# Probing Ultradense Matter with Neutron Stars

### Neutron Star Equation of State

The nature of matter at the highest possible densities, characterized by the equation of state (EOS) or pressure-density relation for ultradense matter, is one of the great unsolved problems in science. Theory predicts a host of possible exotic states. The regime of supernuclear density and low temperature is inaccessible in the laboratory, but the EOS sets the internal structure of NSs and is thus reflected in their mass-radius (M-R) relation. With few-%-accuracy estimates of R and M over a range of masses, we can invert the M-R relation to recover the EOS, resolving major questions in both quantum chromodynamics and astrophysics. Although radio astronomy has delivered precise mass measurements, prospects for precision radius measurements in the radio are limited. Current constraints on R from X-ray spectral fitting of accreting NSs are at the 10% level but are affected by modelling and spectral calibration uncertainties. The X-ray timing capabilities of STROBE-X would be transformative, unlocking three complementary techniques that can be combined to deliver robust high-precision measurements of M and R: waveform modelling, spin measurements, and seismology.

**Figure 2: **Simulated neutron star mass-radius constraints with STROBE-X. The red ellipses
illustrate how 5% measurements of M and R from many neutron stars, as expected from STROBE-X, will
map out the M-R relation and thus tightly constrain the ultradense matter equation of state (EOS).
An earlier RXTE constraint is also shown. The M-R curve for the “true” EOS (blue in this example)
must be consistent with all observations. The other colored curves are M-R relations for other
representative EOS models, and the grey band shows the range of EOS models based on chiral
effective field theory.

We focus on the waveform modeling technique. A “hot spot” on the stellar surface is modulated by the
stellar rotation, giving rise to coherent pulsations. As photons propagate through the curved
spacetime of the relativistic star, information about M and R is encoded into the shape and
energy-dependence of the waveform. Modeling yields M/R, and also allows recovery of both M and R
separately for suciently fast (millisecond) rotators. Simulations show the importance of large
collecting area: the precision with which M and R can be measured scales as 1/R where R = 1.4
f_{rms} N^{1/2}, where N is the number of photons and frms is the fractional rms
amplitude of the oscillation. For precisions of a few %, we need R > 400, typically requiring > 106
counts. NICER, which is scheduled for launch in 2017 and will apply this technique to
rotation-powered pulsars, aims to measure R to ~ 5% precision for four bright sources.

### Mesaurement Feasability

The LEMA and HEMA instruments on STROBE-X will take the next step towards a full recovery of the
ultradense matter EOS, which requires measurements over as wide a mass range as possible (see Fig.
2). LEMA (with an area 8.5x NICER) will more than triple the available sample size of
rotation-powered pulsars, thus also increasing the mass
span. HEMA (with an area 5.5x RXTE) will allow
us to extend the waveform modeling technique to X-ray burst oscillations in accreting NSs, rotating
hot spots that develop during episodic thermonuclear bursts from the stellar surface. These
accreting sources are particularly attractive for M-R measurement in that they are numerous, bright,
very rapidly rotating, have a well-understood thermal X-ray spectrum, and can be cross-checked with
R constraints from continuum spectral modelling to remove systematic errors. However, the short
duration (~20 s) of these bursts requires them to be combined to reach the desired precision in R.
Obtaining the necessary fluence with only a small number of bursts requires a collecting area of >3
m^{2}. Based on burst recurrence times measured with RXTE and the fraction of bursts
containing oscillations, we estimate required HEMA observing times of 100-500 ks per source. There
are 24 known NSs with burst oscillations and/or accretion-powered pulsations (another type of hot
spot that could also be used for waveform modeling), and we expect more to be discovered in the
STROBE-X era, although not all will be suitable for modeling. We estimate that, together, LEMA and
HEMA will map the M-R curve with > 20 NSs and tightly constrain the NS EOS (Fig. 2).