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Move older conventions out to separate files
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‎test/conventions/lal.jl

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@testmodule LAL begin
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"""
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Reproduces the XLALSpinWeightedSphericalHarmonic function from the LALSuite C library:
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https://lscsoft.docs.ligo.org/lalsuite/lal/_spherical_harmonics_8c_source.html#l00042
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"""
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function LALSpinWeightedSphericalHarmonic(
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theta::Float64, # polar angle (rad)
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phi::Float64, # azimuthal angle (rad)
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s::Int, # spin weight
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l::Int, # mode number l
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m::Int # mode number m
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)
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# Sanity checks
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if l < abs(s)
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error("Invalid mode s=$s, l=$l, m=$m - require |s| <= l")
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end
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if l < abs(m)
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error("Invalid mode s=$s, l=$l, m=$m - require |m| <= l")
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end
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if s != -2
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error("Unsupported mode s=$s (only s=-2 implemented)")
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end
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if l < 2 || l > 8
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error("Unsupported mode l=$l (only l in [2,8] implemented)")
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end
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# Compute real factor
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fac = if l == 2
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if m == -2
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sqrt(5.0 / (64.0 * π)) * (1.0 - cos(theta)) * (1.0 - cos(theta))
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elseif m == -1
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sqrt(5.0 / (16.0 * π)) * sin(theta) * (1.0 - cos(theta))
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elseif m == 0
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sqrt(15.0 / (32.0 * π)) * sin(theta) * sin(theta)
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elseif m == 1
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sqrt(5.0 / (16.0 * π)) * sin(theta) * (1.0 + cos(theta))
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elseif m == 2
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sqrt(5.0 / (64.0 * π)) * (1.0 + cos(theta)) * (1.0 + cos(theta))
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end
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elseif l == 3
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if m == -3
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sqrt(21.0 / (2.0 * π)) * cos(theta/2.0) * sin(theta/2.0)^5
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elseif m == -2
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sqrt(7.0 / (4.0 * π)) * (2.0 + 3.0 * cos(theta)) * sin(theta/2.0)^4
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elseif m == -1
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sqrt(35.0 / (2.0 * π)) * (sin(theta) + 4.0 * sin(2.0 * theta) - 3.0 * sin(3.0 * theta)) / 32.0
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elseif m == 0
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sqrt(105.0 / (2.0 * π)) * cos(theta) * sin(theta)^2 / 4.0
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elseif m == 1
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-sqrt(35.0 / (2.0 * π)) * (sin(theta) - 4.0 * sin(2.0 * theta) - 3.0 * sin(3.0 * theta)) / 32.0
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elseif m == 2
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sqrt(7.0 / π) * cos(theta/2.0)^4 * (-2.0 + 3.0 * cos(theta)) / 2.0
