For radiation in cosmology, $\rho \propto a^{-4}$ by the cosmological perfect-fluid continuity equation. The flat Friedmann equation gives $a\propto t^{1/2}$, so $a\propto \eta$ in conformal time. Hence $1+z={\eta }_0/{\eta }_e$.

The opposite galaxies again have separation $2({\eta }_0-{\eta }_e)$. Direct communication requires this to be at most ${\eta }_e$, soFor overlap of their past light cones, the criterion is instead ${\eta }_e\ge {\eta }_0/2$, giving $\boxed{z_c=1\text{ for a shared causal past}}$. The two notions are precisely those distinguished in direct cosmological communication and a shared causal past.

For pressureless matter, the cosmological perfect-fluid continuity equation gives $\rho \propto a^{-3}$. The flat Friedmann equation then gives $a\propto t^{2/3}$, and conformal time $d\eta =dt/a$ gives $a\propto {\eta }^2$. Choose the initial conformal time to be zero. A galaxy seen at emission time ${\eta }_e$ has comoving radius $\chi ={\eta }_0-{\eta }_e$ and cosmological redshift $1+z=({\eta }_0/{\eta }_e{)}^2$.

Two opposite comoving galaxies are separated by $2\chi$. Direct one-way communication between their worldlines before emission requires the particle horizon ${\eta }_e$ to cover this separation: $2({\eta }_0-{\eta }_e)\le {\eta }_e$. Thus the threshold is ${\eta }_e=2{\eta }_0/3$, andThe distinction in direct cosmological communication and a shared causal past matters here. If causal contact means that their past light cones overlap, the condition instead is $2\chi \le 2{\eta }_e$, giving ${\eta }_e={\eta }_0/2$ and $\boxed{z_c=3\text{ for a shared causal past}}$. These are different operational meanings of the horizon criterion; the boundary corresponds to the limiting initial-time contact.

For a fixed point $g_f$, put $g=g_f+\delta g$. Linearizing the renormalization-group flow givesBecause $g$ is proportional to temperature at fixed microscopic stiffness, this derivative is the thermal eigenvalue. The thermal eigenvalues of the spherical nonlinear sigma model areThe first value describes the stable zero-temperature ordered fixed point. At the nonzero critical point the thermal perturbation grows as $e^{ϵ\ell }$, so the stopping scale gives $\xi \propto |g-g_{\ast }{|}^{-1/ϵ}$ and $\nu =1/ϵ$ to leading order in the expansion. Higher-loop terms change this to $y_T=ϵ+O({ϵ}^2)$.

At $ϵ=0$ the linear eigenvalue is zero, so its sign alone does not decide the flow. For $n>2$ the first nonlinear term is positive. Integrating it givesThe flow reaches order-one coupling at a length logarithm of order $2\pi /[(n-2)g(0)]$, explaining the exponentially large correlation length in a two-dimensional spherical sigma model rather than a finite-temperature power-law critical point in this smooth-field regime. For $n=2$, $ϵ=0$, the linear eigenvalue is zero everywhere on the perturbative line and vortex effects require an additional coupling. For $n=2$, $ϵ>0$, the sole zero-coupling fixed point has $y_T=-ϵ$.

Set $A_D=S_D{\Lambda }^{ϵ}/(2\pi {)}^D>0$. The one-loop flow is $\mathcal{B}(g)=-ϵg+(n-2)A_D{g}^2$. For $ϵ>0$ and $n>2$, its physical fixed points areThe zero point attracts small positive couplings, whereas the nonzero point is the separatrix between the low-temperature weak-coupling flow and the strong-coupling regime. This is a perturbative description for small $ϵ$; extrapolation of its nonzero root to large $ϵ$ is not controlled.

At $ϵ=0$ and $n>2$, the only finite one-loop fixed point is $g=0$, a double root. Positive $g$ flows to strong coupling. For $n=2$ the loop coefficient vanishes: at $ϵ>0$ only $g=0$ is fixed, while at $ϵ=0$ every $g$ is fixed in this perturbative spin-wave description. The compact two-component model also admits vortices; their Berezinskii–Kosterlitz–Thouless transition is absent from a calculation retaining only the smooth tangent modes. Thus a zero beta function here does not exclude that transition. The formula should not be interpreted as a continuum result for the discrete $n=1$ Ising target.

