Cosmology · Dark-matter candidates and discriminating tests

Upper limits, exclusions and discovery requirements

Derive an exact zero-count upper limit and follow how a conditional signal model turns it into a bound without declaring a discovery.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Derive a one-sided upper limit for a supplied zero-background count experiment.
Interpret repeated-sampling coverage without assigning a posterior probability.
Separate a count bound, a conditional physical exclusion and a discovery test.

An upper limit is a defined procedure

A search result can constrain a model without identifying the searched-for constituent. To see the distinction, first define a complete ideal count experiment rather than starting with a plotted exclusion curve.

Supply N distributed as Poisson with mean μ ≥ 0: P(N = n|μ) = exp(−μ) μⁿ/n!. The background is exactly zero, and exposure, selection and response are fixed and known.

In this synthetic realization the observed count is n = 0. A positive mean can still produce no events: P(N = 0|μ) = exp(−μ). Zero observed events therefore do not prove that μ equals zero.

No actual detector or observation is represented here. In particular, the exact zero-background assumption is part of the supplied model and must not be inferred merely because no background event was recorded.

Define the one-sided bound for every count

Choose α between zero and one. For every possible observed n, define μ_up(n) by Σ from k = 0 to n of exp(−μ_up) μ_upᵏ/k! = α. This is a specified one-sided upper-limit procedure.

For n = 0 the sum contains only one term: exp(−μ_up) = α. Taking a natural logarithm gives μ_up(0) = −ln α.

At α = 0.10 the endpoint is ln 10 ≈ 2.302585. At α = 0.05 it is ln 20 ≈ 2.995732. These numbers follow from chosen probabilities; they are not empirical particle limits.

Under repetitions of this supplied experiment, the interval from zero to μ_up(N) covers a fixed true μ with probability at least 1−α. The discreteness of the count can make the coverage greater. Defining the procedure for all n is necessary to give that coverage statement meaning.

Probability of the supplied zero-count outcome at selected means
Supplied μ	P(N = 0\|μ)
0	1
1	e⁻¹ ≈ 0.367879
ln 10 ≈ 2.302585	0.1
3	e⁻³ ≈ 0.049787
5	e⁻⁵ ≈ 0.006738

Coverage belongs to the procedure

For the chosen 90 percent procedure and this zero-count realization, means above about 2.302585 lie above the reported endpoint. At the endpoint, the chance to obtain zero events is exactly 0.1.

The frequentist statement concerns how often the procedure’s intervals cover a fixed parameter in repeated experiments. It does not assign a 90 percent probability that this fixed μ lies in the realized interval.

A posterior probability instead requires a prior and a likelihood with their own interpretation. No prior has been supplied here. A different interval construction, such as a central or unified interval, need not give this endpoint.

Likewise, the 90 and 95 percent endpoints are not discovery thresholds. They describe an upper-limit construction for a particular zero-background model, not a comparison of a candidate signal with realistic competing backgrounds.

A physical exclusion retains its assumptions

Now separately supply a fixed-mass signal model μ = κ f_χ σ. Let σ ≥ 0 be a scattering cross section with units of area, f_χ the dimensionless fraction of a relevant population in this constituent, and κ > 0 a known response coefficient with units inverse area.

The 90 percent count bound implies κσ_up = ln 10/f_χ when f_χ > 0. Reducing the assumed fraction weakens the conditional cross-section bound because fewer constituents contribute to the same response.

At f_χ = 0 the mean is zero for every σ under this response. The experiment then has no sensitivity to σ through this channel; dividing by zero does not produce a finite constraint.

For an actual search, κ would encode a specific mass, population, interaction and instrument model. Changing those assumptions changes the physical interpretation. The count endpoint alone cannot exclude every candidate that shares the name dark matter.

