제1부(요약). 공간과 시간(Space and Time)
제2부. 우리 우주의 역학과 기하학(Geometry and Dynamics of our universe)
V2.1 로버트슨-워커 측량(The Robertson-Walker Metric)
V2.2 원주율(π, Pi)
V2.3 휘어진 공간(Curved Space)
V2.4 프리드먼 방정식(The Friedman Equation)
V2.5 임계 밀도(Critical density)
V2.6 밀도 변화(Density Evolution)
V2.7 우주의 진화(Evolution of the Universe)
V2.8 결론(Conclusion)
RN2. 우리 우주의 역학과 기하학(Geometry and Dynamics of our universe)
1. 로버트슨-워커 측량(The Robertson-Walker Metric)
구형 극좌표(Spherical Polar Coordinates)의 세 축은 (r, θ, φ)다.
직교공간(normal space)의 변위(displacement, δs)를 구형 극좌표계(spherical polar coordinates)로 계산하는 방법이다.
로버트슨-워커 측량은 기존의 구형 극좌표계의 변위 측량에 우주가 균일(uniform)하며 등방형(isotropic)이라고 전제를 반영하였다.
로버트슨-워커 측량이 직교공간의 측량과 다른 점은 두가지 인데, 우주의 팽창요소인 배율인자(scale factor,척도인자) a(t)와 기하학적인 변형을 주는 곡률 k를 고려해 넣었다.
k에 따라 원주율 π가 어떻게 변화하는지 보자.(우주는 균일하고 등방성을 갖는다고 전제한다. 등방성을 보여주는 기하학적 도형인 원의 기본 특성인 원주율을 RW측량법으로 계산해볼 것이다. 팽창과 곡율은 우주의 운명에 영향을 주는 요인이 될 것이다) 원주율 π는 원의 둘레를 직경으로 나눈 값이다. 우주에서 원의 반경과 둘레를 계산해 보자. (직경과 둘레는 모두 길이의 차원이다. 공간에서 길이를 측정하는 방법이 바로 측량법이다.) 두가지 길이(반경과 둘레)를 각각 δs로 두고 로버트슨-워커 측량법으로 계산해 보자. 각 길이는 미소 변위 δs를 누적하여 구한다. 미소길이 δs의 누적은 적분법으로 계산한다.
먼저 원둘레의 계산이다. 반지름이 r0로 고정된 원의 미소 둘레를 RW측량법으로 기술하였다. 우주의 등방성을 고려하여 φ를 고정한 원을 계산한다(δr=0, δφ=0).
반경의 계산이다. 역시 등방성을 감안하여 두 각도는 고정되고 반지름의 미소 길이 δr를 RW 측량으로 기술하였다. 구간 [0,r0]에서 적분 하여 원의 반경을 구한다. 반경의 길이는 휜 공간을 감안하면 직선이 아닐 수 있다. 공간이 휜 정도는 곡률 k에 따라 다르다. 공간이 휜 정도를 k=0, k<0, k>0로 구분하여 반지름을 계산하면 다음과 같다. 계산의 편이를 위해 k=-1, k=1 로 하였다.
공간이 휜 우주라면 원주율 π은 상수가 아니다.
곡률이 k>0 인 우주에서 원주율,
곡률이 k<0 인 우주에서 원주율,
π0 는 k=0인 평편한 우주의 원주율(i.e. 3.141592...)이다.
How do these solutions behave? Regardless of the value of k, both give the normal value of when the radius of a circle is much smaller than the radius of curvature of the universe. But as the circles become larger, if k>0, the value of π drops, while if k<0, the value of π rises. So π is not a constant, on large enough scales, unless k=0.
These different geometries can perhaps be best understood through a two-dimensional analogy. A universe with k>0 is analogous to a two-dimensional life-form on the surface of a sphere. If you go far enough in any direction, you will end up back where you started. The universe is finite, but has no edge. π is small on large scales, parallel lines will ultimately meet, and the interior angles of a triangle add up to more than 180 degrees.
