Pressure anisotropy

Pressure anisotropy#

This page describes the beyond-GR TOV solver with phenomenological pressure anisotropy and its implementation in jester.

Physical motivation#

In the standard GR treatment, neutron star matter is assumed to be a perfect fluid with isotropic pressure. Several physical scenarios — superfluid vortex structures, strong magnetic fields, and exotic phase transitions — can produce a local anisotropy between the radial and tangential pressures. Rather than committing to a specific microphysical model, the AnisotropyTOVSolver parametrizes the effect through phenomenological correction terms added directly to the stellar structure equations. This allows Bayesian inference to jointly constrain the EOS and potential deviations from the isotropic GR result.

Modified structure equations#

There are several models for pressure anisotropy in neutron stars, and the AnisotropyTOVSolver includes three of them, which can be turned on or off independently. An overview of the models can be found in [2], Eqs. (12) - (14) (note: in Eq. 12, the power of epsilon should be 2).

The radial pressure gradient in the standard TOV equation is modified by an anisotropy correction \(\sigma(r)\):

(1)#\[\frac{dp_r}{dr} = -\frac{m\varepsilon\left(1 + \frac{p_r}{\varepsilon}\right)\!\left(1 + \frac{4\pi r^3 p_r}{m}\right)}{r^2\!\left(1 - \frac{2m}{r}\right)} - \frac{2\sigma}{r} \, .\]

When \(\sigma = 0\) the equation reduces to the ordinary TOV result.

We have the following models implemented for the jester anisotropy paper [1]:

  • Bowers-Liang model (\(\sigma_\mathrm{BL}\)) (ref: [3])

  • Horvat et al (\(\sigma_\mathrm{DY}\)) (ref: [4])

  • Cosenza model (\(\sigma_\mathrm{HB}\)) (ref: [5])

The correction \(\sigma\) can be decomposed into contributions from these three independently parametrized phenomenological models so that \(\sigma = \sigma_\mathrm{BL} + \sigma_\mathrm{DY} + \sigma_\mathrm{HB}\). The parametrizations have the following form:

(2)#\[\sigma_\mathrm{BL} = -\frac{\lambda_\mathrm{BL}\,\varepsilon^2 r^2}{3} \left(1 + \frac{3p}{\varepsilon}\right)\!\left(1 + \frac{p}{\varepsilon}\right) \frac{1}{1 - 2m/r}\]
(3)#\[\sigma_\mathrm{DY} = \lambda_\mathrm{DY}\,\frac{2m}{r}\,p\]
(4)#\[\sigma_\mathrm{HB} = -\!\left(\frac{1}{\lambda_\mathrm{HB}} - 1\right)\frac{r}{2}\frac{dp}{dr}\]

Setting \(\lambda_\mathrm{BL} = 0\), \(\lambda_\mathrm{DY} = 0\), and \(\lambda_\mathrm{HB} = 1\) recovers the standard GR equations with pressure isotropy.

The tidal deformability equations are also modified consistently by including the derivative of \(\sigma\) with respect to pressure inside solve().

Usage#

Configuration file:

tov:
  type: anisotropy
  min_nsat_TOV: 0.75
  ndat_TOV: 100
  nb_masses: 100

Prior file (excerpt):

Note that these priors have to be defined on top of your priors for the EOS parameters, which are not shown here.

# Only sample lambda_DY, while fixing the other two couplings to their GR values
# This is what was done in Pang:2025fes
lambda_BL = Fixed(0.0,  parameter_names=["lambda_BL"])
lambda_DY = UniformPrior(-0.5, 0.5, parameter_names=["lambda_DY"])
lambda_HB = Fixed(1.0,  parameter_names=["lambda_HB"])

Any of the three coupling constants can be fixed to its GR value or treated as a free parameter depending on the analysis goal. Using Fixed for the couplings you do not want to constrain is more efficient than placing a broad prior on them. Complete working examples are in examples/inference/anisotropy/.

Further resources#

References

[1]

Peter T. H. Pang, Stephanie M. Brown, Thibeau Wouters, and Chris Van Den Broeck. Revealing tensions in neutron star observations with pressure anisotropy. Phys. Rev. D, 113(6):063047, 2026. arXiv:2507.13039, doi:10.1103/6qc8-3k97.

[2]

A. Rahmansyah, A. Sulaksono, A. B. Wahidin, and A. M. Setiawan. Anisotropic neutron stars with hyperons: implication of the recent nuclear matter data and observations of neutron stars. Eur. Phys. J. C, 80(8):769, 2020. doi:10.1140/epjc/s10052-020-8361-4.

[3]

Richard L. Bowers and E. P. T. Liang. Anisotropic Spheres in General Relativity. Astrophys. J., 188:657–665, 1974. doi:10.1086/152760.

[4]

Dubravko Horvat, Sasa Ilijic, and Anja Marunovic. Radial pulsations and stability of anisotropic stars with quasi-local equation of state. Class. Quant. Grav., 28:025009, 2011. arXiv:1010.0878, doi:10.1088/0264-9381/28/2/025009.

[5]

M. Cosenza, L. Herrera, M. Esculpi, and L. Witten. Some models of anisotropic spheres in general relativity. J. Math. Phys., 22(1):118, 1981. doi:10.1063/1.524742.