Finishing the Robertson-Walker Metric
We’ve been following Weinberg’s derivation to show there are discrete metrics. We’ll start with:
Recall that is the curvature constant and is in comoving coordinates. If we define , then:
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:
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 and . We’ll look at light propagation (), and take a radial path () to know from the Robertson-Walker metric that:
Separating out our dependencies and our dependencies and integrating, we get:
- For the flat, , matter-dominated model (, , ), we’ll start with the Friedmann Equation:
Recognizing that , we have:
These integrals evaluate to (in comoving ):
Here, 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 , .
- For the open, , , model, we’ll substitute , for in the integral above:
From this, we can use Mattig’s Formula, which states for , arbitrary , that:
In general, for arbitrary (we’ll derive this in PS#3), one can show that, in comoving :
where is a funny function:
Note that when , for arbitrary , 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 , but we have not seen any perturbations off of that. Similarly, we’ve seen , 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 )