Dark Energy and Modified Gravity

Shinji Tsujikawa(Gunma National College of Technology ）

Collaborations withL. Amendola, S. Capozziello, R. Gannouji, S. Mizuno,D. Polarski, R. Tavakol, K. Uddin, J. Yokoyama

SNe Ia

The current universe isaccelerating!

LSS

CMB

Dark Energy

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ΩDE = 0.72 ± 0.015

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−1.11 < wDE < −0.86Komatsu et al, 0803.0547 (astro-ph)

Simplest model of dark energy

Cosmological constant:

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ΛThis corresponds to the energy scale

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ρΛ =3H0

2

8πG≈10−47GeV 4

If this originates from vacuum energy in particle physics, €

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ρvac ≈ mpl4 ≈1076GeV 4

Huge difference compared to the present value!

(Equation of state: )

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wDE = −1

Cosmological constant problem

　　

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There are two approaches to dark energy.

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Gμν = 8πGTμν

(i) Changing gravity (ii) Changing matter

f(R) gravity models,Scalar-tensor models,Braneworld models,Inhomogeneities, …..

Quintessence,K-essence,Tachyon,Chaplygin gas,…..

Are there some other models of dark energy?

(Einstein equations)

‘Changing matter’ models

To get the present acceleration most of these models are based upon scalar fields with a very light mass:

Quintessence, K-essence, Tachyon, phantom field, …

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mφ ≈ H0 ≈10−33eV

FlatIn super-symmetric theories the severe fine-tuning of the field potential is required.

(Kolda and Lyth, 1999)

The coupling of the field to ordinary matter should lead to observable long-range forces.

(Carroll, 1998)φφφφ

zv

Potential

‘Changing gravity’ modelsf(R) gravity, scalar-tensor gravity, braneworld models,..

Dark energy may originate from some geometricmodification from Einstein gravity.

The simplest model: f(R) gravity

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S = d4∫ x −g f (R) /2 + Lm[ ]

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ΛCDMmodel:

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f (R) = R − Λ

Starobinsky’s inflation model:

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f (R) = R + αR2

Used for early universe inflation

f(R) modified gravity models can be used for dark energy ?

R: Ricci scalar

　

f(R) dark energy: Example

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f (R) = R − μ 2(n +1)

Rn

Capozziello, Carloni and Troisi (2003)Carroll, Duvvuri, Trodden and Turner (2003)

It is possible to have a late-time acceleration as the second term becomes important as R decreases.

In the small R region we have

Late-time acceleration is realized.

(n>0)

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S = d4∫ x −˜ g ˜ R /2 − ( ˜ ∇φ)2 /2 −V (φ) + ˜ L m (φ)[ ]

• The f(R) action is transformed to

Matter fluid satisfies:

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˙ ˜ ρ m + 3 ˜ H ( ˜ ρ m + ˜ p m ) = Q ˙ φ ( ˜ ρ m − 3 ˜ p m )

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Q = −1/ 6

Coupled quintessence

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φ=( 6 /2)lnFwhere

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V = (RF − f ) /2F 2

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F = ∂f /∂R

Dark matter is coupled to the field (curvature).

Is the model (n>0) are cosmologically viable?

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f (R) = R − μ 2(n +1)

Rn

No!This model does not have a standard matter eraprior to the late-time acceleration.

(Einstein frame)

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f (R) = R − μ 2(n +1)

RnThe model

The potential in Einstein frame is

The standard matter era is replaced by ‘phi matter dominated era’

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˜ Ω φ = ˜ w eff = 2Q2 /3 =1/9

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V (φ) = Ae−2 6φ / 3(e 6φ / 3 −1)n /(n +1)

For large field region,

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V (φ)∝ e−λφ

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λ =6

3n + 2n +1

Coupled quintessence withan exponential potential

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˜ a ∝ ˜ t 3 / 5

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φ

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V (φ)

:

(n>0)

Jordan frame:Incompatible with observations

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a∝ t1/ 2

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weff =1/3

L. Amendola, D. Polarski, S.T., PRL (2007)

.

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S = d4∫ x −g f (R) /2κ 2 + Lm + Lrad[ ]

For the FRW background with a scale factor a, we have

Pressure-less Matter: Radiation:

We carried out general analysis without specifying the form of f(R).

What are general conditions for the cosmological viabilityof f(R) dark energy models?

