Hubble time and age of the universe I'm having trouble with the following derivation of the 'age of the universe': http://imgur.com/gRvLWX8
The parts I'm struggling to conceptualize is what a 'universe expanding' means, and also why the derivation assumes the galaxy is receding at slightly less than the speed of light. Moreover, if this is the case, that is the galaxy is receding at slightly less than the speed of light, then what of the period of time where the light from this galaxy has not reached us yet? How would an age of the universe calculation be done in this period of time, since no light from any galaxy has reached us?
Sorry if these questions are not well formulated, I'm very new to physics and have almost no experience.
Any and all help will be much appreciated!  
 A: Expanding Universe: The idea of the Universe expanding is often described using the analogy of an inflating balloon. It is tempting as a new physics student to imagine the expansion as galaxies whizzing away from each other through some 'medium', however in actuality it is spacetime itself that is expanding. If we glue some pieces of confetti (representing galaxies) to the balloon (representing the Universe) and then blow it up, the confetti move away from each other because the space between them on the balloon is expanding, not because they are moving around on the surface of the balloon.
The pieces of confetti themselves do not expand because concentrated distributions of matter such as galaxies gravitationally 'pin' spacetime so as to overcome the rate of expansion within them.
Measuring expansion: In order to measure the expansion of the Universe, we need some kind of information from the Universe. This is not a problem for us because the Universe is around 13.8 billion years old, so we can receive light from galaxies up to around 13.8 billion light-years away. This gives us access to a lot of galaxies (our closest galaxy Andromeda is only 2.5 million light years away). If humans had somehow evolved much earlier in the Universe's history, then it is true that we would not have had a big enough data set to get an accurate value of the Universe's expansion rate or age. But luckily we live in an era where the observable Universe is much larger, and additionally our techniques for measuring Cepheid variables and Type IA supernovae are becoming more and more precise. Therefore we can measure Hubble's constant to within around 5%, which gives a value of
\begin{equation}
H_0 \approx 71 \; \text{km/s/Mpc} \pm 5\%
\end{equation}
today. This is then used as in your textbook to approximately calculate the age of the Universe.
Recession at faster than the speed of light: Any galaxy that is moving away from us faster than the speed of light due to the expansion of the Universe will never be visible to us unless the expansion of the Universe slows, simply because the photons it sends out will never reach us. For a given rate of expansion, this defines a Hubble volume around the Earth from which we can receive information: beyond this we cannot observe the Universe. Therefore any galaxies receding from us faster than the speed of light are irrelevant: we cannot even know whether they exist or not.
I hope this answers your questions! Good luck with the IB.
A: Before Einstein, we thought squared infinitesimal lengths $ds^2=dx^2+dy^2+dz^2$ in space wouldn't change if you switched to a different coordinate system, e.g. by rotating coordinate axes or translating the origin. Relative motion actually does change spatial lengths, but they preserve another quantity. In special relativity, it's $ds^2=-c^2dt^2+dx^2+dy^2+dz^2$, or in spherical polar coordinates $ds^2=-c^2dt^2+dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2$. It takes the even more complicated theory of general relativity to understand what spatial expansion really means. Over scales large enough to make gravity negligible, a better expression would be $$ds^2=-c^2dt^2+a^2(t)(\frac{1}{1-kr^2}dr^2+r^2 d\theta^2+r^2\sin^2\theta d\phi^2),$$where $k$ is a constant (which is probably $0$ or small) and $a$ is a function of time called the scale factor. The universe's large-scale structure scales its lengths over time in proportion to the growing scale factor.
The Hubble parameter $H=\frac{\dot{a}}{a}$ is also in general time-dependent, but let's say its current value is $H_0$, and adopt a scale for which today $a=1$. If the universe's age is $T$, $a$ has gradually increased from $0$ to $1$. Thus $T=\int_0^T dt=\int_0^1 \frac{da}{\dot{a}}=\int_0^1 \frac{da}{aH}$. Since galaxies' relative recession speeds are proportional to $\dot{a}$, what your imgur link does is to approximate this quantity as constant over the universe's history. Then our final formula for $T$ is the area of a width-$1$, height-$H_0^{-1}$ rectangle. To get a more accurate value you need to take the time-dependence of $a$ into account, but you'll still get an answer that approximates $H_0^{-1}$.
(For what it's worth, if we'd instead said the current value of $a$ is $A$ we'd have $T=\int_0^A\frac{da}{aH}\approx A\times\frac{1}{AH_0}=H_0^{-1}$.)
