Pseudo-FIMP dark matter in presence of a SIMP

Dipankar Pradhan; Jahaan Thakkar; Subhaditya Bhattacharya

REVIEW 4 major objections 6 minor 84 references

Pseudo-FIMP dark matter in presence of a SIMP

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that a feebly interacting dark-matter component can become a pseudo-FIMP and freeze out like a thermal relic when its dark-sector partner is a strongly interacting SIMP.

desk verdict A legitimate, mostly sound extension of pFIMP to SIMP partners: the existence proof holds, but the modified-equilibrium formulas need derivation and the kinetic-equilibrium check is done at a different coupling than the scan uses. read the letter →

arxiv 2411.15108 v1 pith:VV7EHQ4V submitted 2024-11-22 hep-ph hep-th

Subhaditya Bhattacharya , Dipankar Pradhan , Jahaan Thakkar This is my paper · ORCID

classification hep-phhep-th

keywords pseudo-FIMPSIMPdarkmattertwo-componentcoupledBoltzmannequationsrelicdensitydark-matterself-interactionZ2xZ3scalarfreeze-out

topics Dark Matter

open problems Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a dark-matter component with only feeble couplings to ordinary matter can nevertheless be a thermal relic if it lives alongside a strongly interacting massive particle (SIMP): once the conversion process that turns two SIMPs into two of the feeble particles is fast enough, the feeble component reaches equilibrium and freezes out like a WIMP, becoming what the paper calls a pseudo-FIMP. The claim is established first model-independently, by solving the coupled Boltzmann equations (3.1)–(3.2) and scanning the conversion rate, and then in a concrete model with a real scalar pFIMP and a complex scalar SIMP stabilised by a $\mathbb{Z}_2\otimes\mathbb{Z}_3$ symmetry. In that model the relic-density-allowed region includes SIMP masses up to about $50$ MeV when the pFIMP is the heavier component, and the dominant constraint is dark-matter self-interaction. If correct, this widens the class of viable dark-matter production mechanisms beyond single-component SIMP or WIMP–FIMP setups, and it ties the detectability of the feeble component to the dark-sector partner's access to the visible sector.

What carries the argument

The engine of the argument is the pair of coupled Boltzmann equations (3.1)–(3.2) tracking the yields of the weak component $w$ and the SIMP $s$, with the cross-component conversion term $\langle\sigma v\rangle_{ss\to ww}$ mediating energy and number exchange. A SIMP is defined by a number-changing $3\to2$ self-annihilation in the dark sector, so the SIMP equation also carries the $\langle\sigma v^2\rangle_{3s\to2s}$ term; when the conversion rate is large enough, the feeble component follows a modified equilibrium distribution before freeze-out, with the form depending on whether the SIMP or the pFIMP is heavier. The concrete model supplies the same physics through a real scalar $\phi$ (pFIMP) and a complex scalar $\chi$ (SIMP) interacting through the portal coupling $\lambda_{\chi\phi}$; varying this single coupling moves the system from pure FIMP behaviour through pFIMP freeze-out to the regime where the heavier component depletes into the lighter one.

What would settle it

Compute $\sum_f \Gamma_{\chi f\to\chi f}/H(T)$ at $T=m_\chi/25$ for a relic-allowed benchmark with $\lambda_{\chi H}=10^{-3}$ and $m_\chi$ in the $10$\textendash$50$ MeV range; if this ratio drops below about one, dark and visible temperatures decouple and the Boltzmann solutions that assume $T_{\rm dark}=T_{\rm SM}$ miscompute freeze-out.

Watch

Extended reading notes

Core claim

The central discovery is that SIMP dark matter can host a pseudo-FIMP. In the model-independent treatment, the yield $Y_w$ of the weakly coupled component and $Y_s$ of the SIMP obey coupled Boltzmann equations whose conversion term $\langle\sigma v\rangle_{ss\to ww}$ is the knob. For negligible conversion the weak component is an ordinary freeze-in FIMP; once the conversion rate $\gamma_{sw}$ becomes comparable to the SIMP's self-annihilation rate $\gamma_{3s\to2s}$ it tracks equilibrium and freezes out with a density locked to the SIMP's, with the modified equilibrium yields given by eqs. (3.3) and (3.4) for the two mass hierarchies. The concrete $\mathbb{Z}_2\otimes\mathbb{Z}_3$ scalar model realizes this with $\phi$ as the pFIMP and $\chi$ as the SIMP; after imposing relic density, unitarity, perturbativity, vacuum stability and self-interaction bounds, the surviving parameter space allows SIMP masses up to about $50$ MeV when the pFIMP is heavier, and it is the self-interaction bounds, not relic density alone, that most tightly fix the portal couplings $\lambda_\phi$ and $\lambda_{\chi H}$.

