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The product team is one of the most critical aspects of any startup. If an organization is to succeed and be self-sustaining beyond initial funding, it must maintain execution speed while simultaneously evolving its workflows to accommodate growth — responsibilities that largely fall on the shoulders of product managers. Pearce Dolan has overcome these challenges twice. At digital banking platform Revolut, he joined as the brand’s first dedicated product manager and launched features like bill-splitting functionalities and GIF payments before they became standard in fintech. He’s now the head of product at payroll and HR service provider Deel, where he built a product team that delivered a global HR platform in just six months (compared to the industry standard of 1–2 years). His experiences offer a wealth of insight into developing product teams that can build and maintain momentum through rapid growth. From Technology Enthusiast to Product Leader From Technology Enthusiast to Product Leader As a child, Pearce Dolan was dreaming about creating technology while his peers preferred to consume it. “At a young age, I would buy technology magazines and read them cover to cover,” he recalls. “My ambition was to one day work for technology companies and help them strategize on how to make the best products in the world.” Pearce Dolan Though he was born in England, his family moved to Hong Kong when he was 10, then to Dubai a couple of years later. He didn’t realize it at the time, but these global experiences would prove invaluable when it came time to build products for worldwide audiences. “Spending my formative years in international settings really influenced my view of the world,” Dolan reflects. “Now, it’s where I feel most comfortable — in an international setting.” He went on to study computer science and business management at King’s College London, planning to work as an engineer before transitioning to product management. But when the opportunity came early, he seized it. Expanding Revolut’s Digital Banking Services When Dolan joined Revolut as its 250th employee, the company was just starting to expand its offerings beyond its core currency exchange service. It was a hire that the organization had needed to make for a long time: While prior engineers had served in the role out of necessity, Dolan was the organization’s first dedicated product manager. He ran the company’s first sprint, conducted its first A/B test, pioneered customer interview processes — he basically created the entire product culture. The focus was speed. With traditional banking notorious for its slow, bureaucratic development cycles, speed would be Revolut’s greatest advantage. To this end, Dolan focused on creating systems that would allow the company to iterate faster than its competitors, including lightweight approval processes, direct customer feedback loops, and rapid testing protocols that garnered feedback in days rather than months. The result was a suite of features that redefined what users expected from financial applications: Bill-splitting functionalities, making dividing expenses frictionless
Shareable payment links, eliminating the need for bank details
QR code payments, launching long before widespread European adoption
Group and username-based payments, simplifying transactions Bill-splitting functionalities, making dividing expenses frictionless Shareable payment links, eliminating the need for bank details QR code payments, launching long before widespread European adoption Group and username-based payments, simplifying transactions But one of Dolan’s most innovative additions, and one that would have ripple effects throughout the sector, was the ability to pay alongside GIFs — an industry first that added an unprecedented emotional connection to financial transactions. “The idea came to me while walking down the streets of London,” Dolan says. “I still have the doodles!” The feature was built in just two weeks, and in testing, Dolan and his team saw an increase in payment frequency and that one-third of all payments were sent with a GIF. After launching and achieving widespread popularity, the feature was later adopted by major payment platforms globally. Over the next two and a half years, in part due to Dolan’s innovations and streamlining of product development, Revolut’s user base exploded from 3.5 million to 10 million. Its revenue nearly tripled, achieving decacorn status with a $10 billion valuation (a figure that would later climb to $45 billion). Building a Global Team at Deel Dolan’s next challenge came at Deel, a SaaS company that empowers businesses worldwide to hire, manage, and pay talent globally — a challenge that became particularly urgent as remote work expanded. When Dolan was brought on as the brand’s 20th employee, the existing infrastructure was minimal: “There was me, a first-time designer they found on Reddit, a CTO, eight engineers, and no QA.” This lean structure demanded breakneck speeds from Dolan and his team. If Revolut’s product culture was fast-paced, Deel’s was practically frenzied in its infancy. “The culture was hyper execution driven,” Dolan says. “We moved fast and broke things. Literally. We would build things in crazy rapid time, push it to production with no QA, and crash the site multiple times a day.” This pace wasn’t recklessness — it was strategy. Deel was solving a completely new problem as it defined what a global HR platform should be from scratch. Speed wasn’t just an advantage; it was fundamental to the organization’s success. It was a strategy that paid off in spades. Deel built the world’s first global human resource information system (HRIS) in just six months — a process that competitors would typically take 1-2 years to complete. Building upon this momentum, Dolan oversaw the rapid development and deployment of other vital features like: Global immigration services, helping migrant workers secure their visas
Employer of record, a service that employs workers on behalf of companies in countries where they don’t have legal entities
Contractor of record, an industry-first solution with advanced compliant controls, shielding employers from risk and empowering them to hire and pay contractors around the world Global immigration services, helping migrant workers secure their visas Employer of record, a service that employs workers on behalf of companies in countries where they don’t have legal entities Contractor of record, an industry-first solution with advanced compliant controls, shielding employers from risk and empowering them to hire and pay contractors around the world As a result, Deel became the fastest-growing SaaS company in history, reaching $100 million in annual revenue just 20 months after hitting $1 million. The product team expanded to over 1,000 product managers and engineers, distributed across 150 specialized teams. Finally, the company scaled to 5,000 employees operating in more than 100 countries — all without establishing any physical offices. fastest-growing SaaS company in history fastest-growing SaaS company in history While this explosive growth speaks to the virtues of speed and feature development, Dolan stresses the importance of adapting your product strategy as you go. At Deel, for example, Dolan began to prioritize high-level strategy as he fine-tuned his vision for the product org. While singular focus and breakneck speeds were what propelled Deel to initial success, it wouldn’t be a sustainable practice at scale. This meant the critical transformation from a monolithic product to specialized teams — that is, evolving from having a single product team working on one platform to having multiple teams focusing on different verticals. “Each stage of the journey has required a different approach and behavior," Dolan reflects. What works at 50 employees won’t work at 500, and what works at 500 won’t work at 5,000. The best teams, therefore, are the ones that adapt the fastest. Achieving a Balanced Approach to Product Excellence Pearce Dolan’s success demonstrates that the companies that thrive aren’t necessarily those with the best ideas or most resources. Instead, much of the impressive growth that Revolut and Deel achieved can be attributed to Dolan’s work in establishing a strong foundation of product practices early on, prioritizing speed of execution over perfection, and creating adaptable systems that can evolve as the organization scales. The product leaders who master this balancing act create the foundation for sustainable growth that outlasts any single product feature or market trend. That’s what effective product leadership in high-growth companies requires — and it’s something that Dolan has demonstrated throughout his career.

