result many physicists believe that the universe must be closed curved and positively curved. In order for this to be the case in general relativity, there must be much more matter in the universe than we have yet observed. This is called *the missing mass*. Experimentally the universe is found to be too nearly flat to tell for sure. In other words flat K in Eqn.8.1.9 is to small to tell from zero or determine the sign if its not exactly zero.

We know that there is much more matter here than luminous matter. The nonluminous matter is called *dark matter*. This constitutes some of the difference. A positive cosmological constant would account for more. Neutrino mass constitutes some as well, but even considering these, there would still be missing matter left to find.

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If we define a curvature parameter for general relativity lowercase k by k = -1, 0, 1 for negatively curved, zero curved, and Positively curved universes we can write the Robertson-Walker interval as

(8.1.13)

in terms of r and making the definition a(t) = R(t)/R_{0} we can write it

(8.1.14)

or by doing the transformation

(8.1.15)

we arive at the most simple general form

(8.1.16)

Now our universe as observed is very nearly flat, k ~ 0. So a good model for it is

(8.1.17)

Recall Einstein's field equations with the inclusion of the cosmological constant equation 6.3.14.

(8.1.18)

which he originally included in attempt to find steady state solutions [Bondi, H., and T. Gold, 1948] .

The following is an exact solution to equation 8.1.18 for , which is a case of equation 8.1.17

(8.1.19)

This is called the deSitter spacetime expressed in conformally flat coordinates. Note that if the cosmological constant is greater than zero, that not only does the space expand, but the rate of the expansion increases with time and with the "size" of the universe:

(8.1.20)

The cosmological constant term in Einstein's field equations, can be thought of as a stress-energy source term in itself.

(8.1.21)

The stress-energy of the cosmological constant then is

(8.1.22)

And so we note that a positive cosmological constant has a positive energy density. This is nothing mysterious. The only interesting aspect of the stress-energy of this "dark energy" is that the sum of the pressure terms add up to something greater in magnitude than the energy density term and that they correspond to a negative pressure state.

Under a coordinate transformation from 8.1.19, the deSitter spacetime [De Sitter, W., 1917] is often expressed

(8.1.23)

One may consider a negative cosmological constant leading to the following called the anti-deSitter spacetime

(8.1.24ab)

But that would correspond to a negative energy density for the dark energy and would lead to the opposite effect of the accelerated expansion observed.

In particular, the transformation that takes 8.1.19 to 8.1.23 is

8.1.25ab

Exercises

**Problem **8.1.1

Draw an example of a 2d cross section otherwise called a spacelike hypersurface, a cross section of the manifold for

a. A closed positive curved universe - this should be easy now.

b. An open positive curved universe.

c. A open negative curved universe.

**Problem **8.1.2

Does the answer for b look the same or different from a point of view of a bug at different locations on the surface?

**Problem **8.1.3

Given , what is the minimum value for r_{0} to close the universe.

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In this section we will use Eqn. 8.1.16

for general relativity to derive Hubble's Law

v = HL

(8.2.1)

And show that for general relativity wavelengths of light from other nearby galaxies in comoving frames undergos a shift in wavelength given by

(8.2.2)

From the above form of the Robertson-Walker interval of general relativity we see that the rigid ruler distance between here and a far distant galaxy is given by

(8.2.3)

Galaxies associated with comoving frames will have a zero coordinate velocity. .But since a(t) depends on time their ruler velocity v = dL/dt will be nonzero [Hubble, E. P., 1929] .

At this point we make a definition for Hubble's "constant"

(8.2.4)

Note Hubble's "constant" is not really a constant, but is a function of time. Inserting this definition we have Hubble's law [Hubble, E. P., and M. L. Humason, 1931] Eqn. 8.2.1

v = HL

The *current* value of Hubble's constant is not very well known but observations indicate that it is

(8.2.5a)

where 1 parsec = 3.26 ly

and in units of 1 over years this is

(8.2.5b)

94 Chapter 8 The Big Bang

Also note that for large enough distances v is greater than c. This is allowed because there are no *local *faster than c motions to violate special relativity and as long as this is the case, faster than c motions are allowed in general relativity.

