Dorian

dorian

(Even More) Functional Elixir — 1 If you (like me) got into Elixir from an object-oriented language, then concepts like may not be very familiar. Some of them do not apply by default to Elixir. But they are part of the and worth exploring. currying, function composing and pointfree functions general functional programming (FP) concepts I started with this free book recommended by a colleague: . I would recommend it to anybody getting into functional . It uses and a few libraries to exemplify the concepts. Yet, JS is not my main language, so I’m very curious to see how those apply in Elixir. Professor Frisby’s Mostly Adequate Guide to Functional Programming programming JavaScript Anything those days is about cryptocurrencies. To keep up with the trends let’s build a small crypto price checker. Then we will refactor it using FP concepts. Initial Implementation The public function takes the amount in GBP as only argument. It outputs the number of crypto-coins you can buy with. Example: buy_crypto_with_gbp(2000) =>[BTC: 0.18983617051918897,ETH: 1.9337893873932355,LTC: 10.66955239648603] The role of the module is to communicate with the external APIs and return the exchange rates. You can find the implementation in the . It exposes 2 public functions: Support Github repo # exchange rate between 2 currenciesexchange_rate("USD", "GBP") => 0.7463 # exchange rate between crypto and USDlist_crypto_usd() => [BTC: 14256.67, ETH: 1371.76, LTC: 262.51] There are a few simple actions happening in the code above: get a list of the 3 cryptocurrencies prices in USD get an exchange rate between 2 currencies convert the list of crypto prices from USD to GBP. map the quantity for each crypto-coin The challenge is to refactor the code above using the functional concepts of currying, composing and pointfree. Curry, Compose, Pointfree First of all, I must say that those concepts are not provided with Elixir out of the box. We could implement some of them on our own, but it’s safer and easier to make use of this amazing package: . You may say that we can already compose functions in Elixir with the pipe operator. Stay with me, the composing we are going to use is slightly different. Add the Quark package in the mix file, and at the top of the module. Quark |> use Quark Currying Is the ability to call a function that takes many arguments, with one argument at a time. Each time it will return a new function until the last argument is applied and the result is returned. Quark defines currying with or for private functions. defcurry defcurryp Let’s start by determining the exchange rage from USD to GBP. Add a wrapper around our function: Support.exchange_rate/2 defcurry exchange_rate(from, to) doSupport.exchange_rate(from, to)end Now you can start to apply the arguments one by one to our new function. You can assign the partial result (which is a function) to a variable. Or it can become a new function definition: # apply argumentsexchange_rate.("USD").("GBP") => 0.7463 # assign resulting function to a variablefrom_usd = exchange_rate.("USD") => #Function<5.65603100/1 in CryptoCurry.exchange_rate/0> from_usd.("GBP") => 0.7463 # new function definitiondef exchange_rate_from_usd doexchange_rate().("USD")end exchange_rate_from_usd.("GBP") => 0.7463 Next step is to apply the conversion above to the cryptocurrencies info we receive . We use the same function defined in the Initial implementation, but we just currying it: {BTC: 14256.67} apply_conversion/2 defcurry apply_conversion(rate, {crypto, value_usd}) do{crypto, value_usd * rate}end If we follow the same logic as above we can now do the following to the bitcoin value in GBP: apply_conversion_gbp =apply_conversion.(exchange_rate_from_usd.("GBP")) => #Function<7.34465968/1 in CryptoCurry.apply_conversion/0> apply_conversion_gbp.({:BTC, 14265.67}) =>{:BTC, 10646.469520999999} Getting a bit confusing? This is where the next concept may come handy. Composing The Elixir pipe operator takes the output from the expression on its left side and passes it as the first argument to the function call on its right side. |> The composing is very similar, only it passes a function as the argument to another function. It can be: right to left — the mathematical way — represented with <|> left to right — the flow way, as the Elixir pipe — represented with <~> Let’s look at a small example outside our test application: In this case: bazz_rl.("a").("b") == bazz_lr.("a").("b") Back to the crypto example, we are going to keep close to the FP book and use the mathematical way of composing. We want to define a function that will take the crypto data and convert it to GBP. def apply_conversion_gbp do(apply_conversion() <|> exchange_rate_from_usd()).("GBP")end will partially apply as its first argument. The resulting function will be called with the attribute. The result is, of course, another function that can be called with crypto data: apply_conversion exchange_rate_from_usd "GBP" apply_conversion_gbp.({:BTC, 14265.67}) =>{:BTC, 10646.469520999999} We are almost there. Need a function that will return the quantity of coin that we can buy with the amount of GBP. We already have that function from the Initial implementation. Just need to curry it. defcurryp get_quantity({crypto, value_gbp}, amount) do{crypto, amount / value_gbp}end Mapping and composing If you remember from the Initial example, we were iterating twice over the crypto data list. Once to apply the exchange rate and then to get the quantities: Support.list_crypto_usd()|> Enum.map(&apply_conversion(Support.exchange_rate("USD", "GBP"), &1))|> Enum.map(&get_quantity(&1, amount)) This can be inefficient, but math helps us again here: map(f2) <|> map(f1) == map(f2 <|> f1) So we can compose the 2 functions and map them only once. def convert_gbp_and_get_quantity doget_quantity() <|> apply_conversion_gbp()end defcurry buy_crypto_with_gbp(amount) doEnum.map(Support.list_crypto_usd(),&convert_gbp_and_get_quantity().(&1).(amount))end Our application is now in working state. buy_crypto_with_gbp.(2000) =>[BTC: 0.19774571082071027,ETH: 2.0169243958209693,LTC: 11.090413196189877] Pointfree This last step is a cosmetic change in our case. It was the only reason we curried the . buy_crypto_with_gbp/1 Pointfree is the ability to declare the function without mentioning the arguments. We use the Quark package to define a pointfree interface function for our application. defx defx buy_crypto_with_gbp dobuy_crypto_with_gbp().()end Which we can then call with buy_crypto_with_gbp(2000). Putting it all together Quite a long journey, but we’ve made it. Implementing crypto converter using FP concepts. Back to Elixir Yet, keeping some of the learnings from the implementation above, we can refactor our Elixir code, without curry or need for additional libraries. Conclusion If I would need to choose between and implementations above, I would go for the last one. I find the Elixir native way more clear and explicit. CryptoCurry CryptoElixir Currying and composing functions can DRY your code. Can help building specialised functions generic ones. Also, the code may look cleaner as function names are shorter, without so many arguments. But on longer term, or for new people in the project, it can be confusing. Let’s take our example. I see that is called with 2 successive arguments. But I cannot be sure if that is just a partial application or not. I don’t know if it returns a final result or another function waiting for more arguments. convert_gbp_and_get_quantity In the end, it is important to know that we have the option to use those concepts in Elixir as well. Especially if you have a functional background, you may miss them sometimes. I hope the post gave you at least some basic idea about curry, composing and freepoint in Elixir. I would be very curious to find your opinions on Elixir and those FP concepts. If you worked with them in other languages, do you need them in Elixir? If you didn’t, would you consider using them?