How Pearce Dolan Built World-Class Product Teams at Revolut and Deel

كيف بنيت Pearce Dolan مجموعة منتجات عالمية في Revolut و Deel

“Pearce Dolan” “Revolut” və “DELECE” şirkəti ilə dünya standartlarında məhsul komandalarını necə qurub

কিভাবে পিরস ডোলান Revolut এবং Deel এ বিশ্বমানের পণ্য টিমগুলি তৈরি করেছিলেন

Cómo Pearce Dolan construyó equipos de producto de clase mundial en Revolut y Deel

Pearce Dolan が Revolut と Deel で世界クラスの製品チームを構築した方法

Kaip Pearce Dolan sukūrė pasaulinės klasės produktų komandas Revolut ir Deel

Jak Pearce Dolan zbudował zespoły produktów światowej klasy w Revolut i Deel

Cum a construit Pearce Dolan echipe de produse de clasă mondială la Revolut și Deel

Ako Pearce Dolan vybudoval tímy produktov svetovej triedy v spoločnosti Revolut a Deel

Paano Pearce Dolan Bumuo ng World-Class Product Teams sa Revolut at Deel

Pearce Dolan 如何在 Revolut 和 Deel 建立世界一流的产品团队

Ungayifaka kanjani i-Pearce Dolan i-World-Class Product Teams ku-Revolut ne-Deg

Table of Links Abstract


Introduction
Background
Reinforcement Learning
Similar Work


Methodology
DRLAgent Design


Training Procedures


Testing Procedures


Results


SABR Experiments


Conclusions


Appendix A


References REFERENCES Assa, H., C. Kenyon, and H. Zhang. 2021. “Assessing Reinforcement Delta Hedging.” SSRN SSRN 3918375. http://dx.doi.org/10.2139/ssrn.3918375. Black, F., and M. Scholes. 1973. “The Pricing of Options and Corporate Liabilities” 81 (3): 637– 54. https://doi.org/10.1086/260062. Cao, Jay, Jacky Chen, Soroush Farghadani, John Hull, Zissis Poulos, Zeyu Wang, and Jun Yuan. 2023. “Gamma and Vega Hedging Using Deep Distributional Reinforcement Learning.” Frontiers in Artificial Intelligence 6. https://www.frontiersin.org/articles/10.3389/frai.2023.1129370. Cao, Jay, Jacky Chen, John Hull, and Zissis Poulos. 2021. “Deep Hedging of Derivatives Using Reinforcement Learning.” The Journal of Financial Data Science 3 (1): 10–27. https://doi.org/10.3905/jfds.2020.1.052. Chebyshev, P. 1864. “Sur l’interpolation.” Zapiski Akademii Nauk 1 (4): 539–60. Du, Jiayi, Muyang Jin, Petter N. Kolm, Gordon Ritter, Yixuan Wang, and Bofei Zhang. 2020. “Deep Reinforcement Learning for Option Replication and Hedging.” The Journal of Financial Data Science 2 (4): 44–57. https://doi.org/10.3905/jfds.2020.1.045. Fathi, A., and B. Hientzsch. 2023. “A Comparison of Reinforcement Learning and Deep Trajectory Based Stochastic Control Agents for Stepwise Mean-Variance Hedging.” arXiv arXiv:2302.07996. Florescu, Ionuţ, and Cristian Gabriel Pãsãricã. 2009. “A Study about the Existence of the Leverage Effect in Stochastic Volatility Models.” Physica A: Statistical Mechanics and Its Applications 388 (4): 419–32. Giurca, B., and S. Borovkova. 2021. “Delta Hedging of Derivatives Using Deep Reinforcement Learning.” SSRN SSRN 3847272. https://ssrn.com/abstract=3847272. Glau, Kathrin, Micro Mahlstedt, and Christian Potz. 2018. “A New Approach for American Option Pricing: The Dynamic Chebyshev Method.” arXiv. https://doi.org/10.48550/arXiv.1806.05579. Hagan, Patrick, Deep Kumar, Andrew Lesniewski, and Diana Woodward. 2002. “Managing Smile Risk.” Wilmott Magazine 1 (January): 84–108. Halperin, Igor. 2017. “QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds.” The Journal of Derivatives 28 (1): 99–122. https://doi.org/10.3905/jod.2020.1.108. Heston, Steven L. 1993. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” The Review of Financial Studies 6 (2): 327–43. Hull, John, 1946-. 2012. Options, Futures, and Other Derivatives. Eighth edition. Boston : Prentice Hall, [2012] ©2012. https://search.library.wisc.edu/catalog/9910112878402121. Kolm, Petter N., and Gordon Ritter. 2019. “Dynamic Replication and Hedging: A Reinforcement Learning Approach.” The Journal of Financial Data Science 1 (1): 159–71. https://doi.org/10.3905/jfds.2019.1.1.159. Lillicrap, Timothy, Johnathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. 2015. “Continuous Control with Deep Reinforcement Learning.” arXiv arXiv:1509.02971. https://doi.org/10.48550/arXiv.1509.02971. Lin, Long-Ji. 1992. “Self-Improving Reactive Agents Based on Reinforcement Learning, Planning and Teaching.” Machine Learning 8 (3): 293–321. https://doi.org/10.1007/BF00992699. Liu, Peng. 2023. “A Review on Derivative Hedging Using Reinforcement Learning.” The Journal of Financial Data Science. https://doi.org/10.3905/jfds.2023.1.124. Longstaff, Francis A., and Eduardo S. Schwartz. 2001. “Valuing American Options by Simulation: A Simple Least-Squares Approach.” UCLA: Finance. https://escholarship.org/uc/item/43n1k4jb. Mikkilä, Oskari, and Juho Kanniainen. 2023. “Empirical Deep Hedging.” Quantitative Finance 23 (1): 111–22. https://doi.org/10.1080/14697688.2022.2136037. Mnih, V., K. Kavukcuoglu, David Silver, A. Graves, I. Antonoglou, Daan Wierstra, and M. Riedmiller. 2013. “Playing Atari with Deep Reinforcement Learning.” arXiv arXiv:1312.5602. https://doi.org/10.48550/arXiv.1312.5602. Murray, Phillip, B. Wood, H. Buehler, Magnus Wiese, and Mikko Pakkanen. 