Distinguishing boson stars from black holes and neutron stars with tidal interactions
Noah Sennett
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Department of Physics, University of Maryland
in collaboration with T. Hinderer, J. Steinhoff, A. Buonanno, and S. Ossokine Phys. Rev. Dxx xxx (2017)
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 1 / 22 Outline
1. Introduction 2. Tidal deformability of boson stars 3. Distinguishing boson stars from neutron stars and black holes 4. Conclusions (To be fleshed out)
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 2 / 22 Probing new physics with gravitational waves
Gravitational waves provide an unprecedented view into the highly dynamical, strong-field regime of gravity
Detectors may be sensitive to new physics that emerge in this regime as: 1. modifications to general relativity 2. compact objects composed of exotic matter
Measuring (or constraining) new physics in an observed gravitational wave requires accurate waveform models.
(See Chris van den Broeck’s plenary talk on Thursday)
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 3 / 22 Probing new physics with gravitational waves
Can gravitational wave detectors distinguish exotic compact objects from black holes and/or neutron stars? The bodies’ composition impacts each part of the gravitational wave:
[courtesy of T. Hinderer]
Absorption Merger Echoes (Spin-induced) Tidal interactions Quasinormal modes multipole interactions
An abridged bibliography: [Chirenti+, 2007], [Pani, 2015], [Macedo+, 2013], [Cardoso+, 2016], [Cardoso+, 2017], [Krishnendu+, 2017], [Bezares+, 2017], [Maselli+, 2017],...
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 4 / 22 Tidal interactions in compact binary systems
In a binary, the gravitational field from each body deforms its companion.
Decomposed into STF tensors, the response of each body is characterized by dimensionless tidal deformabilities Λ`
(E) 2`+1 Qi1...i` = −Λ` M Ei1...i` (B) 2`+1 Si1...i` = −Λ` M Bi1...i` (E) The dominant effect in the waveform is determined solely by Λ2
(E) 1 (E) 4 For black holes: Λ2 = 0 For neutron stars: 10 . Λ2 . 10
(E) We compute Λ2 for a particular class of exotic compact objects and estimate whether such objects can be distinguished from black holes and neutron stars by measuring their tidal deformability Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 5 / 22 We consider boson star models parameterized by two quantities: 1. The boson mass (µ) determines the overall size of the star. 2. The scalar coupling (λ or σ0) determines the “EOS” of the star. 2 V (|Φ| ) Mmax[M ] Compactness
2 2 85peV Mini BS µ Φ µ 0.08 √ 2 2 2 λ 4 270MeV Massive BS µ Φ + 2 |Φ| λ µ 0.158 Neutron star 2 − 4 0.3 2 2 3 2 2 2|Φ|2 µ 700TeV Solitonic BS µ Φ 1 − 2 0.349 σ0 σ0 µ Black hole ∞ 0.5
Boson stars
Boson stars are self-gravitating configurations of a complex scalar field Z √ R S = d4x −g − ∇αΦ∇ Φ∗ − V (|Φ|2) 16π α
Boson stars behave as anisotropic perfect fluid stars whose properties are determined by the potential V (|Φ|2).
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 6 / 22 Boson stars
Boson stars are self-gravitating configurations of a complex scalar field Z √ R S = d4x −g − ∇αΦ∇ Φ∗ − V (|Φ|2) 16π α We consider boson star models parameterized by two quantities: 1. The boson mass (µ) determines the overall size of the star. 2. The scalar coupling (λ or σ0) determines the “EOS” of the star. 2 V (|Φ| ) Mmax[M ] Compactness
2 2 85peV Mini BS µ Φ µ 0.08 √ 2 2 2 λ 4 270MeV Massive BS µ Φ + 2 |Φ| λ µ 0.158 Neutron star 2 − 4 0.3 2 2 3 2 2 2|Φ|2 µ 700TeV Solitonic BS µ Φ 1 − 2 0.349 σ0 σ0 µ Black hole ∞ 0.5
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 6 / 22 Boson stars
We consider spherically symmetric, non-spinning boson stars in their ground state
2 v(r) 2 u(r) 2 2 2 ds0 = −e dt + e dr + r dΩ
−iωt Φ0(t, r) = φ0(r)e
0.004 0.004 0.04
0.003 0.003 0.03
0.002 0.002 0.02
0.001 0.001 0.01
0.000 0.000 0.00 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 7 / 22 Computing the tidal deformability of boson stars
We consider static perturbations to the metric and scalar field
(0) gαβ = gαβ + hαβ, Φ = Φ0 + δΦ
decomposed into spherical harmonics in the Regge-Wheeler gauge
