General relativity and cosmology

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 - \frac{1}{{c^2 }}\left( {\frac{{dr^2 }}{{\left( {1 - kr^2 } \right)}} + r^2 d\Omega ^2 } \right)}
• Can put in overall scale-factor ("radius of balloon")
  \color{red}{ d\tau ^2 = dt^2 - \frac{{a\left( t \right)^2 }}{{c^2 }}\left( {\frac{{dr^2 }}{{\left( {1 - kr^2 } \right)}} + r^2 d\Omega ^2 } \right)}
  so all the information is in curvature k(t) and scale-factor a(t)
• FRW (Friedmann-Robertson-Walker) metric. Note this is cosmic kinematics: i.e. it doesn't depend on any model for a(t), k(t).
• Can't be measured, but we can now use this to derive measurable quantities.

Warning

• 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.

• Notation: books differ on the metric.

• we'll separate the sign of the curvature from its value:
• \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 }}}

1. a dimensionless scale factor \color{red}{a(t)}
2. two choices for the FRW metric
   1. 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
   2. 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)
   3. or
      \color{red}{ ds^2 = - c^2 dt^2 + a\left( t \right)^2 \left( {dr^2 + S_\kappa \left( r \right)^2 d\Omega ^2 } \right),S_\kappa \left( r \right) = \left\{ {\begin{array}{*{20}c} {R_0 \sin \left( {r/R_0 } \right)(\kappa = 1)} \\ {r(\kappa = 0)} \\ {R_0 \sinh \left( {r/R_0 } \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_p(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) = R\left( t \right)\int_0^{\sigma _e } {\frac{{d\sigma }}{{\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_p \left( t \right) = ra(t)}

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=zc . It emits a photon with wavelength λ0 We see the photon at time t1 with wavelength λ1

• In time t between γ emission and absorption, the distance has increased (universe has expanded) by zct. so γ has to travel further: travel time

   
  t = d1/c
  

• so

  d1 = d₀ + zct = d₀ + zd1 so that   d1 = d₀/(1-z) (non-relativistic)
  

• 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

   d₀ = vt₀ = zct₀ hence  t₀/t1 = d₀/d1  = (1-z)
  

• so that

  t₀ = t1(1-z)
  

• so again if z = .25, the universe was only 75% of its current age
• (Correctly d₁ = d₀(1+z) and t₀ = t₁/(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.

• Emitting galaxy produces photon, with gap \color{red}{\delta t_e } between crests, observing galaxy sees gap of \color{red}{\delta t_o } Both crests travel at speed of light so follow null-geodesic \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) = 0}

• so for the first crest
  \color{red}{ \int_{t_e }^{t_o } {\frac{{dt}}{{R\left( t \right)}}} = \frac{1}{c}\int_0^{\sigma _e } {\frac{{d\sigma }}{{\sqrt {1 - \kappa \sigma ^2 } }}} }
• and for the second,
  \color{red}{ \int_{t_e + \delta t_e }^{t_o + \delta t_o } {\frac{{dt}}{{R\left( t \right)}}} = \frac{1}{c}\int_0^{\sigma _e } {\frac{{d\sigma }}{{\sqrt {1 - \kappa \sigma ^2 } }}} }

• \color{red}{ \int_{t_e }^{t_o } {\frac{{dt}}{{R\left( t \right)}}} = \int_{t_e + \delta t_e }^{t_o + \delta t_o } {\frac{{dt}}{{R\left( t \right)}}} }
  Expanding this (valid since times are short)
• \color{red}{ \frac{{\lambda _o }}{{\lambda _e }} = \frac{{c\delta t_o }}{{c\delta t_e }} = \frac{{R\left( {t_o } \right)}}{{R\left( {t_e } \right)}}= \frac{{a\left( {t_o } \right)}}{{a\left( {t_e } \right)}}}
  Note normally this will happen: the dimensional quantities will cancel
• Hence Red-shift $\color{red}{z = \frac{{a\left( {t_o } \right)}}{{a\left( {t_e } \right)}} - 1}$ is given the ratio of the size of the universe when it was emitted to now (obviously!). We can then expand this
  \color{red}{ a\left( {t_e } \right) = a\left( {t_o } \right) + \left( {t_e - t_o } \right)\dot a\left( {t_o } \right) + \frac{1}{2}\left( {t_e - t_o } \right)^2 \ddot a\left( {t_o } \right) + .....}
• assumes the universe has been expanding smoothly over the time interval between emission and observation.

• Can then use the present value of the Hubble param.
  \color{red}{ H_0 = H\left( {t_o } \right) = \frac{{\dot a\left( {t_o } \right)}}{{a\left( {t_o } \right)}}}
• \color{red}{ a\left( {t_e } \right) = a\left( {t_o } \right)\left( {1 + \left( {t_e - t_o } \right)H_0 - \frac{1}{2}\left( {t_e - t_o } \right)^2 H_0 ^2 q_0 + .....} \right)}
• where
  \color{red}{ q_0 = - \frac{{\ddot a\left( {t_o } \right)a\left( {t_o } \right)}}{{\dot a\left( {t_o } \right)^2 }} = - \frac{{\ddot a\left( {t_o } \right)}}{{a\left( {t_o } \right)H_0 ^2 }}}
  Deceleration Parameter defined in this odd way so that q₀ =1 if the universe is a critical one.

  What is q₀ if universe is empty?
  

• Hubble plot corrections $\color{red}{z = H_0 \left( {t_0 - t_E } \right) + \left( {1 + \frac{1}{2}q_0 } \right)\left( {t_0 - t_E } \right)^2 + .....}$
• Object horizon (distance of most distant object we can see now

  $\color{red}{\int_{t_{\min } }^{t_o } {\frac{{dt}}{{R\left( t \right)}}} = \int_0^{r '} {\frac{{dr }}{{\sqrt {1 - \kappa \sigma ^2 }}}} }$ (note that in some $\kappa =1$ models we can see the whole universe!)

• Apparent luminosity: $\color{red}{l = \frac{L}{{4\pi D^2 }} \Rightarrow l = \frac{L}{{4\pi a\left( t \right)^2 r _E^2 }}}$:

  again can expand result to give $\color{red}{l = \frac{{LH_0 ^2 }}{{4\pi c^2 z_{}^2 }}\left[ {1 + \left( {q_0 - 1} \right)z + ...} \right]}$

  Since we measure magnitudes, better to use $\color{red}{m - M \approx 25 - 5\log H_0 + 5\log \left( {cz} \right) + \left( {1 - q_0 } \right)z}$

• This becomes an observational question: density of galaxies at different distances depends on $\kappa$

• Note that all have same density of galaxies but (e.g.) $\kappa = 1$ has fewer galaxies at large distances (Earth has less land at large distance than a flat plane would have!)

• Observationally this could be done via Hubble plot at very large distance:
• Unfortunately, only individual objects that can be seen at these red-shifts are quasars: these have evolved since the BB and hence cannot be used as constant density markers
• Now we want to put the dynamics in: i.e. what is a(t)?