Case Study 25.1 — GW170817 Revisited: What Gravitational Waves Tell Us About Nuclear Matter

The Event

On August 17, 2017, the LIGO and Virgo gravitational wave detectors recorded a signal that would change nuclear astrophysics. The event — designated GW170817 — was the first detection of gravitational waves from a binary neutron star merger. We encountered this event in Chapter 23, where it confirmed the neutron star merger origin of r-process elements through the associated kilonova. Here, we return to GW170817 with a different question: what does it tell us about the equation of state of dense nuclear matter?

From Gravitational Waves to the EOS

The gravitational wave signal from a binary neutron star inspiral carries information about the masses of the two stars, their orbital parameters, and — crucially — their tidal deformability. During the final $\sim 100$ seconds of the inspiral, as the orbital separation decreases from $\sim 300$ km to the point of merger, each star raises tidal bulges on the other. These tidal bulges extract energy from the orbit, causing the inspiral to accelerate slightly faster than for two point masses. The cumulative effect modifies the gravitational wave phase evolution in a way that depends on the dimensionless tidal deformability $\Lambda$.

The tidal deformability is defined as:

$$\Lambda = \frac{2}{3} k_2 \left(\frac{c^2 R}{GM}\right)^5$$

where $k_2$ is the quadrupolar tidal Love number and $R$ is the stellar radius. The Love number $k_2$ depends on the internal density profile of the star, which is determined by the EOS through the TOV equation. A larger star (larger $R$, stiffer EOS) is more easily deformed and has a larger $\Lambda$. A more compact star (smaller $R$, softer EOS) is harder to deform and has a smaller $\Lambda$.

The Measurement

The LIGO/Virgo analysis of GW170817 measured the component masses as $m_1 \in [1.36, 1.60]\,M_\odot$ and $m_2 \in [1.17, 1.36]\,M_\odot$ (at 90% credible level), with a total mass of $2.73^{+0.04}_{-0.01}\,M_\odot$. The mass ratio $q = m_2/m_1$ was constrained to $0.73$--$1.0$.

The key nuclear physics result was the constraint on the effective tidal deformability:

$$\tilde{\Lambda} = \frac{16}{13}\frac{(m_1 + 12m_2)m_1^4 \Lambda_1 + (m_2 + 12m_1)m_2^4 \Lambda_2}{(m_1 + m_2)^5}$$

The initial LIGO/Virgo analysis (2017) found $\tilde\Lambda \leq 800$ at the 90% credible level, assuming low-spin priors. A refined analysis (2018) tightened this to $\tilde\Lambda = 300^{+420}_{-230}$ (at 90% credibility). Under the assumption that both stars obey the same EOS (as they must, being made of the same material), the constraint on $\Lambda$ for a $1.4\,M_\odot$ star is:

$$\Lambda_{1.4} = 190^{+390}_{-120}$$

What the Numbers Mean for Nuclear Physics

The tidal deformability measurement excludes the stiffest EOS models. In the language of the mass-radius diagram:

• EOS models predicting $R_{1.4} > 13.5$ km (very stiff) are ruled out because they predict $\Lambda_{1.4} > 800$, well above the measured upper bound.

• EOS models predicting $R_{1.4} < 10.5$ km (very soft) were already ruled out by the $2\,M_\odot$ mass constraint — they cannot support the observed heaviest neutron stars.

• The "allowed band" is $R_{1.4} \approx 11$--$13.5$ km, corresponding to $\Lambda_{1.4} \approx 70$--$800$.

Several specific nuclear physics models can be tested:

The exclusion of MS1 and similar very stiff models is significant: it means the nuclear force cannot be too repulsive at high density. Combined with the $2\,M_\odot$ constraint (which means the force cannot be too soft), the GW170817 measurement squeezes the EOS into a remarkably narrow range.

The Electromagnetic Counterpart: Additional Constraints

The kilonova (AT 2017gfo) and short gamma-ray burst (GRB 170817A) associated with GW170817 provided additional constraints:

1. Threshold mass. The fact that the merger produced a short-lived hypermassive neutron star (rather than promptly collapsing to a black hole) constrains $M_\text{max}$ to be above approximately $2.15\,M_\odot$. If $M_\text{max}$ were below $\sim 2.15\,M_\odot$, the total mass of $2.73\,M_\odot$ would have produced a prompt collapse, inconsistent with the observed kilonova properties (which require a delay before black hole formation to allow mass ejection).

2. Ejecta mass and velocity. The kilonova light curve constrains the mass and velocity of the ejecta, which depend on the EOS through the tidal disruption dynamics. Models favor $R_{1.4} \gtrsim 11$ km.

