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Testing Astrophysical Black Holes Cosimo Bambi Fudan University http://www.physics.fudan.edu.cn/tps/people/bambi/ Black Holes and Friends 2 11-13 April 2016, Fudan University (Shanghai, China) Plan of the talk Introduction

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Testing Astrophysical Black Holes Cosimo Bambi

Fudan University http://www.physics.fudan.edu.cn/tps/people/bambi/ Black Holes and Friends 2 11-13 April 2016, Fudan University (Shanghai, China)

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Cosimo Bambi (Fudan University) 2

Plan of the talk

  • Introduction
  • Continuum-fitting method
  • X-ray reflection spectroscopy
  • Transfer function
  • Conclusions
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Cosimo Bambi (Fudan University) 3

Introduction

  • Introduction
  • Continuum-fitting method
  • X-ray reflection spectroscopy
  • Transfer function
  • Conclusions
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Cosimo Bambi (Fudan University) 4

Tests of general relativity

  • 1915 → General relativity (Einstein)
  • 1919 → Deflection of light by the Sun (Eddington)
  • 1960s-present → Solar System experiments
  • 1970s-present → Binary pulsars

Today:

  • Cosmological tests (dark matter/dark energy)
  • Black holes
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Cosimo Bambi (Fudan University) 5

Tests of general relativity

  • 1915 → General relativity (Einstein)
  • 1919 → Deflection of light by the Sun (Eddington)
  • 1960s-present → Solar System experiments
  • 1970s-present → Binary pulsars

Today:

  • Cosmological tests (dark matter/dark energy)
  • Black holes

Weak fields Strong fields Large scales

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Cosimo Bambi (Fudan University) 6

Black holes in GR (Theory)

  • Final product of the gravitational collapse → Black hole
  • 4D General Relativity → Kerr black hole
  • Only 2 parameters: the mass M and the spin J (a* = J/M2)
  • Kerr bound: |a*| < 1
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Cosimo Bambi (Fudan University) 7

Black hole candidates (Observations)

  • Stellar-mass BH candidates in X-ray

binary systems (5 – 20 Solar masses)

  • Supermassive BH candidates in galactic

nuclei (105 – 1010 Solar masses)

  • Intermediate-mass BH candidates in

ULXS (102 – 104 Solar masses?)

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Cosimo Bambi (Fudan University) 8

Stellar-mass BH candidates

  • Dark objects in X-ray binary systems
  • Mass function:
  • In general, a good estimate of MC and i is necessary
  • Maximum mass for relativistic stars about 3 Solar masses (see Rhoades

& Ruffini 1974 and Kalogera & Baym 1996)

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Cosimo Bambi (Fudan University) 9

From Remillard & McClintock 2006

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Cosimo Bambi (Fudan University) 10

From Remillard & McClintock 2006

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Cosimo Bambi (Fudan University) 11

Supermassive BH candidate in the Galaxy

  • We study the orbital motion of individual stars
  • Point-like central object with a mass of 4x106 Solar masses
  • Radius < 45 AU (600 RSch)

From Ghez et al. 2005

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Cosimo Bambi (Fudan University) 12

Solar System experiments: Schwarzschild solution in the weak field limit

  • Parametrized Post-Newtonian formalism (PPN formalism)
  • Weak field limit (M/r << 1)
  • Solar System experiments
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Cosimo Bambi (Fudan University) 13

Testing the Kerr solution around black hole candidates

  • No satisfactory formalism at present
  • Strong gravity, no expansion in M/r
  • Proposals: Johannsen-Psaltis (2011), Johannsen (2013), Cardoso-Pani-

Rico (2014), Konoplya-Rezzolla-Zhidenko (2016), etc.

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Cosimo Bambi (Fudan University) 14

Important remarks

  • The study of the properties of the electromagnetic radiation emitted by

the gas in the accretion disk can test the Kerr metric, not the Einstein equations

  • The Kerr metric is the unique uncharged BH solution of GR, but it is a

solution of many other theories of gravity

  • If we want to test the Einstein equations, we need to study the

perturbations around the Kerr background (see Barausse & Sotiriou 2008)

  • It is not enough to observe relativistic features absent in Newtonian

gravity (common misunderstanding in the literature). In order to test the Kerr BH hypothesis it is necessary to check that observational data exclude deviations from the Kerr solution. Non-Kerr BHs typically look like Kerr BHs with different spin

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Cosimo Bambi (Fudan University) 15

