Why Was the Universe Smaller Than a Proton Pre-Big Bang?

[megacal]

I'm a microbiologist by training, and only have the basics of physics & math, so please bear with me.

Why do cosmologists assert that all the matter in the universe
was contained in a volume many times smaller than e.g. a proton
(or infinitely small?) prior to the Big Bang?

Why couldn't it have been a more "reasonable" size, e.g. the sun?

Everything we see that explodes has a reasonable volume...super novas,
sticks of dynamite, a nuclear bomb, etc. Isn't it reasonable to predict the
mass of the universe would only be stabile in a reasonable sized volume?

Just because you can extrapolate (theoretically) to an incredibly small volume
doesn't mean it was so.

(I have another question as about the assertion that everything expanded at
much greater than C, but will ask in another thread if necessary).

Thanks in advance! =)

 

[Chronos]

That figure ignores quantum gravity effects, which are not yet understood, and only applies to the observable universe, not the UNIVERSE

 

[megacal]

That figure ignores quantum gravity effects, which are not yet understood, and only applies to the observable universe, not the UNIVERSE

Do you believe all the matter was contained in a volume infinitely smaller than a proton,
or is it possible it was something larger, e.g. the Milky Way, or a cubic lightyear, or the sun?

 

[Chronos]

I think it is irrelevant without quantum gravity corrections.

 

[Chalnoth]

I'm a microbiologist by training, and only have the basics of physics & math, so please bear with me.

Why do cosmologists assert that all the matter in the universe
was contained in a volume many times smaller than e.g. a proton
(or infinitely small?) prior to the Big Bang?

Why couldn't it have been a more "reasonable" size, e.g. the sun?

Everything we see that explodes has a reasonable volume...super novas,
sticks of dynamite, a nuclear bomb, etc. Isn't it reasonable to predict the
mass of the universe would only be stabile in a reasonable sized volume?

Just because you can extrapolate (theoretically) to an incredibly small volume
doesn't mean it was so.

(I have another question as about the assertion that everything expanded at
much greater than C, but will ask in another thread if necessary).

Thanks in advance! =)

In order for dark matter to be produced efficiently, the temperature of the early universe had to be above what we've probed in particle accelerators so far, or above about 1TeV. Our universe would have been at a temperature of 1TeV when the observable universe was contained within a radius of about 100 million kilometers, or a bit less than the distance from the Earth to the Sun.

That is the maximum possible minimum size of the visible universe when it was hottest. The natural expectation is that the starting temperature (at the end of inflation) of the very early universe would have been much, much higher than that. Some models, for example, put the temperature at around a hundred billion times that, which would have meant the observable universe would have fit within about a meter radius.

 

[megacal]

Chalnoth,
thanks...you read my mind.
I was going to ask what is the smallest volume that could contain all the
visible & dark matter (another problem for me, but will assume it exists for the purpose of this thread) in the universe prior to the BB.

Even a 2 meter diameter sphere is much more reasonable than something infinitely dense and infinitely smaller than a proton, which is what I've heard stated confidently as fact from several cosmologists without any qualification.

Do cosmologists consider what is "reasonable"? I.e. if something seems impossible, couldn't there
be a different explanation? E.g. that maybe there isn't any "dark matter" if we can't see it or measure it directly even though we're supposedly surrounded by it? (should I ask that question
in another thread?)

Chronos,
thanks for your reply,
"I think it is irrelevant without quantum gravity corrections",
but Chalnoth's answer was more in the ballpark of my understanding.

 

[Chalnoth]

Even a 2 meter diameter sphere is much more reasonable than something infinitely dense and infinitely smaller than a proton, which is what I've heard stated confidently as fact from several cosmologists without any qualification.

I think the distinction here is that the observable universe was indeed very much smaller than a proton in the very early universe. But at that time, it didn't have any matter around.

This would have been during cosmic inflation. The matter came about when inflation ended, and the field that drove inflation (the inflaton field) decayed into various particles. It is at this event that I'm talking about our universe being the hottest.

 

[Mordred]

Chalnoth,
thanks...you read my mind.
I was going to ask what is the smallest volume that could contain all the
visible & dark matter (another problem for me, but will assume it exists for the purpose of this thread) in the universe prior to the BB.

