Inflation: How the Universe Became Flat

by Chris Sterpka

Observations of the large scale structure of the universe indicate that it is essentially flat,

isotropic, and homogeneous. This is determined from calculations of the overall curvature of space-

time from the Einstein equation paired with the Friedman-Robinson-Walker model (i.e. FRW model).

These equations accurately model our universe as isotropic, homogeneous, and flat; however, in order

to explain these characteristics, an inflationary model of the universe is required.

The FRW model is defined by the relation:

2
ds 2 =−dt 2a2  t∗
[
dr 2
1−kr 2
r 2 dθ 2sin 2 θ dφ 2 
]
where a t  is the scale factor, and k is a constant which is determined by the large scale curvature

2 8π ρ 2
of the universe. Both k and a 2 t  are determined by the Friedman equation: ȧ − a =−k .
3

The Friedman equation is an orthonormal projection of the Einstein equation in coordinate basis for the

time-time component of the FRW metric (1: Hartle 484). The constant k can take on one of three

values, {1,0,-1}. Which value k takes is determined by the actual curvature of the universe. If the

universe is positively curved, the value is 1, if it is flat, the value is 0, and if it is negatively curved, the

value taken is -1. What determines the actual value of the curvature is the other three non-vanishing

components of the Einstein equations for the FRW metric:

[ ä 1
]
Gr r =Gθ θ =G φ φ =− 2  2  k ȧ2  =8 π p (1: Hartle 484).
a a

In order to be able to plug in values to the Einstein and Friedman equations, it is necessary to

8  0 −k
relate these quantities to observables in our universe. The equation: H 02 − = 2 is found by
3 a0
dividing the Friedman equation by a 2 t  and evaluating it for the present moemnt in time (2: Hartle

388). H 0 is the Hubble constant, 0 is the current density of the universe. The denisity of the

0
universe is most often expressed as the the dimensionless parameter: ≡ where crit is the
crit

density required for the universe to be exactly flat. The constants have been determined in current

observations.

Since the universe is such a big complicated place, many approximations have to be made. In

order to estimate the overall structure of the universe against the average density of the universe,

galaxies are approximated as cosmological fluid. This cosmological fluid approxmates these gallaxies

as a pressuless gas which is the average density of the galaxy over the whole area of the galaxy. This

model is defined by the relation:

There are three versions of the FRW equation: the flat metric, the closed metric, and the open

metric. Each is a possible representation of our universe. What determines whether one metric is

representative of our universe is based on the parameter omega. Omega is a relative measure of the

current energy density of the universe verses the critical density. The critical density is found from the

Friedman equation for a flat universe:

The Friedman equation is a derivation from the Einstein equation:

The Einstein equation takes the mass-energy density of the universe as input and produces the
space-time curvature as output. From this, the FRW metric specifies the relationship between distance

elements, the expansion coefficient, and the rate of acceleration of the universe.

(Provide in detail these calculations)

If the universe is flat, then the measure of the relative density to current density (omega) is one.

If the measure is greater than one, then the universe is closed. Finally, if it is less than one, the universe

is open.

If the Einstein equation and the FRW metric are a valid method of modeling our universe on the

large scale, then flatness is just one value predicted from a whole spectrum of possible values for the

curvature of our universe. So, it is very coincidental that the universe is homogeneous, isotropic, and

flat. Data to justify these calculations is taken from many recent observations and calculations of

curvature via the Einstein equation. These data include: recession of stars and galaxies with respect to

distance from earth, large scale curvature implied from the mass of visible and dark matter, and

estimations of the curvature from dark energy.

The current density is found via experimental data and observations of matter and energy. This

density is determined by these observations and data - this is not just visible matter and energy, it takes

into account several forms of matter and energy.

Omega is formed by four different parameters: omega baryonic (which includes all the common

forms of matter such as protons, neutrons, and other forms of visible matter), omega m (which is

determined from dark matter), omega v (which is determined by the density of dark energy), and

omega r (which is the density of visble radiation in the form of gravitons, photons, and neutrinos)

H0 is determined via redshifts – 1/H0 gives Hubble time. This is the approximate age of

universe. Not real age, since t must be extracted from FWR. (18.43)
Of the four parameters that make up the FWR model, only two are determined by direct observation:

H0 = 72 +/- 7 km/s/Mpc and Omega(r) = Omega(cmb) = (h^-2) *2.5 x 10 ^ -5 ~= 5 x 10 ^ -5

where h = H0 / (100km/s/Mpc)

With incorporating massless neutrinos (blasphemy! EXO experiment) above becomes 8 x 10 ^ - 5

Since gravitons have not been observed, they are unknown, but likely small contributions since....

