We investigate whether the 4.4 \sigma tension on H _ { 0 } between SH _ { 0 } ES 2019 and Planck 2018 can be alleviated by a variation of Newton ’ s constant G _ { N } between the early and the late Universe .
This changes the expansion rate before recombination , similarly to the addition of \Delta N _ { eff } extra relativistic degrees of freedom .
We implement a varying G _ { N } in a scalar-tensor theory of gravity , with a non-minimal coupling of the form ( M ^ { 2 } + \beta \phi ^ { 2 } ) R .
If the scalar \phi starts in the radiation era at an initial value \phi _ { I } \approx 0.3 M _ { Pl } and with \beta \approx - 0.8 , a dynamical transition occurs naturally around the epoch of matter-radiation equality and the field evolves towards zero at late times .
As a consequence the H _ { 0 } tension between SH _ { 0 } ES ( 2019 ) and Planck 2018+BAO decreases , as in \Delta N _ { \text { eff } } models .
However , mostly due to late-time constraints from Post-Newtonian ( PN ) local gravity , the tension is reduced only to 3.5 \sigma level .
When including also the SH _ { 0 } ES data in the fit , the varying G _ { N } model has H _ { 0 } = 69.2 _ { -0.75 } ^ { +0.62 } and an improvement of \Delta \chi ^ { 2 } = -3.6 compared to \Lambda CDM , at the cost of 2 extra parameters .
This corresponds to a decrease of 7 _ { -6 } ^ { +3 } percent in the value of G _ { N } from the radiation era to the present time .
For comparison , we update the fit of the \Delta N _ { eff } model to the same dataset .
We find that the \Delta N _ { eff } model performs better than the simplest varying G _ { N } scenario , with H _ { 0 } = 70 _ { -0.95 } ^ { +0.93 } and \Delta \chi ^ { 2 } = -5.5 .
The \Lambda CDM limit of the \Delta N _ { \text { eff } } model is disfavored at slightly more than 2 \sigma , since \Delta N _ { eff } = 0.316 _ { -0.15 } ^ { +0.15 } .