Correlated important remarks

  • Technically, a black hole is a region causally disconnected to future null

infinity and the event horizon is its boundary

  • Observationally, we can test the existence of an apparent horizon. To

test the existence of an event horizon we should know the future, which is impossible. A long-living apparent horizon behaves like an event horizon

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Cosimo Bambi (Fudan University) 16

Continuum-fitting method

  • Introduction
  • Continuum-fitting method
  • X-ray reflection spectroscopy
  • Transfer function
  • Conclusions
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Cosimo Bambi (Fudan University) 17

Today

  • Continuum-fitting method → only stellar-mass black hole candidates

(Zhang, Cui & Chen, 1997)

  • Iron line → stellar-mass and super-massive black hole candidates

(Fabian et al., 1989) From Gou et al. 2011

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Cosimo Bambi (Fudan University) 18

Continuum-fitting method

  • The soft X-ray component of the spectrum of stellar-mass BH

candidates is the thermal spectrum of a geometrically thin and optically thick accretion disk From Gou et al. 2011

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Cosimo Bambi (Fudan University) 19

Novikov-Thorne Model

  • Geometrically thin and optically thick accretion disk
  • Relativistic generalization of the Shakura-Sunyaev model

Assumptions:

  • Disk on the equatorial plane
  • Gas's particles move on nearly geodesic circular orbits
  • No magnetic fields
  • No heat advection; energy radiated from the disk surface
  • Inner edge of the disk at the ISCO, where stresses vanish

→ Efficiency = 1 – EISCO

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Cosimo Bambi (Fudan University) 20

Novikov-Thorne Model

  • Geometrically thin and optically thick accretion disk
  • Relativistic generalization of the Shakura-Sunyaev model

Assumptions:

  • Disk on the equatorial plane
  • Gas's particles move on nearly geodesic circular orbits
  • No magnetic fields
  • No heat advection; energy radiated from the disk surface
  • Inner edge of the disk at the ISCO, where stresses vanish

→ Efficiency = 1 – EISCO Selection criterion: 0.08 LEDD < L < 0.30 LEDD

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Cosimo Bambi (Fudan University) 21

Continuum-fitting method in Kerr background

  • 5 parameters (BH mass, BH spin, BH distance, viewing angle, mass

accretion rate)

  • BH mass, BH distance, viewing angle → BH spin, mass accretion rate
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Cosimo Bambi (Fudan University) 22

Continuum-fitting method results to date

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Cosimo Bambi (Fudan University) 23

Step 1: computation of the image

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Cosimo Bambi (Fudan University) 24

Step 2: calculation of the disk's spectrum

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Cosimo Bambi (Fudan University) 25

Constraints from the continuum-fitting method [Kong, Li & Bambi (2014)]

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Cosimo Bambi (Fudan University) 26

Constraints from the continuum-fitting method [Kong, Li & Bambi (2014)]

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Cosimo Bambi (Fudan University) 27

Constraints from the continuum-fitting method [Kong, Li & Bambi (2014)]

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Cosimo Bambi (Fudan University) 28

Constraints from the continuum-fitting method [Bambi (2014)]

  • Cardoso-Pani-Rico parametrization
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Cosimo Bambi (Fudan University) 29

Constraints from the continuum-fitting method [Bambi (2014)]

  • Cardoso-Pani-Rico parametrization

Conclusion: The continuum-fitting method is currently the most robust technique, but the shape of the spectrum is simple. We can only measure one parameter of the background geometry

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Cosimo Bambi (Fudan University) 30

X-ray reflection spectroscopy

  • Introduction
  • Continuum-fitting method
  • X-ray reflection spectroscopy
  • Transfer function
  • Conclusions
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Cosimo Bambi (Fudan University) 31

Today

  • Continuum-fitting method → only stellar-mass black hole candidates

(Zhang, Cui & Chen, 1997)

  • Iron line → stellar-mass and super-massive black hole candidates

(Fabian et al., 1989) From Gou et al. 2011

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Cosimo Bambi (Fudan University) 32

Disk reflection

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Cosimo Bambi (Fudan University) 33

Reflected spectrum

From Garcia et al. 2013

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Cosimo Bambi (Fudan University) 34

Model parameters

  • Photon index of the power-law component
  • Ionization parameter
  • Iron abundance
  • Inclination angle of the disk
  • Emissivity profile...
  • Spacetime metric (spin + possible deformation parameters)

[Assumption: inner edge of the disk at the ISCO radius]

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Cosimo Bambi (Fudan University) 35

Iron line

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Cosimo Bambi (Fudan University) 36

Novikov-Thorne Model

  • Geometrically thin and optically thick accretion disk
  • Relativistic generalization of the Shakura-Sunyaev model

