Whatever helps you sleep at night....

It's two in the morning. Two a.m. I can't sleep. Whatever I try my mind won't switch off, this is paradoxically, despite the fact I am exhausted.  And I have work in the morning, and I know my two beautiful children will not sleep past six, they never do. Never. Looks like a day propped up by Lava Java again (other coffee brands are available). The things that keep me awake at night now tend to be very different than in the past. My childhood was cursed by night terrors of ghosts and vampires and the like. I'm over that stage, to say the least. Bumps in the night now are more likely to bring thoughts of rodents, or my three-year-old falling out of bed. There are no bumps tonight. Nor are there any particular work stressors, money worries or the other things that tend to keep peers of a similar age up at night.

I know from experience that the only way I'll get any sleep is by occupying myself somehow. I search the internet for potential blog topics. Nothing particularly jumps out at me. Frustrating. Paranormal stories are thin on the ground, plus what there is out there has been covered by other bloggers. There are topics I want to cover, but they require far more research than my tired brain can handle.

Maybe some TV. I haven't watched the last season finale of Doctor Who yet.

So I start to watch. It's fairly enjoyable and has a happily upbeat and nostalgic, if somewhat low-key, conclusion, but there's something distracting about it. At one point the Doctor meets a young girl, Ashilidir, played by Maise Williams, who became immortal earlier in the series, at the end of the universe.As the episode concludes Ashilidir leaves in another Tardis with departing companion, Clara Oswald. But what if that hadn't been available as an escape route?

The Doctor, being compassionate, wouldn't leave Ashilidir to be eradicated with what remains of the universe. His only option is to return her to the creation of the universe to continue her life. As he is making a fundamental change to the initial conditions of the universe, the Doctor labels this universe 2. As he leaves in his Tardis, a worrying thought occurs to the Doctor. He travels to the end of universe 2 to confirm his suspicions: and finds two Ashilidirs waiting for him. He concludes that one is the initial Ashilidir, whom he left at the beginning of the universe and the other is the Ashilidir who he will make immortal during the course of that universe's history.

This leads him to his first two assumptions:

1.The Ashilider I place at the beginning of each consecutive universe is unable, or unwilling, to change the course of that universe's history.

2.Also, Ashilidirs are truly immortal  any created  or deposited during that universe will survive until it's endpoint. 

He returns these two Ashilidirs to the beginning of the universe, naming it Universe 3, and decides to travel back to the end of the Universe to confirm his first assumptions. If it is incorrect he will find less than three Ashilidirs. The Doctor is shocked to find not three Ashilidirs at the end of Universe 3 but four.

This leads him to a third assumption:

3. Each subsequent universe begins not with the initial conditions of  Universe 1, but with the initial conditions of the universe which preceded it. 
Therefore, when he dropped the two Ashilidirs from Universe 2 into Universe 3, there was already an Ashilidir from Universe 1 present. He chides himself for the error, quickly drops the four Ashilidirs in Universe 4, predicting that if assumption three is correct he will find eight Ashilidirs at the end of Universe 4 and indeed he does.

At this point the Doctor he can formulate some simple relations between the number of the universe (n) and the number of initial Ashildirs (Ai) and number of final Ashilidirs (Af). The relationship between the initial and final Ashilidirs is very simple, and should be immediately clear;

Likewise, the formula relating the Universe number (n) and the final Ashilidirs is pretty simple. 

From these formulas, the Doctor decides to find out in which universe the total mass of the Ashilidirs at the end point will be greater than the mass of the Earth ( Me = 5.98 x 10^24 kg). He estimates the mass of one Ashilidir is 51 kg, and uses this to find the number of  final Ashilidirs equal to Earth's mass to 3 significant figures:

So to find n he rearranges the formula relating Af  to n by taking the logarithm of both sides of the equation to base 2.

So rounding this to a whole integer, as there are no partial universes, the Doctor sees that by Universe 78 the mass of the Ashilidir's will be greater than that of Earth itself. At this point, he begins to panic. At what point will there be more Ashilidirs in the universe than there were atoms in Universe 1? Taking the estimates of 21st century Earth cosmologists of atoms in the visible universe, the Doctor uses the value of 4 x 10^79 atoms. Using his above rearrangement and this value for final Ashilidirs:

So by Universe 266, matter will be dominated by Ashilidirs, he reasons, deciding perhaps to come back to his endeavor of shifting Ashilidirs back to the origin of the universe at another time... that's the beauty of time travel I suppose.

As for me, figuring out that dark matter, the dominant mass in the universe, is likely composed of free-floating, immortal Maisie Williams is enough to persuade me it's time for bed.