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Abundance of Cosmological Relics in Low-Temperature Scenarios

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Manuel Drees, Hoernisa Iminniyaz, Mitsuru Kakizaki

High Energy Physics - Phenomenology · Submitted 21 Mar 2006 · 21 references

Pith's verdict: A genuinely useful resummed approximation for low-reheat relic densities, but the '3% for all parameters' claim outruns the numerical evidence.

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arXiv IDhep-ph/0603165 (version 2)
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CitationHoernisa Iminniyaz, Manuel Drees, Mitsuru Kakizaki. "Abundance of Cosmological Relics in Low-Temperature Scenarios." arXiv:hep-ph/0603165 (2006). https://pith.science/paper/PXDSIWGI
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The paper

The paper claims

A single resummed analytic expression reproduces the final relic density of non-relativistic cold relics to within about 3% in both low-temperature and standard high-temperature scenarios, including full equilibrium.

The reading

This paper aims to provide an analytic description of the relic abundance of stable or long-lived non-relativistic particles $\chi$ in cosmologies where the reheat temperature is too low for $\chi$ to have been in full chemical equilibrium. It derives a first-order perturbative solution to the Boltzmann equation and then 're-sums' it into the ratio ansatz $Y_\chi = Y_0/(1-\delta/Y_0)$. The paper argues that this resummed expression reproduces the exact numerical relic density to about 3% or better for all initial temperatures and annihilation cross sections, even where full equilibrium is reached and the standard freeze-out formula applies. A version including non-thermal production from decays of a heavier particle is also covered. If correct, the result supplies a single semi-analytic tool spanning the whole range from out-of-equilibrium production to full freeze-out.

What carries the argument

The machinery is the resummed ratio ansatz $Y_{1,r} = Y_0/(1-\delta/Y_0)$ for the abundance $Y = n_\chi/s$ as a function of scaled inverse temperature $x = m_\chi/T$. $Y_0$ includes thermal production and, in the decaying-heavy-particle case, non-thermal production from $\phi$ decay; $\delta$ is the first-order annihilation correction computed by integrating $-f(a+6b/x)Y_0^2/x^2$. The denominator $(1-\delta/Y_0)$ turns the perturbative series into a form whose large-cross-section limit is $1/\langle\sigma v\rangle$, matching the standard freeze-out result. The ansatz solves the annihilation-only Boltzmann equation exactly, and the error in the full equation is driven by a source term of order $(\delta/Y_0)^2$ multiplied by $Y_{\chi,\mathrm{eq}}^2$, which decays exponentially at late times.

Extended reading notes

Core claim

The central discovery is that the intermediate regime, where thermal production and annihilation are both significant but equilibrium is not complete, can be captured by replacing the first-order expansion $Y_1 = Y_0 + \delta$ with the resummed form $Y_{1,r} = Y_0/(1-\delta/Y_0)$. Here $Y_0$ is the zeroth-order solution containing only production, and $\delta$ is the first-order correction from annihilation. The resummation is exact when thermal production is negligible but the initial abundance is nonzero, and numerically accurate when both sources are present; it fails only for the temperature history in the transient region when $\chi$ comes close to full equilibrium, where it still yields the correct asymptotic relic density. Applied to the standard single-particle Boltzmann equation, and to the two-particle system with a decaying heavy particle, the ansatz gives final relic densities within about 3% of direct numerical solutions for all tested cross sections and initial temperatures.

Load-bearing premise

The argument assumes $\chi$ stays in kinetic equilibrium, kept at the plasma temperature by frequent elastic scattering, throughout the whole time its abundance is changing, so the standard Boltzmann equation with a thermally averaged cross section and equilibrium density applies; if that contact is lost, the equation itself would need modification.

Editorial extensions

If this is right

  • Low-reheat cosmologies in which $\chi$ never reaches chemical equilibrium can be treated with the same analytic machinery as standard freeze-out, by replacing $x_0$ with $\max(x_0, x_{0,\max})$ while keeping accuracy near 3%.
  • The relic density no longer forces a unique annihilation cross section: small cross sections can match the observed abundance because production freezes before equilibrium is reached, and large cross sections can match it if a heavier particle decays into $\chi$.
  • Combining low reheat temperature with non-thermal sources decouples the WIMP relic density from the annihilation cross section, widening the range of predicted detection rates in both directions.
  • For vanishing initial abundance and $a$- or $b$-dominated cross sections, the final relic density scales universally as $e^{-2(x_0-x_F)}$ relative to its value for $x_0 \ll x_F$, independent of the cross-section size.
  • The semi-analytic formula requires only $x_{0,\max} \simeq 20$ rather than iterative freeze-out temperature calculations, simplifying parameter scans in particle-cosmology models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The resummation is exact in the annihilation-only case, suggesting that the nonlinear structure of the full Boltzmann equation is dominated by the same ratio form; an order-by-order extension beyond $\delta$ might converge quickly, though the paper does not construct it.
  • The same ratio ansatz should apply to any non-relativistic species whose abundance obeys a Boltzmann equation quadratic in $Y$, including asymmetric dark matter with a chemical potential, provided kinetic equilibrium with the plasma is maintained.
  • The universal exponential scaling of the final abundance with initial temperature could serve as a probe of low-reheat cosmologies: measuring the relic density as a function of the reheat temperature in a class of models would confirm or rule out the predicted $e^{-2(x_0-x_F)}$ dependence.
  • Extending the method to resonance or coannihilation scenarios should be possible whenever the product of the thermally averaged cross section and the squared equilibrium density is analytically integrable, which is exactly the condition the paper itself identifies.
Original abstract

