How physicists come to the conclusion that the cosmological constant and the other constants are really fine-tuned in a way that if they are changed just a bit, then stars and life won't exist?
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The Standard Model of modern physics includes over a hundred constants that just have to have their numerical values measured in the lab and plugged in to the equations, we have little or no idea why they are what they are.
Using these measured values in our equations, we can run complex and sophisticated computer simulations which accurately model many aspects of the physical world, from the subatomic to the cosmic.
But if we feed just slightly different numbers into these simulations, they invariably descend into a minimalist chaos with insufficient structure to support the complexity we see around us.
This provides strong evidence that almost all the numbers have to be pretty much exactly what they are, which we call fine-tuning, to sustain the kind of world we find ourselves in.
Guy Inchbald
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I'm not going to enter in why fundamental constants such as the electric charge or the mass of the proton are fine-tuned to favour life in the universe, since it requires a long explanation and I'm not an expert on particle physics. However, since you mention the cosmological constant, I'm going to try to explain why cosmologists think this quantity is fine-tuned.
Different observations, coming from the Cosmic Microwave Background or the Supernovae type Ia, tell us that around 70 % of the total energy content in the universe corresponds to a mysterious component called dark energy, which also explains why the current expansion of the universe is accelerating. This dark energy is well described by a fluid of constant energy density, proportional to the cosmological constant $\Lambda$.
However, dark energy has not always been the dominating component. In fact it has started to dominate only recently (around 9 billion years after the Big-Bang). The time at which dark energy starts to dominate depends directly on its present energy abundance, that is, on the value of $\Lambda$. Some careful calculations show that, when dark energy starts to dominate, this inhibits the growth of matter overdensities (that will lead to the formation of large scale structure, such as galaxies or clusters). The current epoch is dark energy dominated, so this is actually preventing new structure to form. If the value of $\Lambda$ was bigger, then dark energy would have started to dominate earlier in the history of the universe, blocking the formation of galaxies, stars, planets... and consequently not allowing life to appear.
