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### Course: Cosmology and astronomy > Unit 1

Lesson 4: Big bang and expansion of the universe- Big bang introduction
- Radius of observable universe
- Radius of observable universe (correction)
- Red shift
- Cosmic background radiation
- Cosmic background radiation 2
- Hubble's law
- A universe smaller than the observable
- How can the universe be infinite if it started expanding 13.8 billion years ago?

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# Radius of observable universe

Radius of Observable Universe. Created by Sal Khan.

## Want to join the conversation?

- Am I wrong in thinking that this example is logically impossible? The distance between the photon and the Earth can never increase with time. If it did, it would mean that the distance would continually increase and the photon would never reach Earth. For example, it starts off thirty million light years away, and after 100 million years it is eighty million light years away. If this is the case the Earth and the photon would get further and further away for eternity.(70 votes)
- Well, if you count this in porpotions, it will seems like it's possible! After 10 million years, the photon travelled 20% of the whole distance. After another 40 million years, it travelled another 40%. Now it might looks possible since the porpotion is always increasing. I don't believe it at the first time, either. But it IS mathematically possible.(4 votes)

- We perceive the time as 13.7 billion years but since time slows as one approaches the speed of light, how much actual time has passed from the photon's perspective? It's been said that if a spaceship from Earth went to the nearest star traveling near the speed of light, the journey would only take years for those on the spaceship while those on Earth would see it as millenniums. If I were hitching a ride with the photon, would I perceive the universe as far less than 13.7 billion years old?(50 votes)
- From the perspective of the photon, the age of the Universe is zero. It arrives at the moment it is created. For the photon, not only is the time zero, the distance is zero. The observation about the spaceship is true. Given a destination 10 light years away, at 83.2% of the speed of light the distance is 5.6 light years and the time is 6.7 years, not 12.0 years. At 86.6% of the speed of light, the distance is 5.0 light years and the time is 5.8 years, not 11.6 years. At 90.0% of the speed of light, the distance is 4.4 light years and the time is 4.8 years, not 11.1 years. Two observations: While the speed increases 6,400 miles per second between each example, the rate of decrease of both distance and time is accelerating. The velocity (distance divided by time) measured inside the spaceship is exactly the same as would be measured by an observer at rest outside the spaceship even though the distance and time are different.

To derive these numbers, start with the fraction of v over c where v is equal to the velocity (speed) and c is the speed of light. Formula 1 is v divided by c. If the observer is at rest (v = 0) then the value of this fraction is zero (0). If an observer is traveling at the speed of light (v = c), then the value of this fraction is one (1). Next multiply Formula 1 by itself. The result is the fraction v squared over c squared. Formula 2 is v squared divided by c squared. Note that the value both at rest and at the speed of light remains the same as Formula 1. The next step is to subtract Formula 2 from one (1). Formula 3 is 1 minus the fraction v squared over c squared. Note that Formula 2 and Formula 3 are complimentary (the sum being one) and that the values at rest and at the speed of light are opposite. The last step is to take the square root of Formula 3. Formula 4 is the square root of the entire expression 1 minus the fraction v squared over c squared.

To calculate the distance and time measured inside the spaceship, simply multiply their value (as would be observed at rest) by Formula 4. Note that as the velocity approaches the speed of light, both distance and time approach zero (0).

To generate the numbers in the second example (86.6%), start with the velocity 161,000 miles per second (light being 186,000 miles per second). Formula 1 is 86.6%. Formula 2 is 3/4. Formula 3 is 1/4. Formula 4 is 1/2. At a distance of 10 light years, the spaceship will measure a distance of 5 light years, half of the original 10 light years, and the elapsed time will be measured at 5.8 years, half of the original 11.6 years.(39 votes)

- If the universe is expanding, are the atoms in our bodies increasing in distance between each other?(23 votes)
- No because they are held together by forces that counteract the expansion of the universe. Galaxies themselves as well as solar systems are held together by gravity that counteracts the expansion. The expansion of the universe is only noticed at intergalactic distances.(59 votes)

- If the space itself is expanding with a speed faster than light then how can a photon traveling with speed of light from edge of universe ever reach us?(15 votes)
- Good question! The answer is that it's quite likely that it can't, and that there are parts of the universe which we will never know about, because light from those parts will never reach us. We make the distinction between "the universe", which is everything that exists, and "the visible universe", which is everything we can see.(14 votes)

- What is the estimated size of the universe? What is inflation?(7 votes)
- The estimated size of the entire Universe (not just Observable) is 10^60 light years. Inflation was an exponential expansion of space at the very early stages of the Big Bang. To understand how fast the expansion of space was, there are some units that you should know of. A Planck length is a unit of distance that equals roughly 1.616 times 10^-35 meters. This unit is incredibly small. To scale it, if an atom were the size of the observable universe, a Planck length would be the size of a tree. Next is Planck time. Just like light travels one light year in one year, light travels one Planck length every Planck time. Planck time is roughly 10^-43 seconds.