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elseif m == 3
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-sqrt(21.0 / (2.0 * π)) * cos(theta/2.0)^5 * sin(theta/2.0)
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end
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elseif l == 4
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if m == -4
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3.0 * sqrt(7.0 / π) * cos(theta/2.0)^2 * sin(theta/2.0)^6
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elseif m == -3
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3.0 * sqrt(7.0 / (2.0 * π)) * cos(theta/2.0) * (1.0 + 2.0 * cos(theta)) * sin(theta/2.0)^5
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elseif m == -2
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3.0 * (9.0 + 14.0 * cos(theta) + 7.0 * cos(2.0 * theta)) * sin(theta/2.0)^4 / (4.0 * sqrt(π))
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elseif m == -1
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3.0 * (3.0 * sin(theta) + 2.0 * sin(2.0 * theta) + 7.0 * sin(3.0 * theta) - 7.0 * sin(4.0 * theta)) / (32.0 * sqrt(2.0 * π))
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elseif m == 0
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3.0 * sqrt(5.0 / (2.0 * π)) * (5.0 + 7.0 * cos(2.0 * theta)) * sin(theta)^2 / 16.0
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elseif m == 1
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3.0 * (3.0 * sin(theta) - 2.0 * sin(2.0 * theta) + 7.0 * sin(3.0 * theta) + 7.0 * sin(4.0 * theta)) / (32.0 * sqrt(2.0 * π))
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elseif m == 2
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3.0 * cos(theta/2.0)^4 * (9.0 - 14.0 * cos(theta) + 7.0 * cos(2.0 * theta)) / (4.0 * sqrt(π))
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elseif m == 3
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-3.0 * sqrt(7.0 / (2.0 * π)) * cos(theta/2.0)^5 * (-1.0 + 2.0 * cos(theta)) * sin(theta/2.0)
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elseif m == 4
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3.0 * sqrt(7.0 / π) * cos(theta/2.0)^6 * sin(theta/2.0)^2
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end
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elseif l == 5
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if m == -5
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sqrt(330.0 / π) * cos(theta/2.0)^3 * sin(theta/2.0)^7
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elseif m == -4
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sqrt(33.0 / π) * cos(theta/2.0)^2 * (2.0 + 5.0 * cos(theta)) * sin(theta/2.0)^6
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elseif m == -3
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sqrt(33.0 / (2.0 * π)) * cos(theta/2.0) * (17.0 + 24.0 * cos(theta) + 15.0 * cos(2.0 * theta)) * sin(theta/2.0)^5 / 4.0
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elseif m == -2
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sqrt(11.0 / π) * (32.0 + 57.0 * cos(theta) + 36.0 * cos(2.0 * theta) + 15.0 * cos(3.0 * theta)) * sin(theta/2.0)^4 / 8.0
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elseif m == -1
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sqrt(77.0 / π) * (2.0 * sin(theta) + 8.0 * sin(2.0 * theta) + 3.0 * sin(3.0 * theta) + 12.0 * sin(4.0 * theta) - 15.0 * sin(5.0 * theta)) / 256.0
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elseif m == 0
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sqrt(1155.0 / (2.0 * π)) * (5.0 * cos(theta) + 3.0 * cos(3.0 * theta)) * sin(theta)^2 / 32.0
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elseif m == 1
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sqrt(77.0 / π) * (-2.0 * sin(theta) + 8.0 * sin(2.0 * theta) - 3.0 * sin(3.0 * theta) + 12.0 * sin(4.0 * theta) + 15.0 * sin(5.0 * theta)) / 256.0
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elseif m == 2
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sqrt(11.0 / π) * cos(theta/2.0)^4 * (-32.0 + 57.0 * cos(theta) - 36.0 * cos(2.0 * theta) + 15.0 * cos(3.0 * theta)) / 8.0
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elseif m == 3
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-sqrt(33.0 / (2.0 * π)) * cos(theta/2.0)^5 * (17.0 - 24.0 * cos(theta) + 15.0 * cos(2.0 * theta)) * sin(theta/2.0) / 4.0
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elseif m == 4