Load-bearing premise

The load-bearing assumption is that the dark sector and the Standard Model bath share a single temperature throughout freeze-out; the paper's kinetic-equilibration check uses a Higgs-portal coupling of order $0.1$, while the relic scan fixes $\lambda_{\chi H}\sim10^{-3}$, so the shared-temperature condition is not demonstrated at the value used in the scan.

Editorial extensions

If this is right

A feebly coupled dark-matter candidate does not have to be produced by freeze-in; in multicomponent models it can freeze out after equilibrating through dark-sector conversion with a SIMP partner.
In the two-component model the relic-allowed SIMP mass range reaches about $50$ MeV when the pFIMP is heavier, and the self-interaction bound, rather than relic density alone, sets the strongest limits on the parameter space.
Self-interaction constraints from Bullet and Abell clusters become the deciding phenomenological test, constraining couplings like $\lambda_\phi$ and $\lambda_{\chi H}$ that barely affect the relic abundance.
Direct and indirect detection of the pFIMP is hard unless the SIMP communicates with the visible sector through a light mediator; the vector-like-lepton extension discussed in the paper opens electron-scattering and annihilation channels.
The four dynamical regions identified by the conversion-rate ratio give a classification scheme: pure SIMP plus FIMP, converted FIMP, pFIMP freeze-out, and conversion-dominated depletion.

Reading between the lines

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

If the shared-temperature assumption fails at the scan value $\lambda_{\chi H}\sim10^{-3}$, the freeze-out temperatures and relic abundances computed in Sec. 4.2 would need to be redone with two independent dark-sector and visible-sector temperatures; a dedicated kinetic-equilibration scan over the whole relic-allowed region would settle this.
The same coupled-equation structure should apply to pFIMP partners other than scalars, such as fermionic SIMPs or dark vector mesons, whenever a $3\to2$ process sets the bath density.
In the large-conversion regime IV the 'SIMP' stops being defined by its own $3\to2$ freeze-out, so one should expect its self-interaction phenomenology to be diluted; comparing halo-shape predictions between regions III and IV could serve as a model-independent test.
A measurement of the dark-matter momentum distribution, or of dark radiation, at MeV scales could distinguish a pFIMP from a freeze-in FIMP even when the total relic density is fixed, because the two production histories give different phase-space and temperature evolutions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper does what it says—demonstrates that a feebly coupled DM component can be brought into equilibrium by a SIMP partner, both in a model-independent setup and in a concrete two-scalar Z2 x Z3 model. The existence proof holds up. The main weaknesses are a missing derivation of the modified-equilibrium formulas (3.3)-(3.4) and a kinetic-equilibrium assumption checked at a different coupling than the scan uses. Both are fixable.

The novelty is real. pFIMP was previously studied with a WIMP partner; this is the first application to SIMP, and the coupled Boltzmann treatment plus the Z2 x Z3 benchmark give concrete results, including the 50 MeV SIMP mass reach when the pFIMP is heavier. The paper is honest about using fixed numbers in the model-independent section, and the self-interaction constraints are properly applied.

I disagree with the stress-test note I was handed: it claims eqs. (3.3) and (3.4) are dimensionally inconsistent. That is not right. In (3.3), n_s/neq_s is dimensionless, so neq'_w has number-density units. In (3.4), the terms inside the numerator and denominator each have cm^3/s units—the (n_w/neq_w)^2 factor is dimensionless, and n_s has cm^3/s because carries cm^6/s for a 3->2 process. So the ratio is dimensionless and the square root is well-defined. The formulas may be questionable, but on dimensional grounds they are fine.

The real problem is that (3.3)-(3.4) are asserted without derivation. They are the basis for identifying the pFIMP regime: the claim that the FIMP follows a modified equilibrium curve before freeze-out rests on these equations. A referee should ask the authors to derive them from the cBEQs in the limit where conversion dominates, or at least to state the approximations clearly. Without that, the analytic interpretation is under-supported, though the numerical curves in figs. 2-3 stand on their own.

