Neutron stars as a probe of the theory of gravity

Raissa F. P. Mendes Universidade Federal Fluminense

XXXIX Encontro Nacional de Física de Partículas e Campos, 27.09.2018

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Pulsars • +2500 in the Galaxy; 10% in binaries

• Precise masses for ~40 NS, from 1.17 to 2𝑀⊙

Neutron stars

Credit: ESO/L

Neutron stars

X-ray binaries • Simultaneous (but less precise)

measurements of masses and radii

• Radii of ~12 stars in the 9.9-11.2 km interval.

Credit: ESO/L

Credit: NASA

Pulsars • +2500 in the Galaxy; 10% in binaries

• Precise masses for ~40 NS, from 1.17 to 2𝑀⊙

Neutron stars

X-ray binaries • Simultaneous (but less precise)

measurements of masses and radii

• Radii of ~12 stars in the 9.9-11.2 km interval.

Credit: ESO/L

Credit: NASA

Pulsars • +2500 in the Galaxy; 10% in binaries

• Precise masses for ~40 NS, from 1.17 to 2𝑀⊙

Gravitational waves • 1 event: GW170817

• Rich physics: GRB, kilonova, etc.

Neutron stars

10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 [𝜖]

𝜖 =𝐺𝑀

𝑟𝑐2

NS

Neutron stars

10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 [𝜖]

𝜖 =𝐺𝑀

𝑟𝑐2

NS

BH

Introduction

‘Weak field’ vs ‘strong field’ tests of GR

– Example: scalar-tensor theories

Neutron star phenomenology in modified theories of gravity

– Challenge: degeneracy with nuclear equation of state

New results (Mendes & Ortiz, PRL 120, 201104 (2018))

– New families of NS QNM may help break degeneracy

Perspectives

Outline

Introduction

‘Weak field’ vs ‘strong field’ tests of GR

– Example: scalar-tensor theories

Neutron star phenomenology in modified theories of gravity

– Challenge: degeneracy with nuclear equation of state

New results (Mendes & Ortiz, PRL 120, 201104 (2018))

– New families of NS QNM may help break degeneracy

Perspectives

Outline

‘Weak field’ vs ‘strong field’ tests

10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 [𝜖]

𝜖 =𝐺𝑀

𝑟𝑐2

NS

BH

Post-Newtonian regime: 𝜖 ≪ 1, 𝑣 ≪ 𝑐

PPN formalism

‘Weak field’ vs ‘strong field’ tests

Will, Living Rev. Relativ. 9, 3 (2006)

If the theory of gravity is constrained to be so similar to GR in the weak field regime, what freedom remains in the regime of strong

gravitational fields?

‘Weak field’ vs ‘strong field’ tests

Example: scalar-tensor theories:

‘Weak field’ vs ‘strong field’ tests

𝑆 =1

16𝜋𝐺 𝑑4𝑥 −𝑔 𝑅 − 2𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 + 𝑆𝑚 Ψ𝑚; 𝑎 𝜙 2𝑔𝜇𝜈

Example: scalar-tensor theories:

– Expand (𝜙0 = 𝜙 𝜏0 = 𝑐𝑡𝑒):

• GR: 𝛼0 = 𝛽0 = ⋯ = 0;

• FJBD: 𝛽0 = ⋯ = 0, 𝛼0~1

𝜔𝐵𝐷

• NMC (𝜉𝑅𝜙2): 𝛼0 = 0, 𝛽0 = 2𝜉, 𝛽0′ = 0, 𝛽′′0 = 8 1 − 12𝜉 𝜉2, …

‘Weak field’ vs ‘strong field’ tests

𝑆 =1

16𝜋𝐺 𝑑4𝑥 −𝑔 𝑅 − 2𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 + 𝑆𝑚 Ψ𝑚; 𝑎 𝜙 2𝑔𝜇𝜈

𝛼 𝜙 =𝑑 ln 𝑎 𝜙

𝑑𝜙= 𝛼0 + 𝛽0 𝜙 − 𝜙0 + 𝑂[ 𝜙 − 𝜙0

2]

Example: scalar-tensor theories:

– Expand (𝜙0 = 𝜙 𝜏0 = 𝑐𝑡𝑒):

• GR: 𝛼0 = 𝛽0 = ⋯ = 0;

• FJBD: 𝛽0 = ⋯ = 0, 𝛼0~1

𝜔𝐵𝐷

• NMC (𝜉𝑅𝜙2): 𝛼0 = 0, 𝛽0 = 2𝜉, 𝛽0′ = 0, 𝛽′′0 = 8 1 − 12𝜉 𝜉2, …

– PPN parameters:

1 − 𝛾 =2𝛼0

2

1 + 𝛼02 , 𝛽 − 1 =

𝛽0𝛼02

2 1 + 𝛼02 2

To all orders: ∝ 𝛼02 (Damour, Esposito-Farèse, 1996)

‘Weak field’ vs ‘strong field’ tests

𝑆 =1

16𝜋𝐺 𝑑4𝑥 −𝑔 𝑅 − 2𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 + 𝑆𝑚 Ψ𝑚; 𝑎 𝜙 2𝑔𝜇𝜈

𝛼 𝜙 =𝑑 ln 𝑎 𝜙

𝑑𝜙= 𝛼0 + 𝛽0 𝜙 − 𝜙0 + 𝑂[ 𝜙 − 𝜙0

2]

‘Weak field’ vs ‘strong field’ tests

Some STTs (with 𝛼0 = 0) can be perturbatively indistinguishable from GR (in the sense of a PN expansion), but still allow for O(1) deviations from GR

in the strong field environment of NS.

