DavidFirth
Well-Known Member
I agree wholeheartedly.
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Big Bang Theory Primer
- Thread starter sayak83
- Start date
I agree wholeheartedly.
By the way. I had an informative thread about the inflation theory and what it says. Check it out of you haven't seen it. Best.
Singularities and beginning of the universe
DavidFirth
Well-Known Member
OK, I'll check it out.
Does anyone have any good suggestions on how to explain to aboriginalists that the people in photos are not real? I am in a disscuasion with aboriginalists about the dawn of the cosmos and they are convinced their brain caused it!!! It's like holy **** batman.
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DavidFirth
Well-Known Member
LOL exactly.Good luck.
Continued from previous post
Big Bang Theory Primer
In this post I will use natural units to simplify the equations of cosmology derived in the last post.
Our ordinary system of units (Meter, second, kg) is created for human convenience and is arbitrary. Thus there is nothing special about 1 kg mass, 1 meter length or 1 second of time. In contrast, natural units use universal constants of nature to create the unit system. Hence in natural units we have
Speed of light c=1, as the fundamental units of velocity. Thus all velocities are expressed in terms of fractions of speed of light, which is an universal constant.
The unit of mechanical action (product of energy and time) is called the Planck constant h and is also set to unity. So h=1.
The proportionality constant relating energy and temperature is called the Boltzmann constant κ (Kappa) and is set equal to 1. So κ=1.
Finally the unit of energy is chosen to be the energy required to drive 1 electron through 1 volt of electric potential. Thus unit of energy is called 1 electron volt. So E=1eV.
Using these natural units the equations I derived earlier becomes (since speed of light c=1),
Friedmann equation,
(a'/a) ^2 = ( 8π/3)ρG - k/a^2
Fluid equation
ρ' + 3*(a'/a)*(ρ + p) = 0
The next step is to relate the equation above with Hubble's law of recession of galaxies. Astronomer Edwin Hubble discovered that all the galaxies were receding away from each other at a speed proportional to the distance separating them. The Hubble law is given by
v = H°*r
where v is the velocity of recession of the galaxies, r is the distance between the galaxies and H° is Hubble constant. The astronomers have measured the recession speed vs galaxy distance and they plot on a straight line,
And the slope of this line will be the Hubble constant H°
Since velocity v= dr/dt, what the Hubble law says is,
dr/dt = H° r
or, H° = (dr/dt) / r
In the comoving coordinate system that expands with the universe we have,
r = a(t)*x and dr/dt = a'*x where a(t) is the expansion scale factor and a' = da/dt
Thus (dr/dt)/r = a'/a
Hence we find that the Hubble constant,
H° = a'/a
Thus we have the first observational data which we can plug in the cosmological equation, the Hubble parameter H.
The Friedmann equation then is,
H^2 = ( 8π/3)ρG - k/a^2
And the fluid equation is,
ρ' + 3H(ρ + p) = 0
Where, now the Hubble parameter H is determined from the slope of the galactic distance to recession speed line shown in the figure.
On a lighter note, we have new excuse for procrastination
Big Bang Theory Primer
In this post I will use natural units to simplify the equations of cosmology derived in the last post.
Our ordinary system of units (Meter, second, kg) is created for human convenience and is arbitrary. Thus there is nothing special about 1 kg mass, 1 meter length or 1 second of time. In contrast, natural units use universal constants of nature to create the unit system. Hence in natural units we have
Speed of light c=1, as the fundamental units of velocity. Thus all velocities are expressed in terms of fractions of speed of light, which is an universal constant.
The unit of mechanical action (product of energy and time) is called the Planck constant h and is also set to unity. So h=1.
The proportionality constant relating energy and temperature is called the Boltzmann constant κ (Kappa) and is set equal to 1. So κ=1.
Finally the unit of energy is chosen to be the energy required to drive 1 electron through 1 volt of electric potential. Thus unit of energy is called 1 electron volt. So E=1eV.
