LPPL_pileup model definition

[SpaceMan013]

The exponential cut-off factor in LPPL_pileup dist (distr_func_lppl_pileup) is only applied to the LPPL part, which is usually fine in cases where gamma_eq is large, but if gamma_eq<1E7 and alpha is also small (<0.5), this can produce a jump discontinuity at gamma_inj in n(gamma), even worse for n(gamma)*gamma^3

def distr_func_lppl_pileup(gamma, gamma0_log_parab,  gamma_inj, r,  s, gamma_eq,  ratio_pile_up , alpha, gamma_cut_acc):
    b = np.zeros(gamma.shape)
    a = np.zeros(gamma.shape)
    m = gamma < gamma_inj
    s1=s+0.5
    b[m] = np.power(gamma[m]/gamma0_log_parab, s1)
    
    
    f1 = np.zeros(gamma.shape)
    m1 = np.logical_and(gamma < gamma0_log_parab,gamma>gamma_inj)
    f1[m1] = np.power(gamma[m1]/gamma0_log_parab, -s)
    f1[~m1] = np.power(gamma[~m1]/gamma0_log_parab, (-s -r*np.log10(gamma[~m1] / gamma0_log_parab )))
    b[~m] = np.power(gamma_inj/gamma0_log_parab,s1+s)*f1[~m]*np.exp(-np.power((gamma[~m]/gamma_cut_acc),alpha)/alpha)
    
    pile_up=gamma*gamma*np.exp(-np.power((gamma/gamma_eq),alpha)/alpha)
    pile_up=pile_up/(gamma_eq*gamma_eq*np.exp(-np.power((gamma_eq/gamma_eq),alpha)/alpha))
    g_ID=np.argmin(np.fabs(gamma-gamma_eq))
    c=b[g_ID]*ratio_pile_up
    a=pile_up*c
    return a+b

[SpaceMan013]

Additionally, the default range vmin and vmax of parameters alpha and ratio_pile_up are a bit too restrictive and should be alpha==[0-10] and ratio_pile_up==[0, 1], for more testing freedom