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Assuming we shoot a photon out into space, it would go 1 light year after 1 year, and 1 million light years after 1 million years.

Because after that the expansion of space is noticeable, would photon be farther than 1 billion light years after 1 billion years?

Does the expansion of space (Hubble flow) carry the photon forward?

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Technically the universe is not expanding into anything neither it affects the motion of the objects.Just the metric which governs the geometry of spacetime is stretching.So the photon is not going forward.