Astronomy · Learning from more than light

Nondetections and limits on an interpretation

Calculate what a zero selected count means under a stated response model, then show how an uncovered direction leaves a probability floor that greater assumed fluence cannot remove.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Specify the response, coverage, energy and time conditions behind a nondetection.
Calculate and compare direction-conditioned and mixed zero-count probabilities.
Explain why an uncovered possibility and an unspecified statistical construction prevent an automatic universal limit.

No selected counterpart is a statement about a defined search

A nondetection can be scientifically informative. If a proposed source would almost always have produced a selected event in a well-characterized search, a missing event challenges that proposal. The strength of the argument depends on what the experiment could actually have detected.

Specify the time interval, observing coverage, energy or frequency range, source direction, assumed spectrum, interaction channel, response and selection. “No selected counterpart in this search” is different from “nothing of this kind was emitted at any time or energy.” The latter reaches far beyond the recorded data.

A reported upper limit also needs the quantity being constrained and the statistical procedure that gives the limit its interpretation. A confidence bound, a posterior credible bound and a detector’s prospective sensitivity are related tools with different definitions. A threshold drawn on a probability curve does not name its own construction.

We will calculate an original idealized zero-count model, then change its coverage. The supplied areas, source-direction weights and probabilities are synthetic teaching inputs and results. They are not an experiment’s published upper limit or a real sky localisation.

Effective area connects fluence to the expected selected count

Let Φ ≥ 0 be incident particle fluence in particles m⁻² for a fixed known spectral shape, declared energy interval and observing time window. Let A_eff be the corresponding fixed effective area in m² for one source direction. All interaction and event-selection effects relevant to this toy are included in that area.

Stipulate a Poisson selected count N with mean μ = A_effΦ and a known background mean b = 0. Then P(N = 0 | Φ, direction, model) = e^(−A_effΦ). The exponent is dimensionless. For zero effective area the probability is one for every nonnegative Φ in this zero-background model.

A large fluence can therefore be incompatible with a likely zero count in a covered direction, while the same fluence remains invisible to a zero-response direction. Response, rather than the word “nondetection” by itself, supplies the connection to incident emission.

This probability is a sampling statement under a proposed fluence and model. It is not P(no source | N = 0). Real background, uncertain response, different spectra or different coverage change the calculation and must enter their own statistical treatment.

Average the conditional probabilities before using the criterion

Now stipulate two mutually exclusive possible source directions, with assumed weights 0.7 and 0.3. The weights sum to one and are fixed independently of Φ in this model. They are neither observed sky probabilities nor fractions of an instrument’s geometric field of view.

In the covered configuration, take A1 = 2 m² and A2 = 0.5 m². The zero-count probability after averaging over the two possible directions is P0(Φ) = 0.7e^(−2Φ) + 0.3e^(−0.5Φ), where Φ is entered numerically in particles m⁻² with the displayed area units understood.

For Φ = 1 particle m⁻², the exact model probability is approximately 0.276693896. The weighted mean area is 0.7 × 2 + 0.3 × 0.5 = 1.55 m², but substituting it into one exponential gives e^(−1.55) ≈ 0.212247974. That shortcut does not describe the same zero-count probability.

The mean selected count does average linearly to 1.55 m² × Φ. The probability of zero is nonlinear in that count mean, so averaging the response inside its exponential loses the mixture information. Both directions matter even if one is less sensitive.

More assumed fluence cannot illuminate a direction the search misses

For a separate configuration, keep the first area at 2 m² and set the second area to zero. Keep the same fixed weights and known b = 0. The mixed probability is now P0_gap(Φ) = 0.7e^(−2Φ) + 0.3e^0 = 0.7e^(−2Φ) + 0.3.

As Φ grows without bound, the exponential term tends to zero but the uncovered contribution remains 0.3. That is an assumed source-location weight with no response, not a 30% probability that no source exists. Removing it from the calculation would introduce coverage information that the search does not have.

