Extracting my response from this post.

Claims and Assumptions (not exhaustive)

Self-replicating probes for colonizations could be launched to a fraction of lightspeed using fixed launch systems such as coilguns or quenchguns as (opposed to rockets).
Only six hours of the sun's energy (3.8x10^26W) are required to commence the colonization of the entire universe.

Since the procedure is conjunction of designs and yet each of the requirements have multiple pathways to implementation, the whole construction is robust.
Humans have generally been quite successful at copying or co-oping nature. We can assume that anything done in the natural world can be done under human control, e.g. self-replicators and AI.
Any task which can be performed can be automated.
It would be ruinously costly to send over a large colonization fleet, and is much more efficient to send over a small payload which builds what is required in situ, i.e. von Neumann probes.
Data storage will not be much an issue.

Example: can fit all the world's data and upload of everyone in Britain in gram of crystal.

500 tons is a reasonable upper bound for the size of a self-replicating probe.
A replicator with mass of 30 grams would not be unreasonable.
Antimatter annihilation, nuclear fusion, and nuclear fission are all possible rocket types to be used for deceleration.

Processes like magnetic sail, gravitational assist, and "Bussard ramjet" are conceivable and possible, but to be conservative are not relied on.

Nuclear fission reactors could be made 90% efficient. Current reactor designs could reach efficiencies of over 50% of the theoretical maximum.

Any fall-off in fission efficiency results in a dramatic decrease in deceleration potential.
They ignore deceleration caused by the expansion of the universe.

Assume probe is of sturdy enough construction to survive a grenade blad (800kJ)
Redundancy required for a probe to make it to a galaxy is given by R = exp(dAρ ) where is d is distance to be travelled (in comoving coordinate), A is cross-section of the probe, and ρ is the density of dangerous particles.

Dangerous particle size given as a function of speed of the probe by equation in the paper.
From slower probes (80%c and 50%c) redundancy required is low, two probes are enough to ensure one survives.
If you have a 500T replicator, you have more cross-section but also better ability to shield.
Density of matter in space is much higher in interstellar space compared to intergalactic space. Might not be possible to launch universe-colonization directly from our sun.

Dyson spheres are very doable. Assumed to have 1/3 efficiencies over sun's output (3.8x10^26)

Launch systems could achieve energy efficiency of 50%.
Apart from risks of collision, getting to the further galaxies is as easy as getting to the closest, the only difference is a longer wait between the acceleration and deceleration phases.
Travelling at 50c% there are 116 million galaxies reachable; at 80% there are 762 million galaxies reachable; at 99%c, you get 4.13 billion galaxies.

For reference, there are 100 to 400 billion stars in the Milky Way, and from a quick check it might be reasonable to assume 100 billion is the average galaxy.

The ability to colonize the universe as opposed to just the Milky Way is the difference between ~10^8 stars and ~10^16 or ~10^17 starts. A factor of 100 million.

On a cosmic scale, the cost, time and energy needed to commence a colonization of the entire reachable universe are entirely trivial for an advanced human-like civilization.
Energy costs could be cut by a factor of hundred or thousand by aiming for clusters or superclusters [of galaxies] and spreading out from there.