If space were expanding at the speed of light in the early stages of the Big Bang, then space would be expanding at 1 Planck length per Planck time. However, this was not the case as space was expanding at 1 to maybe even 1000's of light years per Planck time in the early stages. Because of this rapid expansion, the entire Universe will most likely always be bigger than the Observable universe. Hope that answers your question.(8 votes)

- If light travels at a finite speed, how do we know that the universe is not now contracting?(4 votes)
- We don't.

The object farthest away from us is a galaxy (UDFy-39546284) which is 13.4 billion light years away. If it started contracting towards us 13.3 billion years ago, we would start to see that in 100 million years. Then I'll post an update ;-).(11 votes)

- If the space between galaxies expands over time, how come people say that in the very far future our galaxy will collide with Andromeda?(4 votes)
- Milky Way and Andromeda are being pulled apart by expansion but they are close enough together such that their gravitational attraction is able to overcome the repulsion from expansion.(8 votes)

- How fast is the Universe expanding?(6 votes)
- You can not truly measure the expansion of the universe itself. Speed is a representation of space traveled over a certain time period. Well when the universe is expanding, it is "creating" space, not expanding through it so you can't really give it a speed. However you can get a general idea by measuring how quickly celestial objects are separating from each other, as Andrew did.(3 votes)

- Isn't this just a theory?(2 votes)
- I never understood that expression 'Just a theory'.

A scientific law is a description of an observed phenomenon. Kepler's Laws of Planetary Motion are a good example. Those laws describe the motions of planets. But they do not explain why they are that way. If all scientists ever did was to formulate scientific laws, then the universe would be very well-described, but still unexplained and very mysterious.

A theory is a scientific explanation of an observed phenomenon. Unlike laws, theories actually explain why things are the way they are. Theories are what science is for. If, then, a theory is a scientific explanation of a natural phenomena, ask yourself this: "What part of that definition excludes a theory from being a fact?" The answer is nothing! There is no reason a theory cannot be an actual fact as well.(7 votes)

- Does that mean that each day (since the universe is expanding in every direction) we could observe further than "the observable universe" that we know today ? In a day matter and light could fill up a huge volume of "void" which the universe is expanding into.

Do you understand me ?(2 votes)- It depends on the rate of expansion at the "edge": slower than light, then yes, each day we see more; same speed as light - see the same each day; faster than light - each day we see less. The easiest way to understand this is to think about a galaxy just beyond the edge of the current observable universe and then imagine what happened to the light it emits right now, does it ever get inside the visible bubble or not?(6 votes)

