--------- #1346 - How Hot Was It at the Big Bang?
- Attachment: Big Bang
- The temperature of the Universe is inversely proportional to its size. As the Universe expands the temperature decrease. Today it has expanded and cooled to a temperature of 2.725 degrees Kelvin. That is 2.725 degrees above Absolute Zero. (see footnote 1). What would happen if we reversed the expansion and sent time collapsing back to the Big Bang. The temperature would get hotter and hotter. The pressure would get greater and greater.
- It turns out that the temperature / pressure relationship is a linear relationship. Plotting temperature versus pressure we get a straight line all the way back to the Big Bang. Therefore, we should be able to use the laws of physics and extrapolate back to the Big Bang to determine how hot it was. The math used is called “dimensional analysis.” It is a simple concept. To learn it we will try it out on a pendulum before we try it out on the Universe.
- Measurements in physics and astronomy can involve 3 dimensions: length (space), time, and mass (energy). We refer to these as dimensions and they are measured in units: meters, seconds, kilograms. Starting with the pendulum let’s use these three dimensions to calculate the period of the pendulum swing. The pendulum is a mass hanging freely on a string and swinging back and forth. The period is the time for one cycle of one swing.
- The period is “time“, it is possibly a function of the three dimensions ,length, time, and mass to some degree. The force on the mass is a function of gravity. If we are on the surface of the Earth the acceleration of gravity is 9.8 meters / second^2 ( 32 feet per second per second.). We might also guess that the period might be a function of the length of the strength, and the weight of the mass. Guess:
------------------ Time = mass * length * acceleration of gravity
---------------- Acceleration of gravity = “g” = (length / time ^2) = (m / sec^2)
-------------- Working with the units of each dimension, try to derive the equation for the period of a pendulum: To get to the equality seconds = seconds.
--------------- second = kilograms * meters * meters / second^2
---------------- Working to cancel the units on the right side of the equation to equal the left side of the equation, which is seconds. Let’s divide by the acceleration of gravity , (meters / seconds^2)^-1
------------- sec = kg * m / m / sec^2
-------------- Squaring the left side to get sec^2. There are no kilograms on the left side of the equation , so, mass must not be a factor. The period of a pendulum is the same regardless of the weight of the mass. Therefore, we can guess the equation for the period of a pendulum to be:
-------------- ( period )^2 = length / acceleration of gravity
--------------- T^2 = L / g
-------------- sec^2 = m^ / m /sec^2
- And, that works. The period is directly proportional to the length of the string. The longer the string the longer the period. It is inversely proportional to the acceleration of gravity. The pendulum would have a longer period swinging on the Moon. The period would be independent of the mass, as long as the string did not break.
- The pendulum illustrates how you can solve a problem just by thinking through the dimensions involved and the units of measurement. Now, you are ready to derive the equation for the temperature at the instant of the Big Bang:
- The three fundamental dimensions are length, time and mass. Our plan will be to use the 3 fundamental constants in Nature: “h“, “c“, and “G“. Use these to calculate the Energy of the Big Bang from E= mc^2. Then use Boltzman’s Constant to convert Energy to temperature. First let’s define the 3 fundamental constants of Nature:
-------------------- h = Planck’s Constant of Action. Action is the product of Energy and Time.
-------------------- h = 6.6*10^-34 kilograms * meters^2 / second
-------------------- h units = kg * m^2 / sec
--------------------- c = (speed of light)
--------------------- c = 3*10^8 meters / second
-------------------- c units = (m / sec)
-------------------- G = Gravitational Constant
--------------------- G = 6.7*10^-11 meters^3 / ( kilogram * seconds^2)
--------------------- G units = m^3 / ( kg*sec^2)
- Using E = mc^2 all we need is Mass to calculate Energy. We know the speed of light. Putting Mass in terms of the units of the fundamental constants: To get the equation to balance we need kilograms in the numerator. One way to do that is to start with “ G” in the denominator:
--------------------- kilograms = 1/ G = ( kg * sec^2) / m^3
- Now, to cancel out sec / meter we multiply by “c” which is in m / sec:
------------------ kilograms = c / G = ( kg * sec^2) / m^3 * m / sec
- Now, we still need to cancel more sec / meter so multiply this by “h” which is kilograms * meters^2 / sec.
------------------kilograms = c * h / G = ( kg * sec) / m^2 * kg * m^2 / sec
--------------- kilograms = c * h / G = ( kg ^2)
- Now, we need to square the left side of the equation in order to get the equality and have all the units cancel out:
--------------- (kilograms)^2 = c * h / G = ( kg ^2)
--------------- M^2 = c * h / G
- Now, we have the Mass in terms of the fundamental constants so we can substitute the values for the constants and solve for the fundamental Mass:
------------ M^2 = (3*10^8 meters / second) * (6.6*10^-34 kilograms * meters^2 / second) / (6.7*10^-11 meters^3 / ( kilogram * seconds^2))
---------------- M^2 = 2.955*10^-15 kg
----------------- M = 5.4 *10^-8 kg
- Now, we can calculate E = mc^2:
------------------ E = (5.4 *10^-8 kg) * ( 9 * 10^16 m^2/sec^2)
------------------ E = (50*10^8 kg * m^2/sec^2)
------------------ E = (5 *10^9 Joules.
- Now, using Boltzman’s constant for converting Energy to Temperature: ( See Review # 1072 to learn about Boltzman’s Constant.
--------------- T = (Kelvin / 1.381 *10^-23 Joules ) * Energy
-------------- T = (Kelvin / 1.381 *10^-23 Joules ) * (5 *10^9 Joules)
----------------- T = 3.5 * 10^32 Kelvin
- Therefore the temperature at the time of the Big Bang according to the fundamental constants was 350,000,000,000,000,000,000,000,000,000,000 degrees Kelvin. I think you needed to know that.
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(1) Other Reviews that have background on this same topic:
# 1345 - Absolute zero temperature strangeness.
# 1234 - What happens when you cool an atom
# 1072 - Boltzman’s Constant, Avogadro’s Number.
# 727 Absolute zero _ TEMPERATURE
# 406 Zero is nothing, but Absolute Zero is something else.
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(2) The same dimensional analysis method can be used to calculate fundamental time:
------------------ T^2 = G * h / c^5
------------------ T = 1.4 * 10^-43 seconds
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(3) To calculate fundamental distance:
----------------- L^2 = G * h / c3
------------------- L = 1.62*10^-35 meters
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