## Thursday, March 22, 2012

### age of the Universe

imagine you're stuck in a discussion with a creationist: what evidence is there that the Universe is more than ten billion years old? of course there are multiple answers, try to think of as many as you can.

bonus question (a tough one, this time): what's the derivative $\frac{\mathrm{d}}{\mathrm{d}x}x^{x^x}$?

$\frac{\mathrm{d}}{\mathrm{d}x}\,x^x=x^x[1+\ln x]$
and
$\frac{\mathrm{d}}{\mathrm{d}x}\,x^{x^x}=x^{x^x}\,\left[x^{x-1}+\ln x \, \frac{\mathrm{d}}{\mathrm{d}x} \, x^x\right]$
so I get:
$\frac{\mathrm{d}}{\mathrm{d}x}\,x^{x^x}=x^{x^x+x-1}\,\left[1+x \, \ln x \, \left(1+\ln x \right)\right]$

2. yea, i phrased it differently,
$$\frac{\mathrm{d}}{\mathrm{d}x} x^{x^x}= x^{x^x}x^x\left(\frac{1}{x}+\ln^2(x)+\ln(x)\right)$$
but it seems fine!

3. a time-scale for the age of the Universe would be given by the inverse Hubble constant. then, naturally, the Universe needs to be older than the oldest objects inside it. we know quite old objects such as low-mass stars, and we know that galaxies and clusters take a long time to form. there are even very old rocks on can find on earth, and we know their age by radiometric dating.

4. So, I came up with 3 estimates (which are also mentioned above):

1) The inverse Hubble constant. This loosely corresponds to requiring that the matter of two galaxies we pick randomly was at the same spot during the Big Bang. We could take into account temporal variation of $\frac{1}{H}$, by integrating it.

2) By finding some red giant for which we have reason to believe that its initial mass was below solar mass. The time spent on the main sequence is:

$\tau_{ms}\approx 10^{10}\left(\frac{M}{M_{solar}}\right)^{-2.5}$

3) By measuring the age of Globular clusters. Either by the main sequence turning point, in which case we have, I think, a more refined version of 2), or by examining the luminosity function (LF). The LF tells us howe many stars have a given luminosity. As stars with different initial mass (and luminosity) evolve with different speeds, this has to change over time.

Of course there is no convincing a creationist! In 1) he would question our cosmological scenario, in 2) and 3) our stellar evolution paradigm...

5. There is another independent argument to estimate the age of the universe, which is quite neat and based on observations of white dwarfs in globular clusters.

Basically, white dwarfs are already pretty old objects because they're the final product of the (looong) life of low-mass (<8 M_sun) stars. After they form, they start cooling. So the coolest a white dwarf, the oldest.

Furthermore, the first stars to produce white dwarfs in cosmic history are the most massive among low-mass stars (remember that the more massive a star, the shorter its life). So these first white dwarfs are expected to be the most massive among white dwarfs, with masses of the order of M_Chandrasekhar. Stars with even lower masses would start dying later, and giving rise to less massive white dwarfs.

Hence, the most massive *and* coolest/lowest in luminosity among white dwarfs are early fossils of the demise of low-mass stars.

By pushing Hubble into the low-luminosity regime, some really old white dwarfs have been detected, with ages > 10 billion years.