Spontaneous scalarization (Damour & Esposito-Farèse, 1993)

– Nonperturbative strong-field effect

– Phase transition ∼ spontaneous magnetization

‘Weak field’ vs ‘strong field’ tests

Some STTs (with 𝛼0 = 0) can be perturbatively indistinguishable from GR (in the sense of a PN expansion), but still allow for O(1) deviations from GR

in the strong field environment of NS.

GR

Mendes & Ortiz, 2016

𝜶 𝝓 = 𝜷 𝝓 − 𝝓𝟎 with 𝛽 = −6

Mendes & Ortiz, 2016

𝜶 𝝓 = 𝜷 𝝓 − 𝝓𝟎 with 𝛽 = −6

Mendes & Ortiz, 2016

𝜶 𝝓 ∝ 𝒕𝒂𝒏𝒉[ 𝟑 𝜷 (𝝓 − 𝝓𝟎)] with 𝛽 = 100

If the theory of gravity is constrained to be so similar to GR in the weak field regime, what freedom remains in the regime of strong

gravitational fields?

‘Weak field’ vs ‘strong field’ tests

The strong field regime may hold surprises!

Introduction

‘Weak field’ vs ‘strong field’ tests of GR

– Example: scalar-tensor theories

Neutron star phenomenology in modified theories of gravity

– Challenge: degeneracy with nuclear equation of state

New results (Mendes & Ortiz, PRL 120, 201104 (2018))

– New families of NS QNM may help break degeneracy

Perspectives

Outline

Modified theories of gravity may:

– Alter the internal structure of neutron stars

Example: scalar-tensor theories of gravity

Neutron stars in modified gravity

𝛽 = −6 𝛽 = 100

gravity weaker than in GR gravity stronger than in GR

If the EoS was known, constrain these theories would be a ‘simple’ matter!

Neutron stars in modified gravity

Ozel & Freire 2016

Neutron stars in modified gravity

Neutron stars in modified gravity

Tidal deformability (Λ)

Neutron stars in modified gravity

LIGO & Virgo (2017)

Tidal deformability (Λ)

Neutron stars in modified gravity

LIGO & Virgo (2017) Yazadjiev, Doneva & Kokkotas (2018)

Tidal deformability (Λ)

𝑘2=3Λ

2𝑅5

𝑓 𝑅 = 𝑅 + 𝑎𝑅2

Neutron stars in modified gravity

Merger time

Neutron stars in modified gravity

Merger time

Neutron stars in modified gravity

Barausse et al. (2013)

Merger time

Neutron stars in modified gravity

Quasinormal modes

Neutron stars in modified gravity

Ferrari & Gualtierri (2008)

Quasinormal modes

Neutron stars in modified gravity

Ferrari & Gualtierri (2008)

Sotani & Kokkotas (2004)

Quasinormal modes

Introduction

‘Weak field’ vs ‘strong field’ tests of GR

– Example: scalar-tensor theories

Neutron star phenomenology in modified theories of gravity

– Challenge: degeneracy with nuclear equation of state

New results (Mendes & Ortiz, PRL 120, 201104 (2018))

– New families of NS QNM may help break degeneracy

Perspectives

Outline

QNM typical frequencies and damping times:

– Fundamental 𝑙 = 2 mode of a Schwarzschild BH: 𝑓 ≈ 12𝑘𝐻𝑧𝑀⊙

𝑀, 𝜏~𝑚𝑠

• For GW150914, 𝑓 ≈ 250𝐻𝑧, 𝜏 ≈ 4𝑚𝑠.

– Fundamental 𝑙 = 2 mode of a 1.4𝑀⊙ neutron star: 𝑓 ≈ 1.6𝑘𝐻𝑧, 𝜏~0.3𝑠.

