Cosmology Lecture 05

Finishing the Robertson-Walker Metric

We’ve been following Weinberg’s derivation to show there are discrete metrics. We’ll start with:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ds^{2}=c^{2}dt^{2}-a^{2}(t)\left[{dr^{2} \over 1-kr^{2}}+r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\right]\,\!}

Recall that $k$ is the curvature constant and $r$ is in comoving coordinates. If we define Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle U\equiv \sinh \theta } , then:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}dU&=\cosh \theta \cdot d\theta \\d\theta &={dU \over {\sqrt {1+U^{2}}}}\\\end{aligned}}\,\!}

The beauty of this metric is that we derived it only using symmetry (no dynamics). There is an alternate way of writing the metric above:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ds^{2}=c^{2}dt^{2}-a^{2}\left[d\chi ^{2}+S^{2}(\chi )(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}\right]\,\!}

where:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S(\chi )\equiv {\begin{cases}\chi &\,(for\ k=0,\ \chi =r)\\k^{-{\frac {1}{2}}}\sin(k^{\frac {1}{2}}\chi )&\,(for\ k>0)\\(-k)^{-{\frac {1}{2}}}\sinh({\sqrt {-k}}\chi )&\,(for\ k<0)\\\end{cases}}\,\!}

Comoving Radial Distance vs. Redshift: the Hubble Diagram

This is the fundamental diagram behind using a standard candle (supernova) to infer the curvature of the universe. What we want here is an algebraic expression relating $r$ and $z$. We’ll look at light propagation ($ds^{2}=0$), and take a radial path (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d\theta =d\phi =0} ) to know from the Robertson-Walker metric that:

$c^{2}dt^{2}=a^{2}{dr^{2} \over 1-kr^{2}}\,\!$

Separating out our $r$ dependencies and our $t$ dependencies and integrating, we get:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{0}^{r_{1}}{dr \over {\sqrt {1-kr^{2}}}}=\int _{t_{1}}^{t_{0}}{c\cdot dt \over a(t)}\,\!}

• For the flat, $k=0$, matter-dominated model ($w=0$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega _{0,\Lambda }=0} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega _{0,M}=1} ), we’ll start with the Friedmann Equation:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle H={{\dot {a}} \over a}=H_{0}{\sqrt {{\Omega _{0} \over a^{3+3w}}+{1-\Omega _{0} \over a^{2}}}}\,\!}

Recognizing that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a={1 \over 1+z}} , we have:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle H_{0}dt=-{dz \over (1+z){\sqrt {\Omega _{0}(1+z)^{3+3w}+(1-\Omega _{0})(1+z)^{2}}}}\,\!}

These integrals evaluate to (in comoving $r$):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r_{1}=\int _{t_{1}}^{t_{0}}{c\cdot dt \over a(t)}\,\!} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {r_{1}={2c \over H_{0}}\left[1-(1+z)^{-{\frac {1}{2}}}\right]}\,\!}

Here, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {2c \over H_{0}}} is called the Hubble distance and is the definition of how far away we can possibly see–how far light could have traveled since the beginning of time. Notice that as $z\to \infty$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r={2c \over H_{0}}\approx 10000h^{-1}Mpc} .

• For the open, $k<0$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega _{0}<1} , $w=0$ model, we’ll substitute Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\sqrt {-k}}r=\sinh \chi } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dr={\cosh \chi d\chi \over \sqrt{-k}}} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} in the integral above:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} r& =\int _-^{\chi _1}{\cosh \chi d\chi \over \sqrt{-k}\sqrt{1+\sinh ^2\chi }}\\ {\chi _1\over \sqrt{-k}}& ={\sinh ^{-1}(\sqrt{-k}r_1)\over \sqrt{-k}}\\ \end{align} \,\!}

From this, we can use Mattig’s Formula, which states for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w=0} , arbitrary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega _{0,M}= \Omega _0} , that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r={2c\over H_0\Omega _0^2}{1\over 1+z}\left[\Omega _0z+(\Omega _0-2)(\sqrt{1+\Omega _0z} -1)\right] \,\!}

In general, for arbitrary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega _{0,M}, \Omega _{0,\Lambda }} (we’ll derive this in PS#3), one can show that, in comoving Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {r={1\over \sqrt{|k|}}sinn\left({c\over H_0}\sqrt{|k|}\int _0^ z{dz^\prime \over \sqrt{\Omega _{0,M}(1+z)^3+\Omega _{0,\Lambda }+(1-\Omega _{0,M}-\Omega _{0,\Lambda })(1+z)^2}} \right)} \,\!}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sinn} is a funny function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sinn\equiv \begin{cases} \sin & \, for\ k>0\\ absent& \, for\ k=0\\ \sinh & \, for\ k<0\\ \end{cases} \,\!}

Note that when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w=0} , for arbitrary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega _{0,M}} , we recover Mattig’s Formula.

Angular Diameter Distance

The angular diameter distance is a useful quantity which relates the physical size or separation of objects to the angular size on the sky. For normal, Euclidean geometries, this is trivial trigonometry. For a curved universe, this is not trivial. For example, in some universes, an object pulled far enough away may actually start looking larger (have a larger angular diameter) than a closer object!

This brings us to the end of the smooth universe. We’ve seen Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(t)} , but we have not seen any perturbations off of that. Similarly, we’ve seen $\rho (t)$, but no spatial components of density. We will begin to talk about perturbations off of the Smooth Universe, and we will call this:

The Bright Side of the Universe

Let’s do a quick tour of the the particles out there to give some context to what we’re talking about. Take a look at Review of Particle Physics, which is also at http://pgd.lbl.gov, for some more detailed information.

First, we’ll talk about fermions. Fermions come in two varieties: Leptons and Quarks. Quarks are hadrons and group together to from baryons (made of 3 quarks) and mesons (made of quark-antiquark pairs). Elementary Particles: Fermions (spin Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac12} )

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} Leptons & Charge & Mass & Mean\ Lifetime \\ e & -1 & 0.51099892\pm 0.00000004 MeV & \infty \\ \nu _ e & 0 & <3 eV & \infty \\ \mu & -1 & 106 MeV & 2.2\mu s,\ \mu ^-\to e^-\bar\nu _ e\nu _ p \\ \nu _\mu & 0 & <190 keV & \infty \\ \tau & -1 & 1.78 GeV & 2.9\cdot 10^{-13} \\ \nu _\tau & 0 & <18.2 MeV & \infty \\ \end{matrix} \,\!} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} Quarks\ (3\ colors\ each) & Charge & Mass \\ U & {2\over 3} & 1.5 \to 4 MeV \\ d & -{1\over 3} & 4 \to 8 MeV \\ s & -{1\over 3} & 80 \to 130 MeV \\ c & {2\over 3} & 1.15 \to 1.35 GeV \\ b & -{1\over 3} & 4.1 \to 4.4 GeV \\ t & {2\over 3} & 174.3 \pm 5.1 GeV \\ \end{matrix} \,\!}