The average space metric is flat for any stationary observer on its proper time path. Hence all local gravitational sinks compensate the general expansion.
A more clearer view is revealed by GR spacetime equations without lambda.
Gravity does no have any relation with expanding universe. Our universe is a complete well-organize and most accurate entity, that by nature is growing Yet, nothing is working with mechanically push/pull (gravity) as our icons thought.
Article Quantum Mechanics Gravity, Mechanical Gravity of Newton is Myth
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The expansion of the universe depends on the passage of time.
Without the arrow, time expansion and gravity do not exist.
All objects are moving at the speed of light in the real dimension of time.
Heavier objects move at a slower speed in the time dimension.
The gravitational field is like a pit on the 2D surface of an expanding sphere.
The cause of the expansion of the universe is the existence of heterogeneity in space and time.
Objects cause heterogeneity in the space-time structure.
The universe has the structure of a 3-sphere. A 3-sphere has circular geodesics. This structure determines gravitation on the large scale without any expansion.
See:
Preprint What gravitation really is
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