Small Cosmological Constant from Running Gravitational

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

http://arxiv.org/abs/1101.4995


WHAT’S THE MATTER WITH COSMOLOGY?


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


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)



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)



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


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


http://arxiv.org/abs/astro-ph/0303041




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










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










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








... BUT NOT QUITE IN THIS WAY!


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



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



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



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



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



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.



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?


SO WHAT DOES THIS MEAN FOR HIERARCHY?..


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 )


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


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


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)



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)



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



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)


3

4

FΛ



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)


3

4

FΛ


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,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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




φ,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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




φ,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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




φ,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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




φ,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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




φ,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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




φ,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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




φ,π, a , H


φ =π, a = a H

H =R

6−2H 2


ρ−3p

3


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


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


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


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


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


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)



N-BODY SIMULATIONS WITH F(R) DARK ENERGYO

yaizu,Lima

&H

u(0807.2462

)



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


http://arxiv.org/abs/gr-qc/9611031

Documents

Small Cosmological Constant from Running Gravitational