## Does Space Expand?

The the LCDM model scale factor is defined as:
$$a(t) = \left[ \frac{\Omega_m}{\Omega_v} \sinh^2 \left( \frac{3}{2} \sqrt{\Omega_v} H_0 t \right) \right]^{\frac{1}{3}}$$

Differentiating the scale factor function with respect to t:
$$\frac{d a(t)}{dt} = \frac{d}{dt} \left[ \frac{\Omega_m}{\Omega_v} \sinh^2 \left( \frac{3}{2} \sqrt{\Omega_v} H_0 t \right) \right]^{\frac{1}{3}} = \frac{ \Omega_m H_0 \cosh \left(\frac{3}{2} \sqrt{\Omega_v} H_0 t \right) \sinh \left(\frac{3}{2} \sqrt{\Omega_v} H_0 t \right)}{\sqrt{\Omega_v} \left(\frac{\Omega_m \sinh^2 \left(\frac{3}{2} \sqrt{\Omega_v} H_0 t \right)}{\Omega_v} \right)^{2/3}}$$

The scale factor derivative function:
$$\boxed{\frac{d a(t)}{dt} = \frac{ \Omega_m H_0 \cosh \left(\frac{3}{2} \sqrt{\Omega_v} H_0 t \right) \sinh \left(\frac{3}{2} \sqrt{\Omega_v} H_0 t \right)}{\sqrt{\Omega_v} \left(\frac{\Omega_m \sinh^2 \left(\frac{3}{2} \sqrt{\Omega_v} H_0 t \right)}{\Omega_v} \right)^{2/3}}}$$

Is this equation correct?

Attachments: plot a(t), plot a'(t)

Reference:
LambdaCDM geometry - mathematical details - Wikipedia
Attached Thumbnails

 I'm not good enough at the math to quickly check your solution but I believe there is no analytic solution when there is more than one phase involved. The graph is initially matter dominated but becomes energy dominated so I think you have to perform an integration to get the curve. However, the section "mixtures" here suggests it can be solved: http://en.wikipedia.org/wiki/Friedma...eful_solutions The "Lecture notes on Astrophysics" looks comprehensive too, though beyond my level.
 i was thinking about light. a particle and/or a wave ? but what about darkness.? does darkness move at the speed of light? does light move at the speed of darkness? does light really bend around corners or is it pulled around by darkness? in a dark universe, does the universe expand when light appears? does light "push" darkness away ? but what if the universe is in a "bubble"? would the darkness get squashed? is light a constant in the universe.? does the universe expand at different speeds at different times depending on how much is being added at that particular time?

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 Quote by lostprophets i was thinking about light. a particle and/or a wave ?
The best description I've heard is that light is an electromagnetic wave that transfers energy only in packets we call photons.

 but what about darkness.? does darkness move at the speed of light? does light move at the speed of darkness?
Darkness is nothing but the absence of light, similar to how a vacuum is the absence of matter in a volume of space.

 does light really bend around corners or is it pulled around by darkness? in a dark universe, does the universe expand when light appears? does light "push" darkness away ?
Light really does diffract (bend if you want to call it that) around corners to a certain extent that depends on the wavelength. The rest of the quote doesn't make any sense.

 but what if the universe is in a "bubble"? would the darkness get squashed? is light a constant in the universe.? does the universe expand at different speeds at different times depending on how much is being added at that particular time?
I think you have a misunderstanding on how we view the universe. I cannot answer these questions because they don't even make sense with current cosmological models. My suggestion is to read up on the subject. There are plenty of websites including wikipedia that will help you understand. Here's two articles that will greatly help you if you read them and follow all the links around. Don't be surprised if it doesn't make much sense first, as unless you understand the basics of light and matter the terms won't mean much.

http://en.wikipedia.org/wiki/Universe
http://en.wikipedia.org/wiki/Physical_cosmology

