def sigma_geon_n(y, n, RA, RB, rh, l, pm1, deltaphi, isP):
if isP:
return RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) - 1\
+ fp.sqrt(RA**2-rh**2)*fp.sqrt(RB**2-rh**2)/rh**2 \
* fp.mpf(mp.cosh(rh/l/mp.sqrt(RA*RB-rh**2) * y) )
else:
return RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) - 1\
+ fp.sqrt(RA**2-rh**2)*fp.sqrt(RB**2-rh**2)/rh**2 \
* fp.mpf(mp.cosh(rh/l**2 * y) )
def integrand_deltaPdot_n(s, tau, n, R, rh, l, pm1, Om, lam, sig, deltaphi):
Zm = mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*(n+1/2))) - 1))
Zp = mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*(n+1/2))) + 1))
K = lam**2 * rh / (2 * fp.pi * fp.sqrt(2) * l *
fp.sqrt(R**2 - rh**2)) * fp.exp(-tau**2 / 2 / sig**2)
return K * fp.exp(-(tau-s)**2/2/sig**2) * fp.exp(-fp.j*Om*s)\
* ( 1/fp.sqrt(Zm + mp.cosh(rh/l/mp.sqrt(R**2-rh**2) * s))\
- pm1*1/fp.sqrt(Zp + mp.cosh(rh/l/mp.sqrt(R**2-rh**2) * s)) )
def XGEON_denoms_n(y, n, RA, RB, rh, l, pm1, Om, lam, sig, deltaphi):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
Zm = rh**2 / l**2 / bA / bB * (
RA * RB / rh**2 * mp.cosh(rh / l * (deltaphi - 2 * mp.pi *
(n + 1 / 2))) - 1)
# Z-minus
Zp = rh**2 / l**2 / bA / bB * (
RA * RB / rh**2 * mp.cosh(rh / l * (deltaphi - 2 * mp.pi *
(n + 1 / 2))) + 1)
# Z-plus
return 1 / mp.sqrt(Zm + mp.cosh(y)) - pm1 / mp.sqrt(Zp + mp.cosh(y))
def f02(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zp = mp.mpf((R**2 + rh**2) / (R**2 - rh**2))
if Zp - mp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zp - mp.cosh(y)))
elif Zp - mp.cosh(y) < 0:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zp))
else:
return 0
def fn1(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zm = mp.mpf(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(2 * fp.pi * rh / l * n) - 1))
if Zm == mp.cosh(y):
return 0
elif Zm - fp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zm - mp.cosh(y)))
else:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zm))
def f02(y,n,R,rh,l,pm1,Om,lam,sig):
K = lam**2*sig/2/fp.sqrt(2*fp.pi)
a = (R**2-rh**2)*l**2/4/sig**2/rh**2
b = fp.sqrt(R**2-rh**2)*Om*l/rh
Zp = mp.mpf((R**2+rh**2)/(R**2-rh**2))
if Zp == mp.cosh(y):
print("RIP MOM PLSSS")
#print(Zp, y, fp.cosh(y))
if Zp - mp.cosh(y) > 0:
return K * fp.exp(-a*y**2) * fp.cos(b*y) / fp.mpf(mp.sqrt(Zp - mp.cosh(y)))
elif Zp - mp.cosh(y) < 0:
return -K * fp.exp(-a*y**2) * fp.sin(b*y) / fp.mpf(mp.sqrt(mp.cosh(y) - Zp))
else:
return 0
def T_P_int_geon(x, tau, tau0, n, R, rh, l, pm1, Om, lam, sig, deltaphi=0):
K = lam**2 / (np.sqrt(2) * np.pi * l * Om) * rh / np.sqrt(R**2 - rh**2)
Gm = rh**2 / (R**2 - rh**2) * (R**2 / rh**2 * fp.mpf(
mp.cosh(rh / l * (deltaphi - 2 * np.pi * (n + 0.5)))) - 1)
Gp = rh**2 / (R**2 - rh**2) * (R**2 / rh**2 * fp.mpf(
mp.cosh(rh / l * (deltaphi - 2 * np.pi * (n + 0.5)))) + 1)
if tau < tau0:
return 0
elif x < (tau + tau0):
return K * fp.sin(Om*(2*tau0+x))\
* (1/fp.sqrt(Gm + fp.mpf(mp.cosh(rh/l**2 * x))) - 1/fp.sqrt(Gp + fp.mpf(mp.cosh(rh/l**2 * x))))
else:
return K * fp.sin(Om*(2*tau-x))\
* (1/fp.sqrt(Gm + fp.mpf(mp.cosh(rh/l**2 * x))) - 1/fp.sqrt(Gp + fp.mpf(mp.cosh(rh/l**2 * x))))
def z_Zolotarev(N, x, m):
r"""
Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param x: The argument at which one would like to evaluate the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, the value of Zolotarev polynomial at x
"""
M = -ellipk(m) / N
x3 = ellipfun('sn', u=-M, m=m)
xbar = x3 * mp.sqrt(
(x**2 - 1) / (x**2 - x3**2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar),
m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if f.imag / f.real > 1e-10:
print("imaginary part of the Zolotarev function is not negligible!")
