Testing strong-field gravity with quasi periodic oscillations

Andrea Maselli

Institute for Astronomy and Astrophysics, University of Tubingen

Feb 7th, 2017High-throughput X-Ray Astronomy in the eXTP era

A.M., P. Pani, L. Gualtieri, V. Ferrari, L. Stella + ApJ 801 115 (2015), PRD 92 083014 (2015), arXiv:1612.00038 +

The strong gravity regimeHow does gravity behave in the strong field regime?

Does General Relativity accurately describe gravity in the strong field regime?

eXTP may provide crucial answers

Singularities, dark energy/matter, quantum connection +

Can other forms of matter/fields form ultra dense objects?

102 years of General RelativityGR represents the best description of gravity we have, and has passed all observational and experimental tests

Eventually Gravitational Waves come to the game

weak - field has been probed extensively, what about the strong - gravity regime?

�10�6 0.2 0.5

NS BH

GMc2R

PPN parameters from Cassini spacecraft � � 1 ⇠ 10�5

The strong gravity regime

Baker et al., ApJ 802:63, (2015)

GMc2R

GMc2R3

(potential)

(cur

vatu

re)

The strong gravity regime

Baker et al., ApJ 802:63, (2015)

Horizon of stellar mass BHs

The strong gravity corner

GMc2R

GMc2R3

(potential)

(cur

vatu

re)

Alternative theories: EDGB

S =1

2

Zd

4x

pg R

General Relativity

A natural way to modify the strong field regime is to include quadratic curvature invariant in the action

Alternative theories: EDGB

S =1

2

Zd

4x

pg R

�12@µ�@

µ�+↵e�

4R2GB

�

General Relativity Einstein-Dilaton- Gauss-Bonnet

A natural way to modify the strong field regime is to include quadratic curvature invariant in the action

is the Dilaton field�

enters as coupling dimensionful parameter ~ ↵⇥`2⇤

QPOs and black holes Quasi Periodic Oscillations in the flux emitted from accreting LMXBs are though to originate in the innermost region of the accretion flow

⌫nod

= ⌫� � ⌫✓ ⌫per = ⌫� � ⌫r

X-ray emission modulated by the azimuthal , periastron and 
nodal precession frequencies⌫per

⌫� ⌫per

The Relativistic Precession Model associates 3 types of QPOs to a combination of the epicyclic frequencies

Stella and Vietri, Phys. Rev. Lett. 82, 17 (1999)

In GR particles on circular-equatorial orbits, will oscillate under small perturbations and �✓�r

RPM

⌫GR� =1

2⇡

M1/2

r3/2 + a?M3/2

⌫GRr =⌫GR'

✓1� 6M

r+ 8a?

M3/2

r3/2� 3a?2M

2

r2

◆1/2

⌫GR✓ =⌫GR'

✓1� 4a?M

3/2

r3/2+ 3a?2

M2

r2

◆1/2

System of 3 equations in there unknown variables

M = (5.31± 0.07)M�

r = (5.68± 0.04)rg

a? = (0.290± 0.003)

Complete application of RPM: J1665-40 discovery by RXTEMotta et al., MNRAS 437, 2554-2565 (2014)

RPM

⌫GR� =1

2⇡

M1/2

r3/2 + a?M3/2

⌫GRr =⌫GR'

✓1� 6M

r+ 8a?

M3/2

r3/2� 3a?2M

2

r2

◆1/2

⌫GR✓ =⌫GR'

✓1� 4a?M

3/2

r3/2+ 3a?2

M2

r2

◆1/2

System of 3 equations in there unknown variables

M = (5.31± 0.07)M�

r = (5.68± 0.04)rg

a? = (0.290± 0.003)

Complete application of RPM: J1665-40 discovery by RXTEMotta et al., MNRAS 437, 2554-2565 (2014)

RPM

⌫GR� =1

2⇡

M1/2

r3/2 + a?M3/2

⌫GRr =⌫GR'

✓1� 6M

r+ 8a?

