A homogeneous scalar with $V=m^2{\varphi }^2/2$ obeys $\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$. If it is to provide potential-dominated dark energy today rather than rapidly oscillating matter, $m\lesssim H_0\sim 10^{-33}\mathrm{eV}$ is required. Its present density gives ${\varphi }_0\simeq \sqrt{6{\Omega }_X}{M}_{\mathrm{Pl}}{H}_0/m$. Taking $m\sim H_0$ gives a Planck-scale field; a much smaller mass requires a larger field value, so density alone does not fix a unique mass.

// Target: cosmology.bigb

The queue workload of a queue is the total remaining service work of all present customers, including residual work in service. With a work-conserving server of rate $C$ and stationary input $A$, the stationary queue workload is represented byThis is the amount by which past offered work exceeds available service, maximized over possible starts of a busy period. It is the reflection of net input at zero. Under first come first served, an arriving infinitesimal customer waits $W/C$; a priority customer instead waits for the remaining work ahead of it under its service discipline.

The queueing delay is the time between a customer's arrival and the start of its service, excluding its own service time. Under first come first served with service rate $C$, its queueing delay is the queue workload ahead of it divided by $C$. The customer sojourn time also includes the customer's own service. In a preemptive priority queue with a shared buffer, a voice customer's initial waiting time depends only on the voice queue workload already present.

The amount of work a queue can serve per unit of time. In a slotted model the same units can describe work served per slot.

The maximum stored work or number of waiting jobs, according to the model's convention. A bufferless queue has no room to retain excess input for a later slot.

An arrival exceeds the available service and storage capacity, causing work to be lost or blocked. For a bufferless queue with capacity $c$ per slot, overflow means input work strictly greater than $c$. Stationary queue workload tails in an infinite-buffer comparison model are often used to estimate finite-buffer overflow probabilities.

A queue which retains no unfinished work between slots. Given nonnegative input $A$ and capacity $c$, its bufferless queue output is $\min (A,c)$ and its lost work is $(A-c{)}^{+}$.

The size and frequency of unusually large arrivals or sustained overload episodes. It depends on the time and threshold scales being examined, rather than just on the one-slot variance. A cap can leave smaller rare deviations unchanged while removing larger positive bursts.

The population $N(t)$ of a queue counts every admitted customer still inside the selected system at time $t$, including customers receiving service when the server is inside that boundary. Its long-run time average is $L$. The waiting population $N_q(t)$ excludes customers currently receiving service, and its long-run time average is $L_q$. For a single server that serves whenever work is present,where $\rho$ is the server utilization, assuming at most one customer receives service at a time. In a stationary process with ergodicity and finite expected values, these time averages equal $\mathbb{E}N(t)$ and $\mathbb{E}{N}_q(t)$. The workload of a queue instead counts unfinished work, so two queues with the same population can have very different remaining workloads.

The displayed complex matrices, with real coefficients, form a four-dimensional real associative algebra. Their three imaginary basis matrices square to minus the identity and satisfy $JK=L$, $KL=J$, $LJ=K$, with the reversed products negative. For $\alpha =aI+bJ+cK+dL$, quaternionic conjugation gives $\alpha \overline{\alpha }=(a^2+b^2+c^2+d^2)I$. Every nonzero matrix therefore has inverse $\overline{\alpha }/(a^2+b^2+c^2+d^2)$ inside this algebra. This realizes the real division algebra of quaternions within $M_2(\mathbb{C})$; it is not a complex vector subspace.

The Gibbs--Thomson relation depresses a spherical interface's equilibrium temperature by $\Gamma /a$. For $a>a_c$, its interface is warmer than the supercooled ambient and heat loss drives growth. For $a<a_c$, its interface is colder than the ambient and heat gain drives melting. The equilibrium radius is unstable because the derivative of the quasistatic radius velocity at $a_c$ is positive.

For $V=\lambda {\varphi }^4/4$ and unreduced Planck mass $M_P$, the potential slow-roll parameter is ${ϵ}_V=M_P^2/(\pi {\varphi }^2)$. Setting it to one gives the displayed conventional endpoint. The exact end of scalar-dominated accelerating expansion is ${ϵ}_H=1$, equivalently ${\dot{\varphi }}^2=V$. Since slow-roll is failing near that point, a precise endpoint requires evolving the full inflaton equation of motion; potential and kinetic criteria evaluated with a failing approximation need not give the same order-one coefficient.