The other substantive gap is kinetic equilibrium. Appendix D shows a dark-SM common temperature holds for lambda_chiH ~ 0.1, but the relic scan fixes lambda_chiH ~ 1e-3. The paper acknowledges this in Sec. 5, but does not verify it at the scan value. That matters because the freeze-out conclusions assume T_dark = T_SM, and the fix should be easy.

Who is this for? Phenomenologists working on multi-component DM, especially SIMP models. It is a useful existence proof and a clean benchmark. It deserves a serious referee; it is not ready for acceptance as-is, but the fixes are straightforward. I would send it to review with a request for the derivation and the kinetic-equilibrium check.

Referee Report

4 major / 6 minor

Summary. The paper studies a two-component dark matter setup in which one component is a SIMP and the other has only feeble couplings to the visible sector but a sizeable coupling to the SIMP, making the latter a pseudo-FIMP (pFIMP). After reviewing the single-component SIMP, the authors present a model-independent analysis based on solving the coupled Boltzmann equations (3.1)-(3.2), with the DM-DM conversion rate varied from negligible to large. They identify four regimes, including a pFIMP regime in which the feeble component reaches thermal equilibrium through conversion and freezes out. They then construct a concrete two-scalar model with Z2 x Z3 symmetry, solve the coupled equations, scan the parameter space under relic density, unitarity, and self-interaction constraints, and discuss detection prospects through a vector-like lepton extension. The paper claims that the SIMP mass range is extended up to about 50 MeV when the pFIMP is heavier.

Significance. If correct, the paper extends the pFIMP mechanism from a WIMP partner to a SIMP partner and provides the simplest scalar realization. The numerical solution of coupled Boltzmann equations is a credible method, and the paper includes useful appendices: a semi-analytic SIMP solution compared with numerical results (Appendix A), cross-section formulas (Appendix B), self-interaction expressions (Appendix C), and a kinetic-equilibration estimate (Appendix D). The concrete model and the parameter-space scan, with self-interaction and unitarity constraints, give falsifiable predictions for SIMP masses and couplings. However, the central pFIMP identification rests on the modified-equilibrium formulas (3.3)-(3.4), which are presented without derivation, and the kinetic-equilibrium assumption is verified only for parameters different from those used in the main scan. These issues are load-bearing and require a major revision.

major comments (4)

[Sec. 3, Eqs. (3.3)-(3.4)] The pFIMP regime is characterized by the statement that the feeble component 'follows equilibrium before freeze out,' but the modified equilibrium number densities in Eqs. (3.3) and (3.4) are asserted without derivation. Please derive these expressions from the coupled Boltzmann equations in the limit of large conversion rate, stating all approximations (e.g., neglect of SM production/destruction terms, steady-state condition dY/dx ~ 0), and validate them by comparing with the numerical solutions in Fig. 2. Note that, contrary to a purely dimensional objection, the formulas are dimensionally consistent if n_s and n_w denote number densities: in Eq. (3.4) the numerator second term (n_s^2/n_eq_s)<sigma v2> has units cm^3 s^-1, matching the other terms. However, the asymmetric appearance of n_s^2/n_eq_s in the numerator and n_s in the denominator needs clarification, as does the meaning of a 'modified equilibrium' for a species that is not itself in chemical equilibrium. Without this derivation, the quantitative identification of region III and the claim that the pFIMP tracks the modified equilibrium are not established.
[Sec. 4.2 and Appendix D] The coupled Boltzmann equations assume that the dark and visible sectors share a single temperature. Appendix D demonstrates that the SIMP kinetic-equilibration condition Gamma_{chi f -> chi f} > H is satisfied for lambda_chiH ~ 0.1, whereas the relic-density scan in Sec. 4.2 fixes lambda_chiH ~ 1e-3. Since the elastic scattering rate scales approximately as lambda_chiH^2, a reduction by two orders of magnitude may invalidate the kinetic-equilibrium assumption at the benchmark points used in the scan. Please compute Gamma_{chi f -> chi f}/H at the freeze-out temperature for representative scan points with lambda_chiH = 1e-3, or impose the kinetic-equilibrium condition in the scan. This is load-bearing because a dark-sector temperature different from the SM bath temperature changes the form of the Boltzmann equations and the freeze-out conclusions.
[Sec. 3, Fig. 2] The model-independent analysis relies on hand-picked numerical values for <sigma v>_{ss->SM SM}, <sigma v>_{ww->SM SM}, <sigma v2>_{3s->2s}, and the conversion cross-section, with no exploration of how the four-region classification and the pFIMP threshold depend on these inputs. Since the paper claims a model-independent conclusion, please show the robustness of the pFIMP regime under order-of-magnitude variations of these cross-sections, or clearly state that the conclusions are illustrative. At minimum, specify how the conversion cross-section maps to the rate ratios gamma_sw/gamma_3s->2s used to define regions III and IV.
[Sec. 4.1, Eqs. (4.2)-(4.3)] In the concrete model, the conversion term has a factor 1/4 in the pFIMP equation (4.2) and a factor 1/2 in the SIMP equation (4.3). This is consistent with the definition Y_s = 2Y_chi and the process chi chi* -> phi phi, but the reasoning is not stated. Please spell out the connection between Y_s, Y_chi, and the symmetry factors so that the density-balance between the two equations is transparent to the reader.

