Laboratory and astronomical searches

Specify the response before interpreting a result

A dark-matter search tests a proposed physical response. A nuclear recoil, conversion of a field, missing collider momentum and a gravitational lensing event are different observations with different connections to a candidate model.

For each search, specify the population, interaction or gravitational effect, instrument selection and competing sources. A feature in an instrument becomes evidence for a candidate only through this chain of prediction and comparison.

A recoil experiment, for example, needs the local population and velocity distribution as well as scattering physics. A missing-momentum collider event also requires an event model and does not alone establish a cosmological abundance or lifetime.

The following supplied recoil calculation isolates kinematics and an energy threshold. It contains no measured events, fitted background, detector data or empirical cross-section limit.

Mass matching sets an energy endpoint

Take nonrelativistic incident particles of one speed v ≪ c and mass m_χ striking stationary target nuclei of mass m_T. Write r = m_χ/m_T and μ = m_χm_T/(m_χ+m_T) = m_Tr/(1+r) for the reduced mass.

For elastic scattering, the largest possible nuclear recoil energy is E_max = 2μ²v²/m_T. It occurs in a head-on reversal of the relative motion. In units m_Tv², E_max is 2r²/(1+r)².

Relative to the incident kinetic energy m_χv²/2, the maximum transferred fraction is 4r/(1+r)². It reaches one at equal masses and falls for large mismatch in either direction.

Supply an isotropic center-of-mass angular distribution and a common total cross section σ₀ for all three comparison masses. With no form factor, these choices give a uniform recoil-energy density on [0,E_max]. Other scattering laws need not have that density.

Flux and acceptance both enter a count rate

Choose a recoil threshold E_th = m_Tv²/4. With the uniform recoil density, the accepted fraction is A = max(0,1−E_th/E_max). Thus the first row gives no accepted recoils despite having a nonzero total cross section.

Hold the incident mass density ρ_χ and the target count N_T fixed. The incident number density is ρ_χ/m_χ, so its flux is (ρ_χ/m_χ)v.

The accepted event rate in this ideal reference is R = N_T(ρ_χ/m_χ)vσ₀A. Define C = N_T(ρ_χ/m_T)vσ₀, which has units of inverse time. Then R/C = A/r.

At r = 4, A = 103/128 ≈ 0.804688, but R/C = 103/512 ≈ 0.201172. Better acceptance does not by itself imply a larger count rate: increasing individual mass at fixed mass density lowers the number flux.

Other searches observe other responses

Some coherent-field searches seek conversion to an electromagnetic signal in an apparatus. Their response involves the field, coupling and instrument conditions; the nuclear-recoil threshold formula does not describe them.

Collider searches can look for momentum imbalance associated with unobserved particles. Linking such a response to the gravitating cosmological component requires further information about identity, stability, interactions and production.

Compact-object lensing searches instead use gravitational light propagation and a population distributed between sources and observer. Their event response depends on geometry, motions, source properties and selection.

These routes can complement one another when a common physical model predicts each response. Agreement has to be assessed with the distinct assumptions and backgrounds of each search intact.

Include backgrounds and uncertainty

An actual recoil analysis must use the relevant speed distribution, interaction response, detector efficiency and energy measurement. The simple common σ₀ and sharp threshold above were chosen inputs.

Conventional backgrounds can also produce events. A statistical comparison therefore needs predicted background shapes or rates and nuisance parameters, rather than treating every accepted event as signal.

A limit, expected sensitivity and discovery significance answer different questions. Each must be tied to a stated statistical procedure and a specific set of physical assumptions.

The table teaches why the response changes with mass and threshold within one ideal model. It does not rank real candidates or experiments, and it supplies no present-day exclusion or discovery claim.

Worked reasoning

Try the reasoning

1. Why is the r = 1/4 row’s accepted rate zero?

   Show the answer and reasoning

   Its maximum recoil energy is 0.08m_Tv², below the supplied threshold 0.25m_Tv².

   A nonzero collision probability need not yield a recorded recoil above the threshold.

2. Which supplied r transfers the largest possible fraction of incident kinetic energy?

   Show the answer and reasoning

   r = 1, equal incident and target masses.

   The fraction 4r/(1+r)² reaches its maximum value one there.

3. Does a momentum-imbalance event alone determine the cosmological abundance of a new constituent?

   Show the answer and reasoning

   No.

   The event response must be connected to a physical identity, stability and production model before making that inference.

What to carry forward

• A candidate prediction must pass through the response of the particular search.
• Threshold acceptance and incident number flux both affect event counts.
• Complementary observations require separate backgrounds and a common physical explanation.