print("f_imaginary = ", f.imag)
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def P_BTZn(n,
RA,
RB,
rh,
l,
pm1,
Om,
lam,
tau0,
width,
deltaphi=0,
eps=1e-8,
isP=True):
f = lambda y: integrandof_P_BTZn(y, n, RA, RB, rh, l, pm1, Om, lam, tau0,
width, deltaphi, isP)
if n == 0 and deltaphi == 0 and RA == RB:
# P0m
P0m = 2 * fp.quad(
lambda x: integrandPBTZ_0m(x, RA, rh, l, pm1, Om, lam, tau0, width,
eps), [0, width])
intlim = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) + 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
if intlim < width:
return fp.quad(f, [0, intlim]) + fp.quad(f, [intlim, width]) + P0m
else:
return fp.quad(f, [0, width]) + P0m
else:
intlim1 = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) - 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
intlim2 = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) + 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
#print('%.4f, %.4f'%(intlim1,intlim2))
if intlim2 < width:
return fp.quad(f, [0, intlim1]) + fp.quad(
f, [intlim1, intlim2]) + fp.quad(f, [intlim2, width])
elif intlim1 < width:
return fp.quad(f, [0, intlim1]) + fp.quad(f, [intlim1, width])
else:
return fp.quad(f, [0, width])
def Xn_plus(n, RA, RB, rh, l, pm1, Om, lam, tau0, width, deltaphi, isP):
intlim2 = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) + 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
#print('plus, %.4f ;'%intlim2)
f = lambda y: integrandof_Xn_plus(y, n, RA, RB, rh, l, pm1, Om, lam, tau0,
width, deltaphi, isP)
if intlim2 < width:
return fp.quad(f, [0, intlim2]) + fp.quad(f, [intlim2, width])
else:
return fp.quad(f, [0, width])
def wightman_btz_n(s, n, R, rh, l, pm1, Om, lam, sig, deltaphi=0, eps=1e-4):
if n == 0:
Bm = (R**2 - rh**2) / rh**2
Bp = (R**2 + rh**2) / rh**2
else:
Bm = R**2 / rh**2 * fp.cosh(rh / l * (deltaphi - 2 * fp.pi * n)) - 1
Bp = R**2 / rh**2 * fp.cosh(rh / l * (deltaphi - 2 * fp.pi * n)) + 1
Scosh = (R**2 - rh**2) / rh**2 * mp.cosh(rh / l**2 * (s - fp.j * eps))
# if s == 0:
# return 0
# elif Bm == Scosh or Bp == Scosh:
# return 0
# else:
return 1/(4*fp.sqrt(2)*fp.pi*l) * (-fp.j/fp.sqrt(2*Bm)/\
mp.sinh(rh/2/l**2 * (s - fp.j*eps)) - pm1/fp.sqrt(Bp - Scosh))
def z_Zolotarev(N, x, m):
"""Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_)."""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def z_Zolotarev(N, x, m):
"""Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_)."""
M = -ellipk(m) / N
x3 = ellipfun('sn', u=-M, m=m)
xbar = x3 * mp.sqrt(
(x**2 - 1) / (x**2 - x3**2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar),
m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print("imaginary part of the Zolotarev function is not negligible!")
print("f_imaginary = ", f.imag)
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def z_Zolotarev(N, x, m):
r"""
Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param x: The argument at which one would like to evaluate the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, the value of Zolotarev polynomial at x
"""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def f1(x):
return mp.cos(x) * mp.cosh(x) - 1
def df1(x):
return mp.cos(x) * mp.sinh(x) - mp.sin(x) * mp.cosh(x)
def sigma_geon(x, n, R, rh, l):
return R**2/rh**2 * fp.mpf(mp.cosh(2*mp.pi * rh/l * (n+1/2))) - 1 + (R**2-rh**2)/rh**2\
* fp.mpf(mp.cosh(rh/l**2 * x))
def gammap_btz_n(s, n, R, rh, l, deltaphi, eps):
Zp = mp.mpf(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(rh / l *
(deltaphi - 2 * fp.pi * n)) + 1))
return Zp - mp.cosh(rh / l / mp.sqrt(R**2 - rh**2) * (s - fp.j * eps))
def integrandPBTZ_0m_GAUSS(y, R, rh, l, pm1, Om, lam, sig, eps):
#print(mp.sqrt(1 - mp.cosh( y -fp.j*eps )))
return fp.exp(-fp.j*fp.sqrt(R**2-rh**2)*Om*l/rh * y)\
/ mp.sqrt(1 - mp.cosh( y -fp.j*eps ))\
* lam**2*sig/(2*fp.sqrt(2*fp.pi))\
* fp.exp(-(R**2-rh**2)*l**2/(4*sig**2*rh**2) * y**2)