When we look at the Andromeda galaxy, we are seeing photons that are 2 million years old (that is how long it took the photons, traveling at the speed of light, to reach us since the distance between our galaxy and the Andromeda galaxy is 2 million light-years); since the scaling factor \(a(t)\) was smaller then, the photons were “hotter” and had more energy. If we look back at the light coming from galaxies 10 billion light-years away, we are seeing photons that are 10 billion years old and that are even hotter and more energetic. If we look back about 13.4 billion light-years, we are seeing photons that are 13.4 billion years old and that were buzzing along through space when the Universe was just 300,000 years old; we are seeing photons right at the time when the Universe transitioned from being opaque to being transparent. Since these photons are 3000K, when we look up at the night sky it should be blindingly bright. Indeed, if we ignore the expansion of space due to \(ρ\), this would be the case. However, because these photons are traveling through a space that is expanding, they become redshifted to the microwave range.
Boltzmann’s relationship between the temperature \(T\) and energy \(E\) of radiation is given by \(E=kT⇒E∝T\). We showed earlier that the total energy \(E\) of a collection of photons is \(E=NE_γ=\frac{Nhc}{λ}=\frac{Nhc}{a∆r}=\frac{k}{a}\) and \(E∝\frac{1}{a}\). Using Boltzmann’s relationship, we also see that \(T∝\frac{1}{a}\). As the Universe expands and \(a\) grows with time \(t\) since the Big Bang, the wavelengths of photons become stretched, there energy decreases, and there temperature decreases. But if we imagine running the clocks backwards and watching the Universe as time runs backwards, the Universe is contracting and \(a\) is decreasing as we run \(t\) backwards, the wavelengths of photons decreases, their energy increases, and their temperature increases. Today we can measure that the temperature of radiation is \(~3K\). During some time period of the very young Universe, the temperature was so hot that light and photons couldn’t pass through it (they became scattered). The hydrogen and helium atoms were ionized; this means that the electrons and nuclei (or protons and neutrons were split too, not sure) were split apart and buzzing passed one another at tremendous speeds. Such a state of matter is called a plasma. During this time period, all of the matter in the Universe looked like a giant, glowing ball of plasma (in other words, it looked like a giant Sun). When the temperature of a collection of particles become \(~3,000K\), it becomes opaque (which means that light and photons cannot pass through it). We will call this temperature \(T_{\text{last scattering}}\).
This video was produced by David Butler. For the transcript of this video, visit: http://howfarawayisit.com/documents/.
The temperature \(T_{\text{last scattering}}≈3,000K\) is the “turning point” so to speak; at the time (which we will call \(t_{\text{last scattering}}\)) when atoms and radiation were at this temperature, this is the instant of time just before the Universe became transparent. For any \(t>t_{\text{last scattering}}\), the temperature \(T\) of radiation and atoms was \(T<T_{\text{last scattering}}≈3,000K\). We are interested in solving for the time \(t_{\text{last scattering}}\): that moment of time when the Universe transitioned from being opaque to transparent. To do this, we take advantage of the fact that the Universe was still matter dominated at \(t_{\text{last scattering}}\); assuming that throughout the entire time interval from \(t_{\text{last scattering}}\) all the way until our present time \(t\) the dominant form of energy density in the Universe was \(ρ_M\), the scaling factor is \(a(t)=Ct^{2/3}\) for this entire interval of time. (It is an enormous idealizations and simplification to assume the only contribution to the energy density \(ρ\) throughout the Universe is \(ρ_M\) and that \(ρ≈ρ_M\); later on, we shall come up with a model from which we can derive a \(ρ\) that includes both radiation energy density \(ρ_r\) and mass-energy density \(ρ_M\).) At our present time (which we will call \(t_{today}≈10^{10}\text{ years}=10\text{ billion years}\)), the scaling factor is given by \(a(t_{today})=Ct^{2/3}_{today}\); at the time \(t_{\text{last scattering}}\), the scaling factor was \(a(t_{\text{last scattering}}=Ct^{2/3}_{\text{last scattering}}\). By taking the ratio of the two scaling factors and doing some algebra, we can find \(t_{\text{last scattering}}\):
\begin{equation}
\frac{a(t_{today})}{a(t_{\text{last scattering}})}=\frac{Ct^{2/3}_{today}}{Ct^{2/3}_{\text{last scattering}}}=1000.
