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How AI and Blockchain are Converging: A Look into IVS Crypto 2024's Innovative Approach

এআই এবং ব্লকচেইন কীভাবে একত্রিত হচ্ছে: IVS ক্রিপ্টো 2024-এর উদ্ভাবনী পদ্ধতির দিকে একটি নজর

Wie KI und Blockchain zusammenwachsen: Ein Blick auf den innovativen Ansatz von IVS Crypto 2024

Cómo convergen la IA y Blockchain: una mirada al enfoque innovador de IVS Crypto 2024

Comment l'IA et la blockchain convergent : un aperçu de l'approche innovante d'IVS Crypto 2024

एआई और ब्लॉकचेन कैसे एक दूसरे से जुड़ रहे हैं: आईवीएस क्रिप्टो 2024 के अभिनव दृष्टिकोण पर एक नज़र

AIとブロックチェーンの融合: IVS Crypto 2024の革新的なアプローチを検証

AI와 블록체인이 융합되는 방식: IVS Crypto 2024의 혁신적인 접근 방식 살펴보기

Como a IA e o Blockchain estão convergindo: uma análise da abordagem inovadora do IVS Crypto 2024

Как ИИ и блокчейн объединяются: взгляд на инновационный подход IVS Crypto 2024

Yapay Zeka ve Blockchain Nasıl Birleşiyor: IVS Crypto 2024'ün Yenilikçi Yaklaşımına Bir Bakış

AI và Blockchain đang hội tụ như thế nào: Nhìn vào phương pháp tiếp cận đổi mới của IVS Crypto 2024

人工智能与区块链如何融合：了解 IVS Crypto 2024 的创新方法

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Nicolas Bernal, New York University Abu Dhabi;
(2) Partha Konar, Physical Research Laboratory;
(3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro
S-matrix: Unitarity and its Consequences
Dark Matter Annihilation and Unitarity Bound
Low-temperature Reheating
Freeze-out with a Low-temperature Reheating
Summary and Conclusion
Acknowledgments and References 4. Low-temperature Reheating where T corresponds to the temperature of the SM bath. It follows that the Hubble expansion rate H is therefore the SM entropy density implies that the temperature of the SM bath scales as However, it is interesting to emphasize that the standard cosmological scenario is not granted and that alternative cosmologies could also have occurred [121]. In the following, we focus on cases characterized by low-temperature reheating. This reheating could correspond to the period just after the end of inflation, or to a secondary period in which an extra component beyond SM radiation dominated the energy density of the universe. In particular, two scenarios will be reviewed: one where the extra component ϕ that dominated the expansion of the universe has an energy density that gets diluted faster than radiation and does not decay (that is, a kination-like scenario), and the other where ϕ scales as nonrelativistic matter and decays into SM particles (that is, an early matter-dominated scenario). These two scenarios will be described below. 4.1. Kination-like In this scenario, the universe was dominated by a component ϕ whose energy density redshifts faster than free radiation [113], as with n > 0. A typical example of this scenario corresponds to kination [122, 123], where n = 2. However, larger values for n are also possible, appearing, for example, in the context of ekpyrotic [124, 125] or cyclic scenarios [126–129]. where we have taken into account that, as ϕ is not decaying, the SM entropy is conserved, and therefore the SM temperature follows the standard scaling shown in Eq. (4.4). 4.2. Early matter domination It follows that the Hubble expansion rate is Having settled the evolution of the background, in the next section the dynamics of the thermal DM in such alternative cosmological scenarios, and in particular the impact on the unitarity limit, will be carefully studied. [4] It is worth mentioning that, in general, the non-adiabatic period could have been preceded by an adiabatic period with ϕ-domination, and by another era dominated by SM radiation. Here, however, we assume that the DM freeze-out happens in the non-adiabatic period so that previous stages of the universe play no role. This is true if ϕ is identified with the inflaton or simply if the nonadiabatic era is long enough. This paper is available on arxiv under CC 4.0 license. Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Acknowledgments and References Abstract and Intro Abstract and Intro S-matrix: Unitarity and its Consequences S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Low-temperature Reheating Freeze-out with a Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Summary and Conclusion Acknowledgments and References Acknowledgments and References 4. Low-temperature Reheating where T corresponds to the temperature of the SM bath. It follows that the Hubble expansion rate H is therefore T H the SM entropy density implies that the temperature of the SM bath scales as However, it is interesting to emphasize that the standard cosmological scenario is not granted and that alternative cosmologies could also have occurred [121]. In the following, we focus on cases characterized by low-temperature reheating. This reheating could correspond to the period just after the end of inflation, or to a secondary period in which an extra component beyond SM radiation dominated the energy density of the universe. In particular, two scenarios will be reviewed: one where the extra component ϕ that dominated the expansion of the universe has an energy density that gets diluted faster than radiation and does not decay (that is, a kination-like scenario), and the other where ϕ scales as nonrelativistic matter and decays into SM particles (that is, an early matter-dominated scenario). These two scenarios will be described below. 4.1. Kination-like In this scenario, the universe was dominated by a component ϕ whose energy density redshifts faster than free radiation [113], as with n > 0. A typical example of this scenario corresponds to kination [122, 123], where n = 2. However, larger values for n are also possible, appearing, for example, in the context of ekpyrotic [124, 125] or cyclic scenarios [126–129]. where we have taken into account that, as ϕ is not decaying, the SM entropy is conserved, and therefore the SM temperature follows the standard scaling shown in Eq. (4.4). 4.2. Early matter domination It follows that the Hubble expansion rate is Having settled the evolution of the background, in the next section the dynamics of the thermal DM in such alternative cosmological scenarios, and in particular the impact on the unitarity limit, will be carefully studied. [4] It is worth mentioning that, in general, the non-adiabatic period could have been preceded by an adiabatic period with ϕ-domination, and by another era dominated by SM radiation. Here, however, we assume that the DM freeze-out happens in the non-adiabatic period so that previous stages of the universe play no role. This is true if ϕ is identified with the inflaton or simply if the nonadiabatic era is long enough.

