Let us now define some of the cosmological parameters which are used to
characterize different cosmological models.
The first of them is the Hubble Constant, or more accurately, Hubble Parameter.
Because it's not constant, it changes in time.
It is defined as the ratio of the time derivative of the scale factor by the
scale factor itself. The normalized slope of R(t)t) curve.
Today, its value is roughly 100 kilometers/MegaParsec.
per second per megaparsec. Remember, this is the slope of the Hubble
Diagram which plots velocity versus distance,
thus the units. Since it's length over time in the
numerator and length in denominator, its dimensions are one over time.
Essentially, Hubble Constant gives the scale of the Universe.
It's a slope of a given time and therefore when you draw a tangent on R0
ticker, its intercepts on the time axis gives you inverse of the slope,
which is one over Hubble Constant which is also known as the Hubble Time.
And that's roughly some ten billion years or so in modern cosmological models.
This is close to but does not have to be equal to the age of the Universe at that
time, T0. If you multiply Hubble Time by the speed
of light, which is length over time, then you get a Hubble Length.
The next one is the density parameter. It characterizes the mean density of the
Universe at any given time. If you look at the Friedmann Equation
assuming that the logical constant lambda is equal to zero, Universe is flat if,
has curvature constant to zero. And that can be also expressed as the
critical density over three times H0^2 divided by eight pi G.
Note that the critical density is determined just by the Hubble Constant of
the time. The ratio of the actual density of the
Universe to this value gives the Omega matter, the density parameter.
If it's exact, if the density is exactly equal to critical, then Omega matter is
one and the Universe is flat. The next would be the dark energy density
parameter, just like mass has a mean density so
would, energy. And we know that, from observations now,
that actually Universe does have a mean energy density.
It is defined from cosmological constant according to this formula.
And the total matter energy density would be then simply a sum of the energy
density of matter and energy density of the vacuum.
We also defined the deceleration parameter, often designated with lower
case q or q0 at the present, which in the absence of cosmological constant reduces
to one half of the density parameter. But, if a cosmological constant is
present, its value is different. So, this is what cosmological parameters
do. Hubble's Constant gives the overall scale
of the Universe, temporal and spatial.
It does not depend on the density of the Universe,
cosmological constant or anything else. The other parameters Omega of matter,
Omega vacuum, and so on, determine the shape of the R(t) curves and do not
depend on Hubble's constant. Therefore, their measurements will be
really done a different way and completely independent.
You can determine scale of the Universe without knowing what its density or, or,
or if it has dark energy or not. Conversely, you can find out the mix of
the densities in the Universe without knowing accurately solar scale.
This is done through cosmological tests which we will describe later.
So, let's have a few notes here. Hubble Parameter is often, or always
called Hubble constant, and is sometimes written in the following normalized form.
We divide actual value of Hubble Constant which is usually expressed in kilometers
per second per megaparsec by 100, and that is denoted with lower case h,
normalized Hubble Constant. Then, you can scale for other values of
the Hubble's constant as you wish. Sometimes, it's divided by 70, which is
actually very close to it's present value.
And that's denoted as little h with the subscript 70.
And that is Hubble Constant in units of 70 kilometers per second per megaparsec.
The current value of the critical density, which is determined entirely by
the value of the Hubble Constant is approximately 10^-29 grams per cubic
centimeter, including bulk matter, matter and energy.
So, we can write density parameters in the following form, the Omega of the
matter, the Omega of the energy density or vacuum,
plus Omega curvature all add up to one. Omega curvature really is defined simply
as a deviation of the other two from unit, but it makes a more elegant formula
this way. So now, when we look at the definitions
of these parameters as shown here, we can plug those in Friedmann's
Equation. And here is the Friedmann Equation
expressed through cosmological parameters.
As we can see, as we can see on the left, its value of Hubble Constant and other
parameters participate in the equation. Alright.
Now, next time, we will actually try to solve Friedmann Equation and actually
reach the cosmological models.