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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.

Let $\mathcal{H}=a^{\prime }/a=\dot{a}$, where a prime denotes differentiation with respect to conformal time $d\tau =dt/a$. Multiplying the Friedmann equation by $a^2$ givesThe critical density is the value obtained when $k=0$, so

Since ${\mathcal{H}}^{\prime }=a\ddot{a}$, the pressure-free Raychaudhuri equation givesorDifferentiating $\Omega =1+k{c}^2/{\mathcal{H}}^2$ and substituting this equation yields

For an expanding universe $\mathcal{H}>0$. The Einstein-de Sitter universe $\Omega =1$ is an unstable constant solution: curves starting with $0<\Omega <1$ decrease towards $0$, while curves starting with $\Omega >1$ rise away from $1$ and diverge as the closed model reaches maximum expansion. This is the flatness problem.

Write the neutrino degeneracy parameter as ${\xi }_{\nu }={\mu }_{\nu }/T_{\nu }$. Requiring one degenerate species not to exceed today's critical density givesUsing $T/T_{\nu }=(11/4{)}^{1/3}$,Several equally degenerate species would strengthen the bound by the fourth root of their number.

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 $ϵ=\rho c^2$ 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, $K=k{c}^2$ 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 $c^2$.

Continuity and the first law. Differentiating the first Friedmann equation with constant $K$ and $\Lambda$, and then eliminating $\ddot{R}/R$ with the second, givesThus, on an expanding or contracting branch,The relation extends continuously to a regular turning point. For a physical volume $V(t)=V_{\ast }{R}^3(t)$ bounded by fixed comoving coordinates, $\dot{V}=3HV$. Multiplication by $c^{2}V$ givesThis 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. 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.

Constant scalar-field energy density. Setting $\dot{\rho }=0$ in the continuity equation, with $H\ne 0$, givesFor a homogeneous canonical scalar field, in units $c=1$, $ϵ={\dot{\varphi }}^2/2+V(\varphi )$ and $P={\dot{\varphi }}^2/2-V(\varphi )$. 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.

Expansion at the stipulated Higgs energy density. With zero explicit $\Lambda$ and zero $K$, constant ${ϵ}_H$ givesThe 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 $1\mathrm{GeV}=1.602176634\times 10^{-10}\mathrm{J}$ gives ${ϵ}_H=2.4033\times 10^{45}\mathrm{J}{\mathrm{m}}^{-3}$, ${\rho }_H=2.6740\times 10^{28}\mathrm{kg}{\mathrm{m}}^{-3}$ and $H_H\simeq 3.87\times 10^9{\mathrm{s}}^{-1}$. ThereforeThis 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 $3\times 10^{54}$, 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.

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 isAfter 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.^[ref]

For a ruler perpendicular to the line of sight and one parallel to it,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,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. The evidence is the measured expansion history within an explicit cosmological framework, rather than the existence of an acoustic peak alone.

supersymmetry relates bosons and fermions through fermionic symmetry generators. A main motivation is the Lorentz scalar hierarchy problem: related boson and fermion loop contributions cancel particular ultraviolet sensitivities in an exactly supersymmetric theory. soft supersymmetry breaking leaves residual sensitivity to the superpartner scale, so a very heavy spectrum does not automatically solve naturalness. Supersymmetric matter content can also improve gauge-coupling unification, and a stable neutral lightest superpartner can be a dark-matter candidate. None of these motivations is itself experimental evidence; supersymmetry must be broken if ordinary particles and their partners have different observed masses.

If R-parity is conserved, superpartners are produced in pairs and cascades end in a stable lightest supersymmetric particle. A neutralino LSP can escape the detector, giving missing transverse momentum. Colored squarks and gluinos can generate energetic jets, sometimes b-tagged jets and leptons, plus that imbalance. An inclusive supersymmetry search classifies broad event samples by jet and lepton multiplicities, transverse momentum sums and missing momentum, without requiring a fully reconstructed decay chain.

Background controls are part of the physics argument. Lost-lepton W and top events can be constrained using lepton collider control regions; invisible-Z production can be related to visible-Z or photon samples with appropriate transfer factors. Instrumental imbalance in multijet events needs detector-quality checks and data sidebands. An excess in a collider signal region should survive uncertainty in those extrapolations, efficiencies and luminosity, with independently defined validation regions and attention to the number of searches performed.

An exclusive supersymmetry search targets a more specific production mode and decay pattern. Electroweak chargino-neutralino production can yield three leptons plus missing momentum; a cascade through a slepton can yield a dilepton invariant-mass edge that constrains mass combinations. Such searches often have cleaner backgrounds but depend more strongly on branching fractions and intermediate masses. An endpoint does not necessarily determine every absolute mass without additional observables.

Different spectra require different strategies. For a compressed supersymmetric spectrum, small parent-daughter mass differences make the visible decay products soft. initial-state radiation can recoil against the pair and provide a trigger, supplemented by soft-lepton reconstruction. The ATLAS compressed-spectrum study explicitly uses small neutralino mass differences and initial-state radiation to define a dedicated search rather than applying a hard-lepton selection indiscriminately.^[ref] If R-parity is violated, the LSP can decay visibly, so high-multiplicity jet or lepton channels may matter more than large missing momentum. CMS separately studied hadronic and leptonic final states for this reason.^[ref] Long-lived particles instead motivate displaced vertices, disappearing tracks, abnormal ionization or time-of-flight measurements. Thus there is no single model-independent missing-energy cut that covers every supersymmetric spectrum.