Unitarity Bound on Dark Matter: Low-temperature Reheating

ডার্ক ম্যাটারে ঐক্যবদ্ধতা: নিম্ন-তাপমাত্রা পুনরায় গরম করা

Unitaritätsgrenze für Dunkle Materie: Wiedererwärmung bei niedrigen Temperaturen

Unitaridad unida a la materia oscura: recalentamiento a baja temperatura

Unitarité liée à la matière noire : réchauffage à basse température

डार्क मैटर पर यूनिटैरिटी बाउंड: कम तापमान पर पुनः गर्म करना

暗黒物質のユニタリー性限界：低温再加熱

암흑물질의 단일성: 저온 재가열

Unitariedade vinculada à matéria escura: reaquecimento em baixa temperatura

Унитарность, связанная с темной материей: низкотемпературный разогрев

Karanlık Maddeye Bağlı Birlik: Düşük Sıcaklıkta Yeniden Isıtma

Sự thống nhất bị ràng buộc trên vật chất tối: Làm nóng lại ở nhiệt độ thấp

暗物质的幺正性界限：低温再加热

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Nicolas Bernal, New York University Abu Dhabi;
(2) Partha Konar, Physical Research Laboratory;
(3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro
S-matrix: Unitarity and its Consequences
Dark Matter Annihilation and Unitarity Bound
Low-temperature Reheating
Freeze-out with a Low-temperature Reheating
Summary and Conclusion
Acknowledgments and References 2. S-matrix: Unitarity and its Consequences For the sake of simplicity, here we will focus on the collisions of nonidentical spin-0 particles. However, the case of other spins can be generalized simply by using Refs. [13, 114]. We carry out the basis transformation from the momentum eigenstates to a general basis state by using rotational invariance. Now, we derive the total cross section, the sum of the elastic and inelastic cross sections, in the channel n with the help of the optical theorem in Eq. (2.1) and the general expression of the matrix element in Eq. (2.2) Then, one can easily obtain the inelastic cross section by subtracting the elastic in Eq. (2.3) from the total cross sections in Eq. (2.4) Finally, we note that for a collision among identical particles, an additional multiplicative factor of 2 in the cross section is required to avoid double counting [114]. This paper is available on arxiv under CC 4.0 license. Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Acknowledgments and References Abstract and Intro Abstract and Intro S-matrix: Unitarity and its Consequences S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Low-temperature Reheating Freeze-out with a Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Summary and Conclusion Acknowledgments and References Acknowledgments and References 2. S-matrix: Unitarity and its Consequences For the sake of simplicity, here we will focus on the collisions of nonidentical spin-0 particles. However, the case of other spins can be generalized simply by using Refs. [13, 114]. We carry out the basis transformation from the momentum eigenstates to a general basis state by using rotational invariance. Now, we derive the total cross section, the sum of the elastic and inelastic cross sections, in the channel n with the help of the optical theorem in Eq. (2.1) and the general expression of the matrix element in Eq. (2.2) Then, one can easily obtain the inelastic cross section by subtracting the elastic in Eq. (2.3) from the total cross sections in Eq. (2.4) Finally, we note that for a collision among identical particles, an additional multiplicative factor of 2 in the cross section is required to avoid double counting [114].

Unitarity Bound on Dark Matter: S-matrix - Unitarity and its Consequences

ডার্ক ম্যাটারে একতা আবদ্ধ: এস-ম্যাট্রিক্স - একতা এবং এর পরিণতি

Unitaritätsgrenze für Dunkle Materie: S-Matrix - Unitarität und ihre Konsequenzen

La unitaridad unida a la materia oscura: S-matrix - La unitaridad y sus consecuencias

L'Unitarité liée à la matière noire : matrice S - L'Unitarité et ses conséquences

डार्क मैटर पर यूनिटैरिटी बाउंड: एस-मैट्रिक्स - यूनिटैरिटी और इसके परिणाम

暗黒物質のユニタリー性限界: S 行列 - ユニタリー性とその結果

암흑 물질에 결합된 단일성: S-행렬 - 단일성과 그 결과

Unitariedade vinculada à matéria escura: matriz S - Unitariedade e suas consequências

Унитарность, связанная с темной материей: S-матрица - Унитарность и ее последствия

