Effects of Phase Transition induced density fluctuations on pulsar dynamics

We show that density fluctuations during phase transitions in pulsar cores may have non-trivial effects on pulsar timings, and may also possibly account for glitches and anti-glitches. These density fluctuations invariably lead to non-zero off-diagonal components of the moment of inertia, leading to transient wobbling of star. Thus, accurate measurements of pulsar timing and intensity modulations (from wobbling) may be used to identify the specific pattern of density fluctuations, hence the particular phase transition, occurring inside the pulsar core. Changes in quadrupole moment from rapidly evolving density fluctuations during the transition, with very short time scales, may provide a new source for gravitational waves.


Our Proposals & Predictions
The core density of a pulsar might change during its evolution by accretion of matter to become supercritical and undergoes phase transition via nucleation of bubbles with macroscopic core size. The rapid phase conversion by expanding bubbles might have observable effects on sudden change of Moment of Inertia (MI) of the Pulsar.
There are various sources (e.g., bubbles, topological defects etc.) of density fluctuations arising during phase transitions may also alter the MI of the pusar. In fact, these density inhomogeneities might cause wobbling of the pulsar (due to develoment of off-diagonal components of MI tensor).
Accurate measurements of pulsar timings and intensity modulations may be used to pin down particular phase transition ocurring inside the pulsar.
The change of MI caused by density change and density inhomogeneities may account for glitches and anti-glitches.
Rapidly evolving density fluctuations may also produce Quadrupole Moment (QM) and may be a new source of gravitational waves.
The formation and evolution of topological defects shows universal characteristics and should lead to reasonably model independent predictions for changes in MI & QM and subsequent relaxation.

Case-I: Effects of Density Change During Phase Transitions
Consider the scenario where the density of the core of a pulsar becomes supercritical by accreting matter [2] (in few million years) and bubbles are nucleated in a macroscopically large core of radius R 0 . The subsequent phase conversion is expected to be very fast as is governed by the relativistic speed of bubble walls.
Assume, the density of the star changes from ρ 1 to higher density ρ 2 inside the core. The fractional change of MI of the pulsar (of radius R ≃ 10 km) is then estimated to be [3] : Observation from glitches : ∆I/I ≤ 10 −5 ⇒ R 0 ≤ 0.3 km (for QCD transition) (considering about 30 % change in density) Note : For a superfluid transition, R 0 may be as large as 5 Km with superfluid condensation energy density ≃ 0.1 MeV/f m 3 .

Hadron to QGP Transitions : Nucleation Rate
The equation of state for the phases (assuming zero temperature) [4]: For quantitative description of nucleation rate at zero temperature we took the model of quantum tunneling mediated by O(4) symmetric instantons, having action The critical density and the critical radius are estimated as :

Density Profile
Density profile of the neutron star with supercritical core, (a) Number of bubbles nucleated in 300 meter core radius in one million year as a function of core density for a QCD transition, taking the acceration rate of 10 17 grams/sec.
Blue : When the supercritical core size (with ρ > ρ c ) has increased to about 300 meter and neutron star, mass M 2 = 1.567M 0 .

Case-II : Effects of Density Fluctuations During Phase Transitions
Density inhomogeneities can be produced by nucleation of large number of bubbles or via formation of Topological defects (string, domain wall etc.).
Various defects generate different density fluctuation, with specific evolution patterns. High precision measurements of pulsar timings, intensity modulations (wobbling) and its relaxation may be used to identify different sources of fluctuations, thereby pinning down the specific phase transition occurring.

Deconfinement-Confinement Transition : Interfaces & Strings
D-C transition can be described by the effective Lagrangian [6], Domain walls (three of them) interpolate between different vacua. Topological string (QGP string [7]) arises at the junction of three different Z Z(3) interfaces.

Results
Fractional Change in MI and QM During Phase Transitions :

Studying the Effects by Static Approach
In the static approach, we model string and wall by forming correlation domains in a cubic lattice, with lattice spacing ξ representing the correlation length [8].
The mass density of the string and the domain wall tension were taken [9] as 3 GeV/fm and 7 GeV/fm 2 , respectively.
We confine defect network within core radius R c = 0.3 10 R.

Gravitational wave generation due to density fluctuation
For conservative estimates we have taken : Expected strain amplitude from a pulsar at a distance r, With △Q/I 0 ≃ 10 −10 and △t = 10 −6 − 10 −5 sec, strain amplitude comes out to be h ≃ 10 −24 − 10 −22 for a pulsar at 1kpc distance.

Concluding Remarks
The net fractional change in the MI is noted to be dominated by the phase change and is of order 10 −6 whereas the string induced fractional change in MI is about 3 orders of magnitude smaller, of order 10 −10 .
QM and off-diagonal components of MI are also found to be of order 10 −10 .
Note that these numbers are not far with the values for a glitch (or anti-glitch). Thus, the change of MI caused by density change and density inhomogeneities may account for glitches and anti-glitches.
The transient change in the MI decays away when the string system coarsens. Thus one expects a net rapid change in the spinning rate and restoration of only few percent of the original value.
Ratio of the quadrupole moment to MI is though very small, however note that gravitation power depends on the (square of) third time derivative of the quadrupole moment. Here, phase transition dynamics will lead to changes in density fluctuations occurring in time scales of microseconds. This may more than compensate for the small amplitude and may lead to these density fluctuations as an important source of gravitational wave emission from neutron stars.