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Dark matter is generally considered to be stable or at least long-lived (age of the universe) and I understand that is necessary to have such abundance today. My questions are 3:

  • Is it 100% necessary to have this long-lived condition? Both for a positive and negative answer could you please maybe elaborate on the necessity of this condition?

  • How can we build a model with unstable/decaying DM?

  • What is the role of discrete symmetries? Are they usually assumed in the model to have a stable particle? Is it, therefore, possible to construct DM models without discrete symmetries? Maybe using gauge symmetry?

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Is it 100% necessary to have this long-lived condition?

We measure the baryonic density at ~3 minutes using Big Bang Nucleosynthesis. Then we have a measurement of baryonic and total densities 300,000 years after using the Cosmic Microwave Background. Then we get the total density through a time-integrated measurement based on simulations of Structure Formation evolving from the tiny CMB fluctuations. And then we measure the dark matter abundance again "today", cosmologically speaking, in galaxy clusters and galaxies. All these measurements are consistent. But we don't e.g. have a measurement of the dark matter density 5725 years after the Big Bang.

So, to answer your question, no, it is not 100% necessary. First, these measurements come with error bars, so there's a bit of wiggle room if you only want, say, 90% of your dark matter to be stable. More interestingly, you can have multi-component dark matter and shuffle density from one component into the other, depending on your model, as long as that doesn't influence these measurements, you're free to dream.

As a particular example, consider Primordial Black Holes. These would evaporate through Hawking radiation. But if you create them big and fat prior to Big Bang Nucleosynthesis, they are long-lived and don't mess up BBN, so you can get around all these constraints, and voila, you have a model. Well, it's rather disfavored now due to various MACHO searches (including a recent measurement using Pulsar Timing Arrays), but it gives you an example of how a model can get around these constraints with a not-100%-stable dark matter particle.

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There are models of DM which have more than one type of particle as candidates for DM. However, models need a particle candidate that exists today and does not interact with ordinary matter (OM) even if when the universe was hotter the intermediary particle between DM and OR was abundant. That solves a problem because it would explain why we don't see that interaction today, the intermediary is very difficult to produce in a cold universe and DM has decoupled from OM (only in rare places where the temperatures allow it you would see that interaction again). For DM to exist in the form QFT would consider it you need a stable state. The Discrete Symmetry ($\mathbb{Z}_2$ for example) is to make that particle stable in many models.

Building a DM model without a stable final state that remains dark would not help understand why we don't interact with it today.

And yes I believe you can find models without such discrete symmetries... a quick search brought this one, I am not familiar with it, just pointing it to you, https://arxiv.org/abs/1704.01107

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