Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 310 / 1 / b

For a constant equation-of-state parameter $w=P/\rho$, the cosmological perfect-fluid continuity equation integrates to $\rho \propto a^{-3(1+w)}$. Comparing the given powers yieldsThis constant-equation-of-state density scaling describes a cosmic string network and a cosmological domain wall network. An intuitive check is that the energy in a fixed comoving cell grows in proportion to the total string length, $a$, or total wall area, $a^2$, while the physical cell volume grows as $a^3$. Thus the densities scale as $a^{-2}$ and $a^{-1}$ respectively. These effective equations of state use the noninteracting, statistically isotropic network assumptions of the problem.

Choose a reference epoch with $H_0=\dot{a}(t_0)/a_0\ne 0$ and let $x=a/a_0$. Define the critical density and each cosmological density parameter byFor a closed universe ${\Omega }_{K,0}<0$. Take the dark-energy component to be a cosmological constant, as indicated by its $\Lambda$ label. The constant-equation-of-state density scaling givesSubstitution into the Friedmann equation givesAt the reference epoch, ${\Omega }_{r,0}+{\Omega }_{m,0}+{\Omega }_{s,0}+{\Omega }_{w,0}+{\Omega }_{\Lambda ,0}+{\Omega }_{K,0}=1$. The Friedmann acceleration equation isStrings have ${\rho }_s+3P_s=0$ and make no direct acceleration contribution, although they affect the Friedmann constraint. The normalized density parameters are undefined at a static epoch, where $H_0=0$; the static solution must therefore be described directly with dimensional densities.

For a single positive-density component, staticity requires $\rho +3P=0$ from the Friedmann acceleration equation. Among the listed components, only the cosmic string network has $w=-1/3$. Put $a(t)=a_E>0$ and use the Friedmann equation to obtainAll other components are absent. This is the static closed cosmic-string universe.

There is a normalization condition hidden in this expression. String conservation fixes $C_s={\rho }_s{a}^2$, and a static solution exists only whenOnce this condition is imposed, every positive radius $a_E$ is a static solution, with its density determined as above. One cannot independently prescribe an arbitrary string normalization and then choose a radius to compensate: both the string density term and curvature term scale as $a^{-2}$.

First perturb the radius consistently with the conserved string normalization. If $a_E\mapsto a_E(1+ϵ)$ while $C_s={\rho }_s{a}^2$ stays at its static value, the density becomesThe Friedmann equation still has an exact cancellation between the string and curvature terms, and the Friedmann acceleration equation still vanishes. ThusThe homogeneous stability of a static cosmic-string universe is neutral in this restricted sense: there is no restoring force and no exponentially growing displacement. Holding the density fixed while changing the radius would instead change $C_s$ and would not be this pure displacement perturbation.

The distinction between the evolution equation and the constraint is essential. For strings alone, the exact equations reduce toLinearizing the acceleration equation gives $\delta a=A+Bt$. At the exactly tuned static $C_s$, however, the full Friedmann equation requires ${\dot{a}}^2=0$, so $B=0$ for an admissible exact perturbation.

A small nonzero velocity $v$ can be admissible if its initial density also changes:Arbitrarily small $v$ and the corresponding second-order density change therefore produce a secular departure. There is no Lyapunov stability against all nearby constrained initial data for the static closed cosmic-string universe. Neutral radius shifts at fixed normalization and drift when the normalization changes are different perturbation classes.

Increase the density at the initial radius by a small fraction $ϵ>0$, maintaining the string equation of state. Then $C_s=C_E(1+ϵ)$ and the Friedmann equation fixesBecause $\ddot{a}=0$, the exact expanding and contracting branches areThe expanding branch grows indefinitely; the contracting branch reaches zero scale factor after $a_E/\sqrt{Kϵ}$. Thus a positive density-normalization perturbation destroys staticity. The departure speed is proportional to $\sqrt{ϵ}$, reflecting the fact that the Friedmann constraint is quadratic in the velocity at the static point. This is another aspect of homogeneous stability of a static cosmic-string universe.

If the initial density is decreased by $ϵ<0$ at the same radius, the string-only Friedmann equation would requirewhich has no real solution. It is not an allowed initial perturbation of a closed string-only universe at fixed curvature normalization. In particular, it should not be interpreted as an oscillating or automatically recollapsing solution.

Likewise, increasing the density while keeping both $a$ and $\dot{a}=0$ fixed violates the constraint; the admissible velocity is the one obtained above. The homogeneous stability of a static cosmic-string universe must always be assessed using both the acceleration equation and the Friedmann constraint. Allowing extra components or changing the spatial curvature would define a different perturbation problem.