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[sci.astro] Cosmology (Astronomy Frequently Asked Questions) (9/9)
Section - I.08. If the Universe is only 10 billion years old, how can we see objects that are now 30 billion light years away? Why

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	isn't the most distant object we can see only 5 billion light
	years away?

When talking about the distance of a moving object, we mean the
spatial separation NOW, with the positions of both objects specified
at the current time. In an expanding Universe this distance NOW is
larger than the speed of light times the light travel time due to the
increase of separations between objects as the Universe expands. This
is not due to any change in the units of space and time, but just
caused by things being farther apart now than they used to be.

What is the distance NOW to the most distant thing we can see?  Let's
take the age of the Universe to be 10 billion years. In that time
light travels 10 billion light years, and some people stop here.  But
the distance has grown since the light traveled.  Half way along the
light's journey was 5 billion years ago.  For the critical density
case (i.e., flat Universe), the scale factor for the Universe is
proportional to the 2/3 power of the time since the Big Bang, so the
Universe has grown by a factor of 22/3 = 1.59 since the midpoint of
the light's trip.  But the size of the Universe changes continuously,
so we should divide the light's trip into short intervals.  First take
two intervals: 5 billion years at an average time 7.5 billion years
after the Big Bang, which gives 5 billion light years that have grown
by a factor of 1/(0.75)2/3 = 1.21, plus another 5 billion light years
at an average time 2.5 billion years after the Big Bang, which has
grown by a factor of 42/3 = 2.52.  Thus with 1 interval we get 1.59*10
= 15.9 billion light years, while with two intervals we get
5*(1.21+2.52) = 18.7 billion light years.  With 8192 intervals we get
29.3 billion light years.  In the limit of very many time intervals we
get 30 billion light years.

If the Universe does not have the critical density then the distance
is different, and for the low densities that are more likely the
distance NOW to the most distant object we can see is bigger than 3
times the speed of light times the age of the Universe.

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