Next consider light waves comming to us from another galaxy in a comoving frame. ds = 0 for light so for general relativity we have

which results in

Integrating this over the time that it takes a wave crest to reach us results in

If the period of the wave is Ti where it is emitted and the period is Tf here where it is recieved then the equation for the following wave crest will also be

because a galaxy of a comoving frame doesn't change in its coordinate position. Therefor

A little manipulation results in

For small periods this becomes

In terms of wavelength this can be rewritten

(8.2.6)

If the cosmological transit time for general relativity for the wave is written so that the initial ruler distance to the galaxy is L, then this becomes

8.2 Hubble's Constant 95

If the galaxies are nearby then the transit time is small so and recalling the definition for H(t) we have

(8.2.7)

At this point we can make a relation between the rigid ruler velocity and the wavelength shift for nearby comoving galaxies.

(8.2.8)

Consider for a moment the wavelength Doppler shift of special relativity. Eqn. 3.3.14.

A little manipulation results in

Next look at the case for . It becomes.

(8.2.9)

This is in exact agreement. Thus on local scales it doesn't matter whether we say that the galaxies are "still" within an expanding space or if we say the galaxies are moving apart from each other within a static space. But beware, this is only a *local equivalence*. Since the universe is not a steady state universe, there is no frame for which the expanding universe, Robertson-Walker *metric*, is *globally *static. This means that globally speaking, of the two, only the expanding space paradigm is descriptive.

The time in any of the above expressions of the Robertson-Walker interval is called *cosmological time*. As we know by now, time is relative and so one might wonder how meaningful it is to discuss the age of the universe. It turns out that the typical average path of galaxies follow comoving frames and so cosmological time is meaningful for us traveling in a typical galaxy. The age of the universe is defined as the amount of time that has gone by since a(ct) was zero. To express time in a unitless way one can also define *cosmic time* in terms of cosmological time by t_{cos} = tH_{0}

96 Chapter 8 The Big Bang

Exercises

**Problem **8.2.1

What is the rigid ruler distance L to a galaxy that travels faster than c according to Eqn. 8.2.1 ?

**Problem **8.2.2

If the rigid ruler distance to a comoving galaxy is L = 10 Mpc, What is v and what is the wavelength we see of light that originated at according to Eqn 8.2.8?

**Problem **8.2.3

If a thin wall has a stress energy tensor whose only significant terms are

according to a local observer's frame the vacuum field solution around the wall is

Were we define by

**a.** Use Einstein's field equations to show that the exact stress energy tensor for this exact spacetime interval is

**b.** Show that including the only significant terms at the origin that it does reduce and that even with all terms included it is a vacuum for any other z than 0.

[Hints: and and at the origin any finite terms are insignificant compared to ]

**c.** Calculate the Riemann tensor to show that is is zero off the wall.

**d.** Demonstrate that a test particle released from rest will accelerate away from the wall according to these coordinates which are for an observer who is stationary with respect to the wall.

**Problem **8.2.4

I David Waite discovered the following exact solution of Einstein's field equations for a domain wall of a different pressure state

**a.** Verify that this spacetime corresponds to a stress-energy tensor for a domain wall whose only nonzero terms are

by using Einstein's field equations or a computor programed to calculate the Einstein tensor.

**b.** Find the Riemannian spacetime curvature for this spacetime.

**c.** Demonstrate that a test particle released from rest will accelerate toward the wall according to these coordinates which are for an observer who is stationary with respect to the wall.

**d.** What would happen to the system if walls of different pressure states such as these where released parallel to eachother?