Omega(b) is observed to be ~= .04 since... This gives lower bound to Omega(m) ~= .3 then Omega(v)

~= .7

What makes this unique is that the density is almost one, so our universe could go either way.

But this consequently begs the question of how we got this data.

redshifts and magnitude of flux from distant galaxy:

f/L = H0^2 / (4 pi z^2)

This assumes flat geometry and neglects evolution of that geometry because with distances

small enough, curvature of universe is not significant. But for distances to far away galaxies, the large

scale curvature needs to be taken into account.

<<< see fug 19.1 >>>

f = L / (4 pi d_eff^2) * 1 / (1+z)^2

deff = <<19.5>>

So for small z, H0 holds, but for larger z, it does not:

f / L = 19.6

...

final result is Ki(OmegaR,OmegaM,OmegaV) = OmegaC^ (1/2) INTEGRAL 1/(1+z) to 1 of da~/

(a~{}

<<19.9>>
If z is close to one, it deviates much from flatspace correction, which corresponds to non-zero vaccuum

energy / cosmo constant.

To probe even further, the Cosmic Microwave Background is analyzed for anisotropies to

answer questions of weather our universe is open or closed. Anisotropies are theorized to have

scattered from matter at nearly 3000k when electrons and nuclei formed neutral atoms. CMB photons

caught today are all from what is called the last scattering sphere, or the time when atoms were first

starting to form in the universe. Anisotropies are from temperature fluctuations in last-scattering

surface – which are believed to be what eventually led to galaxies. Angular size depends on physical

size at time of scattering and geometry of universe. <<19.10>>

CMBs anisotropies do not have definite size, but rather a spectrum of sizes. Information about

cosmological parameters is taken from statistics of angular sizes. A correlation function can be formed

from temp of anisotropy's C(theta). Defined as: 19.13.

Despite all this data, this still shows flat universe. Tiny density fluctuations in early universe

explain anisotropies in CMB - if amplified, they produce galaxies and whatnot. This shows that in

order for the universe to be as uniform, isotropic and homogeneous as is, the different potions of the

universe must have been in contact over 400k years ago.

9.15, no physical method before

last scattering explains isotropy of CMB if universe was matter dominated b4 that so FRW breaks

down in early universe times – the assumptions of what and what no longer hold? Welcome high-

energy physics – this now goes into the realm of quantum cosmology. Increase in horizon scale 19.17,

exponentially. Puts whole universe in causal contact at time of last scattering. Data from current and
future experiments will confirm or deny inflationary models. Things will not really work until QM and

GR are unified. Also predicts universe to be flattish, homogeneous, and isotropic: 19.18 Box 19.2 -

method for inflation.

Inflationary models of the initial state of universe are able to explain why the universe is

homogeneous, isotropic, and flat. But in order for these models to be proven, both more precise data is

needed and possibly quantum gravity.

Due to the inconsistencies and failures from the breakdown of the Freidman-Roberston-Walker
model, inflation is the more accurate and therefore preferred model for explaining th behavior of the
universe before baryonic matter started to form. The reason for this is that the exponential increase in
the horizon of the universe stretches out the irregularities inhomogeneities beyond the observable
horizon of the universe, since the initial expansion is greater than the velocity of light. By doing this,
the curvature is also reduced to nearly flat. The transition from the rapid expansion is seen as a phase
change in the universe – which explains why the universe is expanding at a rate of c today. There are
many inflationary models that exist today, however it is unlikely that one will be able to fit all
observable data until resolution is finally made between general relativity and quantum mechanics.

Works Cited:

AN EXPOSITION ON INFLATIONARY COSMOLOGY

Gary Scott Watson

Dept. of Physics, Brown University, P.O. Box 1843, Providence, RI 02912

http://nedwww.ipac.caltech.edu/level5/Watson/Watson5.html