Both Cosmic and otherwise
To those who had the (mis)fortune of having me as a part of their lives, weight gain, and weight loss are all too familiar. if not by first experience, they have probably witnessed my own physique change, all across the range spanning from wirey-thin to brown dwarf ( a so called Failed-Star, or UltraCool Dwarf, which is a stellar body of roughly 13-85 Jupiter masses), and hopefully, one day, back again.
Since my travels through mass-space have been oscillatory in nature (for the most part), it is with pain-laced mirth I give you the following analysis and analogy of myself, and the Cosmos.
Since my travels through mass-space have been oscillatory in nature (for the most part), it is with pain-laced mirth I give you the following analysis and analogy of myself, and the Cosmos.
I know, I know, this seems a bit megalomanic on my part, but I assure you... I am.
Well really, since my current field of study is Cosmology, and specifically inflation models, I found it might be funny to recount my thoughts and findings and relate it to the human condition, or rather, the fatso condition...
To any and all cosmologists out there, who might be reading this right now, be gentle, I am a mere neophyte to this field, and so I appeal to your sense of humor, and sympathy (enter violins please).
In order to first pique your interest, please listen to this number by Monty Python:
Which, by the way have some major faults in cosmological terms, for instance, the universe (acording to leading theories) actually does NOT expand at the speed of light. our EVENT-HORIZON (or simply horizon) does. It's meaningless to ascribe velocity to the expansion of the universe, since very close to us, the universe recedes from us in small velocities and far from us the universe seems to recede with greater velocity, as reflected by Hubble's law - velocity is proprtional to distance from us... \(\left(v=H_{0}D\right)\) .
By the way, this also implies that Galaxies which are far enough from us, recede at velocities greater than the speed of light (yes it IS possible), and thus they "drop off" our horizon, since they fade away faster than their light travels to us.
This also foretells a dark and lonely eventual demise, for our universe.
At any rate, I would like at this juncture, to acquaint the reader with Friedmann's equation, which I will first write down and derive LATER (in two ways by the way...)
\[\left(\frac{\dot{R}}{R}\right)^{2}=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]
Where \(R\) is the radius of an arbitrary sphere in space. this equation describes the evolution of \(R\) due to gravitational forces alone.
For those who have some interest of keeping their sanity, you are invited to skip these derivations.
For those of brave soul and not so sound mind, please take special interest in the general relativity case...
(Important conclusions, in green)
By the way, this also implies that Galaxies which are far enough from us, recede at velocities greater than the speed of light (yes it IS possible), and thus they "drop off" our horizon, since they fade away faster than their light travels to us.
This also foretells a dark and lonely eventual demise, for our universe.
At any rate, I would like at this juncture, to acquaint the reader with Friedmann's equation, which I will first write down and derive LATER (in two ways by the way...)
\[\left(\frac{\dot{R}}{R}\right)^{2}=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]
Where \(R\) is the radius of an arbitrary sphere in space. this equation describes the evolution of \(R\) due to gravitational forces alone.
In layman terms:
The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.
For those who have some interest of keeping their sanity, you are invited to skip these derivations.
For those of brave soul and not so sound mind, please take special interest in the general relativity case...
(Important conclusions, in green)
<derivation - Newton style>
from Newt's 2nd law: \[F=ma\\
\Rightarrow F=m\ddot{R}\]
Gravitational force due to enclosed mass, on the perimeter of a sphere:
\[F=-\frac{GMm}{R^2}\]
Equating these yields:
\[\ddot{R}=-\frac{GM}{R^2} \Rightarrow \dot{R}\ddot{R}=-\frac{GM\dot{R}}{R^2}\]
Integrating, we get:
\[\dot{R}^2=\frac{2GM}{R}-\kappa\]
So, now, for matter in a sphere with a radius of \(R\), \(M=\frac{4\pi R^3 \rho}{3}\), thus:
\[\dot{R}^2=\frac{8\pi G \rho R^2}{3}-\kappa \Rightarrow \left(\frac{\dot{R}}{R}\right)^2=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]
taking \(R=R_0\cdot a(t)\) we get:
\[\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G \rho}{3}-\frac{\tilde{\kappa}}{a^2}\]
</derivation - Newton style>
It's fairly easy to see that, according to Newton, the term \(\frac{\kappa}{R^2}\), is simply a term that is a result of integration, i.e. an integration constant. And so according to this derivation \(\kappa \) is simply given by initial conditions, and given no other incentive, we can gauge it away.
Let's see what old Einei have to say about this...