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sqrt(33.0 / π) * cos(theta/2.0)^6 * (-2.0 + 5.0 * cos(theta)) * sin(theta/2.0)^2
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elseif m == 5
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-sqrt(330.0 / π) * cos(theta/2.0)^7 * sin(theta/2.0)^3
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end
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elseif l == 6
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if m == -6
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(3.0 * sqrt(715.0 / π) * cos(theta/2.0)^4 * sin(theta/2.0)^8) / 2.0
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elseif m == -5
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(sqrt(2145.0 / π) * cos(theta/2.0)^3 * (1.0 + 3.0 * cos(theta)) * sin(theta/2.0)^7) / 2.0
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elseif m == -4
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(sqrt(195.0 / (2.0 * π)) * cos(theta/2.0)^2 * (35.0 + 44.0 * cos(theta) + 33.0 * cos(2.0 * theta)) * sin(theta/2.0)^6) / 8.0
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elseif m == -3
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(3.0 * sqrt(13.0 / π) * cos(theta/2.0) * (98.0 + 185.0 * cos(theta) + 110.0 * cos(2.0 * theta) + 55.0 * cos(3.0 * theta)) * sin(theta/2.0)^5) / 32.0
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elseif m == -2
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(sqrt(13.0 / π) * (1709.0 + 3096.0 * cos(theta) + 2340.0 * cos(2.0 * theta) + 1320.0 * cos(3.0 * theta) + 495.0 * cos(4.0 * theta)) * sin(theta/2.0)^4) / 256.0
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elseif m == -1
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(sqrt(65.0 / (2.0 * π)) * cos(theta/2.0) * (161.0 + 252.0 * cos(theta) + 252.0 * cos(2.0 * theta) + 132.0 * cos(3.0 * theta) + 99.0 * cos(4.0 * theta)) * sin(theta/2.0)^3) / 64.0
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elseif m == 0
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(sqrt(1365.0 / π) * (35.0 + 60.0 * cos(2.0 * theta) + 33.0 * cos(4.0 * theta)) * sin(theta)^2) / 512.0
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elseif m == 1
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(sqrt(65.0 / (2.0 * π)) * cos(theta/2.0)^3 * (161.0 - 252.0 * cos(theta) + 252.0 * cos(2.0 * theta) - 132.0 * cos(3.0 * theta) + 99.0 * cos(4.0 * theta)) * sin(theta/2.0)) / 64.0
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elseif m == 2
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(sqrt(13.0 / π) * cos(theta/2.0)^4 * (1709.0 - 3096.0 * cos(theta) + 2340.0 * cos(2.0 * theta) - 1320.0 * cos(3.0 * theta) + 495.0 * cos(4.0 * theta))) / 256.0
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elseif m == 3
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(-3.0 * sqrt(13.0 / π) * cos(theta/2.0)^5 * (-98.0 + 185.0 * cos(theta) - 110.0 * cos(2.0 * theta) + 55.0 * cos(3.0 * theta)) * sin(theta/2.0)) / 32.0
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elseif m == 4
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(sqrt(195.0 / (2.0 * π)) * cos(theta/2.0)^6 * (35.0 - 44.0 * cos(theta) + 33.0 * cos(2.0 * theta)) * sin(theta/2.0)^2) / 8.0
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elseif m == 5
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(-sqrt(2145.0 / π) * cos(theta/2.0)^7 * (-1.0 + 3.0 * cos(theta)) * sin(theta/2.0)^3) / 2.0
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elseif m == 6
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(3.0 * sqrt(715.0 / π) * cos(theta/2.0)^8 * sin(theta/2.0)^4) / 2.0
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end
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elseif l == 7
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if m == -7
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sqrt(15015.0 / (2.0 * π)) * cos(theta/2.0)^5 * sin(theta/2.0)^9
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elseif m == -6
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(sqrt(2145.0 / π) * cos(theta/2.0)^4 * (2.0 + 7.0 * cos(theta)) * sin(theta/2.0)^8) / 2.0
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elseif m == -5
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(sqrt(165.0 / (2.0 * π)) * cos(theta/2.0)^3 * (93.0 + 104.0 * cos(theta) + 91.0 * cos(2.0 * theta)) * sin(theta/2.0)^7) / 8.0
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elseif m == -4