α β X h v `m 2 u `m 2 2 `m 2i hαβdx dx = Y`m(θ, ϕ) e h0 (r)dt + e h2 (r)dr + r k (r)dΩ `≥|m| X φ`m(r) δΦ = 1 Y (θ, ϕ)e−iωt r `m `≥|m|
We restrict our attention to ` = 2, even-parity perturbations
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 8 / 22 Computing the tidal deformability of boson stars
The BS “exterior” exponentially approaches vacuum, so h0 satisfies
−3 2 lim h0(r) = c1Qˆ22(r/M − 1) + c2Pˆ22(r/M − 1) ≈ c1(r/M) + c2(r/M) r→∞ The tidal deformability is extracted by identifying the second and first terms with the tidal perturbation and the quadrupole moment it induces, respectively c Λ = 1 3c2
By matching the numerical solution to vacuum solutions at different extraction radii rExtract, we estimate our 2.0 precision on Λ at ∼ 0.1% 1.5 1.0 0 1 5 10 50 100 ∞
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 9 / 22 Tidal deformability of boson stars
Mini and massive boson stars 104 " # √ 22.39 143.5 305.6 wM˜ = −0.529 + − + w log Λ (log Λ)2 (log Λ)3 103 " 20.99 99.1 149.7 # + −0.828 + − + (1 − w) log Λ (log Λ)2 (log Λ)3 102 2 2 −1 0 1 2 3 4 where w ≡ (1 + λmPlanck/(64πµ ))
Solitonic boson stars 102 1079.8 10240 log(σ0M˜ ) = −30.834 + − log Λ + 19 (log Λ + 19)2 101
100
0 2 4 6 8 10 12
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 10 / 22 Tidal deformability of boson stars
V (|Φ|2) Compactness Tidal Deformability
Mini BS µ2Φ2 0.08 Λ & 900 2 2 λ 4 Massive BS µ Φ + 2 |Φ| 0.158 Λ & 280 Neutron star 0.3 Λ & 5 2 2 2 2|Φ|2 Solitonic BS µ Φ 1 − 2 0.349 Λ & 1.3 σ0 Black hole 0.5 Λ = 0
(See also [Mendes+, 2016] for mini BSs and [Cardoso+, 2017] for massive and solitonic BSs)
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 11 / 22 Distinguishing black holes/neutron stars from boson stars
Distinguishing black holes/neutron stars from boson stars hinges on our ability to measure the tidal deformability Λ with high precision
We use a Fisher information matrix approximation to estimate the precision ∆Λ achievable by: 1. Advanced LIGO (aLIGO) at design sensitivity 2. Einstein Telescope (ET) 3. Cosmic Explorer (CE)
For each detector, we consider two fiducial systems: A. Binary black holes (BBH) at 400 Mpc B. Binary neutron stars (BNS) at 200 Mpc with soft (SLy) or stiff (MS1b) EOS
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 12 / 22 Fisher information matrix approximation
Let h(f; θ) be a waveform where θ represents the parameters describing a binary system. For a signal s with high signal to noise ratio (SNR), the probability distribution p(θ|s) is given approximately by a Gaussian centered about the true values θˆ with covariance matrix
Σij = (Γ−1)ij
where Z ∞ ∗ ∗ ∂h ∂h a (f)b(f) + a(f)b (f) Γij = i j , (a|b) = 2 df ∂θ ∂θ 0 Sn(f) The root-mean-square measurement errors are given by q i −1 ii ∆θRMS = (Γ )
This approximation only yields a lower bound on the errors.
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 13 / 22 Fisher matrix estimate of tidal parameters
We model an inspiral-only waveform h(f) with TaylorF2 (3.5PN non-spinning) with 1PN tidal corrections. Leading-order tidal effects are parametrized by the symmetric average
5 ˜ 16 m2 m1 Λ(m1, m2, Λ1, Λ2) = 1 + 12 5 Λ1 + (1 ↔ 2) . 13 m1 M
We consider only systems with total mass M ≤ 12M so that merger occurs at frequencies fmerger ≥ 900 Hz. The waveform begins at f = 10 Hz and terminates at the merger frequency inferred from numerical relativity for BBH [Taracchini+, 2012] and BNS [Bernuzzi+, 2015].
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 14 / 22 Fisher matrix estimate of tidal parameters
We compute the Fisher matrix over the parameters θ = {φc, tc, M, ν, Λ˜}. The precision estimates for these observables can be translated into bounds on the statistical uncertainty in {φc, tc, m1, m2, Λ1, Λ2}.
θ Measurable parameters M √ m = Mν−3/5(1 + 1 − 4ν)/2 ν 1 √ m = Mν−3/5(1 − 1 − 4ν)/2 Λ˜ 2
Boundable parameters Assumptions 0 ≤ Λ1 ≤ g1(ν)Λ˜ 0 ≤ Λ1 ≤ Λ2 Λ1 ≤ Λ2 ≤ g2(ν)Λ˜
We constrain the precision of the latter observables by taking the first variation of these relations. Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 15 / 22 Fisher matrix estimate of tidal parameters: BBH
We restrict to 4M ≤ mi ≤ 8M . At 400 Mpc, SNR ∼ 25 with aLIGO. With ET and CE, the precision improves by factors of ∼ 13.5 and ∼ 23.5, respectively. (See also [Cardoso+, 2017]).