<derivation - Einstein style>
I'll try to be brief yet informative:
The metric for a spherical symmetric curved space is given by:
\[ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-\kappa r^2}+r^2d\Omega\right]\]
Or in matrix form:
\[g_{\mu\nu}=\left(\begin{array}{cccc}
-1&&&\\
&\frac{a^2}{1-\kappa r^2}&&\\
&&r^2&\\
&&&r^2\sin^2(\theta)
\end{array}\right)\]
From here it's relatively easy to find the Christoffel connections:
\[\Gamma^{0}_{11}=\frac{a\dot{a}}{1-\kappa r^2}\;;\;\Gamma^{0}_{22}=a\dot{a}r^2\\
\Gamma^{0}_{33}=a\dot{a}r^2\sin^2(\theta)\;;\;\Gamma^{i}_{0j}=\delta_{ij}\frac{\dot{a}}{a}\\
\Gamma^{1}_{11}=\frac{\kappa r}{1-\kappa r^2}\;;\;\Gamma^{1}_{22}=-r(1-\kappa r^2)\\
\Gamma^{1}_{33}-r(1-\kappa r^2)\sin^2(\theta)\;;\; \Gamma^{2}_{12}=\Gamma^{3}_{13}=\frac{1}{r}\\
\Gamma^{2}_{33}=-\sin(\theta)\cos(\theta)\;;\; \Gamma^{3}_{23}=\cot(\theta)\]
all other symbols vanish.
The Ricci scalar is then given by:
\[R=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]\]
The Einstein equation:
\[G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G T_{\mu\nu}\]
or, equivalently :
\[R_{\mu\nu}=8\pi G \left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\]
Taking the 00 term we get:
\[3\left[\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]=8\pi G\rho\Rightarrow \left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G\rho}{3}-\frac{\kappa}{a^2}\]
And, if we use the other notation also we get:
\[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}
(\rho+3p)\]
</derivation - Einstein style>
For now, we'll leave the second equation be, and look at the first.
We'll notice a couple of things first - Although it SEEMS as though the equations (Newt's, and Einei's) are the same, really the real Friedmann equation (derived from General Relativity) is the more general case. that is due to the former derivation relying on dynamics of MATTER only, and the latter does not, though, a careful massage of the former with reletavistic ideas, might give the correct equation for radiation, and other cosmic stuff...
The second thing I wish to emphasize is the existence of \(\kappa\) and it's meaning: In the Newtonian analysis, it was simply an integration constant, but in the Einsteinian analysis this is an intrinsic factor to the very fabric of the universe, this is the CURVATURE signature of the metric, that governs whether the cosmos is positively, negatively or null curved (i.e. flat) like in this picture:
 |
| Curvature types: positive, negative, and null. |
<derivation - Gangnam style>
And this is just for fun:
</derivation - Gangnam style>
Admission of Guilt, errr... Ignorance
But wait!! What does all of this have to do with inflation?
Well, in fact, suppose space is sufficiently dillute, the matter density is sufficiently close to zero, and if it's sufficiently cold, radiation density drops to zero (almost), with an (almost) flat curvature, we are then left with some quantity we'll call \(\rho_0\), ("Rho naught"), and we get a dynamic equation, with an inflationary/deflationary solution:
\[\left(\frac{\dot{a}}{a}\right)=\pm\sqrt{\frac{8\pi G\rho_0}{3}} \Rightarrow a=A\exp\left(\sqrt{\frac{8\pi G\rho_0}{3}}t\right)+B\exp\left(-\sqrt{\frac{8\pi G\rho_0}{3}}t\right)\]
The second part drops off rapidely, and so we are left with an inflationarry solution.
"What is this witchcraft?!?!" you ask - where does this \(\rho_0\) come form?
well, suppose you're a fat guy, and you go into a fasting mode, note that you are actually GAINING weight (at least in the short run)... and by the time you've lost the battle against hunger, and went on a burger binge, guess what? you've now underwent inflation.
Well, actually this has nothing to do with \(\rho_0\), here's the REAL explanation for this:
Oh my god, I can't believe I said that!!! this is the absolute NO NO for physicists!!!
To actually state that I don't know something? to recognize that everything we *THINK* we know is simply an approximate modeling of the awsome and complex reality we live in???!?
I should stop. NOW!!! I hear them knocking already... don't let them take me! NOOOOOOOOOOOOOOOOO!!!!
Ok, done with that gag, are we?
Moving on, there are several possible explanations for this, most are problematic to say the least, but hey, this is what it means to be in the front lines of science. you either get promoted or get diced....
At any rate, Inflation is a widely accepted theory of the adolescent universe, whatever mechanism manifests it.
What about oscillations?
In a nutshell - at the onset of the early universe, after inflation, some major ocillations occured in matter density, affected by ordinary (barionic) matter as well as dark matter, these oscillations are known as BARIONIC ACOUSTIC OSCILLATIONS (or modes) , and they could be seen quite nicely in analysis of the cosmic background radiation.
Moreover, when the universe was matter dominated, the same dynamics might have happened on a cosmic scale - Suppose matter is the dominant part of the universe, the dominant force then is gravitation, thus the universe itself, aspires to CONTRACT, offset only by radiation pressure, there MIGHT have been an epoch of slight contraction on the universe's part.
Sadly I didn't find a sufficiently fascinating animation to show, of the acoustic oscillations, but maybe some other time...
In conclusion, much like myself, the universe was "born", and began inflating.
Undergoing some oscillations, and at a certain point (just about.... now!) the universe moved from matter driven dynamics into moderate inflationary epoch.
In layman terms:
The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.
In the words of our mutual friend:
