1. Recent status of dark energy and beyond
NCTS Annual Theory Meeting,
December, 2015
Recent status of dark energy and beyond
Shinji Tsujikawa
(Tokyo University of Science)

2. Observational evidence of dark energy
Observations showed that about 70 % of the today’s energy density
is dark energy responsible for the cosmic acceleration.
Supernovae type Ia (SN Ia)
2. Cosmic Microwave Background (CMB)
Baryon Acoustic Oscillations (BAO)
4. Large-scale structure (LSS)
Weak lensing
The cosmic expansion history is constrained.
The evolution of matter perturbations is constrained.
This is especially important for modified gravity models.

3. Today’s cosmic recipe: joint analysis of CMB and other data
Main source for
structure formation.
Gravitational attraction
occurs.
Responsible for
cosmic acceleration
Negative pressure

4. Planck constraints on dark energy in the flat FLRW Universe
Equation of state:
Marginalized posterior distribution for
the constant equation of state
For the flat FLRW the bounds are

5. Constraints on time-varying dark energy equation of state
(CPL parametrization)
Reconstruction on the DE
equation of state
We have to caution that the CPL parametrization is not enough to deal with
the models with fast-varying equations of state.

6. Dark energy candidates
Equation of state
The simplest candidate: Cosmological constant
If the cosmological constant originates from the vacuum energy,
its energy scale is enormously larger than the dark energy scale.
Dynamical dark energy models
Quintessence, k-essence, chaplygin gas, coupled dark energy,
f (R) gravity, scalar-tensor theories, DGP model, Galileon,
massive gravity, Lorentz violating model…
Apparent acceleration
Difficult to be compatible
with observations
Inhomogeneous models, metric backreaction

7. Dynamical dark energy models
1. Modified matter models
e.g. Dilatonic ghost condensate:
2. Modified gravity models
Such as

8. Most general single-field scalar-tensor theories with
second-order equations of motion (Horndeski theories)
Horndeski (1974)
Deffayet et al (2011)
Charmousis et al (2011)
Kobayashi et al (2011)
This action covers most of the dark energy models proposed in literature.
Cosmological constant:
Quintessence and K-essence:
f(R) gravity and scalar-tensor gravity:
Galileon:
Gauss-Bonnet coupling :

9. Friedmann equations in Horndeski theories
(Horndeski’s
action) __ __
Non-relativistic
matter
Radiation
The background equations of motion are
The equation of state of dark energy is given by

10. . . Dark energy equation of state: modified matter models (1) LCDM
(2) Quintessence
(b) Thawing models
e.g. PNGB
boson
(a) Freezing models . .
(3) k-essence

11. Quintessence: Theory and Observations
Theoretical prediction
Observational constraints
for
The freezing model is in high tension with the data.
The thawing model is allowed, but it is not particularly favored over the LCDM.

12. Modified gravity allows w smaller than -1.
Hu and Sawicki,
Starobinsky (2007)
f(R) model:
Galileons:
f(R) model is consistent
with the data.
For Galileons, only late-time
tracking solutions are allowed
from the data.
Tracking solutions of Galileons
are disfavored from the data.

13. Constraints from the growth history of the Universe
Dark energy models can be further distinguished from the growth of structures, e.g., large-scale structure, weak lensing, CMB etc. In doing so, we need to study
the evolution of cosmological perturbations.
CMB
LSS
Weak lensing
Perturbed metric:
Non-relativistic matter:
with the four velocity

14. Growth of structures in General Relativity
(including Lambda, quintessence, k-essence)
(Newton’s gravitational constant)
In the matter era, there is the growing-mode solution
In the dark energy dominated epoch, the growth rate is smaller.
For the models in the framework of GR, the difference between models is small.

15. Planck constraints on the effective gravitational coupling
and the gravitational slip parameter (Ade et al, 2015)
today
Strong
gravity
Weak
gravity GR
today

16. Weak gravity ?
The recent observations of redshift-space distortions (RSD) measured the lower growth
rate of matter perturbations lower than that predicted by the LCDM model.
Macaulay et al, PRL (2014)
Planck LCDM fit
Tension between
Planck and RSD data
RSD fit

17. Cosmic growth rate in Horndeski theories
In the observations of redshift space distortions (RSD), the growth rate of matter perturbations is constrained from peculiar velocities of galaxies.
The gauge-invariant density contrast
obeys
where
___

18. Effective gravitational coupling in Horndeski theories
De Felice, Kobayashi,
S.T. (2011).
Schematically
The effect of modified gravity
manifests itself.

19. Simple form of the effective gravitational coupling
ST (2015)
In the massless limit, the effective gravitational coupling in Horndeski theories reads
____
_______
Tensor
contribution
Scalar
contribution
This correspond to the intrinsic
modification of the gravitational part.
Always positive under the no-ghost
and no-instability conditions:
The necessary condition to realize weaker gravity than that in GR is
The scalar-matter interaction always enhances the effective gravitational coupling.

20. Examples (i) f(R) gravity: (ii) Covariant Galileons Hu and Sawicki,
Starobinsky, ST.
(i) f(R) gravity:
(ii) Covariant Galileons
Deffayet et al.
De Felice, Kase, ST (2011)

21. Latest constraints from redshift space distortion measurements
New data from
FastSound
Okumura et al., arXiv:
The strong gravity models like covariant Galileons are disfavored from the data.

22. Summary of the current status of dark energy and beyond
The cosmological constant is consistent with the data of background, but it shows
interesting tension between high-redshift (CMB) and low-redshift (RSD) data.
There is no strong evidence that modified matter models with w>-1 (quintessence)
are favored over the cosmological constant.
Modified gravity models can explain w<-1 consistent with observations.
The strong gravity model like Galileons are in tension with the RSD data.
In the near future we should clarify the following two issues.