2022. “Deep Hedging: Continuous Reinforcement Learning for Hedging of General Portfolios across Multiple Risk Aversions.” In ICAIF ’22: Proceedings of the Third ACM International Conference on AI in Finance, 361–68. Association for Computing Machinery. https://doi.org/10.1145/3533271.3561731. Pham, Uyen, Quoc Luu, and Hien Tran. 2021. “Multi-Agent Reinforcement Learning Approach for Hedging Portfolio Problem.” Soft Computing 25 (12): 7877–85. https://doi.org/10.1007/s00500-021-05801-6. Pickard, Reilly, and Yuri Lawryshyn. 2023. “Deep Reinforcement Learning for Dynamic Stock Option Hedging: A Review.” Mathematics 11 (24). https://doi.org/10.3390/math11244943. Silver, David, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George van den Driessche, Julian Schrittwieser, et al. 2016. “Mastering the Game of Go with Deep Neural Networks and Tree Search.” Nature 529 (7587): 484–89. https://doi.org/10.1038/nature16961. Smith, Leslie N. 2018. “A Disciplined Approach to Neural Network Hyper-Parameters: Part 1 -- Learning Rate, Batch Size, Momentum, and Weight Decay.” https://doi.org/10.48550/arXiv.1803.09820. Sutton, Richard S., and Andrew G. Barto. 2018. Reinforcement Learning: An Introduction. A Bradford Book. Watkins, Christopher John Cornish Hellaby. 1989. “Learning from Delayed Rewards.” King’s College. https://www.cs.rhul.ac.uk/~chrisw/new_thesis.pdf. Xiao, B., W. Yao, and X. Zhou. 2021. “Optimal Option Hedging with Policy Gradient.” In 2021 International Conference on Data Mining Workshops (ICDMW), 1112–19. IEEE. https://doi.org/10.1109/ICDMW53433.2021.00145. Xu, Wei, and Bing Dai. 2022. “Delta-Gamma–Like Hedging with Transaction Cost under Reinforcement Learning Technique.” The Journal of Derivatives 29 (5): 60–82. https://doi.org/10.3905/jod.2022.1.156. Zheng, C., J. He, and C. Yang. 2023. “Option Dynamic Hedging Using Reinforcement Learning.” arXiv arXiv:2306.10743. Authors:
(1) Reilly Pickard, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada (reilly.pickard@mail.utoronto.ca);
(2) Finn Wredenhagen, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada;
(3) Julio DeJesus, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada;
(4) Mario Schlener, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada;
(5) Yuri Lawryshyn, Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canada. This paper is available on arxiv under CC BY-NC-SA 4.0 Deed (Attribution-Noncommercial-Sharelike 4.0 International) license. Table of Links Abstract Introduction
Background
Reinforcement Learning
Similar Work Methodology
DRLAgent Design Training Procedures Testing Procedures Results SABR Experiments Conclusions Appendix A References Abstract Abstract Abstract Introduction Background Reinforcement Learning Similar Work Introduction Background Background Reinforcement Learning Reinforcement Learning Similar Work Similar Work Methodology DRLAgent Design Methodology Methodology DRLAgent Design DRLAgent Design Training Procedures Training Procedures Training Procedures Testing Procedures Testing Procedures Testing Procedures Results Results Results SABR Experiments SABR Experiments SABR Experiments Conclusions Conclusions Conclusions Appendix A Appendix A Appendix A References References References REFERENCES Assa, H., C. Kenyon, and H. Zhang. 2021. “Assessing Reinforcement Delta Hedging.” SSRN SSRN 3918375. http://dx.doi.org/10.2139/ssrn.3918375. Black, F., and M. Scholes. 1973. “The Pricing of Options and Corporate Liabilities” 81 (3): 637– 54. https://doi.org/10.1086/260062. Cao, Jay, Jacky Chen, Soroush Farghadani, John Hull, Zissis Poulos, Zeyu Wang, and Jun Yuan. 2023. “Gamma and Vega Hedging Using Deep Distributional Reinforcement Learning.” Frontiers in Artificial Intelligence 6. https://www.frontiersin.org/articles/10.3389/frai.2023.1129370. Cao, Jay, Jacky Chen, John Hull, and Zissis Poulos. 2021. “Deep Hedging of Derivatives Using Reinforcement Learning.” The Journal of Financial Data Science 3 (1): 10–27. https://doi.org/10.3905/jfds.2020.1.052. Chebyshev, P. 1864. “Sur l’interpolation.” Zapiski Akademii Nauk 1 (4): 539–60. Du, Jiayi, Muyang Jin, Petter N. Kolm, Gordon Ritter, Yixuan Wang, and Bofei Zhang. 2020. “Deep Reinforcement Learning for Option Replication and Hedging.” The Journal of Financial Data Science 2 (4): 44–57. https://doi.org/10.3905/jfds.2020.1.045. Fathi, A., and B. Hientzsch. 2023. “A Comparison of Reinforcement Learning and Deep Trajectory Based Stochastic Control Agents for Stepwise Mean-Variance Hedging.” arXiv arXiv:2302.07996. Florescu, Ionuţ, and Cristian Gabriel Pãsãricã. 2009. “A Study about the Existence of the Leverage Effect in Stochastic Volatility Models.” Physica A: Statistical Mechanics and Its Applications 388 (4): 419–32. Giurca, B., and S. Borovkova. 2021. “Delta Hedging of Derivatives Using Deep Reinforcement Learning.” SSRN SSRN 3847272. https://ssrn.com/abstract=3847272. Glau, Kathrin, Micro Mahlstedt, and Christian Potz. 2018. “A New Approach for American Option Pricing: The Dynamic Chebyshev Method.” arXiv. https://doi.org/10.48550/arXiv.1806.05579. Hagan, Patrick, Deep Kumar, Andrew Lesniewski, and Diana Woodward. 