 Quote by Drakkith Darkness is nothing but the absence of light, similar to how a vacuum is the absence of matter in a volume of space. Light really does diffract (bend if you want to call it that) around corners to a certain extent that depends on the wavelength. The rest of the quote doesn't make any sense. http://en.wikipedia.org/wiki/Universe http://en.wikipedia.org/wiki/Physical_cosmology
:o are you serious?
darkness is nothing but the absence of light..... ooosh
so what came first, the darkness or the light?
but i thought we knew or at least thought ,that there is no such thing as nothing

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 Quote by lostprophets :o are you serious? darkness is nothing but the absence of light..... ooosh so what came first, the darkness or the light?
Light has been around since the earliest moment of the universe. So I would say light. What happened "before" the universe is pure speculation and doesn't belong here. (Just in case you were going to bring that up)

 but i thought we knew or at least thought ,that there is no such thing as nothing
That is more philosophy than science. We have defined darkness to be the absence of visible light, just as we have defined a vacuum to be the absence of matter.

 These are the scale factor equations that I reviewed from reference 1 and 2. Inflation Hubble parameter (end of inflationary epoch): 'ref. 1 p. 34 (167)' $$H_i = \frac{1}{t_{i}} = \frac{1}{10^{-32} \; \text{s}} = 10^{32} \; \text{s}^{-1}$$ $$\boxed{H_i = 10^{32} \; \text{s}^{-1}}$$ Inflation scale factor: 'ref. 1 p.35 (165)' $$a(t) \propto e^{H_i t} \tag{1}$$ Radiation scale factor: 'ref. 1 p. 22 (119)' $$a(t) = (2 H_0)^{\frac{1}{2}} \cdot t^{\frac{1}{2}} \tag{2}$$ Matter scale factor: 'ref. 1 p. 21 (115)' $$a(t) = \left( \frac{4 H_0}{2} \right)^{\frac{2}{3}} \cdot t^{\frac{2}{3}} \tag{3}$$ LCDM matter scale factor: 'ref. 2' $$a(t) = \left[ \frac{\Omega_m}{\Omega_v} \sinh^2 \left( \frac{3}{2} \sqrt{\Omega_v} H_0 t \right) \right]^{\frac{1}{3}} \tag{4}$$ Equations 2 and 3 appear to be describing a universe that is much younger. Attachments: plot 1, plot 2,3,4 Reference: Friedmann equations - useful solutions - Wikipedia Northern Illinois University - Physics 652 - Astrophysics LambdaCDM - geometry - mathematical details - Wikipedia Attached Thumbnails

 Quote by Drakkith Light has been around since the earliest moment of the universe. So I would say light. .
if light was more abundant than darkness at the start where as the reverse is true now, am i to believe then that the universe is getting smaller?
what if it was expanding and contracting
i ask about light "pushing" darkness (light pressure) clearing a path .
so if we had darkness first with energy, then light energy appears,room has to be made for this light.
could light then clear this "room" creating a vacuum redundant of energy once this light has lost its energy and gone..this then takes time to rebuild itself with dark energy matter, un til it over crowds sparking another light source and repeats the process.

this would mean light energy is finite but that does not mean the universe cannot expand..

i could be way off and have no idea what im on about.but ive read some say that the universe is expanding fast than light... how do we measure this.do we measure it with light?
if light is "pushing" then light will always be behind therefore it could be seen that anything infront of it is moving fast when really its not

 Quote by lostprophets i respect your guess. if light was more abundant than darkness at the start where as the reverse is true now, am i to believe then that the universe is getting smaller?
As has been said, darkness is only the absence of light, so in the presence of a single particle of light, the universe is not dark.

 what if it was expanding
It is expanding, and the rate at which it does so is increasing.

That is poetic licence, it has no physical meaning. Light pressure can push a sail around (look up the Ikaros project) but darkness isn't a substance, just the absence of light.

 so if we had darkness first with energy, then light energy appears,
For the first 378,000 years, the whole universe looked like the interior of the Sun, the farther back in time you go, the brighter it was.

 room has to be made for this light.
Space, time and light possibly arose together but we don't know, that is presently beyond our understanding.

 this would mean light energy is finite but that does not mean the universe cannot expand..
It is definitely expanding, it may be finite or infinite, we cannot tell which.

 i could be way off and have no idea what im on about.but ive read some say that the universe is expanding fast than light... how do we measure this.do we measure it with light?
Yes. Surprisingly, the light can still reach us, but I'd need to go into maths to explain why.

 if light is "pushing" then light will always be behind therefore it could be seen that anything infront of it is moving fast when really its not
Light was created everywhere equally and moved in all directions. There was no "in front" or "behind", it always surrounded.

