Cosmology · Primordial nuclei and light-element tests

Why deuterium formation faces a photon bottleneck

Calculate a normalized blackbody photon-number tail per chosen baryon count, and explain why that count alone does not determine actual deuterium survival.

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

Explain why typical photon energy alone does not decide whether deuterium survives.
Distinguish an above-threshold photon number fraction from a reaction rate.
Interpret a supplied photon-count crossing without labelling it an actual nucleosynthesis onset.

A newly formed nucleus can be broken apart

Deuterium is a bound nucleus containing one proton and one neutron. Its formation provides a route toward further light-nucleus reactions, but photons can destroy it through photodissociation.

The thermal photon distribution has a high-energy tail. Even if k_B T is below a binding-energy scale B, some photons still have energies above B. Their numbers must be considered relative to the much smaller baryon population as well as their reaction probabilities.

A typical energy or the spectrum peak therefore cannot by itself determine whether newly formed deuterium accumulates. Formation, destruction, densities, cross sections and expansion all enter the physical bottleneck.

The calculation here isolates the photon-number tail of a supplied ideal spectrum. It deliberately does not compute a deuterium abundance, a photodissociation rate or an actual release temperature.

Count photons rather than substituting an energy fraction

For an ideal zero-chemical-potential blackbody, put y = E/(k_B T). The photon-number spectrum has dimensionless weight y²/(eʸ − 1), whose integral from zero to infinity is 2ζ(3), with ζ(3) ≈ 1.202056903.

Let x = B/(k_B T). The fraction of photon number above B is F_N(x) = [∫ from x to infinity of y²/(eʸ − 1) dy]/[2ζ(3)]. It lies between zero and one. The numerator’s y² weight is for number, rather than the extra energy weighting used for an energy-density fraction.

Choose η = n_b/n_γ = 10⁻⁹ solely for this comparison. Then C(x) = F_N(x)/η counts above-B photons per baryon. A small fraction of all photons can still correspond to many photons per baryon when η is smaller still.

Use 10 ≤ x ≤ 35 and no actual numerical deuterium binding energy. Increasing x means cooling at fixed B. Neither η nor any temperature in this example is presented as a measured cosmological value.

A small photon fraction can remain a large baryon-normalized count

At x = 20, the thermal energy is k_B T = B/20. The normalized photon-number fraction above B is about 3.78946 × 10⁻⁷. Dividing by the chosen η gives about 378.946 above-threshold photons per baryon.

At x = 25 the fraction has fallen to about 3.91085 × 10⁻⁹, yet C is still about 3.91085. At x = 30 the count is about 0.0374443 per baryon. The fall is rapid, but none of these numbers is a reaction rate.

For an optional exact spectrum calculation, expand 1/(eʸ − 1) as a sum of e^(−my). The tail integral becomes Σ from m = 1 to infinity of e^(−mx)[x²/m + 2x/m² + 2/m³], a positive rapidly convergent series on this domain.

The table uses that fully normalized number integral, separately checked by numerical integration. A rough exponential alone omits its polynomial and normalization factors and should not be labelled the exact photon-number fraction.

Exact ideal-blackbody number-tail comparison with chosen η = 10⁻⁹; no reaction rates or empirical abundance inputs are used
x = B/(k_B T)	k_B T/B	F_N above B	C = photons above B per baryon
10	0.1	0.00230392808	2,303,928.08
20	0.05	3.78946246 × 10⁻⁷	378.946246
25	0.04	3.91085396 × 10⁻⁹	3.91085396
30	1/30	3.74442893 × 10⁻¹¹	0.0374442893
35	1/35	3.40155962 × 10⁻¹³	0.000340155962

One photon per baryon is a count threshold only

Within the supplied spectrum comparison, C = 1 occurs at x ≈ 26.47388, or k_B T/B ≈ 0.0377731. That is a mathematically defined count crossing for the chosen η.

It does not establish when actual deuterium begins to survive. A photon above threshold does not automatically hit and dissociate a deuterium nucleus, and the target nucleus population is not simply one target per baryon.

