Happy New Year, my dear readers! I sincerely hope all your wishes will come true during this wonderful time we're all living in! In the spirit of the New Year, I thought that I might as well calculate how much the Universe has expanded last year. It was either this or a group theory problem with groups of cardinal 2019 taken from Gazeta Matematica. I took this simplified version of the Friedmann equation from Brilliant, which is really great if you want to try hard problems in both Maths and Physics. Without further ado, let's get straight into the problem!
The simplified version of Friedmann's equation states that \((\frac{a'}{a})^2=H_{0}^2(\frac{1-𝛀}{a^3}+𝛀)\), where \(𝛀\) is the proportion of dark energy in the Universe, and \(H_0\) is the Hubble Constant. \(a\) is the size factor, taken such that in 2018 it is 1 (the size factor is the distance between us and a very distant galaxy at a time t over the distance between the same very distant galaxy and us now). In order to find how many times the Universe expanded (at least the known Universe), we must calculate the exact value of the size factor.
First, we are going to make some observations. The size factor is strictly increasing, so \(a'\) can't be 0. Similarly, the size factor is also bigger than 0, because it was 0 only in the initial moment of the Universe.
Back to the problem, let us multiply both sides by \(a^3\), to get rid of the fraction (which makes my Latex job easier). Thus, we get that \((a')^2 a=H_{0}^2 𝛀 a^3+(1-𝛀)H_{0}^2\), and, by moving everything with \(a\) on the left hand side and differentiating with respect to \(a\), we get that
$$2a' a'' a + (a')^3 -3H_{0}^2 𝛀 a' a =0$$
$$⇒ 2 a'' a + (a')^2 - 3H_{0}^2𝛀 a^2=0$$
$$⇒ \frac {2a''}{a}+(\frac{a'}{a})^2=3H_{0}^2𝛀$$
Let us now observe that \((\frac{a'}{a})'=\frac{a''}{a}-(\frac{a'}{a})^2)\), so \(\frac {a''}{a}=(\frac{a'}{a})'+(\frac{a'}{a})^2)\). For the simplification of calculations, let \(y=\frac{a'}{a}\).
$$⇒2y'+3y^2=3H_{0}^2𝛀 ⇒ y'=-1.5y^2+\frac{3H_{0}^2𝛀}{2}$$
As I found out on Wikipedia, this is a simplified version of the Riccati equation (https://en.wikipedia.org/wiki/Riccati_equation), and, as it turns out, its solutions are functions of the form
$$y=-\frac{u'}{3u}, u=A*e^{\frac{3𝛀^{\frac{1}{2}}H_{0}t}{2}}+B*e^{\frac{-3𝛀^{\frac{1}{2}}H_{0}t}{2}}$$
By replacing \(y\) with \(\frac{a'}{a}\) and integrating in t, we get that \(∫\frac{a'}{a}dt=∫-\frac{u'}{3u}dt\).
$$⇒ln(a)+k=-\frac{ln(u)}{3}$$
By taking \(t=t_0\), the current year, we have that $$k=\frac{2 ln(A*e^{\frac{3𝛀^{ \frac{1}{2}}H_{0}t_{0}}{2}}+B*e^{\frac{-3𝛀^{\frac{1}{2}}H_{0}t_{0}}{2}})}{3}$$
By letting t approach 0, we get that \(A=-B\) (this is left as an exercise for the reader)
Thus, after calculations, we reach the final result: $$a(t)=(\frac{e^{\frac{3𝛀^{\frac{1}{2}}H_{0}t}{2}}-e^{-\frac{3𝛀^{\frac{1}{2}}H_{0}t}{2}}}{e^{\frac{3𝛀^{\frac{1}{2}}H_{0}t_{0}}{2}}-e^{-\frac{3𝛀^{\frac{1}{2}}H_{0}t_{0}}{2}}})^{\frac{2}{3}}$$
I promised to find out how much the Universe has expanded during the last year, so here it is. We shall consider that 𝛀=0.68, while \(H_0=\frac{1}{14*10^9}\) years. The Universe's age is about \(13.772*10^9\) years old, and we shall consider that number as \(t_0\), and \(t=t_0+1\). After torturing Wolfram Alpha to tell me the result, here it is; the Universe has expanded 1.0000000000702296877033612965438583887033870149765170 times. I wonder how much that means... how about we find out?
The current size of the Universe is 93 billion light years in diameter. If we consider this to be the size of the Universe in 2018, it means that now, in 2019, the Universe has diameter about 93.000000006531360956412600578578830149414992392816081 billion light years, so it grew by about 000000006531360956412600578578830149414992392816081 billion light years. That's about 6 light years. That might not seem like a lot, but trust me, it is. For example, the distance between the Sun and Alpha Centauri, the closest star, is 4.4 light years. Also, considering that the size factor is a more-or-less the distance between points on a segment, it means that the speed at which the Universe increases is bigger than the speed of light (I won't go into detail about this, because I have no idea how to discuss it). That's a bit too much to comprehend on New Year's.
Happy New Year, folks!
Cris.
No comments:
Post a Comment