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Scalar

scalar

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

Let be a cyclic group, is the element in the group, is the scalar, and the Multi-Scalar Multiplication (MSM) is the algorithm for calculating , the sum of multiple scalar multiplications. As the group operation is more complicated than the addition and multiplication of elements in the finite field, the MSM algorithm is employed to reduce the number of times of group operation as much as possible. Usually, is a cyclic group defined on the elliptic curve of on the finite field of , with the order being bits, and . If the group operation is conducted by the plain algorithm of fast exponentiation, the number of group operations needed for every is times on average, and for all is times in total. Next, we will introduce some optimization methods corresponding to larger . G P_i ∈ G a_i ∈ Z ∑_{i=1}^{n} a_{i} P_{i} G y^2=x^3+ax+b F_p | | G b ≈ 256 0 ≤ a_i< 2^b a_i.P_i 1.5bn a_i.P_i 1.5b n Optimization Based on the Windowing Technique We can split the bit scalar into c width windows: reduce the scalar to base system. b 2^c Here, } , so we reduce the original algorithm issue of -bit MSM into a smaller -bit issue. We can put the same scalars from into a bucket and write them in another form: Q_j=∑_{i=1}^{n a_{i j} P_{i} b c ∑_{i=1}^{n} a_{i j} P_{i} For example, when , it is calculated as: c = 3 When calculating , set the partial sum T_l=S_l+⋯+S_k, then ∑kS_k However, the operation results of can be obtained through the recurrence sequence . It means that only one group operation is needed for the operation for each . Under the MSM algorithm with cc-bit scalar, all requires times of group operation. All requires times and the sum of these requires times of group operation. Therefore, each window with the value of c requires times of group operations. The operation of } only requires the normal method of fast exponentiation by times of group operation. T_l T_l=T_l+1+S_l T_l S_k n−2^c+1 T_k 2^ 1 c− T_k ^c−2 2 n+2^c−2 ∑_{j=0}^{b / c-1} Q_{j} 2^{j c (b/c−1)(c+1)=b − c + b/c−1 In summary, we can conclude that the total number of group operations required by the method of the Windowing Technique of a width of is c Set , then the number of group operations is . When , the number of group operations reduces to of its original value. c = logn 2bn/logn n≈10^5 1/12 Optimization Based on Group Endomorphism For the cyclic group on the elliptic curve on the finite field F_p, if the following group endomorphism of φ can be found: there exists , satisfying holds for all the points on , then it is easy to prove that such an endomorphism is a multiplication map, namely there is a making hold for all the points of on , which means that whenever we know the coordinates of one point, we can change it to the coordinates of another point simply by multiplying a number in to both the values of abscissa and ordinate. The algorithm can be further optimized based on this important property. G y^2=x^3+ax+b α,β ∈ F_p φ(x,y)=(αx,βy) G λ φ(P)=λP P G F_p When . The reverse value of one point can be obtained only by taking the opposite number from the ordinate. Based on this feature, in the plain fast exponentiation algorithm, the scalar can be written into the non-adjacent form (NAF), namely and any two adjacent cannot be non-zero at the same time. Compared to the -bit scalars, the group operation number in the fast exponentiation algorithm can be reduced from the original average of times to times. This technique can also be used to optimize the Windowing Technique. After writing to λ = −1, α = 1, β = −1 e_{i}.2^{i}, e_{i} ∈ {−1,0,1} ∑ e_i b 3/2⋅b 4/3⋅b a_i a_i=∑_{j=0}^{b / c-1} a_{i, j} 2^{j c}, 0 <= a_{i, j}<2^{c} conduct the following operation: The result of and - can be obtained. Since the complexity of addition and subtraction in group operation of an elliptic curve is the same, we can put these terms into a bucket according to the absolute value of scalars. Therefore, the number of groups can be reduced from the original 2c to , and the overall number of group operations required is reduced from (1) to: a_i=∑_{j=0}^{b / c-1} a’_{i, j}^2^{j c} 2^{c-1} <= a’_{i, j}^ <= 2^{c-1} 2^c If the parameters of the elliptic curve are special, for example, the BLS curve can be written as , and , take third-order element α from F_p^x, then there is a corresponding that holds y^2=x^3+b p ≡ 1( mod3 ) λ λ( )=( ) x,y αx,y Algorithm 1: Signed Digits In addition, . Then the multiplication operation can be optimized as follows: λ3 ≡ 1(mod |G| ) From [1], we can find a small enough scalar | |, | |<√ to make the above equation true, which gives us a way to reduce the multiplication of bb bits into the multiplication of two bits. Use it into the Windowing Technique, and the number of group operations is reduced to m1 m2 G b/2 In summary, both the above two optimization methods can reduce the number of group operations by when . 5.5% n=10^5 References [1] Francesco Sica, Mathieu Ciet, and Jean-Jacques Quisquater. Analysis of the gallant-lambert-vanstone method based on eﬀicient endomorphisms: Elliptic and hyperelliptic curves. In International Workshop on Selected Areas in Cryptography, pages 21–36. Springer, 2002.

Optimization of Multi-Scalar Multiplication Algorithm: Sin7Y Tech Review (21)

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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.

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