For the hypothetical relic-density calculation, use the supplied distribution and retain its prefactor exactly. Put $M=m{c}^2$, $\Theta =k_{B}T$ and ${\Theta }_0=k_B{T}_0$. Thennumber density, denoted $N$ in the paper, is written $n$ here to distinguish it from a total particle count. The cosmological particle freeze-out temperature in energy units isAssume a stable nonrelativistic relic, negligible further annihilation and $aT=\mathrm{constant}$ after cosmological particle freeze-out, using sharp-freeze-out relic dilution. Its comoving particle number is conserved, soThe particle mass cancels from this present number density under the stated approximation. Its present rest-energy density is $M{n}_0$, not merely $n_0$:This is only 0.371% of the critical density, not a universe-closing component. The correct further comparison is with the measured dark-matter fraction, not an assumption that all critical density must be dark matter. In standard cosmology this abundance would be a small subcomponent, despite a possible collider discovery.

The exponential freeze-out sensitivity makes the result highly sensitive to $P$. Also, thermal entropy release means $aT$ is not exactly constant through the whole cosmic history. If total comoving entropy is conserved, the above number density acquires a factor $g_{\ast s,0}/g_{\ast s,f}$, involving entropy degrees of freedom. Those degrees of freedom and a full annihilation history are not given, so no precise correction is justified from the supplied data.

Establishing the nature of dark matter would require more than measuring a missing particle's collider mass. Measure its spin, charges, interactions, decay lifetime or stability limits, and annihilation or coannihilation processes; compare the resulting relic calculation with cosmological observations. Seek compatible direct nuclear-recoil signals and indirect annihilation or decay products, while controlling astrophysical and detector backgrounds. A collider-invisible lifetime need not be cosmologically long, and nonthermal production could change the abundance. A consistent particle identity, cosmological abundance and independent non-collider evidence are needed to connect the hypothetical discovery to the dark matter.

Comparing photons with the constant vacuum density gives ${\Omega }_{\gamma 0}(1+z{)}^4=0.7$, soMatter dominates both at this crossing: its density in units of today's critical density is $0.3(10.98735{)}^3\simeq 398$, compared with $0.7$ for each of the other components. radiation-vacuum density equality therefore does not itself mark a switch in the dominant component.

The cosmological principle idealizes spatial slices as homogeneous and isotropic. The corresponding Friedmann-Lemaître-Robertson-Walker metric can be writtenHere $t$ is cosmic time, $a(t)$ has dimensions of length in this convention, and $\chi$ is a dimensionless comoving coordinate. The sign of $K$ gives positive, zero or negative spatial curvature. It does not alone determine global topology. A comoving pair at fixed radial coordinates has proper separation scaling with $a(t)$, so the Hubble parameter is $H=\dot{a}/a$, and cosmological redshift satisfies $1+z=a(t_{\mathrm{obs}})/a(t_{\mathrm{em}})$.

For a perfect fluid of mass-equivalent energy density $\rho$ and pressure $p$, Einstein's equations giveThe last equation is local energy conservation for constant $\Lambda$, with vacuum energy kept separate from $\rho ,p$. A component with $p=w\rho c^2$ has $\rho \propto a^{-3(1+w)}$: pressureless matter dilutes as $a^{-3}$ and radiation as $a^{-4}$, the additional radiation factor being photon redshift. With $K=\Lambda =0$, matter domination gives $a\propto t^{2/3}$ and radiation domination gives $a\propto t^{1/2}$. Positive vacuum energy alone gives exponential expansion, $a\propto e^{Ht}$ with $H=c\sqrt{\Lambda /3}$.

Geometry, contents and initial conditions together determine the expansion history. A closed matter-only model recollapses; an open matter-only model expands indefinitely. With a cosmological constant, positive spatial curvature no longer by itself implies recollapse. Present critical density is ${\rho }_c=3H_0^2/(8\pi G)$ and the density parameters obey ${\Omega }_m+{\Omega }_r+{\Omega }_{\Lambda }+{\Omega }_K=1$, where ${\Omega }_K=-K{c}^2/(a_0^2{H}_0^2)$. These models describe the large-scale background, while local structure and inhomogeneities require perturbations or more general metrics.

At $7000\mathrm{K}$, $k_{B}T=0.603\mathrm{eV}$ andThe $2p$ ground term has fine-structure levels $J=1/2$ and $J=3/2$, with energy-level degeneracy $g_1=2$ and $g_2=4$. Detailed balance gives $C_{21}=C_{12}(g_1/g_2)e^{\Delta E/(k_{B}T)}=7.17\times 10^{-14}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The critical density is thereforeWith collisions and spontaneous decay both retained, $N_2/N_1=n_e{C}_{12}/(A_{21}+n_e{C}_{21})$. A Boltzmann distribution, $N_2/N_1\to 2e^{-\Delta E/(k_{B}T)}\simeq 1.97$, requires $n_e\gg n_{\mathrm{crit}}$, not merely equality. For example $n_e\gtrsim 3\times 10^8{\mathrm{m}}^{-3}$ brings the ratio within about ten per cent of that limit.

For halo mean density ${\Delta }_v$ times the critical density at its epoch, the circular virial velocity obeys $v_{\mathrm{vir}}^2=G{M}_h/R_h=({\Delta }_v/2)H^2{R}_h^2$. Thus $v_{\mathrm{vir}}=H{R}_h\sqrt{{\Delta }_v/2}$. Specifying whether overdensity is relative to mean matter or critical density is necessary for numerical inference.