With unreduced Planck mass $M_P$ and initial potential-dominated density $\lambda {\varphi }_i^4/4\sim M_P^4$, one has ${\varphi }_i^2\sim 2M_P^2/\sqrt{\lambda }$. The slow-roll e-fold count is $N\simeq \pi ({\varphi }_i^2-{\varphi }_e^2)/M_P^2$. Thus small positive $\lambda$ produces a very large expansion factor $e^N$ in this classical quartic model. The order-one endpoint correction is negligible compared with $2\pi /\sqrt{\lambda }$; initial kinetic energy, spatial curvature and the model's validity at trans-Planckian field values are independent assumptions.

With interaction $-{\mu }^{ϵ}\lambda {\varphi }^4/4!$ and $d=4-ϵ$, the two-point tadpole diagram has symmetry factor $1/2$. The Euclidean massive loop integral gives its pole through $I_{d,1}=-m^2/(8{\pi }^2ϵ)+O(1)$. The three four-point bubble diagrams each have symmetry factor $1/2$, and the massive scalar bubble pole in four dimensions gives the quartic counterterm. The tadpole is independent of external momentum, so no one-loop wave-function renormalization is required. A vacuum-energy counterterm is additionally needed if vacuum diagrams are retained.

For a connected six-external-leg Feynman diagram made from quartic vertices, $4V=6+2I$ and $L=I-V+1$ give $V=L+2$, $I=2L+1$. A tree therefore has two vertices and one internal line; partitioning six labelled external legs into two triples gives ten tree channels. A one-loop triangle has three vertices, three internal lines and two external legs at each vertex. With external labels fixed, that triangle has symmetry factor one. Other one-loop topologies also contribute to the full amplitude.

For a real scalar field with interaction $-\lambda {\varphi }^4/4!$, the first Dyson series term is $-i\lambda \int {\varphi }^4{d}^{4}x/4!$. Four labeled external legs can attach to the four fields in $4!$ ways, or in $\left(\genfrac{}{}{0ex}{}{4}{2}\right)2!2!=24$ ways after separating incoming and outgoing legs. The factorial cancels and spacetime integration supplies the four-momentum conservation delta function. This gives the connected tree scattering amplitude $-\lambda$. The separate identical final-state symmetry factor belongs in the event phase-space integral and is not another vertex factor.

A heavy quark has a mass large compared with the characteristic low-energy scale of quantum chromodynamics. Charm quarks, bottom quarks and top quarks are usually called heavy; treating charm as asymptotically heavy is a less accurate approximation than the corresponding treatment of bottom.

The separation of scales permits an expansion in inverse quark mass for many properties of a hadron containing one heavy quark. In the leading limit, the light degrees of freedom respond chiefly to the heavy quark's color field, with reduced sensitivity to its spin and precise flavor. This helps relate patterns in heavy-hadron spectroscopy and decays. The top quark is a special case: it decays before forming ordinary long-lived hadrons, so not every bound-state argument for charm and bottom applies to it.

A quark flavor is one of the six Standard Model quark species: up, down, strange, charm, bottom and top. Each flavor occurs in three colour states and is counted as one Dirac species when both chiralities are included. The active flavor number in a low-energy QCD description excludes quarks heavier than the relevant scale, with matching of the running coupling when a heavy threshold is crossed.

A quark field is a Dirac field or a chiral Weyl spinor carrying the colour fundamental representation of $SU(3)$. Its three colour components transform as ${\psi }^i\mapsto U^i{}_j{\psi }^j$ under local gauge transformations. The colour index is independent of the Lorentz spinor index, and the weak representations of its left and right chiral components differ in the Standard Model.

A quark mass is the coefficient of its Dirac mass term. This term couples left- and right-handed quarks, so nonzero quark masses explicitly break independent rotations of the two chiralities. For two equal masses, the vector flavor symmetry survives, while the nonsinglet axial current obeys the flavor-nonsinglet axial Ward identity with equal quark masses. In the renormalized theory the mass parameter depends on the subtraction scale and scheme, while measurable hadron masses are physical observables.