Even a 2 meter diameter sphere is much more reasonable than something infinitely dense and infinitely smaller than a proton, which is what I've heard stated confidently as fact from several cosmologists without any qualification.

Do cosmologists consider what is "reasonable"? I.e. if something seems impossible, couldn't there
be a different explanation? E.g. that maybe there isn't any "dark matter" if we can't see it or measure it directly even though we're supposedly surrounded by it? (should I ask that question
in another thread?)

Chronos,
thanks for your reply,
"I think it is irrelevant without quantum gravity corrections",
but Chalnoth's answer was more in the ballpark of my understanding.

Chalnoth do you have a paper or article referencing the TEV level to form dark matter? If so I would be interested in reading it particularly if that paper defines the energy limits of its formation.


in regards to the quoted questions above, the evidence for dark matter is numerous, these include gravitational lensing, early universe large scale structure formation (without DM large scale structures would form much later than observed) and galaxy rotation curves (missing mass to explain rotation rates) threads on that subject are numerous on this forum.

The infinitely dense, infinitely small universe at the BB is a pop media misunderstanding of the singularity described in the BB model. More accurately the use of the singularity term in the BB model is merely a point at which our mathematics can no longer describe its nature (infinities).
The size of the universe at the BB is unknown and could be finite or infinite.

for the OP you may find these articles will help aid your understanding of current cosmology.

Misconceptions about the big bang by Lineweaver and Davis

http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf

http://arxiv.org/abs/astro-ph/0310808 :"Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe" Lineweaver and Davies

http://arxiv.org/abs/1304.4446 :"What we have leaned from Observational Cosmology." -A handy write up on observational cosmology in accordance with the LambdaCDM model.

the last article is particularly handy as it describes cosmology without any mathematics in a manner of a Cosmology FAQ. However all 3 articles are fairly straight forward. More articles can be found in my signature "cosmology101 link."

 

[phinds]

I was going to ask what is the smallest volume that could contain all the
visible & dark matter (another problem for me, but will assume it exists for the purpose of this thread) in the universe prior to the BB.

Even a 2 meter diameter sphere is much more reasonable than something infinitely dense and infinitely smaller than a proton,

I'd suggest that you make an attempt to use proper terminology. You continue to say "the universe" when presumably you mean the OBSERVABLE universe.

"infinitely smaller than" is the same as "infinitely small", meaning no dimension at all, which I'm sure was not the case even when talking about only the observable universe.

 

[JesseM]

Everything we see that explodes has a reasonable volume...super novas,
sticks of dynamite, a nuclear bomb, etc. Isn't it reasonable to predict the
mass of the universe would only be stabile in a reasonable sized volume?

The only reason astronomical objects like stars don't keep collapsing under their own gravity is that there are other non-gravitational forces that push outwards when the matter becomes too compressed, countering gravity--when these forces are in equilibrium the object can maintain a constant size. But for each of these non-gravitational forces, there is a limit to how much mass you can compress into a given region and have the outward force be strong enough to resist further collapse--for example, the outward pressure from "electron degeneracy pressure" can't resist gravity for a white dwarf (a type of dense remnant of the core of an exploded star) larger than the Chandrasekhar limit, and similarly the even stronger outward pressure from the "neutron degeneracy pressure" can't resist gravity for a neutron star larger than a certain limit. At a certain point there is no possible degeneracy pressure that can resist continued collapse, and in these cases general relativity (Einstein's theory of gravity) predicts that the matter will continue to collapse until the density approaches infinity in a finite time, creating a "singularity" at the center of a black hole. The same sort of idea would apply to the universe as a whole--if space were contracting so that all the matter in the universe was getting compressed into a smaller volume, general relativity predicts the density would become great enough that no form of outward pressure could resist continued collapse, and the whole universe would approach a state of infinite density in a finite time, an idea known as the Big Crunch (current observations suggest this won't happen in reality, that the universe will continue to expand forever because the mass density isn't great enough to overcome the expansion and start it collapsing, but this is a theoretical possibility in general relativity).