We investigate the relic density n_\chi of non-relativistic long-lived or stable particles \chi in cosmological scenarios in which the temperature T is too low for \chi to achieve full chemical equilibrium. The case with a heavier particle decaying into \chi is also investigated. We derive approximate solutions for n_\chi(T) which accurately reproduce numerical results when full thermal equilibrium is not achieved. If full equilibrium is reached, our ansatz no longer reproduces the correct temperature dependence of the \chi number density. However, it does give the correct final relic density, to an accuracy of about 3% or better, for all cross sections and initial temperatures.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,215 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.43 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We investigate the relic density n_\chi of non-relativistic long-lived or stable particles \chi in cosmological scenarios in which the temperature T is too low for \chi to achieve full chemical equilibrium. The case with a heavier particle decaying into \chi is also investigated. We derive approximate solutions for n_\chi(T) which accurately reproduce numerical results when full thermal equilibrium is not achieved. If full equilibrium is reached, our ansatz no longer reproduces the correct temperature dependence of the \chi number density. However, it does give the correct final relic density, to an accuracy of about 3% or better, for all cross sections and initial temperatures.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the relic density n_\chi of non-relativistic long-lived or stable particles \chi in cosmological scenarios in which the temperature T is too low for \chi to achieve full chemical equilibrium. contribution 0.85 0.88 0.06 asserted 0.90
s1 The case with a heavier particle decaying into \chi is also investigated. contribution 1.00 0.89 0.03 asserted 0.94
s2 We derive approximate solutions for n_\chi(T) which accurately reproduce numerical results when full thermal equilibrium is not achieved. contribution 0.49 0.95 0.36 asserted 1.00
s3 If full equilibrium is reached, our ansatz no longer reproduces the correct temperature dependence of the \chi number density. limitation 0.99 0.88 0.08 asserted 1.00
s4 However, it does give the correct final relic density, to an accuracy of about 3% or better, for all cross sections and initial temperatures. result 0.97 0.93 0.37 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.88.