2002. “Managing Smile Risk.” Wilmott Magazine 1 (January): 84–108. Halperin, Igor. 2017. “QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds.” The Journal of Derivatives 28 (1): 99–122. https://doi.org/10.3905/jod.2020.1.108. Heston, Steven L. 1993. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” The Review of Financial Studies 6 (2): 327–43. Hull, John, 1946-. 2012. Options, Futures, and Other Derivatives. Eighth edition. Boston : Prentice Hall, [2012] ©2012. https://search.library.wisc.edu/catalog/9910112878402121. Kolm, Petter N., and Gordon Ritter. 2019. “Dynamic Replication and Hedging: A Reinforcement Learning Approach.” The Journal of Financial Data Science 1 (1): 159–71. https://doi.org/10.3905/jfds.2019.1.1.159. Lillicrap, Timothy, Johnathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. 2015. “Continuous Control with Deep Reinforcement Learning.” arXiv arXiv:1509.02971. https://doi.org/10.48550/arXiv.1509.02971. Lin, Long-Ji. 1992. “Self-Improving Reactive Agents Based on Reinforcement Learning, Planning and Teaching.” Machine Learning 8 (3): 293–321. https://doi.org/10.1007/BF00992699. Liu, Peng. 2023. “A Review on Derivative Hedging Using Reinforcement Learning.” The Journal of Financial Data Science. https://doi.org/10.3905/jfds.2023.1.124. Longstaff, Francis A., and Eduardo S. Schwartz. 2001. “Valuing American Options by Simulation: A Simple Least-Squares Approach.” UCLA: Finance. https://escholarship.org/uc/item/43n1k4jb. Mikkilä, Oskari, and Juho Kanniainen. 2023. “Empirical Deep Hedging.” Quantitative Finance 23 (1): 111–22. https://doi.org/10.1080/14697688.2022.2136037. Mnih, V., K. Kavukcuoglu, David Silver, A. Graves, I. Antonoglou, Daan Wierstra, and M. Riedmiller. 2013. “Playing Atari with Deep Reinforcement Learning.” arXiv arXiv:1312.5602. https://doi.org/10.48550/arXiv.1312.5602. Murray, Phillip, B. Wood, H. Buehler, Magnus Wiese, and Mikko Pakkanen. 2022. “Deep Hedging: Continuous Reinforcement Learning for Hedging of General Portfolios across Multiple Risk Aversions.” In ICAIF ’22: Proceedings of the Third ACM International Conference on AI in Finance, 361–68. Association for Computing Machinery. https://doi.org/10.1145/3533271.3561731. Pham, Uyen, Quoc Luu, and Hien Tran. 2021. “Multi-Agent Reinforcement Learning Approach for Hedging Portfolio Problem.” Soft Computing 25 (12): 7877–85. https://doi.org/10.1007/s00500-021-05801-6. Pickard, Reilly, and Yuri Lawryshyn. 2023. “Deep Reinforcement Learning for Dynamic Stock Option Hedging: A Review.” Mathematics 11 (24). https://doi.org/10.3390/math11244943. Silver, David, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George van den Driessche, Julian Schrittwieser, et al. 2016. “Mastering the Game of Go with Deep Neural Networks and Tree Search.” Nature 529 (7587): 484–89. https://doi.org/10.1038/nature16961. Smith, Leslie N. 2018. “A Disciplined Approach to Neural Network Hyper-Parameters: Part 1 -- Learning Rate, Batch Size, Momentum, and Weight Decay.” https://doi.org/10.48550/arXiv.1803.09820. Sutton, Richard S., and Andrew G. Barto. 2018. Reinforcement Learning: An Introduction. A Bradford Book. Watkins, Christopher John Cornish Hellaby. 1989. “Learning from Delayed Rewards.” King’s College. https://www.cs.rhul.ac.uk/~chrisw/new_thesis.pdf. Xiao, B., W. Yao, and X. Zhou. 2021. “Optimal Option Hedging with Policy Gradient.” In 2021 International Conference on Data Mining Workshops (ICDMW), 1112–19. IEEE. https://doi.org/10.1109/ICDMW53433.2021.00145. Xu, Wei, and Bing Dai. 2022. “Delta-Gamma–Like Hedging with Transaction Cost under Reinforcement Learning Technique.” The Journal of Derivatives 29 (5): 60–82. https://doi.org/10.3905/jod.2022.1.156. Zheng, C., J. He, and C. Yang. 2023. “Option Dynamic Hedging Using Reinforcement Learning.” arXiv arXiv:2306.10743. Authors: (1) Reilly Pickard, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada (reilly.pickard@mail.utoronto.ca); (2) Finn Wredenhagen, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada; (3) Julio DeJesus, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada; (4) Mario Schlener, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada; (5) Yuri Lawryshyn, Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canada. Authors: Authors: (1) Reilly Pickard, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada ( reilly.pickard@mail.utoronto.ca ); reilly.pickard@mail.utoronto.ca (2) Finn Wredenhagen, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada; (3) Julio DeJesus, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada; (4) Mario Schlener, Ernst & Young LLP, Toronto, ON, M5H 0B3, Canada; (5) Yuri Lawryshyn, Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canada. This paper is available on arxiv under CC BY-NC-SA 4.0 Deed (Attribution-Noncommercial-Sharelike 4.0 International) license. This paper is available on arxiv under CC BY-NC-SA 4.0 Deed (Attribution-Noncommercial-Sharelike 4.0 International) license. available on arxiv available on arxiv