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(sqrt(165.0 / (2.0 * π)) * cos(theta/2.0)^2 * (140.0 + 285.0 * cos(theta) + 156.0 * cos(2.0 * theta) + 91.0 * cos(3.0 * theta)) * sin(theta/2.0)^6) / 16.0
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elseif m == -3
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(sqrt(15.0 / (2.0 * π)) * cos(theta/2.0) * (3115.0 + 5456.0 * cos(theta) + 4268.0 * cos(2.0 * theta) + 2288.0 * cos(3.0 * theta) + 1001.0 * cos(4.0 * theta)) * sin(theta/2.0)^5) / 128.0
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elseif m == -2
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(sqrt(15.0 / π) * (5220.0 + 9810.0 * cos(theta) + 7920.0 * cos(2.0 * theta) + 5445.0 * cos(3.0 * theta) + 2860.0 * cos(4.0 * theta) + 1001.0 * cos(5.0 * theta)) * sin(theta/2.0)^4) / 512.0
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elseif m == -1
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(3.0 * sqrt(5.0 / (2.0 * π)) * cos(theta/2.0) * (1890.0 + 4130.0 * cos(theta) + 3080.0 * cos(2.0 * theta) + 2805.0 * cos(3.0 * theta) + 1430.0 * cos(4.0 * theta) + 1001.0 * cos(5.0 * theta)) * sin(theta/2.0)^3) / 512.0
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elseif m == 0
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(3.0 * sqrt(35.0 / π) * cos(theta) * (109.0 + 132.0 * cos(2.0 * theta) + 143.0 * cos(4.0 * theta)) * sin(theta)^2) / 512.0
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elseif m == 1
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(3.0 * sqrt(5.0 / (2.0 * π)) * cos(theta/2.0)^3 * (-1890.0 + 4130.0 * cos(theta) - 3080.0 * cos(2.0 * theta) + 2805.0 * cos(3.0 * theta) - 1430.0 * cos(4.0 * theta) + 1001.0 * cos(5.0 * theta)) * sin(theta/2.0)) / 512.0
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elseif m == 2
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(sqrt(15.0 / π) * cos(theta/2.0)^4 * (-5220.0 + 9810.0 * cos(theta) - 7920.0 * cos(2.0 * theta) + 5445.0 * cos(3.0 * theta) - 2860.0 * cos(4.0 * theta) + 1001.0 * cos(5.0 * theta))) / 512.0
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elseif m == 3
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-(sqrt(15.0 / (2.0 * π)) * cos(theta/2.0)^5 * (3115.0 - 5456.0 * cos(theta) + 4268.0 * cos(2.0 * theta) - 2288.0 * cos(3.0 * theta) + 1001.0 * cos(4.0 * theta)) * sin(theta/2.0)) / 128.0
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elseif m == 4
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(sqrt(165.0 / (2.0 * π)) * cos(theta/2.0)^6 * (-140.0 + 285.0 * cos(theta) - 156.0 * cos(2.0 * theta) + 91.0 * cos(3.0 * theta)) * sin(theta/2.0)^2) / 16.0
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elseif m == 5
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-(sqrt(165.0 / (2.0 * π)) * cos(theta/2.0)^7 * (93.0 - 104.0 * cos(theta) + 91.0 * cos(2.0 * theta)) * sin(theta/2.0)^3) / 8.0
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elseif m == 6
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(sqrt(2145.0 / π) * cos(theta/2.0)^8 * (-2.0 + 7.0 * cos(theta)) * sin(theta/2.0)^4) / 2.0
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elseif m == 7
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-(sqrt(15015.0 / (2.0 * π)) * cos(theta/2.0)^9 * sin(theta/2.0)^5)
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end
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elseif l == 8
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if m == -8
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sqrt(34034.0 / π) * cos(theta/2.0)^6 * sin(theta/2.0)^10
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elseif m == -7
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sqrt(17017.0 / (2.0 * π)) * cos(theta/2.0)^5 * (1.0 + 4.0 * cos(theta)) * sin(theta/2.0)^9
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elseif m == -6
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sqrt(255255.0 / π) * cos(theta/2.0)^4 * (1.0 + 2.0 * cos(theta)) * sin(π/4.0 - theta/2.0) * sin(π/4.0 + theta/2.0) * sin(theta/2.0)^8
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elseif m == -5
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(sqrt(12155.0 / (2.0 * π)) * cos(theta/2.0)^3 * (19.0 + 42.0 * cos(theta) + 21.0 * cos(2.0 * theta) + 14.0 * cos(3.0 * theta)) * sin(theta/2.0)^7) / 8.0
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elseif m == -4
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(sqrt(935.0 / (2.0 * π)) * cos(theta/2.0)^2 * (265.0 + 442.0 * cos(theta) + 364.0 * cos(2.0 * theta) + 182.0 * cos(3.0 * theta) + 91.0 * cos(4.0 * theta)) * sin(theta/2.0)^6) / 32.0