Write $S=H/T$ and take $n\ge 2$. The unit-length constraint prevents an arbitrary multiplicative field rescaling, so the running coupling is found by renormalizing the coefficient of $|\nabla \sigma {|}^2$. We compute that coefficient in a slowly rotating planar background, which is enough because $O(n)$ symmetry makes the two-derivative stiffness isotropic on the sphere.

Let $n_0=(\cos \theta ,\sin \theta ,0,\dots )$, $e_1=(-\sin \theta ,\cos \theta ,0,\dots )$ and $e_a$ for $2\le a\le n-1$ the remaining constant orthonormal tangent vectors. Parameterize the fast tangent fluctuations byThis satisfies ${\sigma }^2=1$ exactly. Put ${\Omega }_{\mu }={\partial }_{\mu }\theta$, and choose it locally constant and small compared with the shell momenta. Since ${\partial }_{\mu }{n}_0=e_1{\Omega }_{\mu }$ and ${\partial }_{\mu }{e}_1=-n_0{\Omega }_{\mu }$, direct differentiation givesAt quadratic order in the fast fields, the cross term is $2{\Omega }_{\mu }{\partial }_{\mu }{\pi }_1$, whose integral is zero for constant ${\Omega }_{\mu }$. The term $(\partial q{)}^2$ is quartic in the fast fields. Remaining cross terms are cubic and have zero Gaussian expectation; their effects on the two-derivative stiffness start at higher loop order. Field-measure corrections independent of the background contribute normalization terms and do not alter the leading gradient coefficient in this calculation.

The quadratic fast action is $(2g{)}^{-1}\int |\nabla \pi {|}^2$. In the shell $\Lambda /b<|p|<\Lambda$ its connected correlation function is $⟨{\pi }_a(p){\pi }_b(-p)⟩=g{\delta }_{ab}/p^2$ with the momentum delta function understood. Thus $⟨{\pi }_a(x{)}^2⟩=g{I}_D$, wherewith the limit $I_2(b)=(2\pi {)}^{-1}\log b$. Here $S_D=2{\pi }^{D/2}/\Gamma (D/2)$ is the angular measure of the unit $(D-1)$-sphere. Averaging the background-gradient coefficient gives the one-loop stiffness of a spherical nonlinear sigma model:The coefficient is $n-2$, not $n-1$: the fluctuation tangent to the rotation plane does not reduce the gradient cost, because its contribution cancels between $q^2$ and ${\pi }_1^2$. All other $n-2$ transverse components do reduce it.

After integrating out the shell, put $x=b{x}^{\prime }$ to restore the cutoff. The constrained field keeps unit norm, so $g^{\prime }=b^{-ϵ}{g}_{\mathrm{eff}}$, or at this orderFor $b=e^{d\ell }$, $I_D=(S_D/(2\pi {)}^D){\Lambda }^{ϵ}d\ell +O(d{\ell }^2)$. Therefore the renormalization-group beta function isThe displayed terms are the requested one-loop flow. They are a weak-coupling expansion; the discrete $n=1$ target has no tangent modes and must instead be treated by domain walls, as in the first part.

A sensitivity analysis for unit and item nonresponse asks how much the population estimate and substantive conclusions change when the untestable response assumptions are weakened. Vary plausible virginity probabilities among the missing groups, and vary the unknown embarrassment mixture among unit nonresponders; identify combinations which materially alter the estimate or reverse a conclusion.

For example, let $a,b$ be the positive probabilities among embarrassed and nonembarrassed item nonresponders, let $q$ be the embarrassed fraction among unit nonresponders, and let $c,d$ be their corresponding positive probabilities. The sensitivity estimate isThe nominal assumptions set $a=c=0.18$, $b=d=0.08$ and $q=0.4$, reproducing $12\%$. Different choices model outcome-dependent refusal or a changed mixture. Sampling confidence intervals measure a different uncertainty and should also be reported where appropriate; they do not validate the missingness assumptions.

The unit and item nonresponse distinction matters: embarrassment was not recorded for the 300 unit nonresponders. Conditional independence of unit response and virginity given embarrassment is therefore not enough unless the embarrassment distribution among unit nonresponders is also supplied.

