I've been using both side by side, as of this lunchtime the S22 Ultra was on the o2 eSIM but got my transfer pack in the post and did the swap over to the S23U so the latter half of the day the 23 has been on the eSIM.
Been using both cameras side by side, exact same apps and settings, same synchros in the BG etc and o2 network is a bit pants here although is 4G, at 15Mbps...
At the moment the S22 Ultra is on 89% battery left, S23 Ultra is on 90%
Both phones were sat on the dual wireless fast charger overnight.
Both are connected to Wifi (5GHz band), and both have the exact same screen brightness and other settings etc.
Keep in mind the 23's battery is still "learning usage patterns" so will take more than a day to do that as Android trains battery use over the course of 7 days.
So with that in mind here's my day 1 battery stats, note that the screen on time between the two, the S22U has had half the SoT yet the battery is lower. That should highlight how much more efficient the SD8 Gen 2 chipset is!
S23 Ultra (left) / S22 Ultra (right)
Edit*
Actually forgot to say that the only diff is WhatsApp was deactivated on the S22 Ultra last night so that's been syncing all day today on the 23, hence the 1.1% usage there. So effectively the 23's battery is even better still.