The dynamics of the Local Group (LG), especially concerning the contributions

of the Milky Way (MW) and Andromeda (M31) galaxies, is sensitive to the

presence of dark energy. This work compares the evolution of the LG by

considering it as a two-body problem in a homogeneous and isotropic expanding

spacetime, i.e. the McVitte spacetime (McV) versus the spherically symmetric

metric for LG dynamics with the Cosmological Constant, i.e. the De

Sitter-Schwarzschild spacetime (DsS). Using the Timing Argument (which links LG

dynamics to LG mass), calibrated by the IllustrisTNG simulations, we find that

the McV spacetime predicts a lower mass for the LG: $left(4.20 pm 0.61right)

cdot 10^{12} M_{odot}$ for McV spacetime vs. $left(4.65 pm 0.75right)

cdot 10^{12} M_{odot}$ for DsS spacetime ($68 % ,$ CL). Due to uncertainties

in tangential velocity measurements, the masses are indistinguishable. However,

with future astrometric measurements, we demonstrate that the predicted masses

will be distinguishable, indicating different LG histories. By independently

estimating the total mass of MW and M31, we compare the possible upper bounds

for the Cosmological Constant in these scenarios. We find a tighter upper bound

for the DsS spacetime model, $Lambda < 3.3 ,Lambda_{text{CMB}}$, compared

to $Lambda < 8.4, Lambda_{text{CMB}}$ for the McV spacetime (where

$Lambda_{text{CMB}}$ is the mean value from Planck). Future astrometric

measurements, such as those from JWST, hold the potential to independently

detect dark energy for both spacetime models independent from Planck’s value.

## Future Roadmap: Challenges and Opportunities

### Potential Challenges:

**Tangential Velocity Measurements:**The current uncertainties in tangential velocity measurements make it difficult to distinguish between the predicted masses of the Local Group (LG) in the McVitte spacetime (McV) and the De Sitter-Schwarzschild spacetime (DsS).**Indistinguishable LG Histories:**Without more precise astrometric measurements, it is not currently possible to determine the different histories of the LG predicted by the two spacetime models.**Upper Bound Estimation:**While the comparison of upper bounds for the Cosmological Constant in the two scenarios is informative, there are uncertainties in independently estimating the total mass of the Milky Way (MW) and Andromeda (M31) galaxies, which could affect the reliability of these bounds.

### Potential Opportunities:

**Future Astrometric Measurements:**With advancements in astrometric measurements, particularly from the James Webb Space Telescope (JWST), there is the potential to overcome the challenges mentioned above. More precise measurements can help distinguish between the predicted masses in the McV and DsS spacetime models, revealing different LG histories.**Independent Detection of Dark Energy:**Future astrometric measurements, such as those from JWST, hold the potential to independently detect dark energy for both spacetime models, regardless of Planck’s value. This could provide valuable insights into the nature and properties of dark energy.

Note:It is important to acknowledge that further research and advancements in observational techniques are necessary to overcome the current challenges in determining the dynamics and contributions of the Milky Way and Andromeda galaxies to the Local Group. Continued investigations into the LG’s evolution and the presence of dark energy will deepen our understanding of the cosmos.