Crypto Curry with Elixir

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

Recasting Three-Peak Exclusion Limits on B − L Dark Matter via Accelerometer Studies

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 Application to Accelerometer Studies However, those works that performed a Fourier space analysis did so by only focusing on the Compton peak. As we discussed in Section 3, such an analysis for an axial sensor is sub-optimal, as it fails to capture the full signal and therefore suffers from a signal loss at a range of latitudes. Moreover, ignoring the randomness of the nuisance variables leads to an overly aggressive constraint, as was illustrated in Fig. 3. Our more holistic threepeak strategy, which also considers this stochasticity, retains the full signal and is largely latitude-independent in its constraining power. Therefore, this choice of sensor and DM model makes for an excellent case study with which to showcase the improved constraining power of our method. Furthermore, a proper vector treatment of the DM field in Fourier space, which includes the stochasticity of the ULDM field variables and the effect of the rotation of the Earth, has not been done. Ignoring the randomness of the nuisance variables leads to an overly aggressive constraint, as was also pointed out in Refs. [42, 79]. This paper is available on arxiv under CC BY 4.0 DEED license. [9] Here we assume that the sensitivity axis of the detector is locally fixed with respect to the zenith. 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 Application to Accelerometer Studies However, those works that performed a Fourier space analysis did so by only focusing on the Compton peak. As we discussed in Section 3, such an analysis for an axial sensor is sub-optimal, as it fails to capture the full signal and therefore suffers from a signal loss at a range of latitudes. Moreover, ignoring the randomness of the nuisance variables leads to an overly aggressive constraint, as was illustrated in Fig. 3. Our more holistic threepeak strategy, which also considers this stochasticity, retains the full signal and is largely latitude-independent in its constraining power. Therefore, this choice of sensor and DM model makes for an excellent case study with which to showcase the improved constraining power of our method. Furthermore, a proper vector treatment of the DM field in Fourier space, which includes the stochasticity of the ULDM field variables and the effect of the rotation of the Earth, has not been done. Ignoring the randomness of the nuisance variables leads to an overly aggressive constraint, as was also pointed out in Refs. [42, 79]. 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 [9] Here we assume that the sensitivity axis of the detector is locally fixed with respect to the zenith.