minor comments (6)

[Sec. 3, after Eq. (3.2)] The symbol mu_sw in the definitions of H(x), s, and Y_eq is not defined. Since the horizontal axis in Fig. 2 is labeled mu_sw/T, please define mu_sw explicitly and clarify the convention for x in a two-mass system.
[Sec. 3] In several places the notation 'sigma^T_{ss->ww}' or 'sigmaT' appears without definition; use a consistent notation for thermally averaged cross-sections.
[Sec. 5] The text 'pFIMP-SMIP model' in the conclusions is a typo; it should read 'pFIMP-SIMP model.'
[Sec. 4.2] The sentence 'allows lambda_chi ~ x 10^-2' is incomplete; it should read 'lambda_chi ~ 10^-2' or give the explicit numerical value.
[Sec. 2] The phrase 'The ncecessary condition' is a typo; it should be 'The necessary condition.'
[Sec. 4.2 and Fig. 4] The text says 'SIMP mass is allowed up to ~ 50 MeV when m_phi > m_chi' in one place and 'Delta m <= 100 MeV' in the conclusions; please ensure the mass-separation statements are consistent and clearly defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pFIMP label is a defined regime, while the coupled-Boltzmann solutions and relic scan are independent numerical content.

full rationale

The paper's central quantities are obtained by solving the coupled Boltzmann equations (3.1)-(3.2) and the concrete model equations (4.2)-(4.3), with conversion couplings and cross-sections treated as scanned inputs; no fitted parameter is renamed as a prediction. Calling the large-conversion regime 'pFIMP' follows from the definition of pFIMP (feeble visible-sector coupling plus sizeable partner coupling), so this is a model classification rather than a derived output recycled as input. The self-citations [36,37] establish the pFIMP concept, but the SIMP-pFIMP dynamics and the mass-splitting/relic bounds are obtained in this paper and are not justified only by those references. Two non-circular caveats are worth recording: eqs. (3.3)-(3.4) are asserted without derivation, and the kinetic-equilibrium check in Appendix D uses lambda_chiH ~ 0.1 while the Sec. 4.2 scan fixes lambda_chiH ~ 1e-3. These are correctness and robustness concerns, not circularity, because no loaded claim reduces to its own input by construction.

Assumptions & free parameters 10 free parameters · 8 assumptions · 3 invented entities

The pFIMP-SIMP claim rests on coupled Boltzmann equations, a single-temperature assumption, and the scalar dark sector defined in Sec. 4. The free parameters are chosen by scan or by hand, and the invented scalars and lepton carry the model's new degrees of freedom.

free parameters (10)