General relativity is a time-symmetric theory, meaning that the equations you'd use to predict the future state of a given system are the same as the ones you'd use to "retrodict" its past state, given knowledge of its present state. So, the Big Bang is just like a theoretical Big Crunch happening in reverse, and the same theoretical statements apply--if you keep extrapolating the present expansion backwards, general relativity predicts that there's no outward pressure that could avoid the conclusion that the density must have approached infinity at a finite time in the past.

So, that's what's predicted in general relativity, which is the current most accurate known theory of gravity that has a long history of its predictions being verified experimentally. But there are theoretical arguments that convince physicists that general relativity is likely to be only an approximation to a future theory of quantum gravity, and that the two theories will start to diverge significantly in their predictions at a scale of very high density of mass/energy (or very small units of distance and time) known as the Planck scale. So, physicists trust general relativity's prediction of increasing density further into the past right up to about the point where the density reaches the Planck density, before that general relativity isn't trustworthy and we can't know what would have happened without a theory of quantum gravity. I would guess the "size of a proton" statements are based on taking the estimated mass of the observable universe and figuring out how big it would be if squeezed to the Planck density.

 

[Chalnoth]

Chalnoth do you have a paper or article referencing the TEV level to form dark matter? If so I would be interested in reading it particularly if that paper defines the energy limits of its formation.

Sorry, but no. I'm just going by the fact that we've probed the behavior of matter up to around 1TeV or so in particle accelerators and not yet found any deviation from the standard model. There needs to be some rather significant deviation from the standard model to produce dark matter (and also for baryogensis, now that I think about it).

So this is just an extremely loose bound. The true maximum temperature was almost certainly much, much higher.

 

[megacal]

Many thanks to all for taking time to answer my questions.

Mordred,

the evidence for dark matter is numerous, these include gravitational lensing, early universe large scale structure formation (without DM large scale structures would form much later than observed) and galaxy rotation curves (missing mass to explain rotation rates) threads on that subject are numerous on this forum.

Yes, BUT... we still haven't got a piece of it in hand.
Thanks for the links...I'll review them before posting again.

JesseM,
thank you for that very clear & straightforward overview (with links!)...very helpful.

phinds,

I'd suggest that you make an attempt to use proper terminology. You continue to say "the universe" when presumably you mean the OBSERVABLE universe.

"infinitely smaller than" is the same as "infinitely small", meaning no dimension at all, which I'm sure was not the case even when talking about only the observable universe.

Thanks for the correction. I assumed "universe" vs "Universe" was understood, but will be careful in the future to specify "observable universe"...presuming "observable" needn't be in all caps.

BTW, found your page on the Balloon Analogy, etc, and will spend time there. I tried to get on the "same page" here a year or so ago, and gave up...will have another go at it...thanks.

 

[JesseM]

I just did a calculation on another thread which suggests where the "size of a proton" estimate probably comes from:

Let's do a back-of-the-envelope style rough calculation. According to this, the current density of all forms of mass and energy (and energy can be treated as a form of mass by E=mc^2) is about 0.85 * 10^-26 kg/m^3. The Planck density is around 5*10^96 kg/m^3. So to squeeze the observable universe to the Planck density, you'd have to divide its volume by about 6*10^122. Since the volume of a spherical region is the cube of its radius, we have to take the cube root of this to find how much to divide the radius, or 8*10^40. The radius of the observable universe is thought to be about 47 billion light years which works out to about 4*10^25 meters, so dividing this by 8*10^40 implies the radius of the observable universe would be about 5*10^-16 meters if it was squeezed to the Planck density. That's fairly close to the size of a proton, about 9*10^-16 meters.

 

[megacal]

Thanks Jesse...the math says "way" but my brain says "no way".

Will have to mull it over with what limited tools I have. ;)

 

[megacal]

I printed out and read through "Misconceptions about the Big Bang" by Lineweaver & Davis.

Very interesting and raises even more questions...

E.g. re: the Big Bang...

."It did not go off at a particular location and spread out from there
into some imagined pre-existing void. It occurred everywhere at once"

...?

How large is "everywhere"?
What's wrong with a pre-existing void? If nothing is there, it's a void, right?
There is no limit or edge to the Universe as I understand it, though there is an
boundary to the observable uiniverse, which is a smaller structure within the Universe...that's my perception anyway.