The table gives all six chosen fluence samples for both configurations. At Φ = 2 particles m⁻² the covered probability is about 0.123184780, while the gap configuration remains about 0.312820947. The figure uses common linear axes beginning at zero and retains both limiting probabilities.

Choose P0 = 0.1 only as an illustrative conditional criterion. The covered curve reaches it at Φ ≈ 2.333391137 particles m⁻² under the exact supplied assumptions. The gap curve never reaches it at any finite nonnegative Φ, because it is always above its 0.3 limiting floor.

The line is not labelled a universal 90% confidence limit or a posterior bound. To report either, specify the full statistical construction and its assumptions. A nonzero background would also alter these zero-count probabilities, so the 0.3 floor belongs specifically to this fixed zero-background coverage example.

All six original zero-count probability samples plus their limits. Fluence uses a fixed known spectral shape and time/energy interval; the background mean is zero and the assumed direction weights are 0.7 and 0.3.
Φ (particles m⁻²)	Covered areas 2 and 0.5 m²: P0	Gap areas 2 and 0 m²: P0
0	1.000000000	1.000000000
1	0.276693896	0.394734698
2	0.123184780	0.312820947
3	0.068674175	0.301735127
4	0.040835409	0.300234824
6	0.014940421	0.300004301
Φ → ∞	0	0.3

Zero counts: coverage changes the inference

Scroll sideways to see the full diagram. Its explanation is below.

Original synthetic conditional zero-count probabilities, not measured event data, a real detector sensitivity curve, an all-sky exclusion or a posterior probability that no source exists. N is the number of selected counts in one fixed observing window. The known background mean is exactly b = 0, fixed for both curves. The source has a fixed known incident spectral shape and energy range. Its fluence Phi is in particles per square metre. There are two mutually exclusive possible source directions with explicitly assumed weights 0.7 and 0.3, which sum to 1; these are not measured sky probabilities or solid-angle fractions. Conditional on a direction, N is Poisson with mean A_eff Phi, so P(N = 0 | Phi, direction) = exp(-A_eff Phi). Effective areas are constant within this declared model, after the fixed spectrum and selection are specified. In the covered case they are 2 and 0.5 square metres. In the gap case they are 2 and 0 square metres: the second possible direction is completely unobserved. The blue solid curve with filled circles is covered P0(Phi) = 0.7 exp(-2 Phi) + 0.3 exp(-0.5 Phi). The orange dashed curve with hollow diamonds is gap P0(Phi) = 0.7 exp(-2 Phi) + 0.3. Every exponent is dimensionless when the numeric effective area is in square metres and Phi in particles per square metre. Both shared axes are linear and start at zero: horizontal Phi from 0 to 6 particles per square metre; vertical P0 = P(N = 0 | Phi, stated model) from 0 to 1. Both curves begin at P0(0) = 1, consistent with weights summing to 1 and zero known background. A dotted orange horizontal line at 0.3 is the gap case limiting floor. A dashed violet line at 0.1 is an illustrative conditional zero-count probability criterion, not a confidence level, posterior probability or automatic 90 percent upper limit. The covered curve crosses 0.1 at Phi = 2.3333911373616543 particles per square metre, shown by an open blue marker and vertical guide. The gap curve cannot reach 0.1 for any finite nonnegative fluence because it stays above 0.3. All six defining rows are: Phi 0, covered 1, gap 1; Phi 1, covered 0.2766938961794189, gap 0.3947346982656289; Phi 2, covered 0.12318477957354662, gap 0.3128209472221139; Phi 3, covered 0.06867417456819538, gap 0.30173512652366646; Phi 4, covered 0.04083540881051557, gap 0.30023482383953176; Phi 6, covered 0.014940421459006512, gap 0.30000430094864733. At infinite fluence the limits are 0 and 0.3 respectively. Displayed decimals are arithmetic precision, not observational uncertainty. The full static fallback additionally gives the mathematically incorrect mean-area shortcut exp(-1.55 Phi), which is not plotted and must not replace the probability mixture. Its six values are 1, 0.21224797382674304, 0.0450492023935578, 0.009561601930543505, 0.002029430636295734 and 0.00009142423147817327. The source-direction probabilities must be averaged after evaluating each direction-specific zero-count probability. With the source direction known in region A alone the illustrative 0.1 crossing is ln(10)/2 = 1.151292546497023 particles per square metre; in region B alone it is ln(10)/0.5 = 4.605170185988092. The mean-area shortcut would give 1.485538769673578 and is not the mixture result. A nonzero or uncertain background, uncertain response, different spectrum or different source-location weights would change the model and potentially its normalization and limiting probability. No bounds are exported to unobserved times, energies, flavors, directions or physical source models. The curve is P(no selected counts given the model), not P(no source given no counts).