## Video transcript

Right now, the best estimate
of when the Big Bang occurred-- and once again, I don't like
the term that much because it kind of implies some
type of explosion. But what it really is
is kind of an expansion of space, when space
started to really start to expand from a singularity. But our best estimate
of when this occurred is 13.7 billion years ago. And even though we're used
to dealing with numbers in the billions,
especially when we talk about large amounts
of money and what not, this is an unbelievable
amount of time. It seems like something that is
tractable, but it really isn't. And in future
videos, I'm actually going to talk about
the time scale. So we can really
appreciate how long, or even start to
appreciate, or appreciate that we can't appreciate how
long 13.7 billion years is. And I also want to emphasize
that this is the current best estimate. Even in my lifetime, even in
my lifetime that I actually knew about the Big Bang and
that I would pay attention to what the best estimate
was, this number's been moving around. So I suspect that in
the future, this number might become more accurate
or might move around some. But this is our best guess. Now with that said, I want to
think about what this tells us about the size of the
observable universe. So if all of the expansion
started 13.7 billion years ago, that 13.7 billion
years ago, everything we know in our
three-dimensional universe was in a single
point, the longest that any photon of light could
be traveling that's reaching us right now-- so let's say that
that is my eye right over here. That's my eyelashes, just like
that-- so some photon of light is just to getting to
my eye or maybe it's just getting to the
lens of a telescope. The longest that that
could have been traveling is 13.7 billion years. So it could be traveling
13.7 billion years. So when we looked
at that depiction-- this I think was two
or three videos ago, of the observable universe--
I drew, it was this circle. And when we see light coming
from these remote objects-- that light is getting
to us right here. This is where we are. This is where I guess
in the depiction the remote object was. But the light from
that remote object is just now getting to us. And that light took 13.7
billion years to get to us. Now, what I'm going
to hesitate to do, because we're talking
over such large distances and we're talking on such large
time scales and time scales over which space
itself is expanding-- we're going to see in this video
that you cannot say that this object over here, this is
not necessarily, this is NOT, I'll put it in caps, this is NOT
13.7 billion light years away. If we were talking about
smaller time scales or I guess smaller
distances, you could say approximately that. The expansion of the universe
itself would not make as much of a difference. And let me make it
even more clear. I'm talking about an
object over there. But we could even talk about
that coordinate in space. And actually, I should say
that coordinate in space-time, because we're viewing it at
a certain instant as well. But that coordinate is not
13.7 billion light years away from our current coordinate. And there's a couple of
reasons to think about it. First of all, think
about it, that light was emitted 13.7
billion years ago. When that light was
emitted, we were much closer to that coordinate. This coordinate was
much closer to that. Where we are in the
universe right now was much closer to that
point in the universe. The other thing
to think about is as this-- let me
actually draw it. So let's go 300,000 years
after that initial expansion of that singularity. So we're just 300,000 years
into the universe's history right now. So this is roughly 300,000
years into the universe's life. I guess we could
view it that way. And first of all, at that point
things haven't differentiated in a meaningful
way yet right now. And we'll talk more
about this when we talk about the cosmic
microwave background radiation. But at this point
in the universe, it was kind of this almost
uniform white-hot plasma of hydrogen. And then we're going
to talk about it. It was emitting
microwave radiation. And we'll talk more about
that in a future video. But let's just think about two
points in this early universe. So in this early universe,
let's say you have that point. And let's say you have the
coordinate where we are right now. You have the coordinate
where we are right now. And in fact, I'll just make
that roughly-- I won't make it the center just because I
think it makes it easier to visualize if
it's not the center. And let's say at that very
early stage in the universe, if you were able to just take
some rulers instantaneously and measure that, you
would measure this distance to be 30 million light years. And let's just say
right at that point, this object over here--
I'll do it in magenta-- this object over here
emits a photon, maybe in the microwave
frequency range. And we'll see that that was the
range that it was emitting in. But it emits a photon. And that photon is traveling
at the speed of light. It is light. And so that photon, says,
you know what, I only got 30 million light
years to travel. That's not too bad. I'm going to get there
in 30 million years. And I'm going to do it discrete. The math is more complicated
than what I'm doing here. But I really just
want to give you the idea of what's
going on here. So let's just say,
well, that photon says in about 10 million years,
in roughly 10 million years, I should be right about
at that coordinate. I should be about one
third of the distance. But what happens over the course
of those 10 million years? Well, over the course of
those 10 million years, the universe has expanded some. The universe has expanded
maybe a good deal. So let me draw the
expanded universe. So after 10 million years, the
universe might look like this. Actually it might even
be bigger than that. Let me draw it like this. After 10 million
years, the universe might have expanded a good bit. So this is 10 million
years into the future. Still on a cosmological time
scale, still almost at kind of the infancy of the
universe because we're talking about 13.7
billion years. So let's say 10 million years. 10 million years go by. The universe has expanded. This coordinate, where
we're sitting today at the present time, is
now all the way over here. That coordinate where the
photon was originally emitted is now going to be
sitting right over here. And that photon, it said, OK,
after 10 million light years, I'm going to get over there. And I'm approximating. I'm doing it in a
very discrete way. But I really just want
to give you the idea. So that coordinate,
where the photon roughly gets in 10 million light years,
is about right over here. The whole universe has expanded. All the coordinates have gotten
further away from each other. Now, what just happened here? The universe has expanded. This distance that was 30
million light years now-- and I'm just making
rough numbers here. I don't know the
actual numbers here. Now, it is actually--
this is really just for the sake of giving you the
idea of why-- well, giving you the intuition of
what's going on. This distance now is no
longer 30 million light years. Maybe it's 100 million. So this is now 100 million light
years away from each other. The universe is expanding. These coordinates, the space
is actually spreading out. You could imagine it's