Using these natural units the equations I derived earlier becomes (since speed of light c=1),
Friedmann equation,
(a'/a) ^2 = ( 8π/3)ρG - k/a^2
Fluid equation
ρ' + 3*(a'/a)*(ρ + p) = 0
The next step is to relate the equation above with Hubble's law of recession of galaxies. Astronomer Edwin Hubble discovered that all the galaxies were receding away from each other at a speed proportional to the distance separating them. The Hubble law is given by
v = H°*r
where v is the velocity of recession of the galaxies, r is the distance between the galaxies and H° is Hubble constant. The astronomers have measured the recession speed vs galaxy distance and they plot on a straight line,
And the slope of this line will be the Hubble constant H°
Since velocity v= dr/dt, what the Hubble law says is,
dr/dt = H° r
or, H° = (dr/dt) / r
In the comoving coordinate system that expands with the universe we have,
r = a(t)*x and dr/dt = a'*x where a(t) is the expansion scale factor and a' = da/dt
Thus (dr/dt)/r = a'/a
Hence we find that the Hubble constant,
H° = a'/a
Thus we have the first observational data which we can plug in the cosmological equation, the Hubble parameter H.
The Friedmann equation then is,
H^2 = ( 8π/3)ρG - k/a^2
And the fluid equation is,
ρ' + 3H(ρ + p) = 0
Where, now the Hubble parameter H is determined from the slope of the galactic distance to recession speed line shown in the figure.
On a lighter note, we have new excuse for procrastination
Please explain why the BB theory was DOA in any sense more than any other mathematical theory is.
How is this any different than any other science?
It *is* self-correcting: it has to be able to predict *new* observations correctly or else it will be modified or eliminated.
I
DavidFirth
Well-Known Member
Seriously, I know one thing you should do. Spend a whole lot of time in prayer and God will provide the answers.
OK so now you know when you try and explain to a creationist that they might be confused? Now you have first hand experience of what it is like for them to have say you are cocoon we are right. I can no more be clear to you than you to a creationist. That's how normals are. You are just normal is all. And so is creationism it's normal too. I hope that was written clearly enough no I am absolutely not in the creationist intelligence design reality nonsense either. I am a bit more like a cranky "it's not even wrong" Wolfgang Pauli.
Continued
@DavidFirth
Big Bang Theory Primer
Shape of the universe.
In the last post I showed that astronomers can plot the rate of recession of the galaxies vs their distance, and the slope of this line provides them with the Hubble parameter H. This H is a measure of the rate of expansion of space in the universe can be related to the mass-energy density of our universe and the curvature of space.
H^2 = ( 8π/3)ρG- k/a^2
The variable k, designates the curvature of the universe. Thus,
if k is positive then space is positively curved, like the surface of a sphere;
if k is negative then space is negatively curved, like the surface of a saddle;
if K is zero, then space has no curvature, like a flat sheet.
But, what does it mean when one says that space is curved (or not)? Imagine you draw three straight lines in space to make a triangle. Then in a positively curved space, the angles of the triangle will sum to more than 180° while for a negatively curved space the sum of angles will be less than 180°. This is shown in the figure below.
So, what is the case for our universe? Astronomers have determined that the universe is flat to the limits of our current ability to detect curvature (± 0.4%).
Space Matter: A Flat Universe
BOSS—The Baryon Oscillation Spectroscopic Survey—is a survey of the sky carried out at a telescope in New Mexico measuring these fluctuations, and using the survey, scientists have been able to make the most accurate map of the universe to date, pinpointing the position of galaxies and other stellar matter with 1% accuracy. It doesn’t cover the whole universe—just a mere 1.2 million galaxies. But with this three dimensional map, we’re able to make new observations about the universe, including the discovery that the universe is likely flat.
Findings from the WMAP spacecraft (Wilkinson Microwave Anisotropy Probe), which used fluctuations in CMB to calculate the geometry of the universe, support this conclusion with a 0.4% margin of error.
So what does this mean? It means that either the universe is completely flat and infinite in extent, or it's size is at least hundreds of times more than than the region we can see. And the region we can see spans a diameter of 90 billion light years.
