# What if cosmological constant was zero?

Physicists always ask why the cosmological constant is not exactly zero! I would ask here, what if cosmological constant was zero? The universe wouldn't expand and matter would exert gravitational force and shrink the universe into a big crunch! So, why physicists want the constant to be zero then? I must have missed something here!

Can cosmological constant be zero since we see the universe already expanding? How would the universe support life further as some claim?

• The universe can expand just fine without a cosmological constant. In fact, it was this fact that made Einstein originally add it to the equations when he was making his first cosmological model: he did not know space-time is expanding, so he used a negative constant as an allowed but ugly fudge-factor to make it static in his model. Later he felt he had made a mistake and should have trusted the math (in an extra heaping of irony current cosmological measurements do find acceleration best described by having a positive constant). But expansion can happen without it. – Anders Sandberg Dec 10 '17 at 18:00
• @AndersSandberg: That should be an answer. – user4552 Dec 10 '17 at 19:18
• @BenCrowell: Thanks, did turn it into a slightly longer answer. – Anders Sandberg Dec 10 '17 at 21:04

## 3 Answers

The universe can expand just fine without a cosmological constant. In fact, it was this fact that made Einstein originally add it to the equations when he was making his first cosmological model: he did not know space-time is expanding, so he used a constant as an allowed but ugly fudge-factor to make it static in his model. Later he felt he had made a mistake and should have trusted the math (in an extra heaping of irony current cosmological measurements do find acceleration best described by having a constant). But expansion can happen without it.

Assuming the universe to be spatially homogeneous and isotropic, and combining this with the Einstein field equations produces the two Friedmann equations $$\frac{\dot{a}(t)}{a(t)} = \frac{8\pi G}{3}\rho - \frac{k}{a^2(t)}+\frac{\Lambda}{3}$$ and $$\frac{\ddot{a}(t)}{a(t)}=-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}$$ where $k=+1,0,-1$ depending on curvature. $\Lambda$ is the cosmological constant.

Note that if we want $\ddot{a}(t)=\dot{a}(t)=0$ (no expansion) and $\Lambda=0$, then the first equation implies $\frac{8\pi G}{3}\rho a^2(t) = k$. This will not work if $k=0, -1$ since the left side is nonzero and positive. The second equation leads to $\rho+3p=0$: for any positive density there has to be negative pressure even if we are just thinking of the contents of the universe as pressure-free dust. So it looks like $\dot{a}(t) \neq 0$... unless one adds a suitable nonzero value of $\Lambda$ to make things stand still.

• Brilliant answer, thank you! But if I may ask, how would the universe expand without the cosmological constant? In other words, "what" would make it expand until now without the constant? – Ali Alskeif Dec 11 '17 at 1:24

To add to Anders Sandberg's answer, the Friedmann equations are really the crucial piece of the puzzle here. These equations assume General Relativity, as well as homogeneity and isotropy (i.e. the universe looks the same in every direction + looks the same at every point). Manipulating the Friedmann equations yields a critical density

$$\rho_c = \frac{3H^2}{8\pi G}$$

The big crunch only happens if the matter density of the universe is larger than this. We observe a matter density that's significantly less than this, which means that even if there were no cosmological constant, the universe