General relativity and cosmology

Calvin & Hobbes

Cosmology: Kinematics

• Homogenous
• Isotropic

• We know how to handle this: space part of metric must be
  \color{red}{ ds^2 = \left( {\frac{{dr^2 }}{{\left( {1 - kr^2 } \right)}} + r^2 d\Omega ^2 } \right)}
• so 4-D metric is
  \color{red}{ d\tau ^2 = dt^2 - \left( {\frac{{dr^2 }}{{\left( {1 - kr^2 } \right)}} + r^2 d\Omega ^2 } \right)}
  so all the information is in curvature k.
• FRW (Friedmann-Robertson-Walker) metric. Note this is cosmic kinematics: i.e. it doesn't depend on any model for k(t).
• Can't be measured, but we can now use this to derive measurable quantities. we'll separate the sign of the curvature from its value:

Warning

Notation: books differ on the metric.

• \color{red}{k } is the value of the curvature: expect this to depend on time
• \color{red}{R_0 } gives the magnitude of the curvature
• \color{red}{\kappa = 1,0, - 1} is the sign of the curvature: (about half the books mix \color{red}k and \color{red}{\kappa }) so \color{red}{k = \frac{\kappa }{{R_0 ^2 }}}

Ryden

1. a dimensionless scale factor \color{red}{a(t)}
2. two choices for the FRW metric
3. r and x are two choices for the "co-moving coord." defined with respect to local frame (draw a grid on the balloon). Both have dimensions of length
4. The metric is defined in terms of proper distance ds
   \color{red}{ ds^2 = - c^2 dt^2 + a\left( t \right)^2 \left( {\frac{{dx^2 }}{{\left( {1 - \kappa \frac{{x^2 }}{{R_0^2 }}} \right)}} + x^2 d\Omega ^2 } \right)}
   (Ω is the solid angle)
5. or
   \color{red}{ ds^2 = - c^2 dt^2 + a\left( t \right)^2 \left( {dr^2 + r^2 S_\kappa \left( r \right)d\Omega ^2 } \right),S_\kappa \left( r \right) = \left\{ {\begin{array}{*{20}c} {R\sin \left( {r/R} \right)(\kappa = 1)} \\ {r(\kappa = 0)} \\ {R\sinh \left( {r/R} \right)(\kappa = - 1)} \\ \end{array}} \right.}

1. a scale factor \color{red}{R(t)} with dimensions of length:
2. the dimensionless σ is the "co-moving coord." defined with respect to local frame.
3. so
   \color{red}{ R(t) = R_0^{} a\left( t \right)}
   and
   \color{red}{ \sigma = \frac{x}{{R_0^{} }}}
4. a metric in terms of the proper time
   \color{red}{ d\tau ^2 = dt^2 - \frac{{R\left( t \right)^2 }}{{c^2 }}\left( {\frac{{d\sigma ^2 }}{{\left( {1 - \kappa \sigma ^2 } \right)}} + \sigma ^2 d\Omega ^2 } \right)}

e.g. d(t) is the proper distance (i.e. the actual distance to a galaxy if you could measure it instantaneously;

1. Berry has $$ \color{red}{ d_p \left( t \right) = a\left( t \right)\int_0^{r '} {\frac{{dr }}{{\sqrt {1 - \kappa \sigma ^2 } }}} }$$ (similar argument to radius of circle. Immediately get
   \color{red}{ d_p \left( t \right) = \left\{ {\begin{array}{*{20}c} {R(t)\sin ^{ - 1} \left( \sigma \right)(\kappa = 1)} \\ {R(t)\sigma (\kappa = 0)} \\ {R(t)\sinh ^{ - 1} \left( \sigma \right)(\kappa = - 1)} \\ \end{array}} \right.}
2. Ryden has
   \color{red}{ d_p \left( t \right) = \left\{ {\begin{array}{*{20}c} {a(t)R_0 \sin ^{ - 1} \left( {x/R_0 } \right)(\kappa = 1)} \\ {a(t)x(k = 0)} \\ {a(t)R_0 \sinh ^{ - 1} \left( {x/R_0 } \right)(k = - 1)} \\ \end{array}} \right.}
3. or \color{red}{d(t) = ra(t)}

• Note (even more confusing!) there are several "distances" in cosmology

• coordinate distance (x in Ryden's notation: not measurable) Proper Distance dp (r in Ryden's notation: not measurable) Luminosity distance ("standard candles") dL Angular size distance dA All are equivalent if universe is flat and distances are small.

FRW

1. the magic of factoring into scale dependent part and co-moving part.
2. in most calculations of observables, the R₀ will cancel out

Time, distance and red-shift

At time t₀, a galaxy is at a distance d₀ from us, and is travelling at v We see the photon at time t1

• In time t between γ emission & absorption, the distance has increased (universe has expanded) by zct. so γ has to travel further: travel time \color{red}{t = \frac{{d_1 }}{c}}
• so \color{red}{d_1 = d_0 + zct}
• so that \color{red}{d_1 = \frac{{d_0 }}{{1 - z}}}
• i.e. if z = .25, the universe was 75% of current size when γ was emitted

• We can also relate red-shift and age: At t = 0, universe had no size so \color{red}{d_0 = vt_o = zct_0 }

  hence \color{red}{\frac{{t_0 }}{{t_1 }} = \frac{{d_0 }}{{d_1 }} = 1 - z}

• so that \color{red}{t_0 = \left( {1 - z} \right)t_1 }
• so again if z = .25, the universe was only 75% of its current age
• (Correctly \color{red}{d_1 = d_0 \left( {1 + z} \right)} and \color{red}{t_0 = \frac{{t_1 }}{{1 + z}}} , because of relativistic effects, but we also need to allow for slowing down of the expansion)
• Note: the red-shift is more fundamental than the velocities: e.g. we'll talk about z = 3000 for the CMBR, which describes the time at which it was emitted, but it's not really sensible to turn this into v.
• We'll do this properly later on, & include more observables, but now we want to put the dynamics in: i.e. what is a(t)?