What was examined, and what was not

sources read
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Predicted present relic density Ωχh 2 as function of the a and b contributions to the total cross section, see Eq.(5); in frame (a), b = 0 whereas in (b), a = 0. We consider two extreme cases: χ particles were in full thermal equilibrium (dotted blue line) or the number density of χ vanished (solid red line) at x0 = 22. The two horizontal double–dotted black lines correspond to the 1σ upper and lower bounds of the dark matter abundance [20]. case where χ annihilation is completely negligible, dY0 dx = f c(ax + 6b) e−2x . (16) This equation can easily be integrated, giving Y0(x) = f
Figure 2. Evolution of the exact solution Yχ (solid red curves), Y1,r of Eq.(25) (dotted blue), the equilibrium density Yχ,eq of Eq.(2) (double–dotted black), and |δ| of Eq.(22) (short–dashed violet) as function of x − x0. The initial abundance is assumed to be Yχ(x0 = 22) = 0. We take (a) a = 10−9 GeV−2 , b = 0, (b) a = 10−8 GeV−2 , b = 0, (c) a = 0, b = 10−8 GeV−2 , and (d) a = 0, b = 10−7 GeV−2 . In frames (a) and (c) the curves for Y1,r practically coincide with the solid lines. Inserting this ansatz into the Boltzmann equation (4) leads to the evolution equation for ǫ: dǫ dx = − fhσvi
Figure 3. Evolution of Yχ (solid red curves), Y1,r (dotted blue), Yχ,eq (double–dotted black) and |δ| (short–dashed violet) as function of x − x0. Here we take (a) a = 10−9 GeV−2 , Yχ(x0) = 10−8 , (b) a = 10−9 GeV−2 , Yχ(x0) = 10−10, (c) a = 10−8 GeV−2 , Yχ(x0) = 10−7 and (d) a = 10−8 GeV−2 , Yχ(x0) = 10−10. The other parameters are as in
Figure 4. Evolution of |δ|/Yχ (upper curves) and ǫ/Yχ (lower curves) as function of x − x0 for a = 3 × 10−8 GeV−2 (solid red), a = 10−8 GeV−2 (dotted blue) and a = 3 × 10−9 GeV−2 (double–dotted black). Here we choose b = 0 and Yχ(x0 = 22) = 0. Let us turn to a discussion of the dependence of the present relic abundance on the initial temperature. In
Figure 5. The present relic density evaluated numerically (solid red curves), the old standard approximation (dotted blue) and our new approximation (double–dotted black) as function of x0. Here we take (a) a = 10−8 GeV−2 , b = 0 and (b) a = 10−9 GeV−2 , b = 0. 10-4 10-3 10-2 10-1 100 101 -10 -5 0 5 10 Ωχ(x0)/ Ωχ(x0<<xF ) x0 -xF a = 10-7 GeV-2 a = 10-8 GeV-2 a = 10-9 GeV-2 (a) 10-4 10-3 10-2 10-1 100 101 -10 -5 0 5 10 Ωχ(x0)/ Ωχ(x0<<xF ) x0 -xF b = 10-6 GeV-2 b = 10-7 GeV-2 b = 10-8 GeV-2 (b)
Figure 6. Ωχ(x0)/Ωχ(x0 ≪ xF ) as function of x0 − xF . In the left frame, a = 10−7 GeV−2 (solid red curves), 10−8 GeV−2 (dotted blue) and 10−9 GeV−2 (double–dotted black) with b = 0, whereas in the right frame, b = 10−6 GeV−2 (solid red), 10−7 GeV−2 (dotted green) and 10−8 GeV−2 (double–dotted black) with a = 0. while, Eq.(13) shows that xF is roughly proportional to ln a. Therefore we obtain the relation Ωχ(x0) Ωχ(x0 ≪ xF ) ∝ ae −2x0 1/a ∝ e −2(x0−xF ) , (32) which has no explicit dependence on the cross section. The same argument is ap￾plicable to the case where b is dominant. In
Figure 7. Ratios of approximate and exact results for the relic density Ω1,r/Ωχ as function of x0 − x0,max, for a 6= 0, b = 0 (left frame) and a = 0, b 6= 0 (right frame). The curves use Y1,r with x0 replaced by max(x0, x0,max), see Eq.(33). In the left frame, a = 10−8 GeV−2 (solid red curves), 10−9 GeV−2 (dotted blue), 10−10 GeV−2 (double–dotted black), 10−11 GeV−2 (short– dashed violet) with b = 0, whereas in the right frame, b = 10−7 GeV−2 (solid red), 10−8 GeV−2 (dotted blue), 10−9 GeV−2 (double–dotted black), 10−10 GeV−2 (short–dashed violet) with a = 0. standard approximation (11) de
Figure 8. Evolution of Yχ (solid red curves), Y1,r (dotted blue), |δ| (double–dotted black), the prediction for purely thermal χ production Yχ,tp (short–dashed violet) and Yeq (triple–dotted or￾ange) as function of x − x0, for Yχ(x0 = 22) = 0, r = 0.1, N = 1 and b = 0. The S−wave cross section and the initial φ density are (a) a = 10−9 GeV−2 , Yφ(x0) = 10−10, (b) a = 10−9 GeV−2 , Yφ(x0) = 10−11, (c) a = 10−8 GeV−2 , Yφ(x0) = 10−9 and (d) a = 10−8 GeV−2 , Yφ(x0) = 10−10 . large cross section and short φ lifetime, the non–thermal production mechanism does not affect the final χ relic density…

Figures reproduced from arXiv:hep-ph/0603165 by the authors.

References
How the literature uses it

18 arXiv papers cite this in 25 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 12 sentences without a confident label

Uses its method or tool 1 paper
  • We proceed with the method as discussed in [ref] , and solve the coupled Boltzman equations shown below to calculate the number densities of DM candidate and $N_1$: [equation] where $N$ is the average number of DM particles produced on the decay of $N_1$, and $Γ_N_1$ is the decay width of $N_1$.

    Scalar dark matter and leptogenesis in the minimal scotogenic model · Dark matter in scotogenic model · 1.00

Compares with it 2 papers
Cites it as background 10 papers
  • DM production during EMD eras has been intensively studied in the literature, usually triggered by a long-lived massive particle [ref] , or by primordial black holes [ref] .

    From WIMPs to FIMPs: Impact of Early Matter Domination · Introduction · 0.92

  • DM production with a low reheating temperature has been intensively studied in the literature, usually triggered by the decay of a long-lived massive particle [ref] , or by Hawking evaporation of primordial black holes [ref] .[For studies on baryogenesis with a low reheating temperature or during an early matter-dominated phase, see Refs. [ref] and [ref] , respectively.

    From WIMPs to FIMPs with Low Reheating Temperatures · Introduction · 0.90

  • DM production with a low reheating temperature has been intensively studied in the literature, usually triggered by the decay of a long-lived massive particle [ref] .[For studies on baryogenesis with a low reheating temperature or during an early matter-dominated phase, see Refs. [ref] and [ref] , respectively.

    Probing low-reheating scenarios with minimal freeze-in dark matter · FIMP Dark Matter · 0.99

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