Well let me start by saying that eastern portion of the Persian Empire is incredibly poorly understood, with an almost complete lack of intact sources dealing with it, and those that exist are of poor quality or are extremely unreliable. That being said, the eastern portion of the Persian Empire does appear to have been the far more important part than the west. Our traditional ideas of the Persians have tended to come from the Greeks which results in an obviously Greco-centric perspective, but from reading between the lines, particularly from Thucydides' "The History of the Peloponnesian War" and Herodotus' various writings on the Persians it becomes clear that the main focus of the Persian Empire was their eastern flank, with large gaps of time where the Persians just leave and have no interest in the Greeks, or where they simply treat the Greeks like tribal border barbarians(In a very similar way to how the Chinese have treated the Steppe Tribes since time immemorial and How the Byzantines treated the Avars, Turks, Bulgars, etc., in short, a fairly standard set of tactics used to pacify dangerous but not wealthy border regions).

With that established, I'll address what I believe you misunderstand about the Indus River Valley civilization(Henceforth abbreviated as the IVC for ease). The IVC had collapsed a VERY long time ago by the time the Persians appeared on the scene. The reasons for this are ill-understood, and I'd be happy to go into more depth if you'd like in the future, but for now I'll just say that it's a VERY controversial topic in early Indian history at the moment, in fact, it's THE topic of discussion among Indian Archaeologists right now. At any rate, the IVC collapsed more than a thousand years before the arrival of the Persians on the scene, and so it's not accurate to say that there was any interactions going on there. In regards to the REGION, which for our purposes we'll define as modern Pakistan as it correlates pretty well considering the border fuzziness of early societies, especially ones with poor to non-existent documentation, the region during the time of the Persian Empire was split up into dozens of minor chiefs, kings, etc., and so while it's technically true to say that they were self-governed, but they were functionally tribal. There's some scant evidence to be found in the Rig Veda regarding the tribes of the Indus River, but the main focus of the Indian Civilization at the time was in the East in Greater Magadha(Johannes Bronkhorst), and so we find that the Western Civilization Center of India was functionally non-existant at the time. In terms of the 'Claim' that Persia had on the region, it was the same claim that any Empire has ever had on a purely Tribal region, technically non-existent but what is anyone going to do about it? I'd expect, though this is unfortunately largely speculation, that the Persian conquest of what regions they controlled(because it wasn't all of them), was largely through their traditional mix of diplomacy and warfare, likely buying off some kings, conquering others, setting up vassal states, etc. I'd suspect that the region was claimed in much the same way as Greece was. There's some early discussions from the Nanda Empire(Greater Magadha) that shows evidence of interactions with Darius the Great and the establishment of a sort of 'no-touching zone' between their empires which was likely in the area of the IVC, which, as a final point, i'll remind you was gone for over a thousand years at this point.

Sources: Thucydides "The History of the Peloponnesian War", Herodotus, Johannes Bronkhorst "Greater Magadha", The Indus River Valley Seals(Not sure how to source these properly for this subreddit, there's hundreds and I've seen them from dozens of sources), and various discussions I've held with Dr. Dominik Wujastyk of the University of Alberta.

Edit: Forgot to Add my Sources.

Thanks very much for your answer yes i am aware that Indus civilisation by the time of Persians was long gone due to unknown things what i am really interested about is after the Indus civilisation who ruled the Indus delta till the Persians got it, if you do know any books on that period or could shed some light i will be thankful.

And if someone could shed some light on this as well thanks in advance.
Alexander did had battle with Pauravas which was battle of Hydaspes, to my knowledge Pauravas only controlled the upper delta of the Indus but after the battle alexander the great did claimed the lower part of Indus which was not controlled by Pauravas and there was no significant battle in that area was it because it was controlled by small tribes rather one empire

So I wouldn't be wrong if i say Pauravas didn't went on to claim the lower delta of Indus long before the Alexanders invasion because they had given sovereignty to Persians and it would bring them in war with both Persians and the small tribes.