Karanlık Maddeye Bağlı Birlik: S-matrisi - Birlik ve Sonuçları

Tính đơn nhất bị ràng buộc trên vật chất tối: Ma trận S - Tính đơn nhất và hệ quả của nó

暗物质的幺正性界限：S 矩阵 - 幺正性及其后果

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Nicolas Bernal, New York University Abu Dhabi;
(2) Partha Konar, Physical Research Laboratory;
(3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro
S-matrix: Unitarity and its Consequences
Dark Matter Annihilation and Unitarity Bound
Low-temperature Reheating
Freeze-out with a Low-temperature Reheating
Summary and Conclusion
Acknowledgments and References 5. Freeze-out with a Low-temperature Reheating In this section, two cases for the DM freeze-out are considered. The first corresponds to the visible freeze-out, where a couple of DM particles annihilate into a couple of SM states, with a total thermally-averaged annihilation cross section ⟨σv⟩. The evolution of the DM number density n can be described with the Boltzmann equation [20] 5.1. Kination-like In the following, Eqs. (5.6) and (5.7) will be analytically solved in the context of a kination-like cosmology. For convenience, we start with the case corresponding to the dark freeze-out. 5.1.1. Dark freeze-out If the freeze-out occurs during the radiation-dominated era, Eq. (5.7) can be analytically solved, from the DM freeze-out until today (i.e., small temperature and therefore large x) To match the whole observed DM relic density, it is required that Alternatively, if the freeze-out happens during reheating the integral has been split into two pieces, to emphasize the two regimes of H in Eq. (4.6). Therefore 5.1.2. Visible freeze-out The case of the visible freeze-out in Eq. (5.6) can be computed following the same procedure presented in the previous subsection. However, one could also derive it by fixing r = 2 in Eqs. (5.10) and (5.13), which gives for the freeze-out in the radiation-dominated era, or during reheating. 5.2. Early matter domination The freeze-out temperature can be estimated by comparing Eqs. (4.8) and (5.4) or (5.5), and is given by Next, analytical solutions are presented for Eqs. (5.16) and (5.17) in the context of an early matter-dominated scenario. We will begin with the case corresponding to the dark freeze-out for convenience. 5.2.1. Dark freeze-out If the freeze-out occurs during the radiation era, the solution of Eq. (5.17), or equivalently of Eq. (5.7), is the one presented in Eq. (5.10). Instead, if it occurs during the reheating period, one has that 5.2.2. Visible freeze-out If the freeze-out occurs during radiation domination, the solution of Eq. (5.16) is the same as the one of Eq. (5.14). Alternatively, if it occurs during reheating, one has instead simply corresponding to the limit r = 2 of Eq. (5.20). This paper is available on arxiv under CC 4.0 license. Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Acknowledgments and References Abstract and Intro Abstract and Intro S-matrix: Unitarity and its Consequences S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Low-temperature Reheating Freeze-out with a Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Summary and Conclusion Acknowledgments and References Acknowledgments and References 5. Freeze-out with a Low-temperature Reheating In this section, two cases for the DM freeze-out are considered. The first corresponds to the visible freeze-out, where a couple of DM particles annihilate into a couple of SM states, with a total thermally-averaged annihilation cross section ⟨σv⟩. The evolution of the DM number density n can be described with the Boltzmann equation [20] 5.1. Kination-like In the following, Eqs. (5.6) and (5.7) will be analytically solved in the context of a kination-like cosmology. For convenience, we start with the case corresponding to the dark freeze-out. 5.1.1. Dark freeze-out If the freeze-out occurs during the radiation-dominated era, Eq. (5.7) can be analytically solved, from the DM freeze-out until today (i.e., small temperature and therefore large x) To match the whole observed DM relic density, it is required that Alternatively, if the freeze-out happens during reheating the integral has been split into two pieces, to emphasize the two regimes of H in Eq. (4.6). Therefore 5.1.2. Visible freeze-out The case of the visible freeze-out in Eq. (5.6) can be computed following the same procedure presented in the previous subsection. However, one could also derive it by fixing r = 2 in Eqs. (5.10) and (5.13), which gives for the freeze-out in the radiation-dominated era, or during reheating. 5.2. Early matter domination The freeze-out temperature can be estimated by comparing Eqs. (4.8) and (5.4) or (5.5), and is given by Next, analytical solutions are presented for Eqs. (5.16) and (5.17) in the context of an early matter-dominated scenario. We will begin with the case corresponding to the dark freeze-out for convenience. 5.2.1. Dark freeze-out If the freeze-out occurs during the radiation era, the solution of Eq. (5.17), or equivalently of Eq. (5.7), is the one presented in Eq. (5.10). Instead, if it occurs during the reheating period, one has that 5.2.2. Visible freeze-out If the freeze-out occurs during radiation domination, the solution of Eq. (5.16) is the same as the one of Eq. (5.14). Alternatively, if it occurs during reheating, one has instead simply corresponding to the limit r = 2 of Eq. (5.20).

