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.