\end{equation} Since \(T≈1/a\), if we are considering a time when \(T\) was 1,000 times greater than it is today, the scaling factor was 1,000 times smaller back at that hotter \(T\)
\begin{equation}
\frac{t_{today}}{t_{\text{last scattering}}}=1000^{3/2}⇒t_{\text{last scattering}}≈\frac{10^{10}years}{1000^{3/2}}≈300,000\text{ years}.
\end{equation} For the first 300,000 years of the life of the Universe, the Universe was opaque. Just after a time of roughly 300,000 years since the Big Bang, the temperature \(T\) of the Universe became cool enough for atomic nuclei to bond with electrons and form electrically neutral hydrogen and helium atoms allowing the Universe to become transparent.
The graph of \(ρ(a)\) vs. \(a\) shows that as \(a\) increases (since \(k=0\), \(a\) will keep growing with \(t\) forever) with time and the Universe expands, during the time interval \(t_0≤t≤t_1\) when the scaling factor increased from \(a(t_0)\) to \(a(t_1)\), the dominant form of energy density was \(ρ_r\) since \(ρ_r>ρ_M\) and \(ρ_r>ρ_0\) over this time period. During the time interval \(t_0≤t≤t_1\) when the scaling factor increased from \(a(t_0)\) to \(a(t_1)\), the dominant form of energy density was \(ρ_M\). For times \(t>t_1\) when \(a(t)>a(t_1)\) (which is the time period we live in today), the dominant form of energy density is \(ρ_0\). From the graph for \(a(t)\) vs. \(t\), we see that after a long enough time (when \(t\) becomes very large), \(a(t)=Ce^{\sqrt{\frac{8}{3}πGρ_0}t}\) will dominate and the Universe will keep expanding at an exponential (and thus accelerating) rate.
How \(a(t)\) changes with \(t\) in a Universe dominated only by dark/vacuum energy only becomes significant over vast distance scales. Substituting \(a(t)\) into \(D=a(t)∆r\) and then substituting \(D\) into \(V=H_0D\), we can calculate that the recessional velocities between our galaxy and very nearby galaxies (small ) is very small (it is only hundreds of kilometers per second which, compared to the speed of light, is extremely small). For very enormous values of \(D\), the recessional velocities of distant galaxies is close to the speed of light. For example, distant quasars that are 10 billion light-years away move away from as at a speed of about half the speed of light. If the distance \(D\) a galaxy is away from us is big enough, it will be moving away from us at a speed greater than the speed of light. As the march of time progresses, Hubble's Parameter \(H(t)\) will continue to grow with time since the scaling factor will continue to increase with time due to dark energy; after an unimaginably long period of time (five billion years), \(H(t)\) will have increased so dramatically that even if one substituted distances for the nearest galaxies only a few million light-years away into Equation (3), they would discover that those galaxies are receding from us faster than light speed. Lawrence Krauss once said that, for this reason, we are living in a very special time in the history of the universe, a time when we can still observe the CMBR and arrive at correct conclusions about the nature of the universe. Five billion years from now, all of the galaxies beyond the Milky Way, and also the CMBR, will be receding from us faster than the speed of light and will become undetectable.
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References
1. Singh, Simon. Big Bang: The Origin of the Universe. New York: Harper Perennial, 2004. Print.
2. Wikipedia contributors. "Cosmic microwave background." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 12 May. 2017. Web. 18 May. 2017.
3. Leonard Susskind. "Vacuum energy". theoreticalminimum.com.