1. Modifying Gravity to Reduce the 𝐻 0 Tension
Meng-Xiang Lin, Marco Raveri, Wayne Hu
arXiv:
at H0 workshop

2. Motivation Current H0 tension between CMB and local H0 measurements
Modified gravity is tested at late times
Study modified gravity at early times

3. Parameterized Phenomenological MG
Background: fixed to LCDM
Perturbation:
𝑘 2 Φ=−4𝜋𝐺 𝑎 2 𝜌Δ
𝑘 2 Φ−Ψ =12𝜋𝐺 𝑎 2 𝜌+𝑃 𝜎
GR:
𝑘 2 Ψ=−4𝜋𝐺 𝑎 2 𝜇[𝜌Δ+3 𝜌+𝑃 𝜎]
𝑘 2 Φ−𝛾Ψ =12𝜋𝐺 𝑎 2 𝜇 𝜌+𝑃 𝜎
MG:
(Hojjati et al. 2011)
𝜇(𝑎,𝑘): the effective strength of gravity (𝐺→𝜇𝐺)
𝛾 𝑎,𝑘 : difference between the gravitational effects on
non-relativistic and relativistic matter

4. 𝜇 ∞ =1.2: the value at beginning
of the Universe
𝝁, 𝜸 in this work:
Step function of time
𝑎 𝑇 : transition time (z=30)
𝜇 0 =1: the value today

5. Late time parameters: 𝝁 𝟎 , 𝜸 𝟎

6. Late time parameters: 𝝁 𝟎 , 𝜸 𝟎

7. Late time parameters: 𝝁 𝟎 , 𝜸 𝟎
It’s very hard to reduce the tension
by only changing the lensing potential
(by only modifying gravity at late times)

8. Early time parameter: 𝝁 ∞
Mention the axis
—— 𝜇 ∞ = GR

9. Early time parameter: 𝝁 ∞
Mention the axis
—— 𝜇 ∞ = GR

10. Early time parameter: 𝝁 ∞
Peak locations in GR
1: phase shift
2: driving effect 10

11. ΛCDM with CMBTT results:
H 0 =67.26±0.99 km/s/Mpc
μ ∞ with CMBTT results:
H 0 =67.80±1.27 km/s/Mpc

12. split
𝐻 0 ≈70 km/s/Mpc

13. split
𝐻 0 ≈70 km/s/Mpc

14. μ ∞ with CMBtension results: H 0 =69.35±0.80 km/s/Mpc
CMBall: CMBTT+CMBlens+CMBpol
CMBtension: CMBall+H0+WL
ΛCDM with CMBTT results:
H 0 =67.26±0.99 km/s/Mpc
μ ∞ with CMBTT results:
H 0 =67.80±1.27 km/s/Mpc
μ ∞ with CMBtension results:
H 0 =69.35±0.80 km/s/Mpc
μ ∞ >1 at 98% CL

15. μ ∞ with CMBtension results: H 0 =69.35±0.80 km/s/Mpc
CMBtension: CMBall+H0+WL
ALL: CMBtension+BAO+SN
μ ∞ with CMBtension results:
H 0 =69.35±0.80 km/s/Mpc
μ ∞ >1 at 98% CL
μ ∞ with ALL results:
H 0 =68.57±0.50 km/s/Mpc
μ ∞ >1 at 95% CL

16. Summary Ability to reduce H 0 tension: μ 0 + γ 0 YES NO μ ∞ NO? γ ∞
CMB TT
CMB TT+
CMB Lensing
CMB Lensing+
CMB polarization
CMB polarization+
H0+WL
H0+WL+
BAO+SN
μ 0 + γ 0
YES NO
μ ∞
NO?
γ ∞

18. CMB anisotropy source functions in k-space

19. Shift on BAO peak

20. TE residual for 𝜇 ∞

21. Relation between lensing and 𝜎 8 𝑊𝐿 Ω 𝑚 0.5

22. Impact on lensing potential