small cosmological constant from running gravitational
TRANSCRIPT
SMALL COSMOLOGICAL CONSTANT FROM
RUNNING GRAVITATIONAL COUPLING
arXiv:1101.4995, arXiv:0803.2500
Andrei Frolov
Department of PhysicsSimon Fraser University
Jun-Qi Guo (SFU)
Steklov Mathematical Institute
Moscow, Russia10 June 2011
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 1 / 20
WHAT’S THE MATTER WITH COSMOLOGY?
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 2 / 20
DARK ENERGY IS TROUBLE!
Value problem:expected ρΛ ∼m 4
pl ←− 10120 −→ ρΛ ∼ρM 6= 0 observed
Coincidence problem:ΩΛ = 0.7 ΩM = 0.3 ΩR = 5 ·10−5
ρΛ ∝ a 0 ρM ∝ a−3 ρR ∝ a−4
100-10-20-30
1
0.8
0.6
0.4
0.2
0
MΩ ΛΩRΩ
Plan
ck s
cale
Ele
ctro
−W
eak
scal
e
Now
Big
−B
ang
Nuc
leos
ynth
esis
log a
UN
KN
OW
Nbaryons
dark energydark m
atter
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 3 / 20
GRAVITY IS AN EFFECTIVE FIELD THEORY! MAYBE...
S[λ] =
∫
(
∞∑
n=0
λ4−2n g (n )(λ)R (n )+ . . .
)
p
−g d 4x
8πG =αm−2pl
µdα
dµ=β (α)
∫
dα
β (α)=
∫
dµ
µ
LGR =R
16πG7→
m 2pl
2
R
α=L f (R)
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 4 / 20
GRAVITY IS AN EFFECTIVE FIELD THEORY! MAYBE...
S[λ] =
∫
(
∞∑
n=0
λ4−2n g (n )(λ)R (n )+ . . .
)
p
−g d 4x
8πG =αm−2pl
µdα
dµ=β (α)
∫
dα
β (α)=
∫
dµ
µ
LGR =R
16πG7→
m 2pl
2
R
α=L f (R)
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 4 / 20
GRAVITY IS AN EFFECTIVE FIELD THEORY! MAYBE...
S[λ] =
∫
(
∞∑
n=0
λ4−2n g (n )(λ)R (n )+ . . .
)
p
−g d 4x
8πG =αm−2pl
µdα
dµ=β (α)
∫
dα
β (α)=
∫
dµ
µ
LGR =R
16πG7→
m 2pl
2
R
α=L f (R)
β (αs ) =−
11−2
3n f
α2s
2π
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 4 / 20
THIS HAS BEEN TRIED BEFORE...
WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD
S =
∫
f (R)16πG
+Lm
p
−g d 4x
UV MODIFICATION:
f (R) =R +R2
M 2
Starobinsky (1980)
IR MODIFICATION:
f (R) =R −µ4
R
Capozziello et. al. [astro-ph/0303041]Carroll et. al. [astro-ph/0306438]
FOR F(R) THEORY TO MAKE SENSE WE NEED:
f ′ > 0 – otherwise gravity is a ghost
f ′′ > 0 – otherwise gravity is a tachyon
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 5 / 20
THIS HAS BEEN TRIED BEFORE...
WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD
S =
∫
f (R)16πG
+Lm
p
−g d 4x
UV MODIFICATION:
f (R) =R +R2
M 2
Starobinsky (1980)
IR MODIFICATION:
f (R) =R −µ4
R
Capozziello et. al. [astro-ph/0303041]Carroll et. al. [astro-ph/0306438]
FOR F(R) THEORY TO MAKE SENSE WE NEED:
f ′ > 0 – otherwise gravity is a ghost
f ′′ > 0 – otherwise gravity is a tachyon
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 5 / 20
THIS HAS BEEN TRIED BEFORE...
WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD
S =
∫
f (R)16πG
+Lm
p
−g d 4x
UV MODIFICATION:
f (R) =R +R2
M 2
Starobinsky (1980)
IR MODIFICATION:
f (R) =R −µ4
R
Capozziello et. al. [astro-ph/0303041]Carroll et. al. [astro-ph/0306438]
FOR F(R) THEORY TO MAKE SENSE WE NEED:
f ′ > 0 – otherwise gravity is a ghost
f ′′ > 0 – otherwise gravity is a tachyon
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 5 / 20
THIS HAS BEEN TRIED BEFORE...
WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD
S =
∫
f (R)16πG
+Lm
p
−g d 4x
UV MODIFICATION:
f (R) =R +R2
M 2
Starobinsky (1980)
IR MODIFICATION:
f (R) =R −µ4
R
Capozziello et. al. [astro-ph/0303041]Carroll et. al. [astro-ph/0306438]
FOR F(R) THEORY TO MAKE SENSE WE NEED:
f ′ > 0 – otherwise gravity is a ghost
f ′′ > 0 – otherwise gravity is a tachyon
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 5 / 20
... BUT NOT QUITE IN THIS WAY!
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 6 / 20
ACCELERATED RG FLOWS OF NEWTON’S CONSTANT
α
β stable?
f ′ =α−βα2 > 0
R f ′′ =−
1−2β
α+β ′
β
α2 > 0
de Sitter?
2 f − f ′R =α+βα2 R = 0
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 7 / 20
ACCELERATED RG FLOWS OF NEWTON’S CONSTANT
forbid
den re
gion (g
hosts
)
α
β
α−β=
0
stable?
f ′ =α−βα2 > 0
R f ′′ =−
1−2β
α+β ′
β
α2 > 0
de Sitter?
2 f − f ′R =α+βα2 R = 0
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 7 / 20
ACCELERATED RG FLOWS OF NEWTON’S CONSTANT
forbid
den re
gion (g
hosts
)
effective Λ
α
β
α−β=
0 α+β=
0
stable?
f ′ =α−βα2 > 0
R f ′′ =−
1−2β
α+β ′
β
α2 > 0
de Sitter?
2 f − f ′R =α+βα2 R = 0
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 7 / 20
ACCELERATED RG FLOWS OF NEWTON’S CONSTANT
forbid
den re
gion (g
hosts
)
effective Λ
α
β
α−β=
0 α+β=
0
UV fixed point
dS in IRG
Rflow
stable?
f ′ =α−βα2 > 0
R f ′′ =−
1−2β
α+β ′
β
α2 > 0
de Sitter?
2 f − f ′R =α+βα2 R = 0
β (α) =−α(α−1)
f (R) =R −2Λ
Einstein’s gravity
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 7 / 20
ACCELERATED RG FLOWS OF NEWTON’S CONSTANT
forbid
den re
gion (g
hosts
)
effective Λ
α
β
α−β=
0 α+β=
0
UV fixed point
dS in IRG
Rflow
stable?
f ′ =α−βα2 > 0
R f ′′ =−
1−2β
α+β ′
β
α2 > 0
de Sitter?
2 f − f ′R =α+βα2 R = 0
β (α) =+α(α−1)
f (R) =R +R2
M 2
Starobinsky
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 7 / 20
ACCELERATED RG FLOWS OF NEWTON’S CONSTANT
forbid
den re
gion (g
hosts
)
effective Λ
α
β
α−β=
0 α+β=
0
UV fixed point
dS in IRunstable dS
GR
flow
stable?
f ′ =α−βα2 > 0
R f ′′ =−
1−2β
α+β ′
β
α2 > 0
de Sitter?
2 f − f ′R =α+βα2 R = 0
β (α) =−2α(α−1)
f (R) =R −µ4
R
Carroll et. al.
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 7 / 20
ACCELERATED RG FLOWS OF NEWTON’S CONSTANT
forbid
den re
gion (g
hosts
)
effective Λ
α
β
α−β=
0 α+β=
0
UV fixed point
dS in IRG
Rflow
dS in IR
f (R) flow
stable?
f ′ =α−βα2 > 0
R f ′′ =−
1−2β
α+β ′
β
α2 > 0
de Sitter?
2 f − f ′R =α+βα2 R = 0
β (α) =−α2
f (R) =R
α0
1+α0 lnR
R0
something new?