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 Quote by lostprophets i respect your guess. if light was more abundant than darkness at the start where as the reverse is true now, am i to believe then that the universe is getting smaller?
No, as has been repeatedly said, the universe is expanding.

 what if it was expanding and contracting
There's no need to ask "what ifs" that aren't real. The universe is expanding, not contracting.

 i ask about light "pushing" darkness (light pressure) clearing a path .
I don't know what you don't understand about darkness simply being the absence of light. Darkness is an abstract concept linked to vision. If a volume of space is completely devoid of EM radiation (light) we do not call it dark, we call it empty of radiation. Light propagates through space and interacts with matter. It cannot interact with empty space as there is nothing to interact with!
 so if we had darkness first with energy
We did not have darkness first. As I said light has existed since the earliest moments of the universe when the density and temperature of the universe was so high that matter and antimatter was continually being created from EM radiation and annihilated, converting back to EM radiation.

 then light energy appears,room has to be made for this light.
It did not "appear". The energy already existed. Furthermore you keep suggesting that "darkness" is something physical and tangible. It is not. Does a vacuum have to make room for particles to exist in it? No!

 could light then clear this "room" creating a vacuum redundant of energy once this light has lost its energy and gone..this then takes time to rebuild itself with dark energy matter, un til it over crowds sparking another light source and repeats the process.
Absolutely not. The earliest moments of the universe was full of interacting particles and radiation. As George said above, imagine being inside the core of the Sun, but a billion billion trillion times denser and hotter. Then go another quadrillion above that. Then you will be getting close to the state of the early universe.

 i could be way off and have no idea what im on about.but ive read some say that the universe is expanding fast than light... how do we measure this.do we measure it with light? if light is "pushing" then light will always be behind therefore it could be seen that anything infront of it is moving fast when really its not
We measure it by looking at the amount of redshift an object presents to us. The further away an object such as a galaxy is, the more its light is redshifted. This is due to the expansion of the universe causing it to recede from us and stretching out the light as it travels over billions of years.

Also, the expansion of the universe is a "rate", not a measurement of velocity. What this means is that objects further away will accelerate away from us quicker than objects closer to us will. The speed at which objects move away from us is called the recession velocity. Currently our measurements show that this recession velocity increases by about 70 km/s per megaparsec (3.26 million light-years) in distance that an object is from us. So a galaxy at 2 megaparsecs in distance from us would be receding at about 140 km/s, while a galaxy at 20 megaparsecs would recede at 1400 km/s. If the rate of expansion were higher, the recession velocity would increase by a larger amount per distance, such as being 100 km/s per megaparsec.

 sorry . i did mean to use the word "front" lightly .ooosh pun ,not so poetic... yes theres no front, back, middle, only edges,curves,surrounding, enclosed ,in a tomb of darkness.. the further we look back the brighter it gets. its logic for it to be so. but is it logic to think that what one is looking at is not the beginning but a random? thanks for the reply ... also how far can my eyes see. meaning when i see light that has come from a far distance.at what distance am i seeing it.? am i seeing the light from the distance of my eye or am i seeing the light light years away. my eyes can see distances.so i ask is it possible to travel down the light to the source and bring it nearer? i no i may like a fruit loop hope you dont mind.. my question is this. is the light seeing me or am i seeing it? also i went to the optitions today.he put a light in my eye .when this light was taken away i had a dark line of vision.i asked why. he said its because the light removes something or other ,sorry cant remember exactly,so there was an empty space .but over time the energy recovers and bring the light back to this dark spot...can space work the same? meaning does light remove matter then once the light has gone this matter returns over a time period...