Lineweaver & Davis's Scientific American article is dated 2005. I didn't see any updates...does that mean their article is still accurate 9 years later?

Does anyone take exception with anything they wrote? I have no way to dispute what they
claim.

BTW, they referred to "the universe" 29 times before specifying "observable universe" (but who's counting?) I'll leave it to someone else here to gently point that out to them. I don't know how well received it would be coming from a noob like me.

 

[phinds]

In the "big bang theory" model, there WAS no pre-existing ANYTHING. There are other theories that say there was but they are talking about the big bang theory.

"Everywhere" means exactly that. Everywhere. If the universe was finite at the time of the singularity, then "everywhere" is that big and if it was infinite at the time of the singularly then "everywhere" is infinite.

They use "universe" because they MEAN "universe". When they start talking about the observable universe, they say so.

 

[megacal]

There are other theories that say there was but they are talking about the big bang theory.

...are or are not?

They use "universe" because they MEAN "universe". When they start talking about the observable universe, they say so.

If it's critical for clarity to distinguish between "the universe" vs "the observable universe", especially
for the general public, it should be made at the beginning of any article, rather than 30 instances later.

 

[Mordred]

I printed out and read through Misconceptions about the Big Bang by Lineweaver & Davis.

Very interesting and raises even more questions...

E.g. re: the Big Bang...?

How large is "everywhere"?
What's wrong with a pre-existing void? If nothing is there, it's a void, right?
There is no limit or edge to the Universe as I understand it, though there is an
boundary to the observable uiniverse, which is a smaller structure within the Universe...that's my perception anyway.

Lineweaver & Davis's Scientific American article is dated 2005. I didn't see any updates...does that mean their article is still accurate 9 years later?

Does anyone take exception with anything they wrote? I have no way to dispute what they
claim.

BTW, they referred to "the universe" 29 times before specifying "observable universe" (but who's counting?) I'll leave it to someone else here to gently point that out to them. I don't know how well received it would be coming from a noob like me.

Although its an older article, its still highly recommended and often cited in current research papers. The hot big bang model today is represented by the $\Lambda$CDM model. Which is the hot big bang with cold dark matter and the cosmological constant added to the energy content of the universe. However as Phinds pointed out there are other models. LCDM is still accurate and yet to be disproved, another contender is LQG (loop quantum gravity) which involves a bounce from a previous universe. I'll let others answer questions on that model as I'm more familiar with LCDM.
There have been countless other models, some more lavish than others such as our universe being inside a black hole, MOND (which tries to model without dark energy) etc.

The problem is that other models with the exception of LCDM and potentially LQC, simply do not fit observational data overall. Most times they work better in specific circumstances but poorly in others. MOND is excellent for predicting rotation curves of specific small galaxies. However loses accuracy when it comes to other galaxies and early large scale structure formation.

Unfortunately most papers and articles assume you know they are referring to the observable universe. Gathering data beyond the observable can only be done via an influence. Thus far no conclusive evidence has been detected. Therefore its assumed to be the same as our observable as there is no evidence to assume its any different. The policy of defining universe as meaning our observable derives from those reasons. Loosely put our universe is defined as "anything we can observe and measure" however that is a loose definition.

the term void unfortunately describes an energy state with space, that energy state being the lowest possible energy density. This has particular influences of surrounding space google ( false vacuum for one example).

In essence we can only speculate on anything outside of the observable universe. Any statements beyond the observable is simply conjecture and speculation.

 

[megacal]

Mordred,

thanks for the helpful explanation.

Unfortunately most papers and articles assume you know they are referring to the observable universe.

Actually, I assumed that when reading Lineweaver & Davis's
article, and when I referred to the universe earlier.

Also, I checked all the links you gave me earlier...much to mull over.

Lineweaver & Davis's paper at Cornell was several orders of magnitude
above my level of comprehension. But I'm grateful they edited it
it for Scientific American so I could digest most of it...choked on just a few concepts...
will chew on them more.

 

[Mordred]

keep at it, this forum is excellent for any questions you may have so feel free to ask for more recommended material and aid.:thumbs:

 

[phinds]

...are or are not?

Sorry. That was confusingly stated. I did mean "are" because the "they" I was referring to was Lineweaver & Davis, not the other theories I was bringing in. Very poor choice of wording on my part.

