jesterTOV.ptov module

jesterTOV.ptov module#

The post-TOV (pTOV) module extends the standard TOV equations to include tidal deformability calculations.

Post-TOV (modified TOV) equation solver with beyond-GR corrections.

This module extends the standard TOV equations to include phenomenological modifications that parameterize deviations from General Relativity. The modifications are implemented through additional terms in the pressure gradient equation.

Units: All calculations are performed in geometric units where \(G = c = 1\).

Reference: Yagi & Yunes, Phys. Rev. D 88, 023009 (2013)

calc_k2(R, M, H, b)[source]#
sigma_func(p, e, m, r, lambda_BL, lambda_DY, lambda_HB, gamma, alpha, beta)[source]#

Compute the non-GR correction term sigma for modified TOV equations.

This function implements various phenomenological modifications to General Relativity that appear as additional terms in the TOV equations. The corrections are parameterized by several coupling constants.

The sigma function includes:

  • Brans-Dicke-like: \(\sigma_{\mathrm{BL}} = -\frac{\lambda_{\mathrm{BL}} r^2}{3}(\varepsilon + 3p)(\varepsilon + p)A\)

  • Dynamical Chern-Simons: \(\sigma_{\mathrm{DY}} = \lambda_{\mathrm{DY}} \frac{2m}{r} p\)

  • Horava-like: \(\sigma_{\mathrm{HB}} = -(\frac{1}{\lambda_{\mathrm{HB}}} - 1) \frac{r}{2} \frac{dp}{dr}\)

  • Post-Newtonian: \(\sigma_{\mathrm{PN}} = \gamma \frac{2m}{r} p \tanh(\alpha(\frac{m}{r} - \beta))\)

Parameters:

  • p (float) – Pressure at current radius.

  • e (float) – Energy density at current radius.

  • m (float) – Enclosed mass at current radius.

  • r (float) – Current radius.

  • lambda_BL (float) – Bowers-Liang coupling parameter.

  • lambda_DY (float) – Doneva-Yazadjiev coupling parameter.

  • lambda_HB (float) – Herrera-Barreto coupling parameter.

  • gamma (float) – Post-Newtonian amplitude parameter.

  • alpha (float) – Post-Newtonian steepness parameter.

  • beta (float) – Post-Newtonian transition point parameter.

Returns:

float – Total sigma correction term.

tov_ode(h, y, eos)[source]#
tov_solver(eos, pc)[source]#

Solve the modified TOV equations for a given central pressure.

This function integrates the modified TOV equations that include beyond-GR corrections. The integration procedure is identical to the standard TOV case, but the differential equations include additional sigma terms.

Parameters:

  • eos (dict) –

    Extended EOS interpolation data containing:

    • p: Pressure array [geometric units]

    • h: Enthalpy array [geometric units]

    • e: Energy density array [geometric units]

    • dloge_dlogp: Logarithmic derivative array

    • alpha, beta, gamma: Post-Newtonian parameters

    • lambda_BL, lambda_DY, lambda_HB: Theory modification parameters

  • pc (float) – Central pressure [geometric units].

Returns:

tuple –

A tuple containing:

  • M: Gravitational mass [geometric units]

  • R: Circumferential radius [geometric units]

  • k2: Second Love number for tidal deformability

Note

The modifications affect the stellar structure but the same integration method and boundary conditions as the standard TOV case are used.

Mathematical Background#

The post-TOV equations include additional perturbative equations for calculating the tidal Love number \(k_2\):

\[\frac{dH}{dr} = \beta(r) H + \alpha(r) b\]
\[\frac{db}{dr} = H + \gamma(r) b\]

where \(H\) and \(b\) are auxiliary functions related to the tidal deformation, and \(\alpha\), \(\beta\), \(\gamma\) are coefficients that depend on the background stellar structure.

The tidal Love number is then:

\[k_2 = \frac{8C^5}{5}(1-2C)^2[2 + 2C(y_R - 1) - y_R] \times \{2C[6-3y_R + 3C(5y_R-8)] + 4C^3[13-11y_R + C(3y_R-2) + 2C^2(1+y_R)] + 3(1-2C)^2[2-y_R + 2C(y_R-1)]\ln(1-2C)\}^{-1}\]

where \(C = GM/Rc^2\) is the compactness and \(y_R\) is related to the tidal response at the surface.

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