One sufficient set of additional assumptions is that unit response is independent of the pair consisting of embarrassment and virginity. Equivalently for this estimation purpose, assume that unit nonresponders have the same embarrassment mixture as the other 750 individuals, and the same virginity probabilities within those strata. The former gives an embarrassed fraction $300/750=0.4$, and the latter gives probabilities $0.18$ and $0.08$. Their estimated positive proportion is consequently $0.4(0.18)+0.6(0.08)=0.12$, or 36 of the 300. Combining the groups givesThe item-response assumption from (ii) supplies the first 90; the added unit-response transport assumptions supply the final 36. A weaker sufficient assumption is simply that the unit nonresponders' marginal virginity proportion equals the estimated $12\%$ in the other group, but it is still untestable from these data.

Without any unit-response transport assumption, retain the item assumption and write the unidentified unit proportion as $p_U$. ThenThus the extrapolation is an assumption-dependent estimate, not a quantity determined by the observed table alone.

Assume that, conditional on the recorded embarrassment category, answering the item is independent of virginity status. Also assume that this category is assessed comparably for both groups and that both response strata have positive observation probabilities. This is covariate-dependent missing completely at random for the item response with embarrassment as the fully observed covariate; it supports MAR standardization over a fully observed covariate.

Use the responder probabilities $0.18$ and $0.08$ within strata, but the stratum sizes in the full group of responders plus item nonresponders: $300$ and $450$. This givesEquivalently, the estimated missing positive counts are $100(0.18)=18$ and $50(0.08)=4$, adding 22 to the 68 observed positives. The marginal estimate changes because the full group contains a larger proportion judged embarrassed. Merely copying the unstratified responder percentage would ignore that difference. The conditional independence assumption cannot be verified using the missing item responses.

The observed positive counts are $200(0.18)=36$ and $400(0.08)=32$. Thus the responder-only estimate isThis estimates the proportion among the responders; it does not yet address either kind of nonresponse.

The Levi-Civita connection is the unique affine connection which is both a metric connection and a torsion-free connection. Uniqueness is encoded by the Koszul formulaIt follows by adding the metric-compatibility identities with derivative fields $X,Y$, subtracting the one with derivative field $Z$, and replacing differences of covariant derivatives by Lie brackets. Nondegeneracy of $g$ determines ${\nabla }_{X}Y$ uniquely from all its pairings with $Z$. The existence and uniqueness of the Levi-Civita connection supplies existence; no existence proof is needed here.

The torsion tensor of an affine connection isThe connection is symmetric, or a torsion-free connection, exactly whenIn a coordinate frame, write ${\nabla }_{{\partial }_i}{\partial }_j={\sum }_k{\Gamma }_{ij}^k{\partial }_k$. Since coordinate fields commute, the condition is ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$; this is the reason for the word symmetric.

A connection on $TM$ is a metric connection for $g$ if for all smooth vector fields $X,Y,Z$,Equivalently, its induced covariant derivative of the metric satisfies $\nabla g=0$. This is also called a metric-compatible connection.

The dual vectors are the Riesz dual basis in an inner product space, and lie in ${\mathcal{U}}_n$ itself. Take the convention that the inner product is conjugate-linear in its first argument; in a real space the conjugations can simply be omitted. Since the $u_j$ form a basis, their Gram matrix $G$ is positive definite and invertible. Let $B=G^{-1}$ and putThe proposed vectors satisfyThey therefore give the unique dual family within ${\mathcal{U}}_n$. Their own Gram matrix isThe inverse of a Hermitian positive-definite matrix is Hermitian, so $B^{\ast }=B$. Using $GB=I$ gives $B^{\ast }GB=B$. ConsequentlyFor a real space this is the familiar identity $B^{T}GB=B$. If the opposite complex inner-product convention is adopted, conjugate the coefficients in the dual-vector formula; the stated identity remains the same. Requiring the dual vectors to lie in ${\mathcal{U}}_n$ is essential: arbitrary additions perpendicular to ${\mathcal{U}}_n$ preserve the pairings with $u_i$ but alter the Gram matrix of the dual vectors.

Write $r=x-u^{\ast }$. If $u^{\ast }$ is a best approximation, then for every $v\in {\mathcal{U}}_n$ and real $t$, the competitor $u^{\ast }+tv$ satisfiesThe linear coefficient must be zero, since otherwise choosing sufficiently small $t$ of the appropriate sign makes this expression negative. Hence $\mathrm{Re}(r,v)=0$. In a real inner product space this is the desired orthogonality; in a complex inner product space, apply the same argument to $iv$ to show that the imaginary part also vanishes. Thus $(r,v)=0$ for all $v\in {\mathcal{U}}_n$.

