Where Does '13.7 Billion' Come From?

[Dorje]

Many articles explain that data from WMAP were used to establish an age of 13.7 billion years for the universe. Could someone explain precisely how '13.7 billion' comes out of the data? What formulas/models/theories/etc. were used, how were they used, etc. ?

 

[Chronos]

Mostly the Hubble flow and Einstein's field equations. If you run the universe in reverse, it vanishes around t=13.7 billion years

 

[Jorrie]

Many articles explain that data from WMAP were used to establish an age of 13.7 billion years for the universe. Could someone explain precisely how '13.7 billion' comes out of the data? What formulas/models/theories/etc. were used, how were they used, etc. ?

I guess to get 'precise answers' you would have to take a course in physical cosmology, because there is no straightforward simple answer!

As far as I know, it is WMAP data together with other observations that converge at 13.7 Gy.

 

[marcus]

Many articles explain that data from WMAP were used to establish an age of 13.7 billion years for the universe. Could someone explain precisely how '13.7 billion' comes out of the data? What formulas/models/theories/etc. were used, how were they used, etc. ?

First here is a practical exercise using the Friedmann equation(s) built into someone's JAVA calculator.
There are ONLINE JAVA CALCULATORS that will do all the work for you so you just have to plug in the numbers for the present and tell it how far back in size you want to go.
You say something like "I want to back to when it was ONE THOUSANDTH of present size" and you type in, like, the number 1000, and press "compute" and it tells you that the LIGHT TRAVEL TIME is something like 13.665 billion years or whatever. It calculates how long it has been since the U was 1/1000 of present.

Ned Wright has a calculator that does that. Someone here named hellfire has programmed his own. It is stuff you could do with a few lines of BASIC language if you do freshman differential calculus stuff with BASIC
or some comparable programming language.

Try Ned Wright's calculator. Essentially it has the Friedmann equation(s) model built in.
http://www.astro.ucla.edu/~wright/CosmoCalc.html

Ned Wright puts in TODAY'S DATA FOR YOU as the default. So you don't even have to enter them. He puts in hubble=71
and matter fraction = 0.27 and Lambda fraction = 0.73.
So you don't have to change anything.

YOU JUST ENTER HOW FAR BACK IN SIZE you want to go.

If you want to go to 1/4 of present size, put in z=3
(it is an astronomer's convention that you put in one less than the reciprocal of the size you want to get to) and it does the rest.

If you want to go to 1/5 of present size, put in z=4

If you want to go to 1/10 of present size, put in z=9

If you want to go to 1/1000 of present size, put in z=999

for large numbers it doesn't matter if you remember to subtract one or not, makes no different to the answer.

Already 1/1000 is close enough to zero that you will see something for the "light travel time" that is like 13.6 billion years.

IN THIS WAY, USING WRIGHT'S JAVA, YOU TOO CAN DETERMINE THE AGE OF THE UNIVERSE. TRY IT TODAY!
======="theoretical" discussion=========
you can still ask WHERE DID TODAY'S DATA COME FROM? where did we get the number 71?
Essentially the values for Hubble (71) and Lambda or dark energy fraction (0.73) and matter fraction (0.27) come from FITTING THE FRIEDMANN MODEL TO A WHOLE PACK OF OTHER OBSERVATIONAL DATA OF VARIOUS KINDS (like cmb and supernovae and galaxy redshift counts and so on). I can't explain all that, but the core thing to understand is the FRIEDMANN MODEL so I will say something about it:

there is a simple differential equation called the Friedmann equation, that is used. much of this should be accessible to someone who has had college calculus

the basics are explained in many online articles. one that comes to mind is Lineweaver Inflation and the Cosmic Microwave Background.
the first few pages has a simple introduction to the mathematical tools used in cosmology.

go here
http://arxiv.org/find
type Lineweaver in the author box (nothing in the other boxes, it will do the rest)

this will get you this
http://arxiv.org/find/grp_physics/1/au:+Lineweaver/0/1/0/all/0/1

reference #9 on that list is Lineweaver's introductory article
================================
IIRC also Eric Linder has an online intro to cosmology (actually two by him, one for people with knowledge of Freshman Calculus and one for zero-math people). Not sure, haven't checked Linder in some time. LOTS of people must.

Oh, there is also Ned Wright site. just google Ned Wright.
================================

here is the gist of what you will learn from first few pages of Lineweaver:

1. cosmologists believe in the Einstein equation (a complicated differential eqn----the core equation of Gen Rel---describing the evolution of a general gravitational field)

2. Friedmann SIMPLIFIED the Einst. eqn. by assuming approximate largescale uniformity of the universe. Cosmologists observe that the universe in fact looks approx. the same in every direction and from every vantage point. So they accept the Friedmann eqn as a working assumption. Actually there are TWO Friedmann equations.

3. The Friedm. eqn. is a real simple diff. eqn. that describes the evolution in time of a(t), the scale factor or "SIZE OF THE UNIVERSE" or "average distance between galaxies". Actually there are TWO equations----one of them is about the time-derivative a'(t)
or if you like you can say da/dt. the result of a small change along time axis----the other is about the SECOND derivative a''(t) or if you like the time derivative of da/dt----the rate of change of the rate of change.

you can lump it all in one Friedmann equation or you can tease it apart into two equations----one gives a formula for the "speed" and one for the "acceleration"-----together they give you a good grip on a(t) the size function, or scale factor, of the universe.

you have to PLUG IN SOME NUMBERS like the Hubble parameter, so the equations give you various plots depending on what parameters you plug in. Cosmologists have an idea of what are the RIGHT numbers to plug in

this a(t) is a function you can plot on graph paper. Lineweaver shows a picture of the plot. IIRC it is figure 7.
A common practice is to normalize a(t) by taking the present value to be ONE.
Both the x-axis and the y-axis are arbitrary so you could take the present time to be zero. you could normalize a(t) by saying
a(0) = 1.

Now having decided to put a(0) = 1, you EVOLVE BACK (like Chronos said) step by step. Each small time-step you take backwards, the Friedmann equation(s) will tell what the change in a(t) is (that is from the formula for a'(t)) and also what the change in a'(t) is. so you keep solving back, or calculating your way back, and you find that THE SIZE KEEPS GETTING SMALLER until at a time t = - 13.7
you find that it gets down to ZERO, namely

a( - 13.7) = 0. assuming your time unit is billion years.