L. Amendola, D. Polarski, R. Gannouji and S.T., PRD75, 083504 (2007)

Autonomous equations

We introduce the following variables:

Then we obtain and

where

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m(r) =Rf,RR

f,R

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r = −Rf,R

f= x3

x2

The above equations are closed.

See the review article:E. Copeland, M. Sami and S.T., IJMPD (2006)

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N = ln(a)

and

,

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ΛCDMmodel:

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f (R) = R − Λ

The parameter

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m(r) =Rf,RR

f,R

characterises a deviation from the model.

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ΛCDM

The constant m model corresponds to

(a)

(b)

(c) The model of Capozziello et al and Carroll et al:

This negative m case is excluded as we will see below.

The cosmological dynamics is well understood by the geometrical approach in the (r, m) plane.

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m(r) =Rf,RR

f,R

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r = −Rf,R

f

(i) Matter point: PM

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(r,m) ≈ (−1,0)

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weff = − m1+ m

≈ 0

From the stability analysis around the fixed point, the existence of the saddle matter epoch requires

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m(r) > 0,

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−1 < m'(r) < 0 at

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r ≈ −1(ii) De-sitter point PA

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r = −2

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weff = −1

For the stability of the de-Sitter point, we require

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0 < m(r = −2) <1

Viable trajectories

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−1 < m'(r) < 0€

bc ≈1

Constant m model:

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f (R) = R1+m − Λ

(another accelerated point)

Amendola and S.T. (2007)

Lists of cosmologically non-viable models(n>0)

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f (R) = R + αR2 − β /R

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f (R) = αRn

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f (R) = R p exp(qR)

…. many !

Lists of cosmologically viable models

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f (R) = R −αR−n

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f (R) = R −αRn (0<n<1) Li and Barrow (2007)

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f (R) = (Rb − Λ)c

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bc ≈1 Amendola and S.T. (2007)

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f (R) = R − λR0(R /R0)2n

(R /R0)2n +1Hu and Sawicki (2007)

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f (R) = R − λR0 1− 1+ R2 /R02

( )−n ⎡

⎣ ⎢ ⎤ ⎦ ⎥ Starobinsky (2007)

More than 200 papers were written about f(R) dark energy!

Local gravity constraints (LGC)The f(R) models need to satisfy constraints coming from solar system and equivalence principle tests.

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S = d4∫ x −˜ g ˜ R /2 − ( ˜ ∇φ)2 /2 −V (φ)[ ] + Sm ( ˜ g μν e2Qφ )

where the coupling between dark energy and dark matter is

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Q = −1/ 6

The f(R) action in the Einstein frame is

(of the order of 1)Even in this case, LGC can be satisfied provided that the mass of the field is sufficiently heavy in high-density regions:

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M 2 ≡ V,φφ ≈ 13 f,RR

>> H02

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f ,RR> 0( is required)

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φ=( 6 /2)lnF

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V = (RF − f ) /2F 2

Chameleon mechanism Khoury and Weltman (2003)

Spherically symmetric body

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ρA

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ρB

where

Inside and outside the body, the effective potential has minima at

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V,φ (φA ) + QeQφA ρ A = 0,

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V,φ (φB ) + QeQφB ρ B = 0

The body has a thin-shell inside it if the field is heavy.

Thin-shell parameter:

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Δrc

rc

= φB − φA

6QΦc

<<1

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Φc :The gravitationalpotential at the surface of the body

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φ(r)

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φ=( 6 /2)lnF

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V (φ) = (RF − f ) /2F 2

f(R) chameleonsIn f(R) gravity the effective gravitational constant and the post-Newtonian parameter are

Faulker, Tegmark et al, Capozziello and S.T.

The tightest solar-system bound is

QuickTime˛ Ç∆TIFFÅiîÒà≥èkÅj êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

.

..€

φB €

Δrc

rc

≈ − φB

6Φc

<1.15 ×10−5

For the Sun (　 ),High-density (massive)

Low-density €

φB < 6.0 ×10−11

This can be satisfiedfor the model where

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M 2 ≈1/(3 f,RR )is large in the region

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R >> H02

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γ−1 < 2.3×10−5€

Geff = G 1+ Δrc

rc

⎛ ⎝ ⎜

⎞ ⎠ ⎟

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γ≈1− Δrc /rc

1+ Δrc /rc

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Φc ≈10−6

(massless)

Models that satisfy local gravity constraints

Hu and Sawicki:

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f (R) = R − λR0(R /R0)2n

(R /R0)2n +1

Starobinsky:

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f (R) = R − λR0 1− 1+ R2 /R02

( )−n ⎡

⎣ ⎢ ⎤ ⎦ ⎥

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R0 ≈ H02

Hu Starobinsky

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f (R = 0) = 0 Cosmological constant disappears in a flat space.