Unitarity Bound on Dark Matter: Freeze-out with a Low-temperature Reheating

ডার্ক ম্যাটারে একতাবদ্ধতা: নিম্ন-তাপমাত্রা পুনরায় গরম করার সাথে ফ্রিজ-আউট

Unitaritätsgrenze für Dunkle Materie: Einfrieren durch Wiedererwärmung bei niedrigen Temperaturen

Unitaridad unida a la materia oscura: congelación con recalentamiento a baja temperatura

Unitarité liée à la matière noire : gel avec réchauffement à basse température

डार्क मैटर पर यूनिटैरिटी बाउंड: कम तापमान पर पुनः गर्म करने से फ्रीज-आउट

暗黒物質のユニタリー性限界：低温再加熱による凍結

암흑 물질에 결합된 단일성: 저온 재가열로 인한 동결

Unitariedade vinculada à matéria escura: congelamento com reaquecimento em baixa temperatura

Унитарность, связанная с темной материей: замерзание с низкотемпературным разогревом

Karanlık Maddeye Bağlı Birlik: Düşük Sıcaklıkta Yeniden Isıtma ile Donma

Sự thống nhất bị ràng buộc trên vật chất tối: Đóng băng khi hâm nóng ở nhiệt độ thấp

暗物质的幺正性界限：低温再加热冻结

Unitarity Bound on Dark Matter in Low-temperature Reheating Scenarios: Acknowledgments and Reference

নিম্ন-তাপমাত্রা পুনঃউষ্ণকরণের পরিস্থিতিতে ডার্ক ম্যাটারে ঐক্যবদ্ধতা: স্বীকৃতি এবং রেফারেন্স

Unitaritätsgrenze für Dunkle Materie in Niedertemperatur-Wiedererwärmungsszenarien: Danksagungen und Referenz

Unitaridad unida a la materia oscura en escenarios de recalentamiento a baja temperatura: reconocimientos y referencias

Unitarité liée à la matière noire dans les scénarios de réchauffage à basse température : remerciements et référence

निम्न-तापमान पुनःतापन परिदृश्यों में डार्क मैटर पर यूनिटैरिटी बाउंड: आभार और संदर्भ

低温再加熱シナリオにおける暗黒物質のユニタリー性限界: 謝辞と参考文献

저온 재가열 시나리오에서 암흑물질에 결합된 단일성: 승인 및 참고자료

Unitariedade vinculada à matéria escura em cenários de reaquecimento de baixa temperatura: reconhecimentos e referências

Унитарность, связанная с темной материей в сценариях низкотемпературного разогрева: благодарности и ссылки

Düşük Sıcaklıkta Yeniden Isıtma Senaryolarında Karanlık Maddeye Bağlı Üniterlik: Teşekkür ve Referans

Tính thống nhất bị ràng buộc đối với Vật chất tối trong các kịch bản hâm nóng ở nhiệt độ thấp: Lời cảm ơn và tài liệu tham khảo