A universe with k<0, on the other hand, is analogous to a saddle-shape. The universe is infinite, π is large on large scales, parallel lines diverge, and the interior angles of a triangle add up to less than 180 degrees.
2. 프리드먼 방정식(Friedmann Equation)
배율인자(scale factor) a(t)는 어떤 의미일까? 이 인자가 어떤 양상을 보일지 에너지 보존법칙(conversation of energy)를 통해 알아보기로 한다. 운동 에너지(kinetic energy)와 잠재 에너지(potential energy) 사이의 변화로 부터 프리드먼 방정식을 유도해 낼 수 있다.
where ρ is the average density of the universe at that moment, and is the rate of change of a. The current rate of the expansion (Hubble's constant) is given by
3. 임계밀도(The critical density)
Using the Friedmann equation, and substituting in H0, we see that to have a geometrically flat universe, we need the density today to be
This is known as the critical density. If the density is larger than this, k>0 and we live in a "spherical", finite universe, while if the density is smaller than this critical value, k<0 and we live in a saddle-shaped open universe. Because this critical density is so important, the average density of various components of our universe is often quoted as the ratio Ω to this critical density:
4. 밀도 진화(density evolution)
To solve the Friedmann equation, we need to know how the density of the universe changes as the universe expands or contracts. For matter, the density is proportional to , because the amount of matter is constant but the volume increases. For radiation (or matter moving very close to the speed of light), the energy of each wave/particle drops due to the expansion of space (and hence increase in wavelength), which means that density is instead proportional to .
This means that the density of radiation drops faster than that of matter. Our universe is currently matter dominated (i.e. the density of matter is much greater than that of radiation), but as we go further back in time, radiation becomes more and more important, and indeed dominated the very early universe.
5. 프리드먼 방정식 풀기(solving the friedmann equation)
The Friedmann equation is a differential equation (i.e. one including derivatives). In general it cannot be analytically solved, but has to be numerically approximated. It can however, be solved exactly for the case where k=0. This solution can be found in the appendix below. For a purely matter dominated universe,
where is the time today. For a radiation-dominated universe,
In general, however, you will need to solve the equation numerically. You can rearrange the Friedmann equation to give the rate of change of a at any given moment:
Starting from today (when a=1 by definition), you can then step forwards or backwards in tiny time steps ( ) and estimate the value of a short time in the future or past, using the equation
(which just comes from the definition of differentiation - this is known as Euler's method). A python program that does this is available here.
We find that if k=0, the universe always expands but the rate of expansion always decreases. If k>0, the expansion of the universe will eventually stop, and the universe will start shrinking, until everything comes together in an almighty "Big Crunch" or "Gnab Gib" (the latter being "Big Bang" spelled backwards). If k<0, the universe will expand forever, with its expansion rate asymptoting to a positive constant.
6. 결론(Conclusions)
So - we have three classes of cosmological model.
k>0, , a finite, spherical universe that will eventually collapse. In this universe, is small on large scales and parallel lines will eventually meet.
k=0, , an infinite flat universe with normal geometry, that will expand forever but at a perpetually decreasing rate.
k<0, , an infinite saddle-shaped universe that will expand forever. In this universe, is larger on large scales and parallel lines diverge.
Which (if any) of these models is true? That is a question to be answered experimentally.
7. 부록: 프리드먼 방정식의 해
appendix: exact solutions to the friedmann equation
The Friedmann eqution can be solved exactly if k=0 and we assume a purely radiation or matter dominated universe. Consider the case of a matter dominated universe. The density is thus
where is the density now and we've chosen to define a(t) = 1 right now ( ). If k=0, the Friedmann equation is thus:
The easiest way to solve this is to guess that the answer might be in the form of a power-law: i.e. that , where q is some unknown index. Substitute this into the above equation, and you will find that the left-hand side depends on while the right-hand side depends on . For a valid solution, these two must behave the same way, so , and hence .
Using exactly the same trick, you can find that for a radiation-dominated universe, .