Assumptions:

  • Disk on the equatorial plane
  • Gas's particles move on nearly geodesic circular orbits
  • No magnetic fields
  • No heat advection; energy radiated from the disk surface
  • Inner edge of the disk at the ISCO, where stresses vanish

→ Efficiency = 1 – EISCO

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Cosimo Bambi (Fudan University) 37

Novikov-Thorne Model

  • Geometrically thin and optically thick accretion disk
  • Relativistic generalization of the Shakura-Sunyaev model

Assumptions:

  • Disk on the equatorial plane
  • Gas's particles move on nearly geodesic circular orbits
  • No magnetic fields
  • No heat advection; energy radiated from the disk surface
  • Inner edge of the disk at the ISCO, where stresses vanish

→ Efficiency = 1 – EISCO Selection criterion: 0.08 LEDD < L < 0.30 LEDD

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Cosimo Bambi (Fudan University) 38

Current spin measurements

From Bambi, Jiang & Steiner 2016

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Cosimo Bambi (Fudan University) 39

Current spin measurements

From Bambi, Jiang & Steiner 2016

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Cosimo Bambi (Fudan University) 40

Constraining power of the iron K-alpha line in the CPR framework

From Jiang, Bambi & Steiner (2015b)

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Cosimo Bambi (Fudan University) 41

Constraining power of the iron K-alpha line in the CPR framework

From Jiang, Bambi & Steiner (2015b) Conclusion: The iron line technique is potentially more powerful than the continuum-fitting method, but we need: 1) a high photon number count, 2) the correct theoretical model

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Cosimo Bambi (Fudan University) 42

Iron K-alpha line (Interior solutions or Boson stars)

Regular solution Singular solution From Bambi & Malafarina (2013)

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Cosimo Bambi (Fudan University) 43

Iron K-alpha line (Traversable wormholes)

Constraint: a < 0.02 Metric

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Cosimo Bambi (Fudan University) 44

Transfer function

  • Introduction
  • Continuum-fitting method
  • X-ray reflection spectroscopy
  • Transfer function
  • Conclusions
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Cosimo Bambi (Fudan University) 45

Transfer function

  • Introduction
  • Continuum-fitting method
  • X-ray reflection spectroscopy
  • Transfer function
  • Conclusions

Project in collaboration with: Sourabh Nampalliwar (Fudan) Alejandro Cardenas-Avendano (Lorentz) Thomas Dauser (Erlangen-Nuremberg) Javier Garcia-Martinez (CfA)

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Cosimo Bambi (Fudan University) 46

Definition

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Cosimo Bambi (Fudan University) 47

Definition

Metric

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Cosimo Bambi (Fudan University) 48

Definition

Reflection process (astrophysics)

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Cosimo Bambi (Fudan University) 49

Definition

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Cosimo Bambi (Fudan University) 50

Definition

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Cosimo Bambi (Fudan University) 51

Hamilton's principal function

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Cosimo Bambi (Fudan University) 52

Carter constant

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Cosimo Bambi (Fudan University) 53

Equations of motion

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Cosimo Bambi (Fudan University) 54

Coniugate momenta

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Cosimo Bambi (Fudan University) 55

Image of the distant observer

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Cosimo Bambi (Fudan University) 56

Calculation of the Jacobian

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Cosimo Bambi (Fudan University) 57

Calculation of the Jacobian

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Cosimo Bambi (Fudan University) 58

Non-Kerr metrics

  • In general, no Carter constant
  • Even if there is a Carter constant, no elliptic integrals
  • Geodesic equations
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Cosimo Bambi (Fudan University) 59

Image plane

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Cosimo Bambi (Fudan University) 60

Photon initial conditions

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Cosimo Bambi (Fudan University) 61

Preliminary results...

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Cosimo Bambi (Fudan University) 62

Preliminary results...

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Cosimo Bambi (Fudan University) 63

Preliminary results...

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Cosimo Bambi (Fudan University) 64

Conclusions

  • Introduction
  • Continuum-fitting method
  • X-ray reflection spectroscopy
  • Transfer function
  • Conclusions
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Cosimo Bambi (Fudan University) 65

Conclusions

  • Testing the Kerr metric via Continuum-fitting and iron line methods
  • For some preliminary studies, we can simply add an iron line profile to a

model (this is what we have done up to now)

  • For precise measurements, we need a consistent reflected spectrum (our

current goal)

  • Formalism of the transfer function
  • Step after: analyze real data with Xspec and our new model
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Cosimo Bambi (Fudan University) 66

Thank you!