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.
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Thank you!
Ismael Ayuso Braga 2019
Questions
Ismael Ayuso Braga 2019
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