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 Quote by lostprophets the further we look back the brighter it gets. its logic for it to be so. but is it logic to think that what one is looking at is not the beginning but a random?
Well, your first sentence is incorrect. It is not brighter the further we look back. After the universe formed it cooled and expanded over millions of years. Finally after the temperature and density dropped beyond a critical point, protons could combine with electrons, forming neutral atoms that are mostly transparent to light. Before this point in time light could not travel more than mere nanometers before interacting with protons or electrons. After this point in time the universe became "transparent", and the radiation that was released from electrons combining with protons could suddenly travel over light-years and is currently seen as the Cosmic Microwave Background. The CMB is literally the furthest back we can see using light. It is not physically possible to see beyond this point unless we can somehow invent a neutrino detector in the future that is a few trillion times more sensitive than current ones.

After this recombination, atoms could finally start to collapse under gravitational attraction to form the first stars and galaxies. Whether the universe is "brighter" now or then is unknown to me.

 also how far can my eyes see. meaning when i see light that has come from a far distance.at what distance am i seeing it.? am i seeing the light from the distance of my eye or am i seeing the light light years away. my eyes can see distances.so i ask is it possible to travel down the light to the source and bring it nearer? i no i may like a fruit loop hope you dont mind.. my question is this. is the light seeing me or am i seeing it?
I don't really know what you are asking. Since photons can travel through space for billions of years, if your eye detects one then you are seeing something billions of lightyears away. The only limit to how far an object can be seen is simply that the universe is only a finite age. The CMB was released over 13 billion years ago, so as time passes the area of space that those photons we see were released from is getting further away.

 im now confused. one of you is saying the further you go back the brighter it gets,and the other is saying not so... the problem may stem from the "no matter which direction we look ,it all looks the same" on a large scale not small... so how do we get around this.? its like looking at a field full of sheep and guessing which one came first. neutrinos collide with things at random points at random times......this to me is very important.

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 Quote by lostprophets im now confused. one of you is saying the further you go back the brighter it gets,and the other is saying not so...
Sorry. Before the CMB was emitted you did in fact get "brighter" the further back you go, but that is kind of inaccurate as the state of the universe was very different from what it is today. I prefer the terms "hotter" and "denser".

 the problem may stem from the "no matter which direction we look ,it all looks the same" on a large scale not small... so how do we get around this.? its like looking at a field full of sheep and guessing which one came first. neutrinos collide with things at random points at random times......this to me is very important.
Get around what? The universe is very homogenous on the large scale.

And may I request that you make specific questions. Much of your posts seem to be ramblings that don't make any sense and don't seem to be asking anything. It would help both us and yourself if you could trim your posts down to clear, concise questions.

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 Quote by lostprophets also i went to the optitions today.he put a light in my eye .when this light was taken away i had a dark line of vision.i asked why. he said its because the light removes something or other ,sorry cant remember exactly,so there was an empty space .but over time the energy recovers and bring the light back to this dark spot...can space work the same? meaning does light remove matter then once the light has gone this matter returns over a time period...
The bright light uses up the chemicals in your eye that respond to light, allowing you to see. These chemicals require time to be replaced in your cells, so it takes a little bit for your vision to return to normal. The light isn't pushing anything out of the way and the chemicals are still there, they are just used up in a reaction that turns them into something else.

 Quote by Drakkith The bright light uses up the chemicals in your eye that respond to light, allowing you to see. These chemicals require time to be replaced in your cells, so it takes a little bit for your vision to return to normal. The light isn't pushing anything out of the way and the chemicals are still there, they are just used up in a reaction that turns them into something else.
yes..
i see. thank you. the light turns them into something else.....1+1 = 3

 Quote by lostprophets im now confused. one of you is saying the further you go back the brighter it gets,and the other is saying not so... the problem may stem from the "no matter which direction we look ,it all looks the same" on a large scale not small... so how do we get around this.? its like looking at a field full of sheep and guessing which one came first. neutrinos collide with things at random points at random times......this to me is very important.
It is summer and you are in a field of sheep, all born in the same week in the spring. If light travels slowly, you see old sheep near you but lambs far off. When we look far away, we see the universe as it was earlier. It was brighter earlier.