Conversely, suppose this orthogonality holds. Every competitor $u\in {\mathcal{U}}_n$ has $u-u^{\ast }\in {\mathcal{U}}_n$. The Pythagorean identity then givesEquality forces $u=u^{\ast }$. We have proved the characterization, as well as uniqueness:This is the orthogonal projection onto a finite-dimensional subspace. Completeness of the ambient space is not needed. Existence follows, for example, by solving the invertible Gram matrix system in a basis of the finite-dimensional linear subspace.

In standard Big Bang nucleosynthesis, with the radiation content and reaction rates fixed, the main adjustable abundance parameter is the baryon-to-photon ratio $\eta$. The baryon-density consistency between nucleosynthesis and the microwave background usesIncreasing cosmological baryon density makes deuterium burn more efficiently, so D/H falls steeply. The primordial helium mass fraction grows only slowly, because its dominant control is the surviving neutron fraction. Near the observationally favored density, lithium-7 production, much of it initially as beryllium-7, rises with density. At lower densities the full lithium curve has a valley; the sketch focuses on the branch relevant to the baryon-density comparison.

For the historical meaning of “current” in this 2008 paper, representative 2007–2008 fits near ${\eta }_{10}\simeq 6$ areThese local fits, with nuclear-rate uncertainties, and representative D/H and lithium ranges are documented in Steigman's 2007 nucleosynthesis analysis. They are not a replacement for a nuclear network outside their calibrated range.

The cosmic microwave background acoustic peak pattern measures baryon loading at recombination, independently of the abundance measurements. For example, the five-year WMAP cosmological analysis gives ${\Omega }_{b,0}{h}^2=0.02273\pm 0.00062$ from WMAP alone, corresponding to ${\eta }_{10}\simeq 6.23\pm 0.17$. For $h\simeq 0.7$, this is ${\Omega }_{b,0}\simeq 0.046$. Galaxy-cluster baryon fractions combined with independent total-matter density provide another, more astrophysically model-dependent cross-check.

At ${\eta }_{10}\simeq 6.2$, the fitted predictions are D/H $\simeq 2.5\times 10^{-5}$, $Y_p\simeq 0.249$, and ${}^{7}Li/H\simeq 4.6\times 10^{-10}$. A representative observed D/H range around $(2.7\pm 0.3)\times 10^{-5}$ overlaps the prediction; its inferred density ${\eta }_{10}\simeq 6.0\pm 0.4$ is compatible with the cosmic microwave background. The helium result $Y_p=0.2477\pm 0.0029$ from the 2007 helium analysis also agrees. Other helium analyses had lower central values, illustrating why ionization and temperature systematics matter.

In contrast, a representative plateau estimate $12+{\log }_{10}({}^{7}Li/H)\simeq 2.1\pm 0.1$ corresponds to roughly $(1.0$–$1.6)\times 10^{-10}$, a factor of several below the prediction at the same density. This is the cosmological lithium problem. The three abundance ranges therefore should not all be described as exact simultaneous agreement: deuterium and helium broadly support the independently measured baryon density, while lithium has a substantial discrepancy. The plotted ranges explicitly refer to the 2007–2008 comparison, not present-day parameter estimates.

A primordial light-element abundance determination seeks gas or stars with little subsequent chemical processing. The three abundances require different observations and corrections.

For helium, measure helium and hydrogen recombination emission in metal-poor extragalactic ionized regions. Correct line ratios for electron temperature, density, extinction, collisional excitation, underlying stellar absorption and unobserved ionization stages. Convert the number ratio $y=n_{\mathrm{He}}/n_{\mathrm{H}}$ to helium mass fraction $Y\simeq 4y/(1+4y)$, allowing for heavy elements when necessary. Since stars produce helium and metals, extrapolate a sample toward zero gas-phase metallicity to infer $Y_p$. The sensitivity to atomic data and these corrections is illustrated by Peimbert and collaborators' helium determination.