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 7 / 20
SO WHAT DOES THIS MEAN FOR HIERARCHY?..
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 8 / 20
f (R)GRAVITY IS A SCALAR-TENSOR THEORY
Einstein equations turn into a fourth-order equation:
f ′Gµν − f ′;µν +
f ′−1
2( f − f ′R)
gµν =m−2pl Tµν
A new scalar degree of freedom φ ≡ f ′−2 appears:
f ′ =1
3(2 f − f ′R)+m−2
pl
T
3
Can rewrite fourth-order field equation as two second order ones!
Gµν =m−2pl Tµν +Qµν , φ =V ′(φ)−F
Qµν =−(1+φ)Gµν +φ;µν −h
φ−3P(φ)i
gµν
P(φ)≡1
6( f − f ′R), V ′(φ)≡
1
3(2 f − f ′R), F ≡
m−2pl
3(ρ−3p )
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 9 / 20
EXAMPLE I: (SAFE) UV MODIFICATION
0
0.5
1.0
1.5
2.0
2.5
–1 –0.5 0 0.5 1
V/M
φ
2
in vacuum
U (φ) =V (φ)+F (φ∗−φ)
f (R) =R +R2
M 2
φ =2R
M 2
V =1
3
R2
M 2=
M 2
12φ2
massive scalar field!
scalar degree of freedom φis heavy and hard to excite
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 10 / 20
EXAMPLE I: (SAFE) UV MODIFICATION
0
0.5
1.0
1.5
2.0
2.5
–1 –0.5 0 0.5 1
in matter
V/M
φ
2
in vacuum
U (φ) =V (φ)+F (φ∗−φ)
f (R) =R +R2
M 2
φ =2R
M 2
V =1
3
R2
M 2=
M 2
12φ2
massive scalar field!
scalar degree of freedom φis heavy and hard to excite
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 10 / 20
EXAMPLE II: (FAILED) IR MODIFICATION
0
0.1
0.2
0.3
0.4
0.5
0.2 0.4 0.6 0.8 1
V/µ2
φ
in vacuum
U (φ) =V (φ)+F (φ∗−φ)
f (R) =R −µ4
R
φ =µ4
R2
V =2
3
µ4
R−µ8
R3
=2
3µ2
φ12 −φ
32
field φ is unstable!Dolgov & Kawasaki
(astro-ph/0307285)
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 11 / 20
EXAMPLE II: (FAILED) IR MODIFICATION
0
0.1
0.2
0.3
0.4
0.5
0.2 0.4 0.6 0.8 1
in matter
V/µ2
φ
in vacuum
U (φ) =V (φ)+F (φ∗−φ)
f (R) =R −µ4
R
φ =µ4
R2
V =2
3
µ4
R−µ8
R3
=2
3µ2
φ12 −φ
32
field φ is unstable!Dolgov & Kawasaki
(astro-ph/0307285)
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 11 / 20
CAN WE COME UP WITH SOMETHING BETTER?..
f (R) =R
α0
1+α0 lnR
R0
α1 =α0
1+α0 ln R1
R0
φ = lnR
R0+
1−α0
α0= ln
R
R∗
R∗ ≡ 4Λ=R0 exp
α0−1
α0
P(φ) =−2
3Λeφ , V ′(φ) =
4
3Λeφφ
φ
VΛ
V (φ) =4
3Λeφ(φ−1)
V ′(φeq) =F =⇒φeq =W
3
4
FΛ
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 12 / 20
CAN WE COME UP WITH SOMETHING BETTER?..
f (R) =R
α0
1+α0 lnR
R0
α1 =α0
1+α0 ln R1
R0
φ = lnR
R0+
1−α0
α0= ln
R
R∗
R∗ ≡ 4Λ=R0 exp
α0−1
α0
P(φ) =−2
3Λeφ , V ′(φ) =
4
3Λeφφ
φ
VΛ
V (φ) =4
3Λeφ(φ−1)
V ′(φeq) =F =⇒φeq =W
3
4
FΛ
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 12 / 20
CAN WE COME UP WITH SOMETHING BETTER?..