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elseif m == -3
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(sqrt(561.0 / (2.0 * π)) * cos(theta/2.0) * (869.0 + 1660.0 * cos(theta) + 1300.0 * cos(2.0 * theta) + 910.0 * cos(3.0 * theta) + 455.0 * cos(4.0 * theta) + 182.0 * cos(5.0 * theta)) * sin(theta/2.0)^5) / 128.0
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elseif m == -2
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(sqrt(17.0 / π) * (7626.0 + 14454.0 * cos(theta) + 12375.0 * cos(2.0 * theta) + 9295.0 * cos(3.0 * theta) + 6006.0 * cos(4.0 * theta) + 3003.0 * cos(5.0 * theta) + 1001.0 * cos(6.0 * theta)) * sin(theta/2.0)^4) / 512.0
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elseif m == -1
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(sqrt(595.0 / (2.0 * π)) * cos(theta/2.0) * (798.0 + 1386.0 * cos(theta) + 1386.0 * cos(2.0 * theta) + 1001.0 * cos(3.0 * theta) + 858.0 * cos(4.0 * theta) + 429.0 * cos(5.0 * theta) + 286.0 * cos(6.0 * theta)) * sin(theta/2.0)^3) / 512.0
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elseif m == 0
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(3.0 * sqrt(595.0 / π) * (210.0 + 385.0 * cos(2.0 * theta) + 286.0 * cos(4.0 * theta) + 143.0 * cos(6.0 * theta)) * sin(theta)^2) / 4096.0
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elseif m == 1
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(sqrt(595.0 / (2.0 * π)) * cos(theta/2.0)^3 * (798.0 - 1386.0 * cos(theta) + 1386.0 * cos(2.0 * theta) - 1001.0 * cos(3.0 * theta) + 858.0 * cos(4.0 * theta) - 429.0 * cos(5.0 * theta) + 286.0 * cos(6.0 * theta)) * sin(theta/2.0)) / 512.0
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elseif m == 2
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(sqrt(17.0 / π) * cos(theta/2.0)^4 * (7626.0 - 14454.0 * cos(theta) + 12375.0 * cos(2.0 * theta) - 9295.0 * cos(3.0 * theta) + 6006.0 * cos(4.0 * theta) - 3003.0 * cos(5.0 * theta) + 1001.0 * cos(6.0 * theta))) / 512.0
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elseif m == 3
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-(sqrt(561.0 / (2.0 * π)) * cos(theta/2.0)^5 * (-869.0 + 1660.0 * cos(theta) - 1300.0 * cos(2.0 * theta) + 910.0 * cos(3.0 * theta) - 455.0 * cos(4.0 * theta) + 182.0 * cos(5.0 * theta)) * sin(theta/2.0)) / 128.0
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elseif m == 4
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(sqrt(935.0 / (2.0 * π)) * cos(theta/2.0)^6 * (265.0 - 442.0 * cos(theta) + 364.0 * cos(2.0 * theta) - 182.0 * cos(3.0 * theta) + 91.0 * cos(4.0 * theta)) * sin(theta/2.0)^2) / 32.0
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elseif m == 5
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-(sqrt(12155.0 / (2.0 * π)) * cos(theta/2.0)^7 * (-19.0 + 42.0 * cos(theta) - 21.0 * cos(2.0 * theta) + 14.0 * cos(3.0 * theta)) * sin(theta/2.0)^3) / 8.0
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elseif m == 6
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(sqrt(255255.0 / π) * cos(theta/2.0)^8 * (-1.0 + 2.0 * cos(theta)) * sin(theta/2.0)^4) * sin(π/4.0 - theta/2.0) * sin(π/4.0 + theta/2.0);
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elseif m == 7
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-(sqrt(17017.0 / (2.0 * π)) * cos(theta/2.0)^9 * (-1.0 + 4.0 * cos(theta)) * sin(theta/2.0)^5)
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elseif m == 8
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sqrt(34034.0 / π) * cos(theta/2.0)^10 * sin(theta/2.0)^6
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end
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end
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# Include complex phase factor
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if m ≠ 0
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ans = cis(m*phi) * fac
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else
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ans = fac
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end
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end
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end # module LAL

‎test/conventions/ninja.jl

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@testmodule NINJA begin
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function Wigner_d(ι::T, ell, m, s) where {T<:Real}
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# Eq. II.8 of Ajith et al. (2007) 'Data formats...'