低温再加热场景中暗物质的幺正性界限：致谢和参考文献

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Nicolas Bernal, New York University Abu Dhabi;
(2) Partha Konar, Physical Research Laboratory;
(3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro
S-matrix: Unitarity and its Consequences
Dark Matter Annihilation and Unitarity Bound
Low-temperature Reheating
Freeze-out with a Low-temperature Reheating
Summary and Conclusion
Acknowledgments and References Abstract Model-independent theoretical upper bound on the thermal dark matter (DM) mass can be derived from the maximum inelastic DM cross-section featuring the whole observed DM abundance. We deploy partial-wave unitarity of the scattering matrix to derive the maximal thermally-averaged cross-section for general number-changing processes r → 2 (with r ≥ 2), which may involve standard model particles or occur solely within the dark sector. The usual upper limit on the DM mass for s-wave annihilation is around 130 TeV (1 GeV) for r = 2 (3), only applies in the case of a freeze out occurring in the standard cosmological scenario. We consider the effects of two nonstandard cosmological evolutions, characterized by low-temperature reheating: i) a kination-like scenario and ii) an early matter-dominated scenario. In the first case, the early freeze-out strengthens the unitarity bound to few TeVs for WIMPs; while in the second case, WIMP DM can be as heavy as ∼ 1010 GeV due to a large entropy dilution. 1. Introduction Consideration of a specific DM production paradigm in the early stage of the universe may further constrain the mass range for a viable DM candidate. For instance, the number-changing pair annihilation of DM to SM particles determines its present mass density, where it maintains the chemical and kinetic equilibrium with the thermal soup in the early universe. Interestingly, the requirement of the unitarity of the S-matrix sets a model-independent upper bound on the DM mass for this scenario [12, 13]. The implication of unitarity offers the maximum inelastic cross-section, which fixes the minimum number density of the frozen-out DM. Using this number density, one can establish the maximum allowed DM mass by fulfilling the observed relic density of it. In the theories of DM with long-range forces, bound states of DM can form and therefore relax the unitarity bound by suppressing the inelastic annihilation rate [14–16]. In addition, dark sectors with particle-antiparticle asymmetry enforce the apprehend of the nonzero equilibrium chemical potential for DM, further constraining the unitarity limits due to the demand for an increased effective DM number density at the time of freeze-out [15, 17]. Furthermore, different indirect searches for DM may put a lower limit on DM mass for some specified scenarios. A strong model-independent lower bound for the thermal DM that is annihilating to visible states through an s-wave process is about 20 GeV [18]. In addition, a more restrictive lower limit has recently been found. It has been shown that the lower bound is 200 GeV, considering H.E.S.S. and other updated observational data [19]. In particular, all the DM scenarios mentioned so far pay attention to the 2 → 2 number-changing process where a DM pair annihilates into a pair of SM particles, that is, the Weakly Interacting Massive Particle (WIMP) paradigm [20–22].[1] Moreover, it is not necessary that the number-changing processes involve SM particles, so they may also occur within the dark sector. The minimalistic realization of this scenario is the 3 → 2 process, where this kind of number-changing reaction involves a single DM species. In general, such processes arise in DM theories with new sizable self-interactions, and in several contexts as self-interacting DM [29–31], the Strongly Interacting Massive Particle (SIMP) paradigm [32–49], or even the ELastically DEcoupling Relic (ELDER) scenario [50, 51]. It is essential to mention that the early history of the universe plays a crucial role in DM genesis, since the decoupling of thermal DM occurred at that time. Generally, the studies of DM consider the standard cosmological picture in which the radiation energy density is assumed to dominate the energy budget before the Big Bang Nucleosynthesis (BBN). However, there is no direct evidence for the energy content at very high temperatures. Therefore, it is vital to look at the effects of modified cosmology on the production of DM. In recent times, the evolution of the DM in the period of non-standard expansion usually triggered by the decay of a long-lived massive particle [48, 52–71] or by Hawking evaporation of primordial black holes [72–95] is receiving increasing attention.[2] All such studies point towards the fact that non-standard cosmology alters the value of the thermally averaged cross section needed to satisfy the observed relic of DM. Such a modification in the thermally averaged cross-section may also change the unitarity mass bound of DM. In a recent article, the authors studied the impact of early matter domination on unitarity limits [112]. This article is decorated as follows. In Sections 2 and 3, we present the detailed derivation of the maximum thermally-averaged cross-section allowed by the unitarity of the Smatrix. We discuss two different non-standard cosmological pictures: kination-like and late-time reheating in Section 4. Section 5 shows the analytical expressions for freeze-out and cross sections for the radiation-dominated universe and the mentioned modified cosmologies, and we also demonstrate our results. Finally, we summarize our findings in Section 6. [1] Alternatively, one can also have in the final state a DM and a SM particle (semi-annihilations) [23–27], or in the initial state a DM and another particle of the dark sector (coannihilations) [28]. [2] For studies on baryogenesis with a low reheating temperature or during an early matter-dominated phase, see Refs. [52, 96–100] and [101–104], respectively. Furthermore, the production of primordial gravitational waves in scenarios with an early matter era has recently received particular attention [105–111]. This paper is available on arxiv under CC 4.0 license. Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Acknowledgments and References Abstract and Intro Abstract and Intro S-matrix: Unitarity and its Consequences S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Low-temperature Reheating Freeze-out with a Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Summary and Conclusion Acknowledgments and References Acknowledgments and References Abstract Model-independent theoretical upper bound on the thermal dark matter (DM) mass can be derived from the maximum inelastic DM cross-section featuring the whole observed DM abundance. We deploy partial-wave unitarity of the scattering matrix to derive the maximal thermally-averaged cross-section for general number-changing processes r → 2 (with r ≥ 2), which may involve standard model particles or occur solely within the dark sector. The usual upper limit on the DM mass for s-wave annihilation is around 130 TeV (1 GeV) for r = 2 (3), only applies in the case of a freeze out occurring in the standard cosmological scenario. We consider the effects of two nonstandard cosmological evolutions, characterized by low-temperature reheating: i) a kination-like scenario and ii) an early matter-dominated scenario. In the first case, the early freeze-out strengthens the unitarity bound to few TeVs for WIMPs; while in the second case, WIMP DM can be as heavy as ∼ 1010 GeV due to a large entropy dilution. 1. Introduction Consideration of a specific DM production paradigm in the early stage of the universe may further constrain the mass range for a viable DM candidate. For instance, the number-changing pair annihilation of DM to SM particles determines its present mass density, where it maintains the chemical and kinetic equilibrium with the thermal soup in the early universe. Interestingly, the requirement of the unitarity of the S-matrix sets a model-independent upper bound on the DM mass for this scenario [12, 13]. The implication of unitarity offers the maximum inelastic cross-section, which fixes the minimum number density of the frozen-out DM. Using this number density, one can establish the maximum allowed DM mass by fulfilling the observed relic density of it. In the theories of DM with long-range forces, bound states of DM can form and therefore relax the unitarity bound by suppressing the inelastic annihilation rate [14–16]. In addition, dark sectors with particle-antiparticle asymmetry enforce the apprehend of the nonzero equilibrium chemical potential for DM, further constraining the unitarity limits due to the demand for an increased effective DM number density at the time of freeze-out [15, 17]. Furthermore, different indirect searches for DM may put a lower limit on DM mass for some specified scenarios. A strong model-independent lower bound for the thermal DM that is annihilating to visible states through an s-wave process is about 20 GeV [18]. In addition, a more restrictive lower limit has recently been found. It has been shown that the lower bound is 200 GeV, considering H.E.S.S. and other updated observational data [19]. In particular, all the DM scenarios mentioned so far pay attention to the 2 → 2 number-changing process where a DM pair annihilates into a pair of SM particles, that is, the Weakly Interacting Massive Particle (WIMP) paradigm [20–22].[1] Moreover, it is not necessary that the number-changing processes involve SM particles, so they may also occur within the dark sector. The minimalistic realization of this scenario is the 3 → 2 process, where this kind of number-changing reaction involves a single DM species. In general, such processes arise in DM theories with new sizable self-interactions, and in several contexts as self-interacting DM [29–31], the Strongly Interacting Massive Particle (SIMP) paradigm [32–49], or even the ELastically DEcoupling Relic (ELDER) scenario [50, 51]. It is essential to mention that the early history of the universe plays a crucial role in DM genesis, since the decoupling of thermal DM occurred at that time. Generally, the studies of DM consider the standard cosmological picture in which the radiation energy density is assumed to dominate the energy budget before the Big Bang Nucleosynthesis (BBN). However, there is no direct evidence for the energy content at very high temperatures. Therefore, it is vital to look at the effects of modified cosmology on the production of DM. In recent times, the evolution of the DM in the period of non-standard expansion usually triggered by the decay of a long-lived massive particle [48, 52–71] or by Hawking evaporation of primordial black holes [72–95] is receiving increasing attention.[2] All such studies point towards the fact that non-standard cosmology alters the value of the thermally averaged cross section needed to satisfy the observed relic of DM. Such a modification in the thermally averaged cross-section may also change the unitarity mass bound of DM. In a recent article, the authors studied the impact of early matter domination on unitarity limits [112]. This article is decorated as follows. In Sections 2 and 3, we present the detailed derivation of the maximum thermally-averaged cross-section allowed by the unitarity of the Smatrix. We discuss two different non-standard cosmological pictures: kination-like and late-time reheating in Section 4. Section 5 shows the analytical expressions for freeze-out and cross sections for the radiation-dominated universe and the mentioned modified cosmologies, and we also demonstrate our results. Finally, we summarize our findings in Section 6. [1] Alternatively, one can also have in the final state a DM and a SM particle (semi-annihilations) [23–27], or in the initial state a DM and another particle of the dark sector (coannihilations) [28]. [2] For studies on baryogenesis with a low reheating temperature or during an early matter-dominated phase, see Refs. [52, 96–100] and [101–104], respectively. Furthermore, the production of primordial gravitational waves in scenarios with an early matter era has recently received particular attention [105–111].

