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General relativity and cosmology

Assume universe is

Homogenous
Isotropic

Means metric cannot depend on θ & φ, and at any time K(t) is the same everywhere. Hence for k > 0, R(t) has obvoius geom interpretation. Now define dimensionless coord σ = r/R(t) (Note that notation is the same as the previous N-R model, but the interpretation is slightly different.) "Co-moving coordinate:" defined with respect to local frame that changes as the universe expands (draw a grid on the balloon) We know how to handle this: space part of metric must be $$ \color{red}{ \Delta s^2 = R^2 \left( t \right)\left( {\frac{{\Delta \sigma^2 }}{{1 - k\sigma ^2 }} + \sigma ^2 \Delta \theta ^2 + \sigma ^2 \sin ^2 \left( \theta \right)\Delta \varphi ^2 } \right)} $$ so 4-D metric is $$ \color{red}{ \Delta \tau ^2 = \Delta t^2 - \frac{{R^2 \left( t \right)}}{{c^2 }}\left( {\frac{{\Delta \sigma ^2 }}{{1 - k\sigma ^2 }} + \sigma ^2 \Delta \theta ^2 + \sigma ^2 \sin ^2 \left( \theta \right)\Delta \varphi ^2 } \right)} $$ so all the information is in R(t). FRW (Friedmann-Robertson-Walker) metric. Note this is cosmic kinematics: i.e. it doesn't depend on any model for R(t). Can't be measured, but we can now use this to derive measurable quantities.e.g. proper distance to a galaxy at $\sigma $ is $$ \color{red}{ D = R\left( t \right)\int_0^{\sigma '} {\frac{{d\sigma }}{{\sqrt {1 - k\sigma ^2 } }}} }$$ (similar argument to radius of circle. Note that this is factored into scale dependent part and co-moving part. Immediately get $$ \color{red}{ H_0 = \frac{{\dot R\left( {t_o } \right)}}{{R\left( {t_o } \right)}} }$$ In similar ways, we get

Red-shift $\color{red}{z = \frac{{R\left( {t_o } \right)}}{{R\left( {t_E } \right)}} - 1}$
Deceleration Parameter $\color{red}{q_0 = - \frac{{\ddot R\left( {t_o } \right)}}{{R\left( {t_o } \right)H_0 ^2 }}}$
What is q₀ if universe is empty?
What is it if ρ = ρ_c?
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
$\color{red}{\int_{t_{\min } }^{t_o } {\frac{{dt}}{{R\left( t \right)}}} = \int_0^{\sigma '} {\frac{{d\sigma }}{{\sqrt {1 - k\sigma ^2 } }}} }$ (note that in some k=1 models we can see the whole universe!)
Apparent luminosity: $\color{red}{l = \frac{L}{{4\pi D^2 }} \Rightarrow l = \frac{L}{{4\pi R\left( t \right)^2 \sigma _E^2 }}}$:
again can expand result to give $\color{red}{l = \frac{{LH_0 ^2 }}{{4\pi c^2 z_{}^2 }}\left[ {1 + \left( {q_0 - 1} \right)z + ...} \right]}$, where q₀ is the "deceleration parameter".

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

Hence importance of 1a supernovae: since we know (maybe) L, we can get $q_0$ directly.
Olber's paradox: total luminosity will depend on density of galaxies (n(t) as well as luminosity of each
$\color{red}{l_{tot} = \frac{{cn\left( {t_o } \right)}}{{R\left( {t_o } \right)}}\int_{t_{\min } }^{t_o } {L\left( t \right)R\left( t \right)dt} }$
so it will be finite for t_min = 0 (i.e. models with a Big Bang) but allows us to rule out other models
Number counts: How can we decide if universe is closed or open? This becomes an observational question: density of galaxies at different distances depends on k

Note that all have same density of galaxies but (e.g.) k = 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 obejcts 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

Dark Energy

Dark Matter is bad enough, but now there is an extra problem.

If we buy the BB, the expansion of the universe should be slowing down, or at worst constant.: i.e. q₀, ≥ 0. LBL & Harvard have been measuring the distance more accurately than ever before by looking at supernovae (Sn1a: all have the same light curve). The implication is that the expansion of the universe is accelerating: q₀,< 0(!)

Confirmed by observations of radio-galaxies: size allows distance to be estimated.

Combining this with data from WMAP gives

Ω_Λ = .7
Ω_CDM = .3
This gives a "best guess" due to Michael Turner

Implies a very different picture for the expansion of the universe
What can dark energy be? We can parametrise the expansion
Ṙ=-^4π/₃Gρ(1+3w) R

where w = P/ρ is the "equation of state parameter". if w<-1/3 we get a positive energy density, but (effectively) a negative pressure which overcomes gravitational attraction at very large distances.

BDM,CDM w ∼ 0
HDM (γ's and ν's) w = 1/3
&Lambda w = -1

This implies a cosmological constant &Lambda (Einstein's "fudge factor")

We don't know (although there are models..................). Note that w need not even be constant with time
However, there are major problems (what, more?). Dark energy implies that the vacuum has an energy density: $$ \color{red}{ \rho _\Lambda \approx 100\rho _B \approx 10^{-10} JM^{ - 3} } $$ If it has a non-zero value, we should be able to find it on dimensional grounds: i.e. construct an energy density with the correct dimensions from G, ħ and c: the "Planck energy density" $$ \color{red}{ \rho _{\dim } = \frac{{c^7 }}{{G^2 \hbar }} \sim 10^{113} JM^{ - 3} } $$ You will notice a discrepancy!
An alternative argument: Can write baryon energy density (units are c=ħ=1)
ρ_BDM≅10^-13 eV⁴.
We can understand ρ_Λ ≡ 0. : The only working theory for particles (the standard model) gives
ρ_Λ ≅ 10¹⁰⁰ eV⁴ - V₀

where V₀ is a (unknown) correction.so we need cancelation to 110 places of decimals. Secondly, ρ_Λ and ρ_Matter are almost equal at present. In the past they would have differed by 10⁴⁰ If w(t) is increasingly negative (which is best fit) universe will accelerate out of control ⇒ Big Rip in ∼ 35 *10⁹ years
To test the Big Bang models, we need to look at the first few minutes