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k_min = max(0, m - s)
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k_max = min(ell + m, ell - s)
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prefactor = √T(
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factorial(big(ell + m))
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* factorial(big(ell - m))
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* factorial(big(ell + s))
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* factorial(big(ell - s))
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)
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sum(
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ifelse(iseven(k), 1, -1)
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* cos(ι / 2) ^ (2 * ell + m - s - 2 * k)
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* sin(ι / 2) ^ (2 * k + s - m)
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* prefactor
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/ T(
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factorial(big(ell + m - k))
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* factorial(big(ell - s - k))
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* factorial(big(k))
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* factorial(big(k + s - m))
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)
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for k in k_min:k_max
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)
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end
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raw"""
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Eq. II.7 of Ajith et al. (2007) 'Data formats...' says
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```math
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{}_sY_{\ell,m} = (-1)^s \sqrt{\frac{2\ell+1}{4\pi}} d^\ell_{m,-s}(\iota) e^{im\phi}
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```
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Below Eq. (2.53) of [Torres del Castillo](@cite TorresDelCastillo_2003), we see
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```math
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{}_sY_{j,m} = (-1)^m \sqrt{\frac{2j+1}{4\pi}} d^j_{-m,s}(\iota) e^{im\phi}
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```
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We can use identities to modify the latter as follows:
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```math
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\begin{aligned}
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{}_sY_{j,m} &= (-1)^m \sqrt{\frac{2j+1}{4\pi}} d^j_{-m,s}(\iota) e^{im\phi} \\
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&= (-1)^m \sqrt{\frac{2j+1}{4\pi}} d^j_{-s,m}(\iota) e^{im\phi} \\
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&= (-1)^{j-s+m} \sqrt{\frac{2j+1}{4\pi}} d^j_{-s,-m}(\pi-\iota) e^{im\phi} \\
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&= (-1)^{j-s+m} \sqrt{\frac{2j+1}{4\pi}} d^j_{m,s}(\pi-\iota) e^{im\phi} \\
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&= (-1)^{2j-s+2m} \sqrt{\frac{2j+1}{4\pi}} d^j_{m,-s}(\pi-(\pi-\iota)) e^{im\phi} \\
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&= (-1)^{s} \sqrt{\frac{2j+1}{4\pi}} d^j_{m,-s}(\iota) e^{im\phi} \\
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\end{aligned}
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```
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The last line assumes that `j`, `m`, and `s` are integers. But in that case, the NINJA
51+
expression agrees with the Torres del Castillo expression.
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54+
"""
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function sYlm(s, ell, m, ι::T, ϕ::T) where {T<:Real}
57+
# Eq. II.7 of Ajith et al. (2007) 'Data formats...'
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# Note the weird definition w.r.t. `-s`
59+
if abs(s) > ell || abs(m) > ell
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return zero(complex(T))
61+
end
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(
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ifelse(iseven(s), 1, -1)
64+
* √((2ell + 1) / (4T(π)))
65+
* Wigner_d(ι, ell, m, -s)
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* cis(m * ϕ)
67+
)
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end
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function sYlm(s, ℓ, m, ιϕ)
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sYlm(s, ℓ, m, ιϕ[1], ιϕ[2])
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end
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# Eqs. (II.9) through (II.13) of https://arxiv.org/abs/0709.0093v3 [Ajith_2007](@cite)
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m2Y22(ι::T, ϕ::T) where {T<:Real} = √(5 / (64T(π))) * (1 + cos(ι))^2 * cis(2ϕ)
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m2Y21(ι::T, ϕ::T) where {T<:Real} = √(5 / (16T(π))) * sin(ι) * (1 + cos(ι)) * cis(ϕ)
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m2Y20(ι::T, ϕ::T) where {T<:Real} = √(15 / (32T(π))) * sin(ι)^2
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m2Y2m1(ι::T, ϕ::T) where {T<:Real} = √(5 / (16T(π))) * sin(ι) * (1 - cos(ι)) * cis(-1ϕ)
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m2Y2m2(ι::T, ϕ::T) where {T<:Real} = √(5 / (64T(π))) * (1 - cos(ι))^2 * cis(-2ϕ)
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m_m2Y2m = [
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(2, m2Y22),
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(1, m2Y21),
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(0, m2Y20),
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(-1, m2Y2m1),
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(-2, m2Y2m2)
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]
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end # module NINJA

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