Conditions for the cosmological viability of f(R) dark energy - PowerPoint PPT Presentation

Conditions for the cosmological viability of f(R) dark energy models Shinji Tsujikawa (Gunma National College of Technology） Collaboration with Luca Amendola (Rome observatory) David Polarski (Montpellier university) Radouane Gannouji (Montpellier university)

Equation of state Dark energy � DE � 0.7 w DE � � 1 SNe Ia w DM � 0 � DM � 0.25 CMB LSS The current universe is accelerating!

What is the origin of dark energy? There are two approaches to dark energy. G µ � = 8 � GT µ � -equations (i) Changing gravity (ii) Changing matter Generally these f(R) gravity models, Quintessence, models give rise to a very light mass: Scalar-tensor models, K-essence, m � � H 0 � 10 � 33 eV DGP braneworld, Tachyon, ….. Chaplygin gas, …..

f(R) gravity, scalar-tensor gravity, ‘Changing gravity’ models DGP braneworld models,.. Dark energy may originate from some modifications from Einstein gravity. The simplest model: f(R) gravity [ ] � d 4 S = x � g f ( R )/2 + L m � CDM model: f ( R ) = R � 2 � Starobinsky’s inflation model: f ( R ) = R + � R 2 Used for early universe inflation f(R) modified gravity models can be used for dark energy ?

M. Turner Example of f(R) dark energy models I want to explain dark energy without using scalar fields… f ( R ) = R � µ 2( n + 1) Capoziello, Carloni and Troisi (2003) R n Carroll, Duvvuri, Troden 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 However this model does not have a standard matter era prior to the late-time acceleration. L. Amendola, D. Polarski and S.T., PRL98, 131302 (2007)

f ( R ) = R � µ 2( n + 1) R n w eff = 1/3 Dark energy is strongly coupled The scale factor to dark matter evolves as in this model. a � t 1/ 2 The absence of a standard matter epoch. Incompatible with observations.

What are general conditions for the cosmological viability of f(R) dark energy models? L. Amendola, D. Polarski, R. Ganouji and S.T., PRD75, 083504 (2007) General analysis x � g f ( R )/2 � 2 + L m + L rad [ ] � d 4 S = In the FRW background we have We wish to carry out general analysis without specifying the form of f(R).

See the dark energy review Autonomous equations E. Copeland, M. Sami and S.T. (2006) We introduce the following variables: and Then we obtain N = ln( a ) , where m ( r ) = Rf , RR r = � Rf , R = x 3 and f x 2 f , R The above equations are closed.

m ( r ) = Rf , RR � CDM model: f ( R ) = R � 2 � = 0 f , R The parameter m(r) characterizes the deviation from the � CDM model. The cosmological dynamics is well understood by the geometrical approach in the (r, m) plane. (i) Matter point: P M m ( r , m ) � ( � 1,0) P : w eff = � M 1 + m The existence of the viable saddle matter epoch requires m ( r ) > 0, � 1 < m '( r ) < 0 r � � 1 at (ii) De-sitter point P A : r = � 2 P w eff = � 1 A For the stability of the de-Sitter point, we require 0 < m ( r = � 2) < 1

Viable trajectories and S.T. (arXiv:0705.396) bc � 1 � 1 < m '( r ) < 0 Constant m model: f ( R ) = R 1 + m � �

Examples of non-viable models m = � 1 + r r m = � r + 1 r

Are there cosmologically viable models satisfying local gravity constraints ? Effective gravitational constant in f(R) models is 1 2 = V , �� � � M � l ] M � G eff = G [1 + e 3 f , RR l : length scale at which gravity experiments are carried out. To satisfy LGC , we require M � l >> 1 2 � S � � l R is the curvature on the s m ( R S ) << � � � 1 H 0 � 0 local structure. � � � s = 10 17 � 0 In the case of Cavendish-type experiment ( ) , we have m ( R s ) << 10 � 43 The quantity m(R) needs to be very small in the 2 high-curvature region: R >> R 0 � H 0

Models that satisfy local gravity constraints Starobinsky Hu ( R / R 0 ) 2 n f ( R ) = R � � R 0 Hu and Sawicki: ( R / R 0 ) 2 n + 1 2 R 0 � H 0 � � n � f ( R ) = R � � R 0 1 � 1 + R 2 / R 0 ( ) 2 Starobinsky: � � � � Cosmological constant disappears f ( R = 0) = 0 in a flat space. m ( r ) = C ( � r � 1) 2 n + 1 and R >> R 0 f ( R ) � R � � R 0 2( n + 1) >> ( H 0 � 1 / l ) 2 The LGC is ( ) � s / � 0 n � 2 is sufficient in most of experiments. These models also satisfy the criterion of cosmological viability and can evolve from the matter point P to the de-Sitter point P M A Another viable model was proposed Appleby and Battye (next talk!).

Conclusions 1. We derived conditions for the cosmological viability of f(R) dark energy models. This is useful to exclude some of the f ( R ) = R � µ 2( n + 1) / R n models, e.g., 2. It is also possible to construct cosmologically viable models that satisfy local gravity constraints. These behave as for R >> R 0 m ( r ) = C ( � r � 1) 2 n + 1 ( n � 2) Now I am considering observational constraints on these models.