Hedging American Put Options with Deep Reinforcement Learning: References

SAN FRANCISCO, United States, October 28th, 2024/Chainwire/--Another set of EasyA hackathon winners (also known as “gigabrains”) are headed to Y Combinator. Artemiy Malyshau and Jeevan Juttla attended their first EasyA x Polkadot hackathon nearly two years ago, where they first started experimenting with the ideas that would later become Gecko Sec. Today, they’ve just been accepted into the world-famous Y Combinator accelerator, which has backed some of the world’s most successful Web3 companies like CoinBase, Filecoin and many more. According to the team, Gecko Sec lets Web3 developers build secure code quickly without wasting time on tools that don’t deliver results, or relying on one-time human pentests that quickly become outdated. As they continue to develop their groundbreaking software, they’re working on rolling this out for teams building on Polkadot. Writing code that is secure and safe is one of the biggest concerns for Web3 developers, with many millions of dollars spent on security audits every month in Web3 alone. EasyA gigabrains Jeevan Jutla and Artemiy Malyshau credit EasyA with helping them get off the ground and giving them the inspiration to succeed. “What started off at an EasyA hackathon has led to us getting backed by Y Combinator. Big shoutout to Phil and Dom for pushing us out of our comfort zone and getting us here!” said Artemiy Malyshau, Co-Founder of Gecko Sec. Jeevan Juttla and Artemiy Malyshau have also got highly accomplished professional and educational backgrounds. Jeevan Juttla, CEO and Co-founder of Gecko Sec, graduated with a First Class Degree in Electrical Engineering from King’s College London. After graduating, Jeevan was hired by the UK government’s National Cyber Security Centre to secure the British Government’s data and subsequently joined Binance as a Security Engineer. Artemiy Malyshau also graduated with First Class Honours in Electrical Engineering from King’s College London, one of the UK’s top universities. Shortly after this, he earned a Master’s Degree with Distinction in Applied Computational Science and Engineering from Imperial College University. GeckoSec joins a long list of EasyA hackathon winners who’ve gone on to achieve storied success in blockchain. Other EasyA hackathon winners like Axal, founded by Harvard grad Ashlan Ahmed, have been backed by a16z and are planning to announce their latest fundraising round later this month. To date, EasyA alumni have already founded companies valued at nearly $3 billion. EasyA has received interest from VCs who want access to its startups pipeline. Although it won’t share the precise details yet, EasyA shares that it has its sights set on launching a fund designed to invest exclusively in its gigabrain community and hackathon winners. "Seeing our EasyA gigabrains go from launching at our hackathons to getting accepted into YC is testament to the talent density and founder spirit within the EasyA community. I’ve never seen anything quite like it.” said Dom Kwok, Co-Founder of EasyA. Much of the success of EasyA hackathon winners can be traced to the company’s co-founders, who have been instrumental in spotting young talent and nurturing them as founders. Users can go to any EasyA hackathon and find Phil and Dom on the ground, inspiring developers to launch the next unicorns. EasyA co-founders Phil and Dom have been active in blockchain for over 10 years. Prior to EasyA, Phil worked at prestigious law firm Sullivan & Cromwell and Dom worked at the world’s largest Private Equity firm, The Blackstone Group. Their educational backgrounds have also given them, and EasyA, a unique pipeline of talented founders across the US and Europe. Phil was a top scholar at Cambridge University, and Dom studied at the Wharton School, University of Pennsylvania, where he graduated Cum Laude as a Joseph Wharton and Benjamin Franklin Scholar. According to Phil and Dom, they’re just getting started. Numerous high-profile fundraising announcements are coming out of the EasyA community over the coming months. About EasyA EasyA is one of Web3’s most popular apps, making it possible for anyone to learn about Web3 right from their phones. Learners earn rewards for mastering new skills, and the best ones are invited to in-person hackathons to launch their startups in world-leading hubs like San Francisco, London and Singapore. EasyA alumni have founded startups valued at nearly $3 billion and have raised from top VCs like a16z, Founders Fund, YC and many more. Launched by brothers Phil and Dom Kwok, top grads from the University of Cambridge and The Wharton School respectively, EasyA has over 1 million users and has won Apple’s highly-coveted App of the Day award. Users can learn more: https://www.easya.io/ Contact Dom Kwok dom@easya.io This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here. SAN FRANCISCO, United States, October 28th, 2024/Chainwire/--Another set of EasyA hackathon winners (also known as “gigabrains”) are headed to Y Combinator. Artemiy Malyshau and Jeevan Juttla attended their first EasyA x Polkadot hackathon nearly two years ago, where they first started experimenting with the ideas that would later become Gecko Sec . Artemiy Malyshau Artemiy Malyshau Jeevan Juttla Jeevan Juttla Gecko Sec Gecko Sec Today, they’ve just been accepted into the world-famous Y Combinator accelerator, which has backed some of the world’s most successful Web3 companies like CoinBase, Filecoin and many more. According to the team, Gecko Sec lets Web3 developers build secure code quickly without wasting time on tools that don’t deliver results, or relying on one-time