kind of a trampoline or the surface of a balloon. It's getting stretched thin. And so this coordinate
where the light happens to be after 10
million years, it has been traveling for
10 million years, but it's gone a much
larger distance. That distance now might
be on the order of-- maybe it's on the order of
30 million light years. And the math isn't exact here. I haven't done the
math to figure it out. So it's done 30
million light years. And actually I shouldn't even
make it the same proportion. Because the distance it's gone
and the distance it has to go, because of the
stretching, it's not going to be completely linear. At least when I'm thinking about
it in my head, it shouldn't be, I think. But I'm going to make a
hard statement about that. But the distance that it
reversed, maybe this distance right here is now 20
million light years because it got there. Every time it moved some
distance, the space that it had traversed is now stretched. So even though its traveled
for 10 million years, the space that it
traversed is no longer just 10 million light years. It's now stretched to
20 million light years. And the space that it
has left to traverse is no longer only 20
million light years. It might now be 80
million light years. It is now 80
million light years. And so this photon might
be getting frustrated. There's an optimistic
way of viewing it. It is like, wow, I
was able to cover 20 million light years
in only 10 million years. It looks like I'm moving
faster than the speed of light. The reality is it's not
because the space coordinates themselves are spreading out. Those are getting thin. So the photon is just moving
at the speed of light. But the distance
that it actually traversed in 10 million
years is more than 10 million light years. It's 20 million light years. So you can't just
multiply a rate times time on these cosmological scales,
especially when the coordinates themselves, the distance
coordinates are actually moving away from each other. But I think you see,
or maybe you might see, where this is going. OK, this photon says, oh, in
another-- let me write this. This is 80 million light
years-- in another 40 million light years, maybe I'm
going to get over here. But the reality is over that
next 40 million light years-- sorry, in 40 million years,
I might get right over here, because this is 80
million light years. But the reality is
after 40 million years-- so another 40 million years
go by-- now, all of a sudden, the universe has
expanded even more. I won't even draw
the whole bubble. But the place where the
photon was emitted from might be over here. And now our current
position is over here. Where the light got after 10
million years is now over here. And now, where the light
is after 40 million years, maybe it's over here. So now this distance,
the distance between these two
points, when we started, it was 10 million light years. Then it became 20
million light years. Maybe now, it's on the order
of-- I don't know-- maybe it's a billion light years. Maybe now it's a
billion light years. And maybe this distance
over here-- and I'm just making up these numbers. In fact, that's probably be too
big for that point-- maybe this is now 100 million light years. This is now 100
million light years. And now, maybe
this distance right here is-- I don't know--
500 million light years. And maybe now the total
distance between the two points is a billion light years. So as you can see, the photon
might getting frustrated. As it covers more
and more distance, it looks back and says, wow,
in only 50 million years, I've been able to cover
600 million light years. That's pretty good. But it's frustrated
because what it thought was it only had to cover
30 million light years in distance. That keeps stretching
out because space itself is stretching. So the reality, just going
to the original idea, this photon that is
just reaching us, that's been traveling
for-- let's say it's been traveling
for 13.4 billion years. So it's reaching us is just now. So let me just fast
forward 13.4 billion years from this point now to
get to the present day. So if I draw the whole visible
universe right over here, this point right
over here is going to be-- where it was emitted
from is right over there. We are sitting right over there. And actually, let me
make something clear. If I'm drawing the whole
observable universe, the center actually
should be where we are. Because we can observe
an equal distance. If things aren't
really strange, we can observe an equal
distance in any direction. So actually maybe we should
put us at the center. So if this was the entire
observable universe, and the photon was emitted from
here 13.4 billion years ago-- so 300,000 years after
that initial Big Bang, and it's just getting
to us, it is true that the photon
has been traveling for 13.7 billion years. But what's kind of nutty about
it is this object, since we've been expanding away from each
other, this object is now, in our best
estimates, this object is going to be about 46 billion
light years away from us. And I want to make
it very clear. This object is now 46 billion
light years away from us. When we just use light to
observe it, it looks like, just based on light
years, hey, this light has been traveling 13.7
billion years to reach us. That's our only way
of kind with light to kind of think
about the distance. So maybe it's 13.4
or whatever-- I keep changing the decimal-- but
13.4 billion light years away. But the reality is if you had a
ruler today, light year rulers, this space here has
stretched so much that this is now 46
billion light years. And just to give
you a hint of when we talk about the cosmic
microwave background radiation, what will
this point in space look like, this
thing that's actually 46 billion light years
away, but the photon only took 13.7 billion
years to reach us? What will this look like? Well, when we say
look like, it's based on the photons that
are reaching us right now. Those photons left
13.4 billion years ago. So those photons are
the photons being emitted from this
primitive structure, from this white-hot
haze of hydrogen plasma. So what we're going to see
is this white-hot haze. So we're going to see this kind
of white-hot plasma, white hot, undifferentiated
not differentiated into proper stable atoms,
much less stars and galaxies, but white hot. We're going to see
this white-hot plasma. The reality today is
that point in space that's 46 billion
years from now, it's probably differentiated
into stable atoms, and stars, and planets, and galaxies. And frankly, if that
person, that person, if there is a civilization
there right now and if they're sitting right
there, and they're observing photons being emitted
from our coordinate, from our point in
space right now, they're not going to see us. They're going to see us
13.4 billion years ago. They're going to see the
super primitive state of our region of
space when it really was just a white-hot plasma. And we're going to talk more
about this in the next video. But think about it. Any photon that's coming
from that period in time, so from any direction,
that's been traveling for 13.4 billion years
from any direction, it's going to come from
that primitive state or it would have been
emitted when the universe was in that primitive
state, when it was just that white-hot plasma,
this undifferentiated mass. And hopefully,
that will give you a sense of where the
cosmic microwave background radiation comes from.