Enhanced Vector Dark Matter Detection in Accelerometer Studies Through a Three-Peak 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 3.2 Projected Exclusions To derive our limits, we construct the one-sided log-likelihood-ratio test statistic, defined as where βˆ is that value of β which maximises the likelihood given the observed data set p, characterising the best-fit model. This statistic is defined as a piecewise function as we only expect excess signals to be disfavoured when excluding a value of β in a one-sided test. This corresponds to values of β greater than the best-fit value. Values below this are deemed under-fluctuations and considered consistent with observation. This statistic tells us how consistent the data are with a signal defined by β compared to the best-fit model, with zero representing perfect consistency and large values indicating high inconsistency. Ultimately, we compute 90% CL limits (αconf = 0.90) via a series of Monte Carlo (MC) simulations, following the above procedure in each MC run[8]. For a given choice of β, we We show the 90% CL limit arising from our three-peak analysis in Fig. 4. Also shown are the corresponding results from two single-peak analyses focusing solely on the Compton and either one of the sum or difference peaks. These results follow the same MC procedure as above but take as the full likelihood only the Compton or the sum/difference likelihood given in Eq. (3.2). We see that the three-peak analysis produces a limit that is largely latitude-independent, rising slightly towards the pole. This is because the sensitivity axis at this latitude only has a component parallel to the Earth’s rotation axis and is thus only able to pick out the Compton peak. This latitude-independent limit is in contrast with the analyses that focus on only single peaks, which are both highly sensitive to where the experiment is placed. For the study focusing on the Compton peak, the constraining power is optimal at the pole, where all of the power is contained in the Compton frequency bin, and it rapidly declines towards the equator, where the Compton peak disappears. Note that this and the three-peak results join at this point, with no difference between the approaches. Conversely, for an analysis focusing on one of the sum or difference peaks, the situation is the opposite. In this case, the results of this and the three-peak methods do not converge since half the power is contained in a single one of the sum or difference peaks at the equator; the threepeak analysis captures all of this power, whereas the single-peak analysis misses half of the power. Thus, the strength of our analysis is that the constraining power is retained no matter where an experiment is placed, such that its latitude is rendered largely irrelevant from the viewpoint of constraining a ULDM signal. We emphasize that the interpretation of Fig. 4 is as a set of exclusion lines whereby background pseudodata is generated and the assumed DM signal strength is constrained. The key point is that the level of this constraint depends on the assumption one makes on the nature of the DM signal given a non-detection. Taking this signal to be only a single peak in Fourier space then leads to constraints that are generally dependent on the latitude of the experiment, rapidly weakening towards latitude extremes. On the other hand, employing the full signal model consisting of three peaks yields stronger constraints that are almost independent of the experiment placement. Throughout the above analysis, we assumed that the sensitivity axis pointed in the zenith direction. However, we can relax this assumption and consider what our results would look like if this axis pointed in some different direction, say for instance directions perpendicular to the zenith—namely, the East/West and North/South directions[9]. If North/South-pointing, all of the curves in Fig. 4 would be flipped about the line ϕ = 45◦ . While the strongest limit for the Compton peak would now occur at the equator instead of the pole (and vice versa for both the sum and difference peaks), crucially we would still retain a largely latitude-independent constraint for our three-peak analysis. This is because, throughout the experiment, the directionality of the detector makes a cone, making it sensitive to the vector DM power in all the three directions. If, instead, the axis pointed towards the East/West, we would only see the sum and the difference peaks. This is because, throughout the experimental expedition, the directionality of the detector is restricted to lie on a plane (which would necessarily be perpendicular to the rotation axis of the Earth). Therefore, it is always insensitive to the power contained across the perpendicular direction, which is tied to the standalone Compton peak. One final possibility is when the directionality of the detector traces out a line throughout the experiment. This is only possible when it points parallel to the Earth’s rotation axis (at any given latitude). In this case, we would naturally be oblivious to the Earth’s rotation and hence to the sum and difference peaks, and we would only be able to resolve the Compton peak. In summary, the best-case scenario is when the detector’s sensitivity axis traces out a cone. In this case, we capture all the three peaks since we are sensitive to the vector DM power across all three directions (Earth’s rotation axis and the two orthogonal directions). This paper is available on arxiv under CC BY 4.0 DEED license. [8] While asymptotic formulae exist for computing exclusion limits, we have opted for an MC approach to ensure proper coverage [81]. Since we are following a hybrid frequentist-Bayesian approach, we found this to be an especially pertinent check. 