 

[petergreen]

I just did a calculation on another thread which suggests where the "size of a proton" estimate probably comes from:

Correct calculation... interesting...

 

[JesseM]

I printed out and read through "Misconceptions about the Big Bang" by Lineweaver & Davis.

Very interesting and raises even more questions...

E.g. re: the Big Bang...?

How large is "everywhere"?
What's wrong with a pre-existing void? If nothing is there, it's a void, right?
There is no limit or edge to the Universe as I understand it, though there is an
boundary to the observable uiniverse, which is a smaller structure within the Universe...that's my perception anyway.

Lineweaver & Davis's Scientific American article is dated 2005. I didn't see any updates...does that mean their article is still accurate 9 years later?

Does anyone take exception with anything they wrote? I have no way to dispute what they
claim.

BTW, they referred to "the universe" 29 times before specifying "observable universe" (but who's counting?) I'll leave it to someone else here to gently point that out to them. I don't know how well received it would be coming from a noob like me.

I think they are discussing how things work in the FLRW metrics which is how cosmologists have traditionally modeled the expanding universe in general relativity, but which doesn't take into account newer ideas like eternal inflation in which new "universes" can inflate out from tiny regions of preexisting ones (a full treatment of such ideas might require a theory of quantum gravity, or at least a better understanding of the quantum fields involved in the inflationary process). Just in the context of these traditional theoretical models, the density of matter/energy is assumed to be perfectly uniform throughout all of space (not just the observable part), which can be either finite (in the case of a "closed" universe with positive spatial curvature) or infinite (in the case of an "open" universe with zero or negative spatial curvature). In this sort of model there is nothing "outside" the matter filling space, the Big Bang is more like an expansion of space itself.

We can understand this more intuitively through a lower-dimensional analogy, imagining a universe with only 2 spatial dimensions as in the famous old book Flatland. In this case, for a finite "closed" universe of positive curvature you can imagine that the 2D surface is curved into the surface of a sphere, and the sphere itself is expanding so all the 2D matter on its surface gets less dense over time. In the case of an infinite universe with zero curvature, you might imagine something like an infinite chessboard drawn on a flat plane where the squares can all simultaneously get larger over time, but the amount of matter in each square stays constant, so the density again decreases with time (and going back in time, each square approaches a size of zero as you approach the Big Bang singularity, so the density of matter in each square approaches infinity--but this happens throughout all of infinite space, with no central location). For an infinite universe with negative curvature, you'd have to picture something like an infinite chessboard drawn on a 2D surface curved into a sort of "saddle shape"--you can see pictures of the 2D analogues of each type of geometry in part 3 of Ned Wright's cosmology tutorial.

 

[Mordred]

This article covers the geometry aspects of the FLRW metric mentioned by JesseM. I tied to keep it as uncomplicated as possible yet still accurately describe Universe geometry in accordance to the FLRW metric. Ned Wright's site (link in JesseM's post) is also an excellent reference

Universe geometry

The origins of the universe is unknown in cosmology. The hot big bang model only covers the history of the universe from 10^-43 seconds forward. Prior to that is described as a singularity. However its important to note that the singularity is not a black hole style. Instead singularity in this case simply means a point in time where our mathematics can no longer accurately describe it. Numerous youtube videos and pop media articles would have you believe our universe exploded from some super particle. This was never predicted by the hot big bang model.

The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite. So why is geometry so important to cosmology if we know the size of the observable universe? The answer to that question lies in how geometry affects the following aspects, Light paths, rate of expansion or collapse and overall shape.

In regards to light paths and geometry a closed universe described as a sphere will have two beams of light emitted at different angles eventually converge. An open hyperbolic universe such as a saddlebag will have those same two light beams diverge. A flat universe will have parallel light paths (provided the beams at emission were parallel to begin with)
You will notice on each image there is a triangle, this triangle represents how the geometry affects our measurements. In a flat curvature the three angles of a equilateral triangle will add up to 180⁰. A positive curvature will add up to greater than 180⁰, a negative curvature will add up to less than 180⁰

Image from http://universeadventure.org

The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula

$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$

P=pressure
c=speed of light
G= gravitational constant.

density is represented by the Greek letter Omega $\Omega$ so critical density is $\Omega crit$
total density is

$\Omega$total=$\Omega$_{dark matter}+$\Omega$_baryonic+$\Omega$_radiation+$\Omega$_{relativistic radiation}+${\Omega_ \Lambda}$

$\Lambda$ or Lambda is the value of the cosmological constant often referred to as "dark energy" more accurately it is the vacuum pressure that attributes to expansion.
the subscript "₀"for $\Omega$ shown in the image above denotes time in the present.