human pentests that quickly become outdated. As they continue to develop their groundbreaking software, they’re working on rolling this out for teams building on Polkadot. Writing code that is secure and safe is one of the biggest concerns for Web3 developers, with many millions of dollars spent on security audits every month in Web3 alone. EasyA gigabrains Jeevan Jutla and Artemiy Malyshau credit EasyA with helping them get off the ground and giving them the inspiration to succeed. “What started off at an EasyA hackathon has led to us getting backed by Y Combinator. Big shoutout to Phil and Dom for pushing us out of our comfort zone and getting us here!” said Artemiy Malyshau, Co-Founder of Gecko Sec. “What started off at an EasyA hackathon has led to us getting backed by Y Combinator. Big shoutout to Phil and Dom for pushing us out of our comfort zone and getting us here!” said Artemiy Malyshau, Co-Founder of Gecko Sec. Jeevan Juttla and Artemiy Malyshau have also got highly accomplished professional and educational backgrounds. Jeevan Juttla, CEO and Co-founder of Gecko Sec, graduated with a First Class Degree in Electrical Engineering from King’s College London. After graduating, Jeevan was hired by the UK government’s National Cyber Security Centre to secure the British Government’s data and subsequently joined Binance as a Security Engineer. Artemiy Malyshau also graduated with First Class Honours in Electrical Engineering from King’s College London, one of the UK’s top universities. Shortly after this, he earned a Master’s Degree with Distinction in Applied Computational Science and Engineering from Imperial College University. GeckoSec joins a long list of EasyA hackathon winners who’ve gone on to achieve storied success in blockchain. Other EasyA hackathon winners like Axal , founded by Harvard grad Ashlan Ahmed, have been backed by a16z and are planning to announce their latest fundraising round later this month. To date, EasyA alumni have already founded companies valued at nearly $3 billion. Axal Axal EasyA has received interest from VCs who want access to its startups pipeline. Although it won’t share the precise details yet, EasyA shares that it has its sights set on launching a fund designed to invest exclusively in its gigabrain community and hackathon winners. "Seeing our EasyA gigabrains go from launching at our hackathons to getting accepted into YC is testament to the talent density and founder spirit within the EasyA community. I’ve never seen anything quite like it.” said Dom Kwok, Co-Founder of EasyA. "Seeing our EasyA gigabrains go from launching at our hackathons to getting accepted into YC is testament to the talent density and founder spirit within the EasyA community. I’ve never seen anything quite like it.” said Dom Kwok, Co-Founder of EasyA. Much of the success of EasyA hackathon winners can be traced to the company’s co-founders, who have been instrumental in spotting young talent and nurturing them as founders. Users can go to any EasyA hackathon and find Phil and Dom on the ground, inspiring developers to launch the next unicorns. EasyA co-founders Phil and Dom have been active in blockchain for over 10 years. Prior to EasyA, Phil worked at prestigious law firm Sullivan & Cromwell and Dom worked at the world’s largest Private Equity firm, The Blackstone Group. Their educational backgrounds have also given them, and EasyA, a unique pipeline of talented founders across the US and Europe. Phil was a top scholar at Cambridge University, and Dom studied at the Wharton School, University of Pennsylvania, where he graduated Cum Laude as a Joseph Wharton and Benjamin Franklin Scholar. According to Phil and Dom, they’re just getting started. Numerous high-profile fundraising announcements are coming out of the EasyA community over the coming months. About EasyA EasyA is one of Web3’s most popular apps, making it possible for anyone to learn about Web3 right from their phones. Learners earn rewards for mastering new skills, and the best ones are invited to in-person hackathons to launch their startups in world-leading hubs like San Francisco, London and Singapore. EasyA EasyA EasyA alumni have founded startups valued at nearly $3 billion and have raised from top VCs like a16z, Founders Fund, YC and many more. Launched by brothers Phil and Dom Kwok, top grads from the University of Cambridge and The Wharton School respectively, EasyA has over 1 million users and has won Apple’s highly-coveted App of the Day award. Users can learn more: https://www.easya.io/ https://www.easya.io/ https://www.easya.io/ Contact Dom Kwok dom@easya.io This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here. This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here . here

EasyA x Polkadot Hackathon Winners Accepted To YCombinator To Secure Web3

تم قبول الفائزين في EasyA x Polkadot Hackathon في YCombinator لتأمين Web3

Los ganadores del hackathon EasyA x Polkadot fueron aceptados en YCombinator para proteger la Web3

EasyA x Polkadot Hackathon -voittajat hyväksyttiin YCombinatoriin suojattuun Web3:een

Pobjednici EasyA x Polkadot hackathona primljeni su u YCombinator kako bi osigurali Web3

Az EasyA x Polkadot Hackathon győzteseit elfogadták az YCombinatorba a biztonságos web3 érdekében

EasyA x Polkadot ハッカソンの優勝者が Web3 のセキュリティ確保のため YCombinator に採用される

EasyA x Polkadot Hackathon жеңімпаздары Web3 қауіпсіздігін қамтамасыз ету үшін YCombinator жүйесіне қабылданды

ຜູ້ຊະນະ EasyA x Polkadot Hackathon ຍອມຮັບກັບ YCombinator ເພື່ອຮັບປະກັນ Web3

Победители хакатона EasyA x Polkadot приняты в YCombinator для защиты Web3

Fituesit e EasyA x Polkadot Hackathon pranohen në YCombinator për të siguruar Web3