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 3.2 Projected Exclusions To derive our limits, we construct the one-sided log-likelihood-ratio test statistic, defined as where βˆ is that value of β which maximises the likelihood given the observed data set p, characterising the best-fit model. This statistic is defined as a piecewise function as we only expect excess signals to be disfavoured when excluding a value of β in a one-sided test. This corresponds to values of β greater than the best-fit value. Values below this are deemed under-fluctuations and considered consistent with observation. This statistic tells us how consistent the data are with a signal defined by β compared to the best-fit model, with zero representing perfect consistency and large values indicating high inconsistency. Ultimately, we compute 90% CL limits (αconf = 0.90) via a series of Monte Carlo (MC) simulations, following the above procedure in each MC run[8]. For a given choice of β, we We show the 90% CL limit arising from our three-peak analysis in Fig. 4. Also shown are the corresponding results from two single-peak analyses focusing solely on the Compton and either one of the sum or difference peaks. These results follow the same MC procedure as above but take as the full likelihood only the Compton or the sum/difference likelihood given in Eq. (3.2). We see that the three-peak analysis produces a limit that is largely latitude-independent, rising slightly towards the pole. This is because the sensitivity axis at this latitude only has a component parallel to the Earth’s rotation axis and is thus only able to pick out the Compton peak. This latitude-independent limit is in contrast with the analyses that focus on only single peaks, which are both highly sensitive to where the experiment is placed. For the study focusing on the Compton peak, the constraining power is optimal at the pole, where all of the power is contained in the Compton frequency bin, and it rapidly declines towards the equator, where the Compton peak disappears. Note that this and the three-peak results join at this point, with no difference between the approaches. Conversely, for an analysis focusing on one of the sum or difference peaks, the situation is the opposite. In this case, the results of this and the three-peak methods do not converge since half the power is contained in a single one of the sum or difference peaks at the equator; the threepeak analysis captures all of this power, whereas the single-peak analysis misses half of the power. Thus, the strength of our analysis is that the constraining power is retained no matter where an experiment is placed, such that its latitude is rendered largely irrelevant from the viewpoint of constraining a ULDM signal. We emphasize that the interpretation of Fig. 4 is as a set of exclusion lines whereby background pseudodata is generated and the assumed DM signal strength is constrained. The key point is that the level of this constraint depends on the assumption one makes on the nature of the DM signal given a non-detection. Taking this signal to be only a single peak in Fourier space then leads to constraints that are generally dependent on the latitude of the experiment, rapidly weakening towards latitude extremes. On the other hand, employing the full signal model consisting of three peaks yields stronger constraints that are almost independent of the experiment placement. Throughout the above analysis, we assumed that the sensitivity axis pointed in the zenith direction. However, we can relax this assumption and consider what our results would look like if this axis pointed in some different direction, say for instance directions perpendicular to the zenith—namely, the East/West and North/South directions[9]. If North/South-pointing, all of the curves in Fig. 4 would be flipped about the line ϕ = 45◦ . While the strongest limit for the Compton peak would now occur at the equator instead of the pole (and vice versa for both the sum and difference peaks), crucially we would still retain a largely latitude-independent constraint for our three-peak analysis. This is because, throughout the experiment, the directionality of the detector makes a cone, making it sensitive to the vector DM power in all the three directions. If, instead, the axis pointed towards the East/West, we would only see the sum and the difference peaks. This is because, throughout the experimental expedition, the directionality of the detector is restricted to lie on a plane (which would necessarily be perpendicular to the rotation axis of the Earth). Therefore, it is always insensitive to the power contained across the perpendicular direction, which is tied to the standalone Compton peak. One final possibility is when the directionality of the detector traces out a line throughout the experiment. This is only possible when it points parallel to the Earth’s rotation axis (at any given latitude). In this case, we would naturally be oblivious to the Earth’s rotation and hence to the sum and difference peaks, and we would only be able to resolve the Compton peak. In summary, the best-case scenario is when the detector’s sensitivity axis traces out a cone. In this case, we capture all the three peaks since we are sensitive to the vector DM power across all three directions (Earth’s rotation axis and the two orthogonal directions). 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 [8] While asymptotic formulae exist for computing exclusion limits, we have opted for an MC approach to ensure proper coverage [81]. Since we are following a hybrid frequentist-Bayesian approach, we found this to be an especially pertinent check.

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