Energy-density is the amount of energy stored per unit volume of space or region. Energy per unit volume has the same physical units as pressure, and in many circumstances is an exact synonym.

$\Omega=\frac{P_{total}}{P_{crit}}$
or alternately
$\Omega=\frac{\Omega_{total}}{\Omega_{crit}}$

Geometry in 2D
In developing a theory of space-time, where curvature is related to the mass-energy density, Scientists needed a way of mathematically describing curvature. Since picturing the curvature of a four-dimensional space-time is difficult to visualize. We will start by considering ways of describing the curvature of two-dimensional spaces and progress to 4 dimensional spaces.
The simplest of two-dimensional spaces is a plane, on which Euclidean geometry holds.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180. Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c² = a² + b²
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to three dimensions as well:
c²= a² + b² + c²
see image 2.0 below


On a plane, a "geodesic" is a straight line(shortest distance between two points). If a triangle is constructed on a flat 2 dimensional plane by connecting three points with geodesics. The curvature can be represented in 2D, if you establish each angle of a equilateral triangle with
$\alpha$,$\beta$,$\gamma$ for a flat geometry this follows the relation

$\alpha$+$\beta$+$\gamma$=$\pi$.

image 1.0

image 2.0 reference (3)

On a plane, (shown above) we can set up a cartesian coordinate system, and assign to every point a coordinate (x; y). On a plane, the distance ds between points (dx and dy) is given by the relation
$d{s^2}=d{x^2}+d{y^2}$

If a triangle is constructed on the surface of the sphere by connecting the angles will obey the relation

$\alpha$+$\beta$+$\gamma$=$\pi+{AR^2}$

image 1.1
where A is the area of the triangle, and R is the radius of the sphere. All spaces in which
$\alpha$+$\beta$+$\gamma$>$\pi$ are called positively curved" spaces. It is a space where the curvature is homogeneous and isotropic; no matter where you draw a triangle on the surface of a sphere, or how you orient it, it must always satisfy the above equation.
"On the surface of a sphere, we can set up polar coordinates "north pole" and "south pole" and by picking a geodesic from the north to south pole to be the "prime meridian". If r is the distance from the north pole, and $\theta$ is the azimuthal. angle measured relative to the prime meridian,"⁽¹⁾ then the distance ds between a point (r; $\theta$) and another nearby point (r+dr+$\theta$+d$\theta$) is given by the relation

${ds^2} = {dr^2} + {R^2} {sin^2}(r/R)d\theta^2$

"An example of a negatively curved two-dimensional space is the hyperboloid, or saddle-shape. A surface of constant negative curvature. The saddle-shape has constant curvature only in the central region, near the "seat" of the saddle."⁽¹⁾ Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices $\alpha$
$\beta$,$\gamma$ obey the relation $\alpha$+$\beta$+$\gamma$=$\pi-{AR^2}$.

${ds^2} = {dr^2} + {R^2} {sinH^2}(r/R)d\theta^2$

image 1.2

A negative curvature is an open topography

If a two-dimensional space has curvature or flat which is homogeneous and isotropic, its geometry can
be specified by two quantities k, and R. The number k, called the curvature constant, R is the radius

k = 0 for a flat space,
k = +1 for a positively curved space,
k = -1 for a negatively curved space

Geometry in 3D
A two dimensional space can be extended to a three-dimensional space, if its curvature is homogeneous and isotropic, must be flat, or have uniform positive curvature, or have
uniform negative curvature. If a three-dimensional space is flat (k = 0), it
has the metric

ds² = dx² + dy² + dz² ;

expressed in cartesian coordinates or

${ds^2} = {dr^2} +{r^2}[d\theta^2 + {sin^2} d\phi^2]$

If a three-dimensional space has uniform positive curvature (k = +1), its
metric is