Thus the universe we see is either flat or infinite, or nearly flat with size much much greater than the small region we can see. In either case, the curvature k is so close to zero that the second term of the cosmology equation drops out. Hence we get a simplified cosmological equation,
H^2 = ( 8π/3)ρ°G
Or,
ρ° = 3H^2/(8πG)
Where ρ° is the mass-energy density of the universe needed to make universe flat. This is called the critical density. Cosmologists define a useful parameter called the density parameter Ω which is defined as,
Ω = ρ/ρ°
Where ρ is the observed mass-energy density and ρ° is the critical density that makes space flat in curvature.
@DavidFirth
Big Bang Theory Primer
Shape of the universe.
In the last post I showed that astronomers can plot the rate of recession of the galaxies vs their distance, and the slope of this line provides them with the Hubble parameter H. This H is a measure of the rate of expansion of space in the universe can be related to the mass-energy density of our universe and the curvature of space.
H^2 = ( 8π/3)ρG- k/a^2
The variable k, designates the curvature of the universe. Thus,
if k is positive then space is positively curved, like the surface of a sphere;
if k is negative then space is negatively curved, like the surface of a saddle;
if K is zero, then space has no curvature, like a flat sheet.
But, what does it mean when one says that space is curved (or not)? Imagine you draw three straight lines in space to make a triangle. Then in a positively curved space, the angles of the triangle will sum to more than 180° while for a negatively curved space the sum of angles will be less than 180°. This is shown in the figure below.
So, what is the case for our universe? Astronomers have determined that the universe is flat to the limits of our current ability to detect curvature (± 0.4%).
Space Matter: A Flat Universe
BOSS—The Baryon Oscillation Spectroscopic Survey—is a survey of the sky carried out at a telescope in New Mexico measuring these fluctuations, and using the survey, scientists have been able to make the most accurate map of the universe to date, pinpointing the position of galaxies and other stellar matter with 1% accuracy. It doesn’t cover the whole universe—just a mere 1.2 million galaxies. But with this three dimensional map, we’re able to make new observations about the universe, including the discovery that the universe is likely flat.
Findings from the WMAP spacecraft (Wilkinson Microwave Anisotropy Probe), which used fluctuations in CMB to calculate the geometry of the universe, support this conclusion with a 0.4% margin of error.
So what does this mean? It means that either the universe is completely flat and infinite in extent, or it's size is at least hundreds of times more than than the region we can see. And the region we can see spans a diameter of 90 billion light years.
Thus the universe we see is either flat or infinite, or nearly flat with size much much greater than the small region we can see. In either case, the curvature k is so close to zero that the second term of the cosmology equation drops out. Hence we get a simplified cosmological equation,
H^2 = ( 8π/3)ρ°G
Or,
ρ° = 3H^2/(8πG)
Where ρ° is the mass-energy density of the universe needed to make universe flat. This is called the critical density. Cosmologists define a useful parameter called the density parameter Ω which is defined as,
Ω = ρ/ρ°
Where ρ is the observed mass-energy density and ρ° is the critical density that makes space flat in curvature.
DavidFirth
Well-Known Member
Fascinating. I have seen something similar to this before but this is better and more informative than what I remember having seen before.
Thanks. If you have questions, please ask.
One technical, mathematical point that even cosmologists tend to skip over: it is possible to have 'flat' space, or even negatively curved space and still be *finite*.
The two dimensional analog is a torus (a donut or bagel shape), even though the usual image of such is curved. It happens like this: suppose that space is flat and there is a huge 'cube' so that if you go out the 'top' boundary, you come in the 'bottom' boundary, and similarly with front/back and left/right. Again, this can *look* like teleportation, but mathematically, it is a single, finite space that is flat.
The main issue with this model is that it is NOT isotropic; there are three 'preferred directions'. Otherwise, though, it is a valid solution of the Einstein field equations.
There are similar constructs for negatively curved spaces. I even saw one report (later refuted) of evidence suggesting one like this.
Anyway, very good exposition, @sayak83 ! Keep up the good work!
The two dimensional analog is a torus (a donut or bagel shape), even though the usual image of such is curved. It happens like this: suppose that space is flat and there is a huge 'cube' so that if you go out the 'top' boundary, you come in the 'bottom' boundary, and similarly with front/back and left/right. Again, this can *look* like teleportation, but mathematically, it is a single, finite space that is flat.
The main issue with this model is that it is NOT isotropic; there are three 'preferred directions'. Otherwise, though, it is a valid solution of the Einstein field equations.
There are similar constructs for negatively curved spaces. I even saw one report (later refuted) of evidence suggesting one like this.
Anyway, very good exposition, @sayak83 ! Keep up the good work!
Thanks! Here is more
The Dark Energy and Accelerating universe
In my OP and following Post I used a rough derivation to get at what is called the Friedmann equation,
(a'/a) ^2 = ( 8π/3)ρG - k*(c/a) ^2
If, however, an exact derivation was made from General Relativity, we get an additional constant term on the right hand side of the equation called the cosmological constant Λ. So full equation becomes (with speed of light c=1 in natural coordinates)
(a'/a) ^2 = ( 8π/3)ρG - k/a ^2 + Λ/3
The value of this constant Λ cannot be determined from general relativity, but for much of the time cosmologists thought its value was zero. However new observations in the last decade has shown that the universe we live in does have a small positive value of the cosmological constant Λ. To understand the impact of Λ, we can differentiate the equation above with respect to time. Setting the acceleration of the scale factor,
a" = da'/dt
and after some algebraic manipulation,
a"/a = - (4πG/3)*(ρ + 3p) + Λ/3
where p is the pressure of the mass and radiation that is distributed in space and ρ is the density of mass and energy in space.
The variable (a"/a) gives the rate at which the expansion of the universe is accelerating or decelerating. Notice that the first term on the right hand side is Negative, causing deceleration of the rate of expansion of the universe. However the second term involving the cosmological constant will be positive if Λ is positive. Hence a positive cosmological constant will cause an acceleration of the rate of expansion of the universe.
What is the physical significance of the cosmological constant?
Remembering from a previous post that the Hubble parameter H is equal to (a'/a) and critical density ρ° is defined as,
ρ° = 3H^2/(8πG)
Where ρ° is the mass-energy density of the universe needed to make universe flat. Cosmologists define a useful parameter called the density parameter Ω which is defined as,
Ω = ρ/ρ°
Where ρ is the observed mass-energy density and ρ° is the critical density.
Then the Friedmann equation can also be written as, after some manipulations,
H^2 = (H^2)Ω - k/a^2 + Λ/3
It is possible to define a density like term for the cosmological constant by setting,
Dark energy density,
Φ = Λ/(8πG)
Dark Energy density parameter
₯ = Φ/ρ° = Λ/(3H^2)
We get the final form of the Friedmann equation,
Ω + ₯ - 1 = k/(aH)^2
For the case of a flat universe, as is the case for our universe, the curvature k=0 and we must have,
Ω + ₯ = 1
Thus for a flat accelerating universe the sum of ordinary mass-energy density parameter and dark energy density parameter from the cosmological constant must equal 1. Thus, knowing the density of common matter and energy, one can deduce the dark energy density or cosmological constant of the universe. Also, observing the rate at which the expansion of the universe is accelerating provides an independent means of determining the value of the cosmological constant.
But here is another take on all this whiz-bang stuff
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DavidFirth
Well-Known Member
Wow, the advanced math may as well be Cantonese to me. Still all very interesting, though.
Just so you know, I have great respect for you. You are very honest about your beliefs and faith and say you don't know when you don't. I am primarily here to remove some misconceptions about why scientists have confidence that certain theories match the reality we live in quite well. I have seen that while some folks believe in science simply based on authority (which I dislike), others dismiss science due to misconceptions about how these theories come about. As long as one knows how science works to create and justify its models of the world correctly, one is free to accept or reject it based on other considerations (personal convictions, philosophical objections, subjective experiences etc). Thus my limited aim is to write posts that provide a correct understanding of how scientists come up with their conclusions as accurately and as clearly as I can. Then, it's upto each person, or more correctly his Karma
.
Meaningless questions are difficult to answer... like what is the smell of blue.