3. Post-merger remnant. The absence of a post-merger gravitational wave signal constrains the remnant lifetime, placing an upper bound on $M_\text{max} \lesssim 2.3\,M_\odot$ — the remnant must have collapsed to a black hole within $\sim 1$ second.

Combining all constraints: $2.15 \lesssim M_\text{max}/M_\odot \lesssim 2.3$ and $11 \lesssim R_{1.4}/\text{km} \lesssim 13.5$.

Quantitative Estimate: From $\Lambda$ to Radius

The strong sensitivity of $\Lambda$ to $R$ can be understood from the definition. For a star of fixed mass $M$, $\Lambda \propto k_2 R^5$. The Love number $k_2$ itself depends on the density profile, but varies only by a factor of $\sim 2$ across realistic EOS models, while $R^5$ varies dramatically. For a $1.4\,M_\odot$ star:

• $R = 11$ km: $(R/R_0)^5 = (11/12)^5 = 0.63$, so $\Lambda \approx 0.63 \times \Lambda_0$
• $R = 12$ km: $\Lambda = \Lambda_0$ (reference)
• $R = 13$ km: $(13/12)^5 = 1.47$, so $\Lambda \approx 1.47 \times \Lambda_0$
• $R = 14$ km: $(14/12)^5 = 2.16$, so $\Lambda \approx 2.16 \times \Lambda_0$

A 27% increase in radius (from 11 to 14 km) produces a factor of $\sim 3.4$ increase in $\Lambda$. This extreme sensitivity is why gravitational wave observations are so powerful for constraining the EOS — even a modest measurement of $\Lambda$ (within a factor of 2, as achieved by GW170817) translates into a useful constraint on the radius.

Using the GW170817 measurement $\Lambda_{1.4} = 190^{+390}_{-120}$ and assuming $k_2 \approx 0.08$--$0.12$, we can back out the radius:

$$R_{1.4} = \left(\frac{3\Lambda_{1.4}}{2 k_2}\right)^{1/5} \frac{GM}{c^2} \approx 11\text{--}13 \text{ km}$$

This is consistent with — and complementary to — the NICER measurements.

The Nuclear Physics Bottom Line

GW170817 demonstrated that gravitational wave astronomy is a tool for nuclear physics. A 100-second gravitational wave signal from objects 130 million light-years away constrained the pressure-density relationship of nuclear matter at $2$--$5$ times nuclear saturation density — a regime that no terrestrial experiment can probe.

The key insight is the chain of reasoning: gravitational wave phase $\to$ tidal deformability $\to$ stellar radius $\to$ EOS $\to$ nuclear force at extreme density. Each link in this chain is now understood well enough to make the inference quantitative.

The impact of GW170817 extends beyond the specific constraints it provided. It established a methodology: gravitational wave signals from neutron star mergers can be systematically used to extract nuclear physics parameters. With the $\sim$50--200 binary neutron star mergers expected to be detected in the next decade by LIGO/Virgo/KAGRA at design sensitivity, the precision on $\Lambda_{1.4}$ will improve by roughly a factor of 5, corresponding to a $\sim 5\%$ measurement of $R_{1.4}$. Third-generation detectors (Einstein Telescope, Cosmic Explorer, 2030s) will improve this by another order of magnitude.

Discussion Questions

1. GW170817 involved two neutron stars with somewhat different masses ($\sim 1.2$ and $\sim 1.5\,M_\odot$). If both masses were equal, how would the tidal deformability constraint change? Would it be more or less constraining?

2. The next observed neutron star merger event (GW190425) had a significantly higher total mass ($\sim 3.4\,M_\odot$) but was detected at lower signal-to-noise ratio, so the tidal deformability was poorly constrained. What total mass range would be most informative for EOS constraints?

3. Future third-generation detectors (Einstein Telescope, Cosmic Explorer) will detect hundreds of neutron star mergers per year with much higher signal-to-noise ratios. How will the precision on $\Lambda_{1.4}$ improve with $N$ observations? (Hint: $\Delta\Lambda \propto 1/\sqrt{N}$ for independent measurements.)

4. GW170817 constrained the EOS at densities of $1$--$5\rho_0$. To probe higher densities (where quark matter might appear), what kind of event would be needed? (Hint: consider more massive neutron stars, or the post-merger phase.)

5. Compare the information content of GW170817 (one event, two tidal deformabilities) to the NICER measurements (two stars, two mass-radius pairs). Which provides tighter constraints on the EOS? Are they complementary?