Ismael Ayuso Marazuela

About the sign of the effective gravitational coupling constant

In collaboration with Jose Pedro Mimoso and Nelson Nunes

Braga 20195th IDPASC/LIP PhD Students Workshop

Ismael Ayuso et al, Galaxies 7 (2019) 38

[arXiv:1903.07604 [gr-qc]]

Outline

1. Introduction

Ø Coupling between gravity and matterØ Modified gravity

2. Is possible to obtain a negative coupling?

Ø What about a negative G?Ø Generalized Brans-Dicke theoryØ Study of the dynamical system

ü Without potentialü With a quadratic potential

3. Mechanism for a positive gravitational coupling

4. Summary and conclusions

Ismael Ayuso Braga 2019

Outline

1. Introduction

Ø Coupling between gravity and matterØ Modified gravity

Ismael Ayuso Braga 2019

A brief history about G

Ismael AyusoIsmael Ayuso

Ø 1687 Newton's work Philosophiæ Naturalis Principia Mathematica is published:

Ø 1798 H. Cavendish measured this proportionality with a torsion balance.Ø 1799 P. S. Laplace introduced the constant for the first time as:

Ø 1890 C. V. Boys introduced the modern notation in which appears GØ 1905 Albert Einstein published the General Relativity

G has the role of coupling the geometry to the matter and is taken to be positive and constant.

F / m1m2

r2

F = �k2m1m2

r2

S =c4

16⇡G

Zd4x

p�gR+

Zd4x

p�gLm Rµ⌫ � 1

2gµ⌫R =

8⇡G

c4Tµ⌫

Braga 2019

Cosmological Model

Ismael AyusoIsmael Ayuso

GR as gravitational theory

The Universe is considered homogeneous and isotropic

à FLRW- metricObservations

• Big Bang

• Cold Dark Matter (CDM) to explain the velocity curves of galaxies and the structure formation

• Dark Energy (DE) related with the Cosmological Constant (CC) to explain the current acceleration of the Universe

• The Universe is (nearly) flat

• There are three epochs dominated by radiation, matter and CC respectively

• Early acceleration to solve the horizon, flatness and monopole problem

ΛCDM model

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Results

Ismael AyusoIsmael Ayuso

ProblemsAchievements

• GR has passed all precision tests

o Anomalous perihelion advanceof Mercury

o Gravitational lensingo Gravitational time dilationo …

• GR predicts gravitational waves

• ΛCDM is able to explain almost allobservations until now

• Singularities in GR

• GR can not be quantized

• CDM has not been detected(directly)

• About the Cosmological Constant:o The CC Problem, related with

theoretical predictions of itsvalue

o The Coincidence Problemo The Fine-Tuning Problem

P. Bull et al. Phys. Dark Univ. 12 (2016) [arXiv:1512.05356 [astro- ph.CO]]

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Attempts to explain the current acceleration of the Universe without a cosmological constant

Ismael AyusoIsmael Ayuso

Introducing a new theory of gravity

Maintaining GR as gravitational theory:

Modified gravityChanging the structure

of the space-time

Introducing inhomogeneities and

anisotropies

Modifying the matter content

Backreaction models

Emanuele Berti et al, Class.Quant.Grav. 32 (2015)

243001[arXiv:0706.2151 [astro-ph]]

Braga 2019

Attempts to explain the current acceleration of the Universe without a cosmological constant

Ismael AyusoIsmael Ayuso

Introducing a new theory of gravity

Maintaining GR as gravitational theory:

Modified gravityChanging the structure

of the space-time

Introducing inhomogeneities and

anisotropies

Modifying the matter content

Backreaction modelsBraga 2019

A possibility: the gravitational

coupling G is not a constant and it

evolves.

G as a dynamical variable

Ismael AyusoIsmael Ayuso

Ø 1938 Dirac put forward that G may evolve with the Hubble rate(motivated by the disparity between gravitational and electromagneticforces) à First time of the variation of some fundamental constant.

Ø 1961 C. H. Brans and R. H. Dicke: 1/G is replaced by a scalar field whichcan vary from place to place and with time:

v The scalar field may be seen as the gravitational permittivity of the space-time.v This coupling could be negative.

S =c4

16⇡

Zd4x

p�g

✓�R� !

�@µ�@

µ�

◆+

Zd4x

p�gLm

G 1/�

Braga 2019

Outline

1. Introduction

Ø Coupling between gravity and matterØ Modified gravity

2. Is possible to obtain a negative coupling?

Ø What about a negative G?Ø Generalized Brans-Dicke theoryØ Study of the dynamical system

ü Without potentialü With a quadratic potential

3. Mechanism for a positive gravitational coupling

Ismael Ayuso Braga 2019

Generalized Brans-Dicke theory

Ismael AyusoIsmael Ayuso

R↵� � 1

2g↵� R+ �(�) g↵� =

!(�)

�2

�;↵�;� � 1

2g↵��;��

;�

�+

1

�[�;↵� � g↵��;�

;� ] + 8⇡T↵�

�

⇤�� 2�2�0(�)� 2��(�)

2!(�) + 3=

1

2!(�) + 3[8⇡ T � !0(�)�;��

;� ]