f (R) =R
α0
1+α0 lnR
R0
α1 =α0
1+α0 ln R1
R0
φ = lnR
R0+
1−α0
α0= ln
R
R∗
R∗ ≡ 4Λ=R0 exp
α0−1
α0
P(φ) =−2
3Λeφ , V ′(φ) =
4
3Λeφφ
φ
VΛ
V (φ) =4
3Λeφ(φ−1)
V ′(φeq) =F =⇒φeq =W
3
4
FΛ
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 12 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
vacuum evolution is constrainedto a (non-trivial) 2D subspace
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
vacuum evolution is constrainedto a (non-trivial) 2D subspace
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
φ
φ
forbiddenfor vacuumsolutions
ΩM ,0 = 0.278ΩQ,0 = 0.397
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
φ
φ
forbiddenfor vacuumsolutions
now
ΩM ,0 = 0.278ΩQ,0 = 0.397
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
φ
φ
dSforbidden
for vacuumsolutions
now
tracker
ΩM ,0 = 0.278ΩQ,0 = 0.397
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
φ
φ
dSforbidden
for vacuumsolutions
now
future
tracker
ΩM ,0 = 0.278ΩQ,0 = 0.397
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
φ
φ
dSforbidden
for vacuumsolutions
now
future
past tracker
ΩM ,0 = 0.278ΩQ,0 = 0.397
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
UNIVERSE AS A DYNAMICAL SYSTEM
2 scalar DoF, 4D phase space
φ,π, a , H
dynamical system equations
φ =π, a = a H
H =R
6−2H 2
π+3Hπ+V ′(φ) =m−2pl
ρ−3p
3
subject to a constraint
H 2(φ+2)+Hπ+P(φ) =m−2pl
ρ
3
P(φ)≡1
6( f − f ′R) =−
2
3Λeφ
V ′(φ)≡1
3(2 f − f ′R) =
4
3Λeφφ
φ
φ
dSforbidden
for vacuumsolutions
now
future
past tracker
ΩM ,0 = 0.278ΩQ,0 = 0.397
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 13 / 20
COSMOLOGICAL EVOLUTION: TRACKER SOLUTION
pres
ent
m−2pl
ρ
3=
ΩM ,0
a 3+ΩR ,0
a 4
H 20 , φtrac 'W
3H 20
4ΛΩM ,0
a 3
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 14 / 20
COSMOLOGICAL EVOLUTION: DISTANCE MODULUS
∆µ= +0.06−0.04 for z ∈ [0, 2]
∆µ= 5 log10
d L, f (R)
d L,ΛCDM, d L =
1
a
1∫
a
d a
a 2H
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 15 / 20
COSMOLOGICAL EVOLUTION: EFFECTIVE Ω& w
ΩRΩM
ΩQ
pres
ent
ΩQ =ρQ
3m 2plH
2,
ρQ = 3m 2plH
2−ρpQ =m 2
pl(H2−R/3)−p
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 16 / 20
COSMOLOGICAL EVOLUTION: EFFECTIVE Ω& w
wQ
wQ→ 1/3
equilibrium approximation
wQ =pQρQ
,ρQ = 3m 2
plH2−ρ
pQ =m 2pl(H
2−R/3)−p
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 17 / 20
INVOKE CHAMELEON TO SATISFY LOCAL TESTS
r/r
ρ (g
cm
-3)
0.1 1 10 100 1000
10-20
10-10
1
R/κ2 (n=4, | fR0|=0.1)
ρ
r/r100 1000 104
R/κ
2 (g
cm
-3)
10-24
10-23
|fR0|=0.001|fR0|=0.01|fR0|=0.05|fR0|=0.1
ρ
n=4
Hu & Sawicki (0705.1158)
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 18 / 20
N-BODY SIMULATIONS WITH F(R) DARK ENERGYO
yaizu,Lima
&H
u(0807.2462
)
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 19 / 20
LONGITUDINAL GRAVITY WAVES FROM COLLAPSE!
Harada, Chiba, Nakao, & Nakamura (gr-qc/9611031)
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
80 100 120 140 160 180 200 220t/M
0
10
20
50
Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration MIAN 2011 20 / 20