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 16 / 22 Fisher matrix estimate of tidal parameters: BNS
We restrict to 1M ≤ mi ≤ mmax. At 200 Mpc, SNR ∼ 15 with aLIGO. With ET and CE, the precision improves by factors of ∼ 13.5 and ∼ 23.5, respectively.
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 17 / 22 Distinguishing black holes/neutron stars from boson stars
We propose two tests to determine the distinguishability of boson stars from conventional sources:
Test 1: Can we distinguish each body in the binary from a boson star? BS For each boson star model, the deformability is bounded below by some Λmin. Can we achieve sufficient precision to constrain tidal deformability BS measurements in the gap 0 ≤ Λ ≤ Λmin?
Test 2: Can we distinguish a binary system from a binary boson star system? In a binary system whose components have deformabilities comparable to those of boson stars, can we rule out the possibility that both objects are boson stars?
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 18 / 22 Test 1: Can we distinguish each body in the binary from a boson star? BS For each boson star model, the deformability is bounded below by some Λmin. Can we achieve sufficient precision to constrain tidal deformability BS measurements in the gap 0 ≤ Λ ≤ Λmin?
For BBH at 400 Mpc, ∆Λi ≤ 100 with aLIGO, and thus can be distinguished from mini and massive boson stars. Black holes cannot be distinguished from solitonic boson stars with aLIGO, but could be marginally distinguished BS (∆Λ ≈ Λmin) with ET and CE for nearly-equal mass systems.
For BNS at 200 Mpc, ∆Λi ≤ 200 with aLIGO for systems with near-maximal mass and nearly-equal mass ratio. These systems can be distinguished from mini and massive boson stars. However, less massive BNS cannot be distinguished because their deformability are comparable to that of boson BS stars, i.e. Λ ≥ Λmin.
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 19 / 22 Test 2: Can we distinguish a binary system from a binary boson star system? In a binary system whose components have deformabilities comparable to those of boson stars, can we rule out the possibility that both objects are boson stars?
∗ BS If an object’s deformability Λ & Λmin, then the boson mass µ can be chosen to produce a boson star with the same 4 mass and deformability (m∗, Λ∗). 10 In this case, single tidal deformability measurement cannot distinguish 103 between boson stars and black holes/neutron stars.
2 To break this degeneracy, one must 10 1. 1.2 1.4 1.6 combine measurements of both objects in the binary.
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 20 / 22 Test 2: Can we distinguish a binary system from a binary boson star system? In a binary system whose components have deformabilities comparable to those of boson stars, can we rule out the possibility that both objects are boson stars?
Consider a (1.55 + 1.35)M BNS at 200 Mpc observed by aLIGO. The shaded regions depict all possible 104 massive boson stars consistent with the measurements of the smaller body. 102 Because the larger body is excluded from this region, we can distinguish 100 this BNS from a massive boson star 1 2 1 2 binary system.
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 20 / 22 Test 2: Can we distinguish a binary system from a binary boson star system? In a binary system whose components have deformabilities comparable to those of boson stars, can we rule out the possibility that both objects are boson stars?
Consider a (6.5 + 4.5)M BBH at 400 Mpc observed by aLIGO.
The shaded regions depict all solitonic 102 boson stars with σ0 = 0.05 mPlanck consistent with the measurement of the 101 smaller body.
With aLIGO, one cannot distinguish 100 this BBH from a solitonic boson star binary. 0 2 4 6 8 10 12
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 20 / 22 Test 2: Can we distinguish a binary system from a binary boson star system? In a binary system whose components have deformabilities comparable to those of boson stars, can we rule out the possibility that both objects are boson stars?
102 With next-generation detectors, this BBH can be distinguished from a 101 solitonic boson star binary.
100
0 2 4 6 8 10 12
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 20 / 22 Conclusions
Tidal interactions occurring during the inspiral of a binary system can be used to distinguish black holes/neutron stars from exotic compact objects We computed the tidal deformability for three models of boson stars and produced fits We estimated the precision with which the tidal deformability can be measured using the Fisher information matrix of TaylorF2 waveforms with tidal effects We outlined two tests to determine whether conventional sources can be distinguished from boson stars in binary systems
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 21 / 22 BNS BBH
Yes for sufficiently Mini Yes Test 1 high-mass systems Can we distinguish each Yes for near-maximal body in the binary from a Massive mass systems Yes boson star? No with aLIGO Solitonic No (Possibly with ET and CE for near-equal mass)
Mini Yes with sufficient Yes Test 2 mass ratio Can we distinguish the Yes with sufficient binary system from a Massive mass ratio Yes binary boson star? No with aLIGO Solitonic (Yes with ET and CE with sufficient mass ratio)
Noah Sennett (AEI) Amaldi 12 Conference (Pasadena, CA) July 10, 2017 22 / 22