${ds^2} = {dr^2} +{R^2}{sin^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

A negative curvature in the uniform portion has the metric (k=-1)

${ds^2} = {dr^2} +{R^2}{sinH^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

Geometry in 4D

Thus far we have discussed the 2 and 3 dimensional components. The Friedmann-Lemaitre-Robertson-Walker metric (FLRW) can be used to describe the 4D dimensions with the use of a(t). a(t) is the scale factor. See the redshift and expansion article for more information or the cosmocalc link on the main page. Scale factor in a homogeneous and isotropic universe describes how the universe expands or contracts with time.
The FLRW metric can be written in the form

$d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]$

references
(1)"Introductory to Cosmology" Barbera Ryden"
images 1.0,1.1 and 1.2 (see (1))
(2)"Modern Cosmology" Scott Dodelson
(3)"lecture notes, Introductory to Cosmology" Dr. Ka Chan Lu

 

[Mordred]

I should mention that eternal inflation is an older but still valid inflationary model, in fact there is over 60 still valid inflation models, many of which derived from "old Inflation" or false vacuum by Allen Guth. The problem with old inflation, new inflation and eternal chaotic inflation. Is that they lacked a mechanism that would stop inflation ("Runaway Inflation"). Later models use a slow roll mechanism to limit inflation and attempt to stop inflation.

the Slow roll approximation is more commonly used as a benchmark model for other inflationary models as it has a tighter fit to observational data. Although eternal chaotic inflation is still within marginal error and therefore still valid.

Personally I prefer the Higg's inflation with 3 goldstones or the zero scalar Higg's inflation, as the Higg's inflation accomplishes similar results without introducing any exotic processes or particles such as the inflaton. Higg's inflation is gaining weight thanks to CERN's discovery of the Higg's boson.

Natural inflation is another personal interest of mine though finding good articles on it specifically has proven daunting.


Keep in mind their are tons of variations of inflation but inflation itself is a specific era in cosmology. Inflation is an extremely rapid expansion of space-time that occurred approximately within the 1st second of the universe. The exact time depends on the model but all models must solve several key problems. Googling each with the term cosmology used in the search will provide numerous links on each problem.

-monopole problem
-flatness problem
-horizon problem

Further information on the problems above can be found in the "What we have learned from Observational cosmology link provided in my first post.

This article has a listing of still valid models. the first section covers the requirements to match observational data. So provides a coverage of those problem. I wouldn't bother trying to read the full article but rather study the specific models your interested in.


http://arxiv.org/abs/1303.3787

the first 4 or 5 models I would however recommend reading, as they cover the history of how the early inflation models were developed historically.

I should also note this article is regularly updated as new research is added

 

[megacal]

phinds,

the "they" I was referring to was Lineweaver & Davis, not the other theories

Thanks for clearing that up...thought it was just a typo.

Mordred,

keep at it, this forum is excellent for any questions you may have so feel free to ask for more recommended material and aid.

Thanks! I'll press on, though you left me in the interstellar dust with those equations! I will see how far I get into the proof before
I hit the wall.

But it's fascinating to see how you all communicate via math that seems so nebulous to
myself. I wish I could join in on your level, but will be happy to be a fly on the wall, and
listen into the conversation.

And I think I am getting a much better understanding of where things stand.
It's good mental excercise.

I enjoy working with various apps for 3D modeling & animation. Maybe that would help me better visualize things, though it would not be anywhere accurate, as I can't set up a proper computer model as those I've seen, e.g. where galaxies collide.

 

[petm1]

if space were contracting so that all the matter in the universe was getting compressed into a smaller volume, general relativity predicts the density would become great enough that no form of outward pressure could resist continued collapse, and the whole universe would approach a state of infinite density in a finite time, an idea known as the Big Crunch (current observations suggest this won't happen in reality, that the universe will continue to expand forever because the mass density isn't great enough to overcome the expansion and start it collapsing, but this is a theoretical possibility in general relativity).

If space were contracting, but space is expanding so could this outward pressure, via big bang, predominate in the early universe or be the cause of inflation?