M3/2

r3/2� 3a?2M

2

r2

◆1/2

⌫GR✓ =⌫GR'

✓1� 4a?M

3/2

r3/2+ 3a?2

M2

r2

◆1/2

System of 3 equations in there unknown variables

M = (5.31± 0.07)M�⇠ 10% rISCO

r = (5.68± 0.04)rg

a? = (0.290± 0.003)

produced near the horizon, probe of the strong field

Complete application of RPM: J1665-40 discovery by RXTEMotta et al., MNRAS 437, 2554-2565 (2014)

Why eXTP? quality and quantityeXTP is expected to measure QPOs frequencies with very high precision

For n>1 redundancy would test GR

eXTP is expected to measure multiple QPOs triplets from the same black hole (⌫�, ⌫nod, ⌫per)i=1,...n

3n quantities equations for 2+n variables (M,a?, ri)

~5 times better than RXTE

Testing gravity with eXTP

Try to interpret the signals with GR and to measure the BH parameters

We compute two set of frequencies within EDGB theory for two emission radii r1,2 = (1.1, 1.4)rISCO

For General Relativity ( ) and M1 = M2 a?1 = a?2↵ = 0

(⌫�, ⌫nod, ⌫per)1 (⌫�, ⌫nod, ⌫per)2

True signals measured by eXTP with (�⌫� ,�⌫nod ,�⌫per)

(M1, a?1, r1) (M2, a

?2, r2)

�M = M1 �M2 �a? = a?1 � a?2 �r = r1 � r2

Verify that the distribution of the variables 
is consistent with a Gaussian distribution with zero mean

~µ = (�M,�a?,�r)

P ⇠ N (~µ,⌃ = ⌃1 + ⌃2)

Testing gravity with eXTP

Rules of the game

Define �2 = (~x� ~µ)T⌃�1(~x� ~µ)

Probability of 32%, 5%, 0.3% to be consistent

1�

2�

3�

�2 = c

Confidence intervalsM = 5.3M�BH parameters: a? = 0.5

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.6

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.4

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0

RPM

Confidence intervalsM = 5.3M�BH parameters:

Consistent with GR

a? = 0.5

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.6

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.4

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0

RPM

Confidence intervalsM = 5.3M�BH parameters:

Consistent with GR

3s threshold

a? = 0.5

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.6

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.4

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0

RPM

Confidence intervalsM = 5.3M�BH parameters:

Consistent with GR

Datasets inconsistent >> 3s

3s threshold

a? = 0.5

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.6

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.4

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0

RPM

Confidence intervalsM = 5.3M�BH parameters:

Consistent with GR

Datasets inconsistent >> 3s

3s threshold

a? = 0.5

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.6

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.4

RPM

××

1σ2σ3σ

-0.06-0.04-0.02 0.00 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0

RPM

In this case would be excluded0.4 . ↵/M2 . 0.6

Confidence intervals

●●

�* = ����* = ����* = ���

-0.06-0.04-0.02 0 0.02 0.04

-0.002

0.000

0.002

0.004

ΔM

Δa*

α/M2 = 0.4

RPM

●●

�σ �����σ ����

-0.04 -0.02 0.00 0.02 0.04 0.06

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

ΔMΔa*

α/M2 = 0.2

RPM

Our ability to distinguish the theory increases with the BH spin

Money in the box: more area more gain

Back up

Epicyclic frequencies

νϕνθν���ν-ν���

1.0 1.2 1.4 1.6 1.8 2.00

100

200

300

400

500

600

700

r/rIsco

ν i[Hz]

ν θ:ν

-=3:2

M = 5 M⊙a* = 0.5

νϕνθν���ν-ν���

1.0 1.2 1.4 1.6 1.8 2.00

100

200

300

400

500

r/rIsco

ν i[Hz]

ν θ:ν

-=3:2

M = 5 M⊙a* = 0.2