Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 57 / 1

Use signature $(-,+,+,+)$ and write $h_{ij}$ for the spatial induced metric, $D_i$ for its spatial covariant derivative, and $q^2=h^{ij}{D}_i\varphi D_j\varphi$. The negative shift vector convention gives $n^{\mu }=N^{-1}(1,N^i)$ and $\Pi =n^{\mu }{\partial }_{\mu }\varphi$. Decomposing the scalar field derivative givesVarying the scalar field matter action with respect to the inverse metric tensor, including the variation of $\sqrt{-g}$, gives the stress-energy tensorRaising both indices gives the requested $T^{\mu \nu }$. Contracting with the unit normal twice, or once with the spatial projection tensor, givesProjecting both indices of the stress-energy tensor onto the spatial hypersurface givesTwo printed identifications require correction. The canonical momentum of the displayed density is ${\pi }_{\varphi }=\sqrt{h}\Pi$, not $\Pi$: explicitly ${\mathcal{L}}_m=N\sqrt{h}[{\Pi }^2/2-q^2/2-V]$ and differentiation with respect to $\dot{\varphi }$ supplies $\sqrt{h}$. Thus $\Pi$ is the normal scalar-field momentum. Also, the printed trace-free stress has an extra factor $1/2$ and uses ${\delta }_{ij}$ where the general spatial induced metric requires $h_{ij}$. In an orthonormal spatial frame the latter becomes ${\delta }_{ij}$, but the coefficient is still one. For a gradient $(q,0,0)$ in that frame, direct projection gives ${\tilde{S}}_{11}=2q^2/3$, rather than the printed $q^2/3$. The PDF's energy-density gradient is spatial ${\partial }_i\varphi {\partial }^i\varphi$; the TeX aid's spacetime index there is an OCR defect.

The unimodular spatial metric separates volume from shape: writing $h_{ij}=a^2{\tilde{h}}_{ij}$ and $a^6=h=\det h_{ij}$ gives $\det \tilde{h}=1$. Contract the extrinsic curvature with a negative shift with $h^{ij}$. Metric compatibility and the Jacobi determinant derivative formula giveConsequentlyFor zero shift vector, $\dot{h}/h=6\dot{a}/a$, soThe infinitesimal spatial volume is $\sqrt{h}{d}^{3}x=a^3{d}^{3}x$; its logarithmic change per unit normal proper time is $3H$. Thus this $H$ is the local volume Hubble parameter, reducing to the usual Hubble parameter in a homogeneous universe. The determinant-defined $a$ refers to fixed spatial coordinates, while this normal volume expansion has a geometric interpretation.

Retain zeroth and first spatial-gradient orders in the long-wavelength approximation in cosmology. Spatial Ricci curvature, $D_i{D}_{j}N$, $q^2$ and scalar anisotropic stress are of second gradient order. Put $A^i{}_j={\tilde{K}}^i{}_j$ and $A^2=A^i{}_j{A}^j{}_i$. The supplied Einstein field equations then reduce toThe scalar-field stress projections show that the momentum constraint is a first-gradient equation and must be retained. The square $A^2$ is not a spatial derivative and cannot yet be dropped merely by invoking the long-wavelength approximation in cosmology. The time-derivative dot on the trace-free evolution equation is visible in the PDF and missing from the TeX aid.

Using $NH=\dot{a}/a$, the last equation becomes ${\partial }_t(a^3{A}^i{}_j)\simeq 0$. HenceThe time-independent spatial tensor must also satisfy the momentum constraint and the symmetry inherited from $K_{ij}$. This is inflationary shear damping: expanding regions lose anisotropic expansion, without requiring the frozen spatial metric shape to vanish. The matter evolution, useful below, is $\dot{\varphi }=N\Pi$ and $\dot{\Pi }\simeq -N(3H\Pi +V_{,\varphi })$, the leading Klein-Gordon equation in each local region.

Now impose the stipulated negligible shear and work where $\Pi \ne 0$. The Hamiltonian constraint, momentum constraint and local expansion evolution giveTime differentiation of the first equation, followed by $\dot{\varphi }=N\Pi$, gives $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$. Spatial differentiation of the same equation and use of the momentum constraint givesThese are the temporal and spatial identities needed for the nonlinear curvature covector. Differentiate its definition, keeping the spatially varying lapse function:The terms proportional to ${\partial }_{i}N$ cancel. The terms $-N{\partial }_{i}H$ and $-4\pi GN\Pi {\partial }_i\varphi$ cancel by the momentum constraint, and the remaining two terms cancel by the spatial momentum identity. Thereforeto the retained gradient order. No homogeneous-lapse assumption was needed.

For linear adiabatic cosmological perturbations, expansion of the local scale and scalar field gives ${\zeta }_i={\partial }_i[\Psi +\bar{H}\delta \varphi /\bar{\Pi }]$. For nonzero spatial modes, linearizing the Hamiltonian constraint and the momentum constraint givesThusThis uses the paper's overall sign for the uniform-density curvature perturbation; the opposite overall sign is also common. For an adiabatic single-field attractor, superhorizon conservation of single-field comoving curvature allows the primordial amplitude to be transported through later epochs without following every short-scale process. This proof specifically discarded the shear contribution to the momentum constraint and assumed a usable field clock. It is not a claim that every single-field background conserves every curvature mode: growing curvature perturbation in ultra-slow-roll inflation is a non-attractor counterexample, and $\Pi =0$ makes this particular covector undefined.