Unitarity Bound on Dark Matter in Low-temperature Reheating Scenarios: Abstract and Intro

নিম্ন-তাপমাত্রা পুনঃউষ্ণকরণের দৃশ্যে ডার্ক ম্যাটারে ঐক্যবদ্ধতা: বিমূর্ত এবং ভূমিকা

Unitaritätsgrenze für Dunkle Materie in Niedertemperatur-Wiedererwärmungsszenarien: Zusammenfassung und Einführung

Unitaridad ligada a la materia oscura en escenarios de recalentamiento a baja temperatura: resumen e introducción

Unitarité liée à la matière noire dans des scénarios de réchauffage à basse température : résumé et introduction

निम्न-तापमान पुनःतापन परिदृश्यों में डार्क मैटर पर यूनिटैरिटी बाउंड: सार और परिचय

低温再加熱シナリオにおける暗黒物質のユニタリー性限界：概要と序論

저온 재가열 시나리오에서 암흑물질에 결합된 단일성: 개요 및 소개

Unitariedade vinculada à matéria escura em cenários de reaquecimento em baixa temperatura: resumo e introdução

Унитарность, связанная с темной материей в сценариях низкотемпературного повторного нагрева: Аннотация и введение

Düşük Sıcaklıkta Yeniden Isıtma Senaryolarında Karanlık Maddeye Bağlı Birlik: Özet ve Giriş

Tính đơn nhất bị ràng buộc đối với Vật chất tối trong các kịch bản hâm nóng ở nhiệt độ thấp: Tóm tắt và giới thiệu