EasyA x Polkadot Hackathon جیتنے والوں کو Web3 کو محفوظ بنانے کے لیے YCombinator کو قبول کر لیا گیا

Những người chiến thắng cuộc thi Hackathon EasyA x Polkadot được YCombinator chấp nhận để bảo mật Web3

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring the Gradient of a Scalar in Dark Matter Velocity Analysis

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar C Covariance Matrix To treat the total likelihood as the product of the individual likelihoods in each frequency bin, we must check that the covariance matrix is diagonal. We will consider a signal-only analysis, discarding the noise, since the noise merely adds to the power and is uncorrelated between different frequency bins. We may write the values of the three peaks as We wish to compute the quantity We can do this using the expression for the raw moments, where, for us, σ = 1/ √ 2. Aside from this, we need to know that We then get that Crucially, we get that the covariance between peaks is 0, allowing us to treat them as statistically independent and hence permitting us to express the total likelihood as the product of the individual likelihoods. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar C Covariance Matrix To treat the total likelihood as the product of the individual likelihoods in each frequency bin, we must check that the covariance matrix is diagonal. We will consider a signal-only analysis, discarding the noise, since the noise merely adds to the power and is uncorrelated between different frequency bins. We may write the values of the three peaks as We wish to compute the quantity We can do this using the expression for the raw moments, where, for us, σ = 1/ √ 2. Aside from this, we need to know that We then get that Crucially, we get that the covariance between peaks is 0, allowing us to treat them as statistically independent and hence permitting us to express the total likelihood as the product of the individual likelihoods. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

nalyzing Covariance Matrix for Independent Likelihoods in Frequency Bins

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Derivation of Marginal Likelihood with Stochastic Field Amplitude

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar A Equipartition between Longitudinal and Transverse Modes The equation of motion for the field Ψ (written in Fourier space) is Now, we wish to decompose the field into transverse and longitudinal pieces. Let us call them ηk and ζk. That is, where Notice that not only are the longitudinal and transverse pieces coupled to one another, but more importantly they source each other. Hence, we expect virialization between them due to such non-linear gravitational dynamics during DM halo formation, ultimately leading to equipartition of power amongst them. That is, 2/3 of the total power would be in the transverse sector, with the remaining 1/3 in the longitudinal sector. Using codes based on the split-Fourier technique [73, 91–93], we have performed simulations of the Schrodinger -Poisson vector system [31, 64, 94] (physical space version of Eq. (A.1)): This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar A Equipartition between Longitudinal and Transverse Modes The equation of motion for the field Ψ (written in Fourier space) is Now, we wish to decompose the field into transverse and longitudinal pieces. Let us call them ηk and ζk. That is, where Notice that not only are the longitudinal and transverse pieces coupled to one another, but more importantly they source each other. Hence, we expect virialization between them due to such non-linear gravitational dynamics during DM halo formation, ultimately leading to equipartition of power amongst them. That is, 2/3 of the total power would be in the transverse sector, with the remaining 1/3 in the longitudinal sector. Using codes based on the split-Fourier technique [73, 91–93], we have performed simulations of the Schrodinger -Poisson vector system [31, 64, 94] (physical space version of Eq. (A.1)): This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Equipartition Between Longitudinal and Transverse Modes

Advancing the Detection of Ultralight Vector Dark Matter through Novel Analytical Strategies

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar 5 Future Directions 5.1 Longer Observation Times 5.2 Expanding the Mass Window This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar 5 Future Directions 5.1 Longer Observation Times 5.2 Expanding the Mass Window This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Longer Observation Times and Expanding the Mass Window

