Why didn't Voldemort recognize that Dumbledore was affected by his curse?

Question

Dumbledore's hand was visibly affected by a curse Voldemort placed on Marvolo Gaunt's ring. Why wasn't Voldemort alerted that perhaps Dumbledore has found one of his Horcruxes based on the appearance of Dumbledore's hand?

Answer 1

Voldemort never knew Dumbledore found his Horcrux

The last time Voldemort saw Dumbledore was when he dueled him in the Ministry of Magic, at the end of Book 5. Dumbledore acquired the Resurrection Stone/Marvolo Gaunt's ring some time in between the end of The Order of the Phoenix and the beginning of The Half-Blood Prince. Remember that Harry saw Dumbledore's withered hand when he came to pick him up from the Dursleys' place.

Let us not also forget that Voldemort was so confident in his own prowess that he did not, even in his wildest dream, dream that someone would know about - much less find - his Horcruxes.

"I believe not. I believe that Voldemort is now so immersed in evil, and these crucial parts of himself have been detached for so long, he does not feel as we do. Perhaps at the point of death he might be aware of his loss..."

-Dumbledore, Harry Potter and The Half-Blood Prince

Voldemort did not even realize that Harry and Dumbledore were after his Horcruxes until after 3 of them had been destroyed (Riddle's diary, Slytherin's locket, and Marvolo's ring) and a 4th captured (Hufflepuff's cup). So it would be logical to assume that during the events of Book 6, Voldemort wasn't paying attention to his Horcruxes, his hubris leading him to believe that they can't even be touched.

Answer 2

It may not be the Dark Lord’s unique curse.

Though the Dark Lord does invent spells, there’s nothing that says that specific curse is unique to him. He also uses more well-known Dark magic that other wizards can cast as well, like the Unforgivable Curses, which predate him by centuries.

“¹⁹ The Cruciatus, Imperius, and Avada Kedavra curses were first classified as Unforgivable in 1717, with the strictest penalties attached to their use.”
_{^{- The Tales of Beedle the Bard}}

The Dark Lord sees Dumbledore himself after he injured his hand when breaking into his tomb, and he doesn’t seem the slightest bit concerned about its appearance despite looking directly at Dumbledore’s hands rested over the Elder Wand.

“The wrappings fell open. The face was translucent, pale, sunken, yet almost perfectly preserved. They had left his spectacles on the crooked nose: he felt amused derision. Dumbledore’s hands were folded upon his chest, and there it lay, clutched beneath them, buried with him.

Had the old fool imagined that marble or death would protect the wand? Had he thought that the Dark Lord would be scared to violate his tomb? The spider-like hand swooped and pulled the wand from Dumbledore’s grasp, and as he took it, a shower of sparks flew from its tip, sparkling over the corpse of its last owner, ready to serve a new master at last.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 24 (The Wandmaker)}}

If the curse is indeed not uniquely the Dark Lord’s own invention, then he has no real reason to suspect it was specifically his curse that injured Dumbledore. He may recognize the effects of the curse without thinking he cast it. Especially since Snape likely told him Dumbledore was injured in a duel, he probably wouldn’t have been suspicious when he saw Dumbledore’s blackened hand. Wizards can recognize the effects of Avada Kedavra, but several different wizards can cast it, so seeing its use doesn’t implicate any one specific individual.

“The Muggle authorities were perplexed. As far as I am aware, they do not know to this day how the Riddles died, for the Avada Kedavra Curse does not usually leave any sign of damage … the exception sits before me,’ Dumbledore added, with a nod to Harry’s scar. ‘The Ministry, on the other hand, knew at once that this was a wizard’s murder. They also knew that a convicted Muggle-hater lived across the valley from the Riddle house, a Muggle-hater who had already been imprisoned once for attacking one of the murdered people.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 17 (A Sluggish Memory)}}

Additionally, while Snape describes it as a curse of extraordinary power, he doesn’t seem surprised by its effects on Dumbledore’s hand at all - it doesn’t seem to be something he’s never seen or heard of happening before.

“It was night-time, and Dumbledore sagged sideways in the throne-like chair behind the desk, apparently semi-conscious. His right hand dangled over the side, blackened and burned. Snape was muttering incantations, pointing his wand at the wrist of the hand, while with his left hand he tipped a goblet full of thick golden potion down Dumbledore’s throat. After a moment or two, Dumbledore’s eyelids fluttered and opened.

‘Why,’ said Snape, without preamble, ‘why did you put on that ring? It carries a curse, surely you realised that. Why even touch it?”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

Further supporting that it may not be a unique spell is that Snape refers to ‘spells of such type’, so there are at least similar curses in existence, whose effects are known.

“It is a miracle you managed to return here!’ Snape sounded furious. ‘That ring carried a curse of extraordinary power, to contain it is all we can hope for; I have trapped the curse in one hand for the time being –’

Dumbledore raised his blackened, useless hand, and examined it with the expression of one being shown an interesting curio. ‘You have done very well, Severus. How long do you think I have?’

Dumbledore’s tone was conversational; he might have been asking for a weather forecast.

Snape hesitated, and then said, ‘I cannot tell. Maybe a year. There is no halting such a spell forever. It will spread, eventually, it is the sort of curse that strengthens over time.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

It may not be possible for the Dark Lord to tell from just looking at Dumbledore’s injury that his own curse caused it, either because the spell he used is itself a known type of Dark magic or because its effects are similar enough to a known piece of Dark magic that they aren’t easily distinguishable. For example, Dumbledore didn’t seem able to tell that a basilisk was specifically what had Petrified Mrs. Norris, just that it was caused by advanced Dark magic.

“No second-year could have done this,’ said Dumbledore firmly. ‘It would take Dark magic of the most advanced –”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 9 (The Writing on the Wall)}}

Therefore, the Dark Lord may have had no reason to suspect that Dumbledore’s injury meant he knew about the Horcruxes. Even if his suspicions were raised, by the time he’d have seen the injury, it’s almost certain he’d already heard an alternate story from Snape, who he trusted. Snape told Bellatrix, and he also had presumably told the Dark Lord, that Dumbledore was injured in a duel.

“Dumbledore has been a great wizard – oh yes, he has’ (for Bellatrix had made a scathing noise) ‘the Dark Lord acknowledges it. I am pleased to say, however, that Dumbledore is growing old. The duel with the Dark Lord last month shook him. He has since sustained a serious injury because his reactions are slower than they once were.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 2 (Spinner’s End)}}

The Dark Lord then likely believed that Dumbledore’s injury was caused in a duel, so had no cause for concern. Additionally, it was likely plausible for an injury like Dumbledore’s to be caused in a duel. Snape was knowledgeable in Dark magic so he’d know what could cause it, and he’d know his lie would have to be plausible because the Dark Lord was also knowledgeable in Dark magic, so would know a bad lie if he saw or learned of the specific nature of Dumbledore’s injury. Therefore, it’s not likely to be obvious from the effects of the curse that Dumbledore was injured by touching the Dark Lord’s Horcrux.