低温再加热场景中暗物质的幺正性界限：摘要和简介

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Nicolas Bernal, New York University Abu Dhabi;
(2) Partha Konar, Physical Research Laboratory;
(3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro
S-matrix: Unitarity and its Consequences
Dark Matter Annihilation and Unitarity Bound
Low-temperature Reheating
Freeze-out with a Low-temperature Reheating
Summary and Conclusion
Acknowledgments and References 6. Summary and Conclusion The requirement of the de Broglie wavelength of dark matter (DM) to hold it inside galaxies and the stability of the stellar galaxy cluster collectively put a broad allowed mass range for DM by providing the lower and upper bound of DM mass, respectively. Specifying some properties of DM can further tighten the mass range. Interestingly, one can put the model-independent upper bound by specifying the thermal production of DM in the early universe. The observed DM abundance and unitarity of partial waves from the scattering matrix jointly place an upper limit on DM mass. Primarily, the upper limits on the inelastic cross section for a general number-changing process 2 → r can be derived with the help of the optical theorem, the matrix elements and the elastic scattering cross section for the process. After that, one can obtain the thermally averaged cross section for the r → 2 process by invoking the principle of detailed balance. Finally, the unitarity bounds on the thermally averaged cross section translate to the upper limits on the DM mass satisfying the relic density constraints. It is known that the maximum allowed DM mass for the 2 → 2 and 3 → 2 DM annihilation processes is around 130 TeV and 1 GeV, respectively. However, these bounds not only depend on the particle physics model, but have a strong dependence on the cosmological evolution of the universe, being valid only if the universe followed the so-called “standard cosmological scenario”. Instead, this article explores the DM mass bound in nonstandard cosmological setups characterized by low-temperature reheating. In particular, we focus on i) kination-like scenarios, where the early universe was dominated by a fluid with an energy density that gets diluted faster than free radiation, and ii) early matter-dominated scenarios, where a component with an energy density that scales as nonrelativistic matter dominates the early universe and eventually decays into SM particles. First, we study the kination-like universe, which demands a larger thermally-averaged annihilation cross section to saturate the observed abundance of DM compared to the standard radiation-dominated picture since, in this case, freeze-out occurs early. As a result, the upper bound on the DM mass becomes more stringent than in the standard case. For example, if the reheating temperature is as low as a few MeVs (corresponding to the start of the Big Bang nucleosynthesis epoch), the usual bound of the DM mass m ≲ 130 TeV can be reduced to a few TeVs for WIMPs. Before closing, we want to emphasize that the evolution of the early universe is largely unknown. The standard assumption of a universe dominated by standard-model radiation from the end of cosmological inflation until matter-radiation equality, together with a transition from an inflaton-dominated to a radiation-dominated universe occurring at a very early time, cannot be taken for granted. Having that in mind, here we have studied the impact of the unitarity bound on DM in the case of low-temperature reheating scenarios. This paper is available on arxiv under CC 4.0 license. Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Acknowledgments and References Abstract and Intro Abstract and Intro S-matrix: Unitarity and its Consequences S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Low-temperature Reheating Freeze-out with a Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Summary and Conclusion Acknowledgments and References Acknowledgments and References 6. Summary and Conclusion The requirement of the de Broglie wavelength of dark matter (DM) to hold it inside galaxies and the stability of the stellar galaxy cluster collectively put a broad allowed mass range for DM by providing the lower and upper bound of DM mass, respectively. Specifying some properties of DM can further tighten the mass range. Interestingly, one can put the model-independent upper bound by specifying the thermal production of DM in the early universe. The observed DM abundance and unitarity of partial waves from the scattering matrix jointly place an upper limit on DM mass. Primarily, the upper limits on the inelastic cross section for a general number-changing process 2 → r can be derived with the help of the optical theorem, the matrix elements and the elastic scattering cross section for the process. After that, one can obtain the thermally averaged cross section for the r → 2 process by invoking the principle of detailed balance. Finally, the unitarity bounds on the thermally averaged cross section translate to the upper limits on the DM mass satisfying the relic density constraints. It is known that the maximum allowed DM mass for the 2 → 2 and 3 → 2 DM annihilation processes is around 130 TeV and 1 GeV, respectively. However, these bounds not only depend on the particle physics model, but have a strong dependence on the cosmological evolution of the universe, being valid only if the universe followed the so-called “standard cosmological scenario”. Instead, this article explores the DM mass bound in nonstandard cosmological setups characterized by low-temperature reheating. In particular, we focus on i ) kination-like scenarios, where the early universe was dominated by a fluid with an energy density that gets diluted faster than free radiation, and ii ) early matter-dominated scenarios, where a component with an energy density that scales as nonrelativistic matter dominates the early universe and eventually decays into SM particles. i ii First, we study the kination-like universe, which demands a larger thermally-averaged annihilation cross section to saturate the observed abundance of DM compared to the standard radiation-dominated picture since, in this case, freeze-out occurs early. As a result, the upper bound on the DM mass becomes more stringent than in the standard case. For example, if the reheating temperature is as low as a few MeVs (corresponding to the start of the Big Bang nucleosynthesis epoch), the usual bound of the DM mass m ≲ 130 TeV can be reduced to a few TeVs for WIMPs. Before closing, we want to emphasize that the evolution of the early universe is largely unknown. The standard assumption of a universe dominated by standard-model radiation from the end of cosmological inflation until matter-radiation equality, together with a transition from an inflaton-dominated to a radiation-dominated universe occurring at a very early time, cannot be taken for granted. Having that in mind, here we have studied the impact of the unitarity bound on DM in the case of low-temperature reheating scenarios.

