Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 66 / 2

For a flat universe containing only the stated fluid, the cosmological continuity equation is $\dot{\rho }+3H(\rho +P)=0$. A constant equation-of-state parameter $w$ therefore gives $\rho ={\rho }_0(a/a_0{)}^{-3(1+w)}$. Combine this with the Friedmann equation $H^2=8\pi G\rho /3$ to findOn the expanding branch, specifying $H(t_0)=H_0>0$ yieldsThe solution is used only where its denominator is positive. For $w\ne -1$, integration also givesFor $w>-1$, choosing the zero of time at the initial singularity gives $H=2/[3(1+w)t]$ and $\rho =1/[6\pi G(1+w{)}^2{t}^2]$. For $w=-1$, $H=H_0$, $\rho$ is constant and $a=a_0{e}^{H_0(t-t_0)}$. For $w<-1$, the displayed initial-value solution instead grows towards a future finite-time singularity; it is not the ordinary positive-time Big Bang branch. With positive density, accelerated expansion requires $w<-1/3$.

Adding and subtracting the canonical scalar field density and pressure givesA real scalar with positive kinetic energy consequently needs $w\ge -1$. On the expanding scalar-dominated branch, divide $\dot{\varphi }$ by $H$ and use the Friedmann equation:Choose the positive rolling sign. Integration and elimination of $a$ using ${\rho }_{\varphi }={\rho }_{\varphi }^0(a/a_0{)}^{-3(1+w)}$ give the scalar-field reconstruction for a constant equation of state:Changing the rolling sign reverses the exponential slope. At $w=-1$, $\varphi$ is constant and $V={\rho }_{\varphi }^0$; elimination by an invertible $\varphi (a)$ then degenerates. The cosmological continuity equation gives $\ddot{\varphi }+3H\dot{\varphi }+V_{,\varphi }=0$ whenever $\dot{\varphi }\ne 0$, so the reconstruction satisfies the field equation as well as the Friedmann equation.

Compare the reconstructed exponent with $V_0{e}^{-\lambda \sqrt{8\pi G}\varphi }$. In this constant-$w$, scalar-dominated scaling solution,Strict ${\rho }_{\varphi }+P_{\varphi }>0$ requires $w>-1$, and acceleration requires $w<-1/3$. HenceThe accelerating branch has $V_0>0$. The zero-slope limit is a constant potential with $\rho +P=0$, so it does not meet the strict inequality. This result characterizes the exponential-potential power-law inflation solution; an arbitrary trajectory with the same potential need not have constant $w$ throughout its evolution.

For equal intrinsic bolometric luminosities $L$, the observed fluxes are $F(z)=L/[4\pi d_L(z{)}^2]$. In the flat model, a null geodesic gives the comoving distance $\chi (z)=c{\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$. Photon energies and arrival rates each bring one factor of $1+z$, yielding the luminosity distance $d_L=(1+z)\chi$. ThusFor the one-component constant-$w$ universe, $H(z)=H_0(1+z{)}^{3(1+w)/2}$. Its luminosity distance iswith the continuous limit $d_L=c(1+z)\log (1+z)/H_0$ for $w=-1/3$. Measurements spanning several redshifts fit the shape of this relation and hence $w$; the common luminosity and $H_0$ cancel in the flux ratio. Equivalently, the relative distance modulus is $5{\log }_{10}[d_L(z_2)/d_L(z_1)]$.

Actual Type Ia supernovae are standardized through their light-curve shape and color rather than being identical standard candles. Scatter, dust extinction, changes in progenitor populations, photometric calibration, spectral corrections between observed bands, flux-limited selection, lensing magnification and low-redshift peculiar velocities all affect inferred distances. The assumed cosmological model is another uncertainty: with a separate matter component, $H^2/H_0^2={\Omega }_m(1+z{)}^3+(1-{\Omega }_m)(1+z{)}^{3(1+w)}$ in the flat case, so $w$ is partly degenerate with ${\Omega }_m$; allowing spatial curvature or varying $w$ adds degeneracies. Flux ratios constrain the expansion history through an integral of $1/H(z)$, rather than measuring $w$ directly without model assumptions.