The photodissociation rate requires the photon distribution together with the energy-dependent cross section and appropriate kinematics. Nuclear formation and reverse reactions must be evolved with the baryon density and expansion history.

The count crossing is therefore useful as a warning against comparing only a typical photon energy with B. Turning it into an exact bottleneck release temperature would claim a kinetic result that the calculation has not supplied.

The ideal zero-chemical-potential blackbody photon-number spectrum has weight y²/[exp(y)−1] and total integral 2ζ(3), with ζ(3) approximately 1.202056903159594. The plotted fraction is F_N(x)=∫_x^infinity y²/[exp(y)−1]dy/[2ζ(3)], where x=B/(k_B T), B is a supplied positive binding scale without a physical numerical value, and 10≤x≤35. C=F_N/η uses chosen η=n_b/n_γ=10^(−9), not a measured cosmological value. The horizontal x axis is linear; the vertical C axis is logarithmic base ten and labels the actual positive photon counts per baryon. Increasing x means cooling only at fixed B. The exact normalized number integral is evaluated using a positive convergent exponential series, rather than a bare exponential or an energy-weighted integral. At x=10, 20, 25, 30, 35, F_N is approximately 0.00230392808, 3.78946246×10^(−7), 3.91085396×10^(−9), 3.74442893×10^(−11), 3.40155962×10^(−13), respectively. C is approximately 2303928.08, 378.946246, 3.91085396, 0.0374442893, 0.000340155962. The orange dashed guide and diamond identify the chosen C=1 count crossing at x approximately 26.47387990, equivalent to k_B T/B approximately 0.03777308063. This is only a number comparison, not the actual deuterium-survival temperature or a photodissociation rate. A photon above B does not automatically encounter and dissociate a nucleus; cross sections, target populations, formation and reverse rates, baryon density and expansion still need evolution. No actual cosmic time, deuterium binding-energy number, nuclear yield, observed photon sample or current fitted η is inserted. Original Ark teaching model.

Deuterium survival opens paths without fixing every yield

When deuterium can accumulate under the actual reaction conditions, it participates in further light-nucleus reactions. The resulting composition depends on a network of channels and their competition, rather than on one binding energy alone.

The expansion rate changes the time available for reactions, while baryon density affects how often nuclei meet. Reverse reactions and later depletion can alter an intermediate’s abundance even after it first becomes viable.

No observed spectrum, fitted nuclear rate, cosmic time or actual binding-energy number has been inserted into the toy account. Its exactness is restricted to the stated ideal photon-number integral and the chosen normalization.

A useful interpretation keeps those layers separate: a spectrum supplies potential projectile counts, reaction physics supplies rates, and an evolving network supplies abundances to compare with observations.

Worked reasoning

Compare the tail with the baryon count

Can above-B photons outnumber baryons when k_B T = B/20 in the supplied example?

The specified temperature gives x = 20. The normalized number-tail fraction is about 3.78946 × 10⁻⁷.
Divide by the chosen η = 10⁻⁹ to obtain C ≈ 378.946.
The above-threshold photon count per baryon is greater than one, although typical thermal energy is only B/20.

Yes, in this ideal number comparison. It does not by itself calculate deuterium destruction or survival.

Try the reasoning

Does C = 1 determine an actual deuterium survival temperature?
Show the answer and reasoning
No.
It is a count threshold; cross sections, target populations, formation and destruction rates and expansion still need calculation.
Can F_N be below one while C is above one?
Show the answer and reasoning
Yes.
C divides the photon fraction by the still smaller chosen baryon-to-photon ratio.
May a blackbody energy fraction be substituted for F_N while retaining the interpretation as a photon-number fraction?
Show the answer and reasoning
No.
Number and energy spectra have different energy weightings.

What to carry forward

A high-energy photon tail can matter even when typical thermal energy is below a binding scale.
An above-threshold photon number fraction and a photon-induced reaction rate are different quantities.
The chosen count crossing is not an actual nucleosynthesis onset prediction.

References

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