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar 4.1 Recasting Generalised Limits onto B − L Dark Matter To recast our limits on β shown in Fig. 4 into one on the parameter of interest for this model, the gauge coupling strength gB−L, we must define four quantities. Firstly, we must make clear what the quantity A is for this experiment, which will depend on both the model and the signal of interest for this sensor. Secondly, we must choose an observation time, Tobs. Thirdly, we must elaborate on what the concrete noise profile, σ(ω), for this type of experiment is. Lastly, and least importantly following from our discussion above, we must choose a latitude for the experiment. Once these are known, Eq. (2.8) can be re-arranged for gB−L, giving us our model- and experiment-specific 90% CL limit. For B − L coupled dark photon dark matter, the relevant signal is a differential acceleration. This is given by Eq. (2.6) in the time domain, with a now concrete choice for A, Here, gB−L is the gauge coupling strength of the model, ∆ij is the differential B −L charge per nucleon between materials i and j, given by For our observation time, we take Tobs = 10T⊕ to be firmly in the regime where the three peaks can be resolved. For light cavities, which may only be able to remain coherent over the scale of hours rather than days, such a runtime is optimistic. However, since our aim here is merely to showcase how our method can be used concretely and compare to the works of Refs. [36, 38], we do not see this as an issue. Our strategy is general and can be applied to any axial sensor and vector-like DM model, and we have settled on this configuration only for the sake of argument; other sensor technologies, such a magnetically levitated sensors, do not have this issue. Moreover, multiple cavities or data-stacking techniques can be employed to mitigate this issue. Our choices for all of the above parameters except for the laser power, which we expand on below, are summarised in Table 1. These are in keeping with the choices made in Refs. [36, 38]. The choice of where to tune the laser power is critical to give us competitive limits for a wide range of dark matter masses. We have found that we can achieve excellent limits for low dark matter masses, which are of most relevance to our work, by tuning the laser power such that the back-action and shot-noise components are minimised at low We note that this choice of power tuning is different from the strategies usually employed in other studies. Typically, the laser power is tuned so that the measurement-added noise is either minimised on resonance or, as LIGO implements it, well above resonance [88]. However, we found that both of these choices detriments the limit that we can draw at low masses, increasing the background at low frequencies beyond the thermal noise floor. For these other strategies to be beneficial at both the respective frequency targets and at low frequencies, the thermal noise would have to be significantly lower than that achieved with our choice of sensor mass, quality factor, and bath temperature. In the treatment of our backgrounds, we have neglected seismic noise, which can become important at frequencies below ∼ 10 Hz. Since we require the use of two materials to measure the differential acceleration peculiar to B − L DM, we can envision constructing two sensors: one with a moveable mirror made of material i and another of material j. By subtracting their signals, we both isolate the differential acceleration signature and remove backgrounds common to both—this includes the seismic noise component [63] We show our limits in Fig. 5 for three choices of resonance frequency: 1 Hz, 10 Hz and 100 Hz. For all but the last of these frequencies, we are able to exclude new regions of the UDLM B − L parameter space, best excluded by the fifth-force satellite experiment MICROSCOPE [89] and torsion-balance experiment E¨ot-Wash [90]. This showcases the power of such sensors in searching for this DM candidate, as was first pointed out in Ref. [63]. In a single-peak analysis focusing solely on the Compton peak, our limits would be weakened by approximately a factor of 2, as can be seen from Fig. 4. This difference becomes more dramatic for sensors located closer to the equator. We note that σ(ω), though mostly slowly varying in the frequency width ∆ω = ω⊕, does exhibit a large gradient around the resonance frequency of the cavity. In the neighbourhood of this frequency, our assumption that σ(ω) does not vary greatly in the above range is incorrect, and a more careful analysis in which all three peaks take on different noise levels would have to be conducted for more representative limits. However, we do not expect that our limits would differ greatly from our calculation and, at any rate, would still smoothly join to the regimes on either side of the resonance frequency, where our assumption holds. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar 4.1 Recasting Generalised Limits onto B − L Dark Matter To recast our limits on β shown in Fig. 4 into one on the parameter of interest for this model, the gauge coupling strength gB−L, we must define four quantities. Firstly, we must make clear what the quantity A is for this experiment, which will depend on both the model and the signal of interest for this sensor. Secondly, we must choose an observation time, Tobs. Thirdly, we must elaborate on what the concrete noise profile, σ(ω), for this type of experiment is. Lastly, and least importantly following from our discussion above, we must choose a latitude for the experiment. Once these are known, Eq. (2.8) can be re-arranged for gB−L, giving us our model- and experiment-specific 90% CL limit. For B − L coupled dark photon dark matter, the relevant signal is a differential acceleration. This is given by Eq. (2.6) in the time domain, with a now concrete choice for A, Here, gB−L is the gauge coupling strength of the model, ∆ij is the differential B −L charge per nucleon between materials i and j, given by For our observation time, we take Tobs = 10T⊕ to be firmly in the regime where the three peaks can be resolved. For light cavities, which may only be able to remain coherent over the scale of hours rather than days, such a runtime is optimistic. However, since our aim here is merely to showcase how our method can be used concretely and compare to the works of Refs. [36, 38], we do not see this as an issue. Our strategy is general and can be applied to any axial sensor and vector-like DM model, and we have settled on this configuration only for the sake of argument; other sensor technologies, such a magnetically levitated sensors, do not have this issue. Moreover, multiple cavities or data-stacking techniques can be employed to mitigate this issue. Our choices for all of the above parameters except for the laser power, which we expand on below, are summarised in Table 1. These are in keeping with the choices made in Refs. [36, 38]. The choice of where to tune the laser power is critical to give us competitive limits for a wide range of dark matter masses. We have found that we can achieve excellent limits for low dark matter masses, which are of most relevance to our work, by tuning the laser power such that the back-action and shot-noise components are minimised at low We note that this choice of power tuning is different from the strategies usually employed in other studies. Typically, the laser power is tuned so that the measurement-added noise is either minimised on resonance or, as LIGO implements it, well above resonance [88]. However, we found that both of these choices detriments the limit that we can draw at low masses, increasing the background at low frequencies beyond the thermal noise floor. For these other strategies to be beneficial at both the respective frequency targets and at low frequencies, the thermal noise would have to be significantly lower than that achieved with our choice of sensor mass, quality factor, and bath temperature. In the treatment of our backgrounds, we have neglected seismic noise, which can become important at frequencies below ∼ 10 Hz. Since we require the use of two materials to measure the differential acceleration peculiar to B − L DM, we can envision constructing two sensors: one with a moveable mirror made of material i and another of material j. By subtracting their signals, we both isolate the differential acceleration signature and remove backgrounds common to both—this includes the seismic noise component [63] We show our limits in Fig. 5 for three choices of resonance frequency: 1 Hz, 10 Hz and 100 Hz. For all but the last of these frequencies, we are able to exclude new regions of the UDLM B − L parameter space, best excluded by the fifth-force satellite experiment MICROSCOPE [89] and torsion-balance experiment E¨ot-Wash [90]. This showcases the power of such sensors in searching for this DM candidate, as was first pointed out in Ref. [63]. In a single-peak analysis focusing solely on the Compton peak, our limits would be weakened by approximately a factor of 2, as can be seen from Fig. 4. This difference becomes more dramatic for sensors located closer to the equator. We note that σ(ω), though mostly slowly varying in the frequency width ∆ω = ω⊕, does exhibit a large gradient around the resonance frequency of the cavity. In the neighbourhood of this frequency, our assumption that σ(ω) does not vary greatly in the above range is incorrect, and a more careful analysis in which all three peaks take on different noise levels would have to be conducted for more representative limits. However, we do not expect that our limits would differ greatly from our calculation and, at any rate, would still smoothly join to the regimes on either side of the resonance frequency, where our assumption holds. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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