**e. **Show that through frame transformation, the following is equivalent:

**Problem **8.2.5

By patching together a transformation of a known vacuum solution with its mirror across z=0, I then went on to discover a domain wall solution, exactly vacuum off of the wall, that in the wall had the pressure terms along x and y the same value and of arbitrary magnitude. Compute the Einstein tensor to obtain the stress-energy tensor for

to verify that it is such a solution and show that the stress-energy tensor exactly corresponding to this line element is

Note that the pressure terms vanish in the limit as . For other wall type solutions see problems 11.1.5-11.1.7

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8.3 From the Big Bang to Dark Energy Era

The mapping of the angular size of fluctuations in the cosmic microwave background radiation agree with modeling the universe as flat. As such the FRW line element becomes

(8.3.1)

Defining

(8.3.2)

The Einstein tensor can be written

(8.3.3)

And the Ricci-scalar given by

(8.3.4)

where is the cosmological constant.

The solution for f from this differential equation from the Ricci-scalar is

(8.3.5)

To put the moment of the big bang at t=0, B=0. So we have for the big bang line element

(8.3.6)

Where T is the age of the universe in this time coordinate. The Einstein tensor for this is

(8.3.7)

Refer to Einstein's field equations

(8.3.8)

where where this latter G is the gravitational constant of the universe.

Here we clearly see two matter dominated eras of the universe. Near the big bang, the energy density terms due to the stress-energy of electromagnetic radiation swamps and Einstein's field equations are dominated by that electromagnetic radiation stress-energy source

(8.3.9)

which for small enough times near the big bang can be approximated by

(8.3.10)

I find it interesting here that near the big bang, the electromagnetic radiation energy density and pressure terms are determined soley by how much time has gone by since the big bang for a flat universe. At a given time only a specific amount of radiation energy per volume can exist and it is only perturbed at larger times by the cosmological constant.

For very large times this source term diminishes and the dark energy swamps as the source in the field equations

(8.3.11)

Using this flat space model we find that Hubble's constant is given by

(8.3.12)

So the current value of Hubble's constant is

(8.3.13)

In units of 1/years Hubble's constant has been measured to be

(8.3.14)

And the age of the universe in years is estimated at

(8.3.15)

So we can use these in the equation for the current value of Hubble's constant and use a function solver to make a prediction for the value of the cosmological constant. Doing this we find it to in units of one over square of years to be

which in SI units is

(8.3.16)

If we consider from the contribution to the Einstein tensor to be "dark energy" and compare to the total of this summed with T^{00} we find that the % Dark energy of the universe's energy that is predicted by this flat space model based on Hubble's constant and age of the universe estemates to be

(8.3.17)

Our measure for this percent is only 70%. If the measure, Hubble's constant and age of the universe estimates are right then this indicates that the universe does have some negative curvature to it, which was not in the flat space model. The universe would be negatively open curved.

The behavior of a(ct) was derived for a Ricci-scalar equation above modeling the universe to contain only dark energy and electromagnetic radiation. In the early history of the real universe, much of the electromagnetic radiation made a phase transition to ordinary matter and dark matter. As such the amount of electromagnetic radiation actually observed left over in the cosmic microwave background radiation is only about 10^{-5} of the electromagnetic radiation energy density represented in this model.

Citations References

Bondi, H., and T. Gold, 1948, "The Steady-State Theory of the Expanding Universe," *Mon. Not. R. Astron. Soc.108, *252-270

Collins, C. B., and S. W. Hawking, 1973, "Why is the Universe Isotropic?" *Astrophys. J. 180, *317-334

De Sitter, W., 1917, "On the Curvature of Space," *Proc. Kon. Ned. Akad. Wet. 20, *229-243

Hubble, E. P., 1929, "A Relation Between Distance and Radial Velocity Among ExtraGalactic Nebulae," *Proc. Nat. Acad. Sci. U.S. 15, *169-173

Hubble, E. P., and M. L. Humason, 1931, "The Velocity-Distance Relation Among Extra-Galactic Nebulae," *Astro-Phys. J. 74, *43-80

Sandage, A., G. A. Tamman, and E. Hardy, 1972, "Limits on the Local Deviation of the Universe from a Homogeneoous Model," *Astrophys. J. 172, *253-263

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