1. Observational Constraints on Viable f(R) Gravity Models
Yow-Chun Chen (National Tsing-Hua University)
Collaborators: Chao-Qiang Geng, Chung-Chi Lee

2. Outline f(R) gravity Viable f(R) gravity models
Observational Constraints on Viable f(R) Gravity Models

3. f(R) gravity
In f(R) gravity, the Ricci scalar R in the Einstein-Hilbert action is extended to an arbitrary function f(R). Modified Einstein-Hilbert action: 𝑆=∫ ⅆ 4 𝑥 −𝑔 𝑓 𝑅 16𝜋𝐺 + 𝑆 𝑚 Taking variation of the action with resect to 𝑔 𝜇𝜈 , we get the modified Einstein equation: 𝑓 𝑅 𝐺 𝜇𝜈 = κ 2 𝑇 𝜇𝜈 𝑚 − 1 2 𝑔 𝑢𝜈 𝑓 𝑅 𝑅−𝑓 + 𝛻 𝜇 𝛻 𝜈 𝑓 𝑅 − 𝑔 𝑢𝜈 □ 𝑓 𝑅 , where 𝑓 𝑅 ≡ ⅆ𝑓 𝑅 ⅆ𝑅 and 𝐺 𝜇𝜈 = 𝑅 𝜇𝜈 − 1 2 𝑔 𝜇𝑣 𝑅

4. Conditions for viable f(R) gravity model
1. Possess positive effective gravitational constants and exhibit stable cosmological perturbations: ⅆ𝑓 𝑅 ⅆ𝑅 >0 and ⅆ 2 𝑓 𝑅 ⅆ 𝑅 2 >0 for 𝑅≥ 𝑅 0 2. Asymptotic behavior to the 𝛬CDM model in the large curvature regime: 𝑓 𝑅 →𝑅−2𝛬 for 𝑅≥ 𝑅 0 3. Presence Stability of the late-time de Sitter point 4. Passing the local system constraints

5. Viable f(R) Gravity Models
Exponential gravity model: 𝑓 𝑅 =𝑅−𝛽 𝑅 𝑐ℎ 𝐸 1− ⅇ −𝑅∕ 𝑅 𝑐ℎ 𝐸 Tsujikawa model: 𝑓 𝑅 =𝑅−𝜇 𝑅 𝑐ℎ 𝑇 tanh 𝑅 𝑅 𝑐ℎ 𝑇 Starobinsky model: 𝑓 𝑅 =𝑅−𝜆 𝑅 𝑐ℎ 𝑆 1− 1+ 𝑅 2 𝑅 𝑐ℎ 𝑆 2 −𝑛 Hu-Sawicki model: 𝑓 𝑅 =𝑅− 𝑅 𝑐ℎ 𝐻𝑆 𝑐 1 𝑅∕ 𝑅 𝑐ℎ 𝐻𝑆 𝑝 𝑐 2 𝑅∕ 𝑅 𝑐ℎ 𝐻𝑆 𝑝 +1

6. Background Evolution
By the continuity equation: 𝜌 𝐷𝐸 +3𝐻 1+ 𝑤 𝐷𝐸 𝜌 𝐷𝐸 =0 we can derive the equation of state for dark energy: 𝑤 𝐷𝐸 ≡ 𝑃 𝐷𝐸 𝜌 𝐷𝐸 =−1− 𝑦 𝐻 ⅆ 𝑦 𝐻 ⅆ ln 𝑎 where the introduced variables: 𝑦 𝐻 ≡ 𝜌 𝐷𝐸 𝜌 𝑚 0 = 𝐻 2 𝑚 2 − 𝑎 −3 −𝜒 𝑎 −4 , 𝑦 𝑅 = 𝑅 𝑚 2 −3 𝑎 −3 with: 𝑚 2 ≡ 𝜅 2 𝜌 𝑚 0 3 , 𝜒≡ 𝜌 𝑟 0 𝜌 𝑚 0

7. Observational data
Code utilized: CAMB and MGCAMB CosmoMC: Markov-Chain Monte-Carlo Data utilized: BAO (baryon acoustic oscillations) data Planck 2015 likelihoods SNLS (Supernova Legacy Survey) data

8. Observational Constraints on Viable f(R) Gravity Models
𝛬CDM model
Exponential
Tsujikawa

9. Observational Constraints on Viable f(R) Gravity Models
Starobinsky (n = 1)
Starobinsky (n = 2)

10. Observational Constraints on Viable f(R) Gravity Models
Hu-Sawicki (p = 2)
Hu-Sawicki (p = 4)

11. Observational Constraints on Viable f(R) Gravity Models 𝛬CDM background Allowed regions:1 𝜎 (68%) confidence level for model parameter 𝜎 (95%) confidence level for the rest
Parameter
𝛬CD M
Exponential
Tsujikawa
Starobinsky (n=1)
100 𝛺 𝑏 ℎ 2
2.23±0.03
2.23 −
𝛺 𝑐 ℎ 2
0.118±0.002
0.117 −
𝛴 𝑚 𝜈
< 0.20 eV
< 0.21 eV
< 0.18 eV
< 0.25 eV
model parameter −
0.685 −
0.377 −
Best fit 𝜒 2
Parameter
Starobinsky (n=2)
Hu-Sawicki (p=2)
Hu-Sawicki (p=4)
𝛺 𝑏 ℎ 2
2.23±0.03
𝛺 𝑐 ℎ 2
0.118±0.002
0.117 −
𝛴 𝑚 𝜈
< 0.20 eV
< 0.25 eV
model parameter −
0.373 − −
Best fit 𝜒 2

12. Observational Constraints on Viable f(R) Gravity Models modify background Allowed regions:1 𝜎 (68%) confidence level for model parameter 𝜎 (95%) confidence level for the rest
Parameter
𝛬CD M
Exponential
Tsujikawa
Starobinsky (n=1)
100 𝛺 𝑏 ℎ 2
2.23±0.03
2.23 −
𝛺 𝑐 ℎ 2
0.118±0.002
0.118 −
𝛴 𝑚 𝜈
< 0.20 eV
< 0.22 eV
model parameter − − −
Best fit 𝜒 2
Parameter
Starobinsky (n=2)
Hu-Sawicki (p=2)
Hu-Sawicki (p=4)
𝛺 𝑏 ℎ 2
2.23±0.03
2.24 −
𝛺 𝑐 ℎ 2
0.118±0.002
0.117 −
𝛴 𝑚 𝜈
< 0.21 eV
< 0.25 eV
< 0.20 eV
model parameter − − −
Best fit 𝜒 2