= Solution
\b[Cosmological variables and conventions.] The <scale factor> $R(t)$ describes the relative separation of <comoving observers>; dots denote <derivatives> with respect to <cosmic time> $t$. The <Hubble parameter> is $H(t)=\dot R/R$, usually time dependent; “<Hubble constant>” conventionally means its present value. Here $\rho$ is mass-equivalent density, so $\epsilon=\rho c^2$ is <energy density>, while $P$ is isotropic <pressure>. $G$ is the <gravitational constant> and $c$ the <speed of light>. The constant $K$ specifies <cosmological spatial curvature>: with $R$ a length, $K=kc^2$ and $k=+1,0,-1$ describe closed, flat and open spatial geometries. Alternative normalizations of $R$ move a constant factor into $K$. In the printed equations $\Lambda$ has units $\mathrm{s}^{-2}$: it is $c^2$ times a <cosmological constant> defined in inverse square metres. This <time-unit convention for the cosmological constant> prevents a missing factor of $c^2$.
\b[Continuity and the first law.] Differentiating the first <Friedmann equation> with constant $K$ and $\Lambda$, and then eliminating $\ddot R/R$ with the second, gives
$$
2H\left(\dot H-\frac{K}{R^2}\right)=\frac{8\pi G}{3}\dot\rho,\qquad
\dot H-\frac{K}{R^2}=-4\pi G\left(\rho+\frac{P}{c^2}\right).
$$
Thus, on an expanding or contracting branch,
$$
\boxed{\dot\rho=-3H\left(\rho+\frac{P}{c^2}\right).}
$$
The relation extends continuously to a regular turning point. For a physical volume $V(t)=V_*R^3(t)$ bounded by fixed <comoving coordinates>, $\dot V=3HV$. Multiplication by $c^2V$ gives
$$
\frac{d}{dt}(\rho c^2V)=-P\dot V,\qquad
\boxed{dU+P\,dV=0,\quad U=\rho c^2V.}
$$
This <adiabatic work in a comoving volume> is the <first law of thermodynamics> for adiabatic evolution with fixed <entropy> and no additional chemical work. It expresses local <energy> conservation, including <pressure> work. \b[It does not assert that $U$ is constant:] <radiation> loses <internal energy> as it does work; negative <pressure> makes $U$ grow during expansion. A generic expanding spacetime has no global time-translation <symmetry> from which a conserved total <energy> follows.
\b[Constant scalar-field <energy density>.] Setting $\dot\rho=0$ in the continuity equation, with $H\ne0$, gives
$$
\boxed{P=-\rho c^2,\qquad w=\frac{P}{\rho c^2}=-1.}
$$
For a homogeneous canonical <scalar field>, in units $c=1$, $\epsilon=\dot\phi^2/2+V(\phi)$ and $P=\dot\phi^2/2-V(\phi)$. A stationary potential-dominated field has precisely this <equation of state>. A general evolving <scalar field> need not have $w=-1$; the constant-density assumption is essential.
\b[Expansion at the stipulated Higgs <energy density>.] With zero explicit $\Lambda$ and zero $K$, constant $\epsilon_H$ gives
$$
H_H=\sqrt{\frac{8\pi G\epsilon_H}{3c^2}},\qquad
\boxed{R(t)=R_i e^{H_H(t-t_i)}}.
$$
The plus sign selects the expanding branch; there is also a contracting solution. The scalar-field <energy> is already included in $\rho$, so adding it again as an explicit <cosmological constant> would double count it. Using $1\ \mathrm{GeV}=1.602176634\times10^{-10}\ \mathrm{J}$ gives $\epsilon_H=2.4033\times10^{45}\ \mathrm{J\,m^{-3}}$, $\rho_H=2.6740\times10^{28}\ \mathrm{kg\,m^{-3}}$ and $H_H\simeq3.87\times10^9\ \mathrm{s^{-1}}$. Therefore
$$
\boxed{t_{\rm double}=\frac{\log2}{H_H}\simeq1.79\times10^{-10}\ \mathrm{s}.}
$$
This <vacuum-dominated doubling time> describes accelerated expansion in <de Sitter spacetime>, with $\ddot R/R=H_H^2>0$. The stipulated <energy density> exceeds the <critical density> in <energy> units used in Question 2 by $3\times10^{54}$, so it cannot persist as the dominant component of the present universe. The calculation exhibits the <cosmological constant problem>: the absolute gravitational <vacuum energy> must be enormously smaller than such a particle-physics scale today. It does not establish that the <Standard Model> <Higgs field> actually remains constant or supplies a viable inflationary epoch.
\b[The acoustic ruler and accelerated expansion.] Before <baryon> drag becomes ineffective, <pressure> in the <photon-baryon fluid> drives acoustic waves around initial <linear cosmological density perturbations>. Their comoving <sound horizon> at the <baryon drag epoch> is
$$
r_d=\int_{z_d}^{\infty}\frac{c_s(z)}{H(z)}\,dz.
$$
After decoupling, this preferred separation survives in <galaxy> clustering as <baryon acoustic oscillations>: a peak in the <two-point correlation function> and oscillations in the <matter power spectrum>. Its value is calibrated by early-universe physics, commonly constrained by the <cosmic microwave background>. The SDSS luminous-red-galaxy measurement established a peak near $100h^{-1}\ \mathrm{Mpc}$; its use as a distance indicator depends on the ruler <calibration> rather than the length of an individual <galaxy>.<Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies by Daniel Eisenstein and others>[Abstract and Sections 2–5, correlation-function peak and distance-scale interpretation]{ref}
For a ruler perpendicular to the line of sight and one parallel to it,
$$
\boxed{\Delta\theta=\frac{r_d}{D_M(z)},\qquad
\Delta z=\frac{H(z)r_d}{c}},
$$
where $D_M=(1+z)D_A$ is the <transverse comoving distance> and $D_A$ the <angular diameter distance>. These measure $D_M/r_d$ and $H r_d$, respectively. Comparing multiple redshifts tests the distance–redshift relation and expansion history. In particular,
$$
q(z)=-\frac{\ddot R R}{\dot R^2}
=\frac{1+z}{H(z)}\frac{dH}{dz}-1,
$$
so sufficiently precise measurements of the shape of $H(z)$ can test $q<0$. In <general relativity>, accelerated expansion requires a sufficiently negative total $\rho+3P/c^2$; a positive <cosmological constant> supplies it.
A single acoustic angular scale does not independently determine acceleration, <cosmological spatial curvature>, the absolute <sound horizon> and the dark-energy <equation of state>. One combines redshift coverage with early-universe <calibration> and independent matter-density or distance information, comparing accelerating and nonaccelerating models. The <galaxy> selection function and redshift completeness must be modeled with random catalogues, survey covariance estimated with mock catalogues, and <redshift-space distortions>, <galaxy bias> and nonlinear broadening controlled. These tests distinguish a cosmological feature from a survey artifact. \b[The evidence is the measured expansion history within an explicit cosmological framework, rather than the existence of an acoustic peak alone.]
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2015/paper-1-acoustic-ruler.png]
{title=Original schematic of transverse and radial acoustic-ruler measurements}
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