def integrand_btz_n(u, tau, n, R, rh, l, pm1, Om, lam, sig, deltaphi=0):
s_int_fnc = lambda s: lam**2 * fp.exp(-s**2/4/sig**2)\
* (fp.exp(-fp.j*Om*s) * wightman_btz_n(s,n,R,rh,l,pm1,Om,lam,sig))
if n != 0 or deltaphi != 0:
pt1 = fp.mpf(
mp.acosh(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(rh / l *
(deltaphi - 2 * fp.pi * n)) - 1)))
pt2 = fp.mpf(
mp.acosh(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(rh / l *
(deltaphi - 2 * fp.pi * n)) + 1)))
uplim = 2 * tau - u
if n == 0:
if uplim < pt2:
output = fp.quad(s_int_fnc, [0, uplim])
else:
output = fp.quad(s_int_fnc, [0, pt2]) + fp.quad(
s_int_fnc, [pt2, uplim])
else:
if uplim < pt1:
output = fp.quad(s_int_fnc, [0, uplim])
elif uplim < pt2:
output = fp.quad(s_int_fnc, [0, pt1]) + fp.quad(
s_int_fnc, [pt1, uplim])
else:
output = fp.quad(s_int_fnc,[0,pt1]) + fp.quad(s_int_fnc,[pt1,pt2])\
+ fp.quad(s_int_fnc,[pt2,uplim])
output *= fp.exp(-u**2 / 4 / sig**2)
return output
def Pdot_n(tau, n, R, rh, l, pm1, Om, lam, sig, deltaphi=0, eps=1e-6):
if n == 0:
if deltaphi == 0:
lim2 = fp.mpf(mp.acosh(mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*n)) + 1))))
#print(lim2)
fmins = lambda s: integrand_Pdotm_n(s, tau, 0, R, rh, l, pm1, Om,
lam, sig, deltaphi, eps).real
fplus = lambda s: integrand_Pdotp_n(s, tau, 0, R, rh, l, pm1, Om,
lam, sig, deltaphi, eps).real
return fp.quad(fmins, [0, tau + 10 * sig]) + fp.quad(
fplus, [0, lim2]) + fp.quad(fplus, [lim2, tau + 10 * sig])
else:
lim1 = fp.mpf(mp.acosh(mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*n)) - 1))))
lim2 = fp.mpf(mp.acosh(mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*n)) + 1))))
fmins = lambda s: integrand_Pdotm_n(s, tau, 0, R, rh, l, pm1, Om,
lam, sig, deltaphi, 0).real
fplus = lambda s: integrand_Pdotp_n(s, tau, 0, R, rh, l, pm1, Om,
lam, sig, deltaphi, 0).real
return fp.quad(fmins,[0,lim1]) + fp.quad(fmins,[lim1,tau+10*sig])\
+ fp.quad(fplus,[0,lim2]) + fp.quad(fplus,[lim2,tau+10*sig])
else:
lim1 = fp.mpf(mp.acosh(mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*n)) - 1))))
lim2 = fp.mpf(mp.acosh(mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*n)) + 1))))
fmins = lambda s: integrand_Pdotm_n(s, tau, n, R, rh, l, pm1, Om, lam,
sig, deltaphi, 0).real
fplus = lambda s: integrand_Pdotp_n(s, tau, n, R, rh, l, pm1, Om, lam,
sig, deltaphi, 0).real
return fp.quad(fmins,[0,lim1]) + fp.quad(fmins,[lim1,tau+10*sig])\
+ fp.quad(fplus,[0,lim2]) + fp.quad(fplus,[lim2,tau+10*sig])
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 fn2(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(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(2 * fp.pi * rh / l * n) + 1))
if Zp == mp.cosh(y):
return 0
elif Zp - fp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zp - mp.cosh(y)))
else:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zp))
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 P_BTZn(n,R,rh,l,pm1,Om,lam,sig):
b = fp.sqrt(R**2-rh**2)/l
lim = 20*sig*rh/b/l**2
print('limit: ',lim)
Zm = fp.mpf(1) #rh**2/(R**2-rh**2)*(R**2/rh**2 * fp.cosh(2*fp.pi*rh/l * n) - 1)
Zp = rh**2/(R**2-rh**2)*(R**2/rh**2 * fp.cosh(2*fp.pi*rh/l * n) + 1)
print('Zm: ',Zm, 'Zp: ',Zp, 'acosh(Zp): ', fp.mpf(mp.acosh(Zp)))
#print(Zm,fp.acosh(Zm))
# plt.figure()
# xaxis = fp.linspace(0,lim/5,50)
# yaxis = [f02(x,n,R,rh,l,pm1,Om,lam,sig) for x in xaxis]
# plt.plot(xaxis,yaxis)
# plt.show()
if pm1==-1 or pm1==1 or pm1==0:
if n==0:
return 0\
- fp.quad(lambda x: f02(x,n,R,rh,l,pm1,Om,lam,sig),[0,fp.mpf(mp.acosh(Zp))])\
- fp.quad(lambda x: f02(x,n,R,rh,l,pm1,Om,lam,sig),[fp.mpf(mp.acosh(Zp)),lim])
#fp.quad(lambda x: f01(x,n,R,rh,l,pm1,Om,lam,sig),[-fp.inf,fp.inf])
# else:
# if fp.cosh(lim) < Zm or Zm < 1:
# return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
# - fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,lim])
# else:
# return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
# - fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,fp.mpf(mp.acosh(Zm))])\
# - fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
# - fn2(x,n,R,rh,l,pm1,Om,lam,sig),[fp.mpf(mp.acosh(Zm)),lim])
else:
print("Please enter a value of 1, -1, or 0 for pm1.")
def P_BTZn(n, R, rh, l, pm1, Om, lam, sig):
b = fp.sqrt(R**2 - rh**2) / l
lim = 20 * sig * rh / b / l**2
#Zp = mp.mpf(rh**2/(R**2-rh**2)*(R**2/rh**2 * fp.cosh(2*fp.pi*rh/l * n) + 1))
#print('Zm: ',Zm, 'Zp: ',Zp)
#print(Zm,fp.acosh(Zm))
if pm1 == -1 or pm1 == 1 or pm1 == 0:
if n == 0:
return fp.quad(lambda x: f01(x, n, R, rh, l, pm1, Om, lam, sig),
[-fp.inf, fp.inf])
else:
Zm = rh**2 / (R**2 - rh**2) * (
R**2 / rh**2 * fp.cosh(2 * fp.pi * rh / l * n) - 1)
if fp.cosh(lim) < Zm or Zm < 1:
return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
- fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,lim])
else:
return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
- fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,fp.mpf(mp.acosh(Zm))])\
- fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
- fn2(x,n,R,rh,l,pm1,Om,lam,sig),[fp.mpf(mp.acosh(Zm)),lim])
else:
print("Please enter a value of 1, -1, or 0 for pm1.")
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))