Conditional 90 percent limit under the separate supplied model μ = κ f_χ σ
Supplied fraction f_χ	κσ_up
1	ln 10 ≈ 2.302585
1/2	2ln 10 ≈ 4.605170
1/10	10ln 10 ≈ 23.025851
0	No sensitivity to σ in this channel

This original ideal repeated-count experiment has N distributed as Poisson with fixed true mean μ ≥ 0, exactly zero known background and fixed known exposure, selection and response. The synthetic observed outcome is n = 0; it is not detector data. For every possible observed n, the one-sided upper endpoint μ_up(n) is defined by the lower-count tail sum from k = 0 to n of exp(−μ_up) μ_up^k/k! = α, with chosen α = 0.1 or 0.05. At n = 0 only one term remains, so μ_up(0) = −ln α. Panel A plots only this zero-count tail P(N = 0|μ) = exp(−μ), with linear μ from 0 to 5 and a separate zero-based probability ordinate from 0 to 1. Solid blue circle samples at μ = 0, 1, ln 10, 3, 5 have probabilities 1, approximately 0.367879, exactly 0.1, approximately 0.049787, 0.006738. The n = 0 branch is a special case of the declared all-count procedure, not a CDF for arbitrary n. Dashed orange guides and a square identify the 90 percent endpoint (ln 10, 0.1), approximately (2.302585, 0.1). Dotted green guides and a diamond identify the 95 percent endpoint (ln 20, 0.05), approximately (2.995732, 0.05). At ln 10 the blue sample and orange endpoint coincide exactly. These are chosen mathematical confidence levels, not discovery significance thresholds. Intervals [0, μ_up(N)] from the whole repeated-sampling procedure cover any fixed true μ with probability at least 1−α; discrete counts can give larger coverage. A realized interval is not assigned a posterior probability and no prior is supplied. Panel B is a separate conditional physical-response law, μ = κ f_χ σ, at a fixed supplied mass and response. Cross section σ ≥ 0 is area, κ > 0 has inverse-area units and the stipulated population fraction f_χ is dimensionless. The 90 percent zero-count endpoint becomes κσ_up = ln 10/f_χ only when f_χ > 0. Dashed violet squares show this law with linear fraction axis 0.1 to 1 and zero-based dimensionless κσ_up ordinate from 0 to 25. Samples f_χ = 1, 1/2, 1/10 give approximately 2.302585, 4.605170, 23.025851. Decreasing a known assumed fraction weakens the conditional cross-section bound. At f_χ = 0 every σ gives μ = 0 through this channel, so there is no sensitivity and no finite endpoint; zero is deliberately outside the plotted positive-fraction domain. Distinct guides, marker shapes, direct formulas and legends carry the meanings without color. No real exclusion, detection status, candidate viability, background subtraction prescription, fitted nuisance model, likelihood posterior or actual discovery is inferred. Original Ark teaching model.

Test a signal against its alternatives

A realistic discovery calculation compares specified signal and background hypotheses using the observed response. Rate, energy or spatial information, calibration uncertainties and nuisance parameters may all matter.

Unexpected events do not identify a new constituent until competing conventional sources and instrument effects have been tested. Searching many channels also affects how an exceptional-looking fluctuation should be interpreted.

An upper limit can be useful even when a discovery comparison finds no persuasive signal. It states which combinations would predict outcomes disfavored under the specified procedure and physical model.

The zero-count construction demonstrates that logic in an exact finite example. It supplies no actual exclusion, detection status or candidate preference, and it is not a recipe for subtracting an uncertain background from an observed count.

Worked reasoning

Carry a population fraction into the bound

At f_χ = 1/2, what is the conditional 90 percent bound on κσ after zero events?

The specified one-sided count procedure gives μ_up = ln 10.
The supplied signal model becomes μ = κσ/2.
Solving gives κσ_up = 2ln 10 ≈ 4.605170.

The bound is κσ ≤ about 4.605170 under the stated model and interval procedure.

Try the reasoning

Can μ = 1 produce zero events in the supplied experiment?
Show the answer and reasoning
Yes, with probability e⁻¹ ≈ 0.367879.
A count realization is not the expectation value.
Does the 90 percent frequentist construction assign a 90 percent posterior probability to the reported interval?
Show the answer and reasoning
No.
It specifies repeated-sampling coverage. A posterior requires a prior and a separate probability interpretation.
May an uncertain background simply be subtracted before applying this exact zero-background formula?
Show the answer and reasoning
No.
Background uncertainty changes the statistical model and requires an appropriate procedure.

What to carry forward

Zero observed events can constrain a positive expected signal without proving its absence.
Confidence coverage describes a specified procedure over repeated data.
Physical exclusions retain the signal and population assumptions; discovery needs a signal-versus-background test.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.