m_chi (SIMP mass) = scanned; relic-allowed up to ~50 MeV in the two-component case
Sets all SIMP cross-sections and freeze-out temperature; scanned subject to relic density and self-interaction bounds.
m_phi (pFIMP mass) = scanned; mass splitting up to ~2 MeV for m_chi > m_phi and ~100 MeV for m_phi > m_chi
Determines the pFIMP yield and the mass hierarchy; scanned in Sec. 4.
lambda_chi (chi quartic coupling) = scanned; ~1e-2 to ~1e-1 for allowed points
Controls SIMP self-interactions and the 3-to-2 cross-section; constrained by unitarity and relic density.
mu_3 (chi cubic coupling) = scanned; mu_3 >= 2 m_chi for allowed points
Drives 3 chi -> 2 chi freeze-out; chosen in the scan with a correlation to the mass.
lambda_chi_phi (chi-phi conversion coupling) = scanned from ~1e-12 to ~1; pFIMP regime near ~1e-6 to ~1e-4
Controls the conversion rate that turns phi into a pFIMP; this is the central parameter of the model.
lambda_chiH (chi-Higgs portal) = fixed at ~1e-3 in the scan; Appendix D uses 0.1
Fixed small to keep chi a SIMP and evade Higgs bounds, but the kinetic-equilibrium check uses a different value.
lambda_phiH (phi-Higgs portal) = fixed at ~1e-12
Fixed tiny to keep phi a FIMP and satisfy invisible Higgs constraints.
lambda_phi (phi quartic) = fixed at 5.25e-2 in the scan
Chosen so that the pFIMP obeys self-interaction bounds while not affecting relic density much.
Thermal cross-section inputs in Sec. 3 = chosen numerical values for <sigma v> and <sigma v^2>
The model-independent cBEQ solutions treat these rates as inputs rather than deriving them from a Lagrangian.
c (analytic SIMP matching constant) = c(c+1)^2 = 4.5
Free constant in Appendix A determined by matching the analytic SIMP solution to the numerical one.

assumptions (8)

standard math FRW cosmology with standard H(x), s(x), and g* evolution
Used in all Boltzmann equations, e.g., eqs. (2.9), (3.1), and (4.2).
domain assumption Boltzmann equations with Maxwell-Boltzmann equilibrium yields
Assumed throughout; standard in DM relic calculations but an approximation.
domain assumption Z2 x Z3 symmetry stabilizes the two dark scalars
Postulated in Sec. 4 to prevent decay; no dynamical origin is provided.
domain assumption CP conservation within the dark sector
Assumed before eq. (4.2), with Y_chi = Y_chi*.
domain assumption Kinetic equilibrium between SM and dark sectors, T_DM = T_SM
Assumed in Sec. 5 and Appendix D; validated only for lambda_chiH ~ 0.1, not for the scan value ~1e-3.
standard math Vacuum stability conditions on the scalar potential
Eq. (2.8) and Sec. 4.2 impose positivity bounds on the scalar couplings.
domain assumption Perturbativity and unitarity bounds on couplings
Used to restrict the scan ranges, eqs. (2.6)-(2.7).
domain assumption g_s* approx g_rho* approx constant during SIMP freeze-out
Used in Appendix A to obtain the semi-analytic solution; approximate for MeV scales.

invented entities (3)

Real scalar phi (pFIMP)
purpose: Second DM component with tiny Higgs coupling that equilibrates through conversion with chi.
Postulated dark sector particle; no direct evidence or predicted observable outside the model.
Complex scalar chi (SIMP)
purpose: Primary DM component with 3 chi -> 2 chi self-annihilation and visible-sector scattering.
Postulated dark sector particle; a standard SIMP field.
Vector-like lepton psi (detection extension)
purpose: Adds a new portal for electron scattering and annihilation signals of the pFIMP-SIMP sector.
Introduced in Sec. 4.3 solely to make detection possible; no signal rate is computed in this paper.

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Cite this review

Pith. "Pith review of Pseudo-FIMP dark matter in presence of a SIMP." pith.science (2026). https://pith.science/paper/VV7EHQ4V

@misc{pith2026241115108,
  author       = {Pith},
  title        = {Pith review of: Pseudo-FIMP dark matter in presence of a SIMP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VV7EHQ4V}},
  note         = {Machine review of arXiv:2411.15108}
}

abstract

Pseudo-feebly Interacting Massive Particle (pFIMP) has been postulated in two component dark matter (DM) scenarios, where it has feeble interaction with the visible sector, but sizeable one with a thermal bath partner. In this work, we study the possibility and dynamics of pFIMP in presence of a Strongly Interacting Massive Particle (SIMP), which is well known to solve too-big-to-fail and core-vs-cusp problems. Our analysis is primarily model-independent via solving coupled Boltzmann equations, with negligible DM-DM conversion adhering to pure SIMP-FIMP limit, and then with larger DM-DM conversion rate pertaining to SIMP-pFIMP limit. We also illustrate the simplest model yielding pFIMP-SIMP set-up having two scalars stabilised under $\mathbb{Z}_2\otimes \mathbb{Z}_3$ symmetry, and explore the accessible parameter space after addressing relic density, unitarity, self interaction constraints etc. pFIMP detectability is limited in such circumstances, but possible via a thermal DM loop when the SIMP has a visible sector interaction via light mediator.