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

G 1/�

Field equations

Braga 2019

Generalized Brans-Dicke theory

Ismael AyusoIsmael Ayuso

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

G 1/�

Field equations for FLRW

3

✓a

a

◆2

+ 3a

a

�

�+ 3

k

a2= �(�) +

!(�)

2

�2

�2+ 8⇡

⇢

�

2d

dt

✓a

a

◆+ 3

✓a

a

◆2

+ 2a

a

�

�+

k

a2= �(�)� !(�)

2

�2

�2� 8⇡

p

�� �

�

�+ 3a

a�+

2�2�0(�)� 2��(�)

2!(�) + 3= � 1

2!(�) + 3

h8⇡(3p� ⇢) + !0(�)�2

i

Braga 2019

Study of the dynamical system

Ismael AyusoIsmael Ayuso

X =�

�0a2 Y 0 =

r2!(�) + 3

3

�0

�

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

We are going to introduce the redefined variables:

where • we have used conformal time:• preserves the sign of i.e.

this allows us to extend the study of the dynamics into the region with a negative coupling

d⌘ = dt/a�X

X < 0 when � < 0 or when G < 0

Braga 2019

Study of the dynamical system

Ismael AyusoIsmael Ayuso

X =�

�0a2 Y 0 =

r2!(�) + 3

3

�0

�

(X 0)2 + 4 kX2 � (Y 0 X)2 = 4M X

✓X

�

◆ 4�3�2

+4

3

✓�(�)

�

◆X3

[Y 0 X]0= M(4� 3�)

r3

2! + 3

✓X

�

◆ 4�3�2

� 2X2

p2!(�) + 3

✓d�

d�� �(�)

�

◆

X 00 + 4 k X = 3M(2� �)

✓X

�

◆ 4�3�2

+ 2X2

✓�(�)

�

◆

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

We are going to introduce the redefined variables:

The FLRW equations are then:

Braga 2019

Study of the dynamical system

Ismael AyusoIsmael Ayuso

X =�

�0a2 Y 0 =

r2!(�) + 3

3

�0

�

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

We are going to introduce the redefined variables:

The FLRW equations for a quadratic potential �(�) = �0�

(X 0)2 + 4 kX2 � (Y 0 X)2 = 4M X

✓X

�

◆ 4�3�2

+4

3�0 X

3

[Y 0 X]0= M(4� 3�)

r3

2! + 3

✓X

�

◆ 4�3�2

X 00 + 4 k X = 3M(2� �)

✓X

�

◆ 4�3�2

+ 2�0 X2

Braga 2019

Without potential With potential

Vacuum and still fluid Radiation Vacuum and still fluid Radiation

Study of the dynamical system

Ismael AyusoIsmael Ayuso

X =�

�0a2

X 00 + 4 k X = 3M(2� �)

✓X

�

◆ 4�3�2

+ 2�0 X2

X 0 = W

W 0 = �4kX

X 0 = W

W 0 = 2M � 4kX

X 0 = W

W 0 = 2�0X2 � 4kX

X 0 = W

W 0 = 2M + 2�0X2 � 4kX

�0 = 0 �0 6= 0

M = 0 � = 2 M = 0 � = 2� = 4/3 � = 4/3

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

Braga 2019

Study of the dynamical system without potential

Ismael AyusoIsmael Ayuso

X 0 = W

W 0 = �4kX

X 0 = W

W 0 = 2M � 4kX

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=+1

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=0

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=+1M=1

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=-1

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=0

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=-1

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

X =�

�0a2

Vacuum and still fluid

Radiation

X 00 + 4 k X = 3M(2� �)

✓X

�

◆ 4�3�2

+ 2�0 X2

M = 0 � = 2

� = 4/3

Braga 2019

Ismael AyusoIsmael Ayuso

X 0 = W

W 0 = 2�0X2 � 4kX

X 0 = W

W 0 = 2M + 2�0X2 � 4kX

S =

Zd4x

p�g

✓�R� ! (�)

�@µ�@

µ�� 2��(�)

◆+ Sm

Study of the dynamical system with a quadratic potential

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=+1

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=0

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

k=-1

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

a tanh HXL

atanhHX

'L

Vacuum and still fluidM = 0 � = 2

Radiation � = 4/3

X =�

�0a2 X 00 + 4 k X = 3M(2� �)

✓X

�

◆ 4�3�2

+ 2�0 X2

Braga 2019

Outline

1. Introduction

Ø Coupling between gravity and matterØ Modified gravity

2. Is possible to obtain a negative coupling?

Ø What about a negative G?Ø Generalized Brans-Dicke theoryØ Study of the dynamical system

ü Without potentialü With a quadratic potential

3. Mechanism for a positive gravitational coupling

4. Summary and conclusions

Ismael Ayuso Braga 2019

Summary and conclusions

Ismael AyusoIsmael Ayuso

Ø We have investigated a cosmological mechanism that induces the value of the gravitational effective coupling “constant” to be positive in the framework of scalar-tensor theories with and without a cosmological potential.

Ø In the absence of the cosmological potential, the presence of matter or radiation favours a positive value of the gravitational “constant”, when the evolution enters a phase of matter domination.

Ø However, it is when a quadratic cosmological potential is present that an attracting mechanism towards a positive value of the gravitational running “constant” becomes manifest.

Braga 2019

Thank you!

Ismael Ayuso Braga 2019

Questions

Ismael Ayuso Braga 2019