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R >> R0

The solar-system constraints are satisfied forThe equivalence principle constraints are satisfied for

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n > 0.5

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n >1(Capozziello and S.T.)

In these models the deviation from the Lambda CDM modelbecomes significant around present on cosmological scales.

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f (R) ≈ R − λR0[1− (R0 /R)2n ]

Observational signatures of viable f(R) modelsTo confront with SN Ia observations, we write the equationsin the form:

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3F0H2 = κ 2(ρ DE + ρ m )

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2F0˙ H = −κ 2(ρ DE + pDE + ρ m )

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F0 = (∂f /∂R)0 present value

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κ 2ρ DE = (1/2)(FR − f ) − 3H ˙ F + 3H 2(F0 − F )where

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κ 2 pDE = ˙ ̇ F + 2H ˙ F − (1/2)(FR − f ) − (2 ˙ H + 3H 2)(F0 − F)

This satisfies

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˙ ρ DE + 3H(ρ DE + pDE ) = 0

The equation of state of dark energy is

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wDE = pDE

ρ DE

=weff

1− (F /F0)Ωm

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weff = −1− (2 ˙ H /3H 2)

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Ωm = ρ m /3FH 2

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wDEThe equation of state diverges at

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Ωm = F0 /F

Viable f(R) models satisfy

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M 2 ≈1/(3 f,RR ) > 0 €

(F = ∂f /∂R)

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dFdR

> 0

F increases toward the past. The divergence of w occurs.DE

The redshift at the divergencecan be as close as z=2.But this behaviour can be still allowed in current SN Ia observations.

Amendola and S.T. (2007)

Are there other observational constraints on f(R) models?

Matter power spectrum

Under the sub-horizon approximation (k>>aH), the matter density perturbation satisfies

where

(i)

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k 2

a2Rm <<1 Standard evolution:

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δm ∝ t 2 / 3

(ii)

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k 2

a2Rm >>1 Non-standard evolution:

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δm ∝ t( 33−1)/ 6

This enhances the growth rate of matter perturbations.

(early time)

(late time)

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Modes relevant to matter power spectrum:

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k 2

a2Rm =1The crossing occurs during the matter era.

The time of crossing has a dependence:

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tk ∝ k−3 /(6n +4 )

This leads to the difference of spectral indices betweenmatter power spectrum and CMB spectrum:

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Δns = 33 − 56n + 4

Likelihood analysis using the LSS and CMB data is necessary. V. Acquaviva, S. Matarrese, S.T., M. Viel, in preparation.

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n ≥ 2 for

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Δns < 0.05

The models of Hu & Sawicki and Starobinsky behave as

StarobinskyS.T.

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m(r) = C(−r −1)2n +1 for

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R >> R0

Generalization to scalar-tensor models

One can generalize the analysis to Brans-Dicke theory with a potential:

Setting , this action is equivalent to

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χ =F = e−2Qφ

where

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3+ 2ωBD =1/(2Q2)

The f(R) gravity corresponds to

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V = (FR − f ) /2 and

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Q = −1/ 6

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ωBD = 0 Chiba (2003)

One can search for viable models for general coupling Q.

S.T., Uddin, Mizuno, Tavakol, Yokoyama, arXiv 0803.1106 [astro-ph]

The local gravity constraints can be satisfied if the field is sufficiently heavy in the large-curvature region.

A representative potential:

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V (φ) = V0[1− C(1− e−2Qφ )p ](0<p<1)

We find that • The divergence of w is generic.• Q and p are constrained by LGC and matter perturbations.

DE

Conclusions 1. We derived conditions for the cosmological viability of f(R) modified models. This is useful to exclude many f(R) models e.g., (n>0).

2. The viable f(R) models show peculiar features for the equation of state of dark energy. It can diverge at the redshift around z=2.

3. We discussed a number of observational and experimental signatures of modified gravity models: SN Ia, LGC, Matter power spectra, CMB, ….

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f (R) = R − μ 2(n +1) /Rn

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4. We also studied the case of general coupling Q and found that the results obtained in f(R) gravity are generic.