For deuterium, fit isotope-shifted hydrogen and deuterium absorption lines in relatively simple, metal-poor quasar absorption systems. The deuterium feature is displaced by about $-82\mathrm{km}{\mathrm{s}}^{-1}$ from the corresponding hydrogen line. Multiple transitions and velocity components help distinguish deuterium from unrelated hydrogen contamination. The fitted neutral-column ratio estimates D/H in predominantly neutral gas. Since stellar processing destroys deuterium, low-metallicity, minimally processed absorbers are favored; the technique is exemplified by Pettini and collaborators' deuterium measurement.

For lithium, measure the ${}^7\mathrm{Li}$ absorption feature near $670.8\mathrm{nm}$ in warm, old, metal-poor halo stars. Stellar-atmosphere models convert line strength to Li/H. The Spite plateau supplies a nearly constant low-metallicity level, but stellar metallicity trends, atmospheric temperature scales, departures from local thermal equilibrium and stellar lithium depletion must be considered. Thus the photospheric level is not automatically the initial abundance. Ryan and collaborators' halo lithium measurements illustrate the measured abundance and chemical-evolution corrections.

For Planck law expressed per unit frequency, put $x=h\nu /(k_{B}T)$. At fixed temperature, the frequency dependence is $B_{\nu }\propto T^3{x}^3/(e^x-1)$. Differentiation givesThe positive interior maximum therefore satisfies $3(1-e^{-x_{\ast }})=x_{\ast }$, with $x_{\ast }=2.821439\dots$ independent of temperature. The zero solution of the rearranged equation is a boundary, not the interior maximum. Hence the frequency form of Wien's displacement law isThis frequency peak is not $c$ divided by the peak wavelength of $B_{\lambda }$: the two spectral densities differ by the frequency-to-wavelength Jacobian.

Choose a parallel-transported orthonormal basis of the screen. In that basis a symmetric trace-free null shear and antisymmetric null twist have formsDirect multiplication givesHenceSince the tidal matrix is symmetric, the antisymmetric part of the Sachs optical equations immediately givesThus an initially null twist-free null geodesic congruence remains null twist-free while its optical description is regular.

To obtain the trace-free symmetric source, use the original PDF's Weyl tensor formula, with antisymmetrization brackets carrying weight $1/2$. The TeX aid has lost indices and brackets. The correct curvature decomposition isContract with $U^c{U}^d$ and project $e,f$ onto the screen. Terms containing $U^2$ or a remaining screen-projected $U$ vanish. The only Ricci term left isThe Weyl term has zero screen trace, by its trace-free property and antisymmetry. Consequently the trace-free symmetric part of the optical equation is the null-shear propagation equationThe two-dimensional matrix identities are essential: in a higher-dimensional screen the null shear square can have a trace-free part and the null shear–null twist anticommutator need not vanish. Choosing the parallel-transported auxiliary null vector makes these tensor transport formulae hold without extra moving-screen terms.

Take the screen trace of the optical matrix of a null congruence equation. The cross-contractions of symmetric null shear with antisymmetric null twist vanish, andThe minus sign in the null twist contribution follows from tracing the square of an antisymmetric matrix. The tidal trace is $P^{ef}{R}_{cedf}{U}^c{U}^d=R_{cd}{U}^c{U}^d$: the additional null terms in $P^{ef}$ vanish by curvature antisymmetry. Thus the Null Raychaudhuri equation isIn particular, null shear focuses the null geodesic congruence while null twist enters with the opposite sign. With vanishing null twist and the null energy condition, the Einstein field equations imply $U\cdot \nabla \theta \le -{\theta }^2/2$. These signs and the factor $1/2$ depend on using the full-trace null expansion in a two-dimensional screen.

First let $B_{ab}={\nabla }_b{U}_a$ without projection. Commute derivatives on this covector and use affine geodesic motion:The last equality relabels the two contracted null-vector indices. Nullness and affine parametrization give $U^a{B}_{ab}=0$ and $B_{ab}{U}^b=0$. Therefore inserting $P$ into the contracted middle index changes no screen-projected quadratic term. Since ${\nabla }_{U}P=0$, screen projection proves the Sachs optical equationsThe optical tidal matrix ${\mathcal{R}}_{ab}$ is symmetric by the Riemann curvature tensor's pair symmetries. Raising one screen index gives the corresponding equation for ${\hat{B}}^a{}_b$.

In terms of the defined optical scalars, the equation is explicitlyThis matrix identity contains the null expansion, null shear and null twist evolution equations as its trace, symmetric trace-free part and antisymmetric part.