Figures

Figures reproduced from arXiv: 2411.15108 by the authors.

**Figure 1.** figs . 1a and 1b represent the relic and unitarity allowed parameter space in mχ − Ωχh 2 plane for a complex scalar SIMP (χ) described by the Lagrangian in eq. (2.2). The variation of the relevant couplings (µ3 , λχ) is shown in the color bar, while the ones fixed is mentioned in the figure heading. Red and green points satisfy the self interaction limits from Bullet and Abell cluster bounds, respectively. The param… view at source ↗

**Figure 2.** Solution to cBEQs 3.1, and 3.2 for pFIMP-SIMP scenario; figs . 2a, 2c represent ms > mw case, while figs . 2b, 2d represent ms < mw case. Figs . 2a, 2b represent the variation of DM yield with dimensionless parameter µsw/T for different values of conversion cross-section illustrated by different colored lines, solid for SIMP, dashed for pFIMP. Figs . 2c, 2d represent DM relics as a function of DM-DM conversion cross… view at source ↗

**Figure 3.** Solution to the cBEQ 4.3, where figs . 3a and 3b represents mχ > mϕ and mχ < mϕ scenarios respectively. The thick, dashed and dotted color lines represent the SIMP, FIMP and pFIMP cases respectively. Different colors show λχϕ variation as mentioned in the figure inset. The thick black and dashed lines represent the SIMP and pFIMP equilibrium yields. Other parameter kept fixed are mentioned in the figure heading. The… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Figs . 4a, 4b and 4c, 4d are represent the DM relic allowed parameter space in mχ − λχ and mχ − µ3 plane, respectively. The variation of portal coupling λχϕ is shown by the BrownCyanTones color bar. The red and green points represent the points within the Bullet and A…

**Figure 5.** Figure 5: A comparison plot analytic vs numeric solution of BEQ where we have considered g s ⋆ = g ρ ⋆ = 10.75, gs = 2 (for complex scalar) and have chosen c(c + 1)2 = 4.5. B Possible Feynman diagrams related to DM phenomenology We have calculated the squared matrix amplitude fo…

**Figure 6.** Figure 6: Self annihilation via χ(p1 )χ(p2 )χ(p3 ) → χ(p4 )χ ∗ (p5 ) process. – 17 – [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: Self-annihilation of SIMP via χχ ∗ χ ∗ → χχ process. |M7 | 2 = µ 2 3 64m 8 χ 117µ 2 3 − 148λχm 2 χ 2 . (B.3) and ⟨σv 2 ⟩χχ ∗ χ ∗→χχ = 1 64πm 3 χ K1 (mχ/T) K2 (mχ/T) 3 √ 5 6 |Mχχ ∗ χ ∗→χχ| 2 . (B.4) χ χ ϕ ϕ [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: Conversion of SIMP to pFIMP via χχ ∗ → ϕϕ process. – 18 – [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Dark matter self scattering relic contribution for a particular DM and the analytic expression could be written as [33], σself mDM = Ωχ ΩDM 2 1 mχ σχχ→χχ + σχ ∗ χ ∗→χ ∗ χ ∗ + σχχ ∗→χχ ∗ + σχχ ∗→ϕϕ + Ωϕ ΩDM 2 1 mϕ σϕϕ→ϕϕ + Ωχ ΩDM Ωϕ ΩDM 2 mχ + mϕ σϕχ→ϕχ + σϕχ…

**Figure 10.** Figure 10: The Feynman diagram represent the DM (χ) and SM fermions (f) scattering. We already discussed that the kinetic equilibrium of SIMP with SM is achieved after considering an appropriate choice of portal coupling. Only relativistic fermions are available during SIMP fre…

**Figure 11.** Figure 11: The evaluation of DM (χ)-fermion elastic scattering rate with temperature. The different color lines represent the different mχ masses in the GeV unit, and coupling is mentioned in the figure inset. the evolution of DM-SM elastic scattering rate with the SM bath tempe…

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