1 Astrophysical Constraints on Dense Matter in Neutron Stars
7
the Hermitian group generators, g is the strong coupling constant, and γ are the
Euclidean gamma matrices.
For a vanishing chemical potential μ = 0, detM(A) is positive definite,
meaning that all contributions add in the same direction and Z is comparatively
straightforward to compute. If instead μ is real and nonzero, then detM(A) is
complex in general. Thus although Z is still real and strictly positive, the integrands
have various phases and the integral takes the form of a cancellation between
large quantities. The bad news is that the general fermion sign problem is NP-hard
[216], but work is proceeding on better approximation methods. If a first-principles
evaluation of Z yields, without ambiguity, the equilibrium state of dense matter at
low temperatures, then measurements of the properties of neutron stars would serve
as important tests of QCD itself and would thus be probes of very fundamental
physics indeed. Until that point, however, it is necessary to use phenomenological
models.
It is very difficult to rule out an entire class of models (e.g., those with only
nucleonic degrees of freedom or those with significant contributions from hyperons). This is because neutron star core densities and the asymmetry in the number
densities of neutrons and protons are significantly greater than those that can be
probed in laboratories. As a result, one could always imagine adding contributions
that involve high powers of the density or asymmetry. These contributions would
have a negligible impact on laboratory matter but would have important effects in
the cores of neutron stars. One can make some general statements: for instance, if
non-nucleonic components become important above some density the equilibrium
radius at a given mass and the maximum mass both tend to be smaller than if
only nucleonic degrees of freedom contribute (because the presence of a new
energetically favorable composition softens the equation of state). Unfortunately,
it is difficult to establish a particular mass or radius that would eliminate such
exotic models. For example, hyperonic and hybrid quark models of neutron stars
have been constructed with maximum masses >2.0 M [117, 122]. Nonetheless,
although neutron star observations cannot entirely rule out model classes in
principle, their role is important because they probe a different realm of matter than
what is accessible in laboratories. Ockham’s razor should then be used to judge
between different model classes: if one class fits all data using a small number of
parameters that have reasonable values and other classes require great complexity
or unreasonable values, the first class would be preferred.
One basic category of models, relativistic mean field theories, is quite phenomenological in nature. In these models the degrees of freedom are nucleons and
mesons (which couple minimally to the nucleons but the coupling could have some
density dependence). The coupling strengths can be adjusted to laboratory data
and/or neutron star observations. In a more microscopically oriented approach, one
starts instead from some given nucleon-nucleon interaction (which can be extended
to more than two nucleons) that is fitted to data including the binding energy of light
nuclei and scattering data (for a recent effort in the context of chiral effective field
theory, see [99]). In both approaches there is considerable freedom about the types
of particles considered, e.g., the particles could be nucleons or the particles could
7
the Hermitian group generators, g is the strong coupling constant, and γ are the
Euclidean gamma matrices.
For a vanishing chemical potential μ = 0, detM(A) is positive definite,
meaning that all contributions add in the same direction and Z is comparatively
straightforward to compute. If instead μ is real and nonzero, then detM(A) is
complex in general. Thus although Z is still real and strictly positive, the integrands
have various phases and the integral takes the form of a cancellation between
large quantities. The bad news is that the general fermion sign problem is NP-hard
[216], but work is proceeding on better approximation methods. If a first-principles
evaluation of Z yields, without ambiguity, the equilibrium state of dense matter at
low temperatures, then measurements of the properties of neutron stars would serve
as important tests of QCD itself and would thus be probes of very fundamental
physics indeed. Until that point, however, it is necessary to use phenomenological
models.
It is very difficult to rule out an entire class of models (e.g., those with only
nucleonic degrees of freedom or those with significant contributions from hyperons). This is because neutron star core densities and the asymmetry in the number
densities of neutrons and protons are significantly greater than those that can be
probed in laboratories. As a result, one could always imagine adding contributions
that involve high powers of the density or asymmetry. These contributions would
have a negligible impact on laboratory matter but would have important effects in
the cores of neutron stars. One can make some general statements: for instance, if
non-nucleonic components become important above some density the equilibrium
radius at a given mass and the maximum mass both tend to be smaller than if
only nucleonic degrees of freedom contribute (because the presence of a new
energetically favorable composition softens the equation of state). Unfortunately,
it is difficult to establish a particular mass or radius that would eliminate such
exotic models. For example, hyperonic and hybrid quark models of neutron stars
have been constructed with maximum masses >2.0 M [117, 122]. Nonetheless,
although neutron star observations cannot entirely rule out model classes in
principle, their role is important because they probe a different realm of matter than
what is accessible in laboratories. Ockham’s razor should then be used to judge
between different model classes: if one class fits all data using a small number of
parameters that have reasonable values and other classes require great complexity
or unreasonable values, the first class would be preferred.
One basic category of models, relativistic mean field theories, is quite phenomenological in nature. In these models the degrees of freedom are nucleons and
mesons (which couple minimally to the nucleons but the coupling could have some
density dependence). The coupling strengths can be adjusted to laboratory data
and/or neutron star observations. In a more microscopically oriented approach, one
starts instead from some given nucleon-nucleon interaction (which can be extended
to more than two nucleons) that is fitted to data including the binding energy of light
nuclei and scattering data (for a recent effort in the context of chiral effective field
theory, see [99]). In both approaches there is considerable freedom about the types
of particles considered, e.g., the particles could be nucleons or the particles could