A useful limit states which interpretation it tests

The ANTARES, IceCube and Pierre Auger search for neutrinos associated with GW170817 is a dated example of complementary searches with direction-, energy- and time-dependent response. Its comparison with emission predictions depends on spectral and source assumptions. A missing selected counterpart did not become proof that the merger emitted no neutrinos.

For an actual report, retain the response and coverage used, the event selection, the background treatment, the assumed spectrum and the convention for flavour or energy integration. A model predicting emission outside the searched band or time range has not been tested merely because another model was constrained inside them.

Additional instruments can reduce a coverage gap if their own response covers the missing possibilities. Combining them requires the joint observing windows, source assumptions and any shared uncertainties; adding detector names is not enough to establish complete coverage.

The chapter has moved from physical messengers through detector response, timing geometry, chance association and joint constraints to nondetections. The next approved chapter begins with calibration pipelines: how recorded signals become reproducible, uncertainty-bearing measurements on which these inferences can rest.

Worked examples

Compare two different probability operations

At Φ = 1 particle m⁻², what zero-count probability follows from the covered two-direction model?

For the two directions, use e^(−2) and e^(−0.5).
Weight and add: 0.7e^(−2) + 0.3e^(−0.5) ≈ 0.276693896.
Using the mean area instead gives e^(−1.55) ≈ 0.212247974, which is a different result.

The mixed probability is approximately 0.276693896. Averaging a mean count before a nonlinear probability transformation changes the model.

Test whether the supplied criterion can be reached

Can either configuration reach the illustrative conditional P0 = 0.1 criterion as Φ increases?

Both covered-direction exponentials decrease toward zero, and the supplied covered mixture reaches 0.1 at about 2.333391137 particles m⁻².
The gap configuration has P0_gap = 0.7e^(−2Φ) + 0.3.
Its uncovered term remains 0.3, so it cannot reach 0.1 at any finite nonnegative Φ.

The covered model can reach the criterion; the gap model cannot. Neither statement assigns a universal confidence or posterior interpretation to the criterion.

Try the reasoning

With known b = 0, what is P(N = 0 | Φ) when the effective area is zero?
Show the answer and reasoning
One for every nonnegative Φ in this model.
The selected-count mean is zero, so zero selected events do not constrain Φ through that zero-response direction.
For uncertain source direction in the supplied model, do we mix zero-count probabilities or insert one weighted mean area into an exponential?
Show the answer and reasoning
Mix the direction-conditioned zero-count probabilities.
The exponential is nonlinear; using one mean area does not retain the same sampling model.
Does the 0.3 gap floor mean a 30% probability that no source exists?
Show the answer and reasoning
No.
It is the stipulated source-direction weight whose response is zero, under the fixed zero-background model.
Can a limit for one assumed spectrum and searched time interval be extended automatically to emission at all energies and times?
Show the answer and reasoning
No.
Different emission conditions require their own response, coverage and statistical analysis.

What to carry forward

A nondetection constrains a specified search and physical model through its response.
Mix direction-conditioned probabilities explicitly; an uncovered possibility can leave an irreducible zero-count probability floor.
State the spectrum, time, coverage and statistical construction before giving a limit a physical interpretation.

References

Teaching resources and research supporting this lesson’s explanations and examples.