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.
• A future human civilization could easily aspire to this amount of energy.
• 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)
• We could disassemble Mercury and turn it into a Dyson sphere.
• 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.