Unitarity Bound on Dark Matter in Low-temperature Reheating Scenarios: Summary and Conclusion

নিম্ন-তাপমাত্রা পুনঃউষ্ণকরণের পরিস্থিতিতে ডার্ক ম্যাটারে একতা আবদ্ধ: সারাংশ এবং উপসংহার

Unitaritätsgrenze für Dunkle Materie in Niedertemperatur-Wiedererwärmungsszenarien: Zusammenfassung und Schlussfolgerung

Unitaridad unida a la materia oscura en escenarios de recalentamiento a baja temperatura: resumen y conclusión

Unitarité liée à la matière noire dans les scénarios de réchauffage à basse température : résumé et conclusion

निम्न-तापमान पुनःतापन परिदृश्यों में डार्क मैटर पर यूनिटैरिटी बाउंड: सारांश और निष्कर्ष

低温再加熱シナリオにおける暗黒物質のユニタリー性限界：要約と結論

저온 재가열 시나리오에서 암흑물질에 결합된 단일성: 요약 및 결론

Limitação de unitaridade na matéria escura em cenários de reaquecimento em baixa temperatura: resumo e conclusão

Унитарность, связанная с темной материей в сценариях низкотемпературного разогрева: итоги и выводы

Düşük Sıcaklıkta Yeniden Isıtma Senaryolarında Karanlık Maddeye Bağlı Üniterlik: Özet ve Sonuç

Tính đơn nhất bị ràng buộc đối với Vật chất tối trong các kịch bản hâm nóng ở nhiệt độ thấp: Tóm tắt và kết luận

低温再加热场景中暗物质的幺正性界限：总结和结论

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Nicolas Bernal, New York University Abu Dhabi;
(2) Partha Konar, Physical Research Laboratory;
(3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro
S-matrix: Unitarity and its Consequences
Dark Matter Annihilation and Unitarity Bound
Low-temperature Reheating
Freeze-out with a Low-temperature Reheating
Summary and Conclusion
Acknowledgments and References 3. Dark Matter Annihilation and Unitarity Bound Now, the Boltzmann equation boils down to the following simplified form Now, we can obtain the maximum thermally averaged rate for the r → 2 process using Eq. (3.2), as where g stands for the internal degrees of freedom of the DM. Therefore, the maximum value of the thermally averaged s-wave annihilation cross section for 3 → 2 can be written as Likewise, one can also express the maximum value of the thermally average s-wave annihilation cross section for 4 → 2 as [3] Although the expression of Eq. (3.5) is derived considering equal mass for all the particles involved in the interaction; it provides maximum thermally-averaged cross section for WIMPs in the case r = 2 where the mass of the final-state particles can be different from the same of the initial-state particles. This paper is available on arxiv under CC 4.0 license. Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Nicolas Bernal, New York University Abu Dhabi; (2) Partha Konar, Physical Research Laboratory; (3) Sudipta Show, Physical Research Laboratory. Table of Links Abstract and Intro S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Acknowledgments and References Abstract and Intro Abstract and Intro S-matrix: Unitarity and its Consequences S-matrix: Unitarity and its Consequences Dark Matter Annihilation and Unitarity Bound Dark Matter Annihilation and Unitarity Bound Low-temperature Reheating Low-temperature Reheating Freeze-out with a Low-temperature Reheating Freeze-out with a Low-temperature Reheating Summary and Conclusion Summary and Conclusion Acknowledgments and References Acknowledgments and References 3. Dark Matter Annihilation and Unitarity Bound Now, the Boltzmann equation boils down to the following simplified form Now, we can obtain the maximum thermally averaged rate for the r → 2 process using Eq. (3.2), as where g stands for the internal degrees of freedom of the DM. Therefore, the maximum value of the thermally averaged s-wave annihilation cross section for 3 → 2 can be written as Likewise, one can also express the maximum value of the thermally average s-wave annihilation cross section for 4 → 2 as [3] Although the expression of Eq. (3.5) is derived considering equal mass for all the particles involved in the interaction; it provides maximum thermally-averaged cross section for WIMPs in the case r = 2 where the mass of the final-state particles can be different from the same of the initial-state particles.

Unitarity Bound on Dark Matter: Dark Matter Annihilation and Unitarity Bound

ডার্ক ম্যাটারে একতা আবদ্ধ: ডার্ক ম্যাটার অ্যানিহিলেশন এবং ইউনিটারিটি বাউন্ড

Unitaritätsgrenze für Dunkle Materie: Vernichtung Dunkler Materie und Unitaritätsgrenze

Unitaridad unida a la materia oscura: aniquilación de la materia oscura y unitaridad unida

L'unité liée à la matière noire : l'annihilation de la matière noire et l'unité liée

डार्क मैटर पर यूनिटैरिटी बाउंड: डार्क मैटर का विनाश और यूनिटैरिटी बाउंड

暗黒物質のユニタリー性限界: 暗黒物質の消滅とユニタリー性限界

암흑 물질에 결합된 단일성: 암흑 물질 소멸 및 결합된 단일성

Limite de unidade à matéria escura: aniquilação da matéria escura e limite de unidade

Унитарность, связанная с темной материей: Аннигиляция темной материи и граница унитарности

Karanlık Maddeye Bağlı Üniterlik: Karanlık Maddenin Yok Edilmesi ve Üniterliğe Bağlı

Giới hạn đơn nhất trên vật chất tối: Sự hủy diệt vật chất tối và giới hạn đơn nhất

暗物质的幺正界限：暗物质湮灭与幺正界限

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