Quasi-normal modes

PRL 116, 061102 (2016)

Modified theories of gravity may:

– Alter the internal structure of neutron stars

– Shift the GR spectrum

Example: scalar-tensor theories of gravity

Neutron stars in modified gravity

Sotani & Kokkotas, 2004

Sotani 2014

o fundamental 𝑙 = 2 mode in the Cowling approximation

o fundamental radial mode in the Cowling approximation

Modified theories of gravity may:

– Alter the internal structure of neutron stars

– Shift the GR spectrum

Example: scalar-tensor theories of gravity

Neutron stars in modified gravity

Sotani & Kokkotas, 2004

Sotani 2014

o fundamental 𝑙 = 2 mode in the Cowling approximation

o fundamental radial mode in the Cowling approximation

Modified theories of gravity may not only shift the frequencies of neutron star quasi-normal modes, but also introduce entirely new families of modes, with no counterpart in GR, and which may be

sufficiently well-resolved in frequency as to allow for a clear detection.

Take home message

Modified theories of gravity may not only shift the frequencies of neutron star quasi-normal modes, but also introduce entirely new families of modes, with no counterpart in GR, and which may be

sufficiently well-resolved in frequency as to allow for a clear detection.

Take home message

In some sense, expected!

Newtonian star General-relativistic star

{𝜔𝑖(𝑁)

} 𝜔𝑖𝑅

= 𝜔𝑖𝑁+ 𝛿𝜔𝑖 + 𝑖Δ𝑖

+ w-modes!

Setting

Action:

𝑆 =1

16𝜋𝐺 𝑑4𝑥 −𝑔 𝑅 − 2𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 + 𝑆𝑚 Ψ𝑚; 𝑎 𝜙 2𝑔𝜇𝜈

Coupling functions:

Background: (spherical) equilibrium solutions

Perturbations: only radial

– In GR: information about (in)stability

– In STTs: scalar sector is dynamical even in spherical symmetry!

– Our approach is general: no Cowling approximation

Model 1: 𝛼 𝜙 =1

3tanh[ 3 𝛽 (𝜙 − 𝜙0)]

Model 2: 𝛼 𝜙 = 𝛽 𝜙 − 𝜙0

Mendes & Ortiz, 2018

Master equations:

𝑒𝜆 𝜖 + 𝑝 𝜉 −Γ1𝑝

𝑎4𝑟2𝑒𝜆+3𝜈 𝑒−𝜈𝑎4𝑟2𝜉 ′

′

+ 𝐴𝜉𝜉 + 𝐴𝛿𝜙𝛿𝜙 + 𝐴𝛿𝜙′𝛿𝜙′ = 0

𝑒2𝜆−2𝜈𝛿𝜙 − 𝛿𝜙′′ + 𝐵𝛿𝜙′𝛿𝜙′ + 𝐵𝛿𝜙𝛿𝜙 + 𝐵𝜉′𝜉′ + 𝐵𝜉𝜉 = 0

Frequency domain calculation:

𝜉 𝑡, 𝑟 = 𝜉 𝑟 𝑒𝑖𝜔𝑡, 𝛿𝜙 𝑡, 𝑟 = 𝛿𝜙 𝑟 𝑒𝑖𝜔𝑡

– Boundary conditions: regularity, outgoing BC for 𝛿𝜙:

lim𝑟→∞

𝛿𝜙(𝑡, 𝑟) → 𝑒𝑖𝜔 𝑡−𝑟

Time domain calculation: 𝛿𝜙 0, 𝑟 ∝ exp[−(𝑟 − 𝑟 )/𝜎2]

𝛿𝜙 0, 𝑟 = 0, 𝜉 0, 𝑟 = 𝜉 0, 𝑟 = 0

New class of QNM in STTs Mendes & Ortiz, 2018

GR

New class of QNM in STTs Mendes & Ortiz, 2018

𝛽 = −5

New class of QNM in STTs Mendes & Ortiz, 2018

𝛽 = −5

New class of QNM in STTs Mendes & Ortiz, 2018

new 𝜙-modes!

𝛽 = −5

New class of QNM in STTs Mendes & Ortiz, 2018

Comparison between eigenfunctions of 𝜉 and 𝛿𝜙 at r=R: • 𝜙-modes predominantly scalar

𝛽 = −5

New class of QNM in STTs Mendes & Ortiz, 2018

Star with 𝑀/𝑅 = 0.26 in GR

Star with 𝑀/𝑅 = 0.26 in M1 with 𝛽 = −5

Introduction

‘Weak field’ vs ‘strong field’ tests of GR

– Example: scalar-tensor theories

Neutron star phenomenology in modified theories of gravity

– Challenge: degeneracy with nuclear equation of state

New results (Mendes & Ortiz, PRL 120, 201104 (2018))

– New families of NS QNM may help break degeneracy

Perspectives

Outline

Nonradial oscillations of spherical stars and quasi-radial oscillations of rotating systems

Extension to other modified theories of gravity

Detectability analyses in various astrophysical scenarios

– Binary neutron star systems

• Inspiral phase (eccentric encounters?): could become resonant with orbital motion, draining energy from the system;

• Post-merger phase, if a neutron star forms.

– Quasi-periodic oscillations of magnetars (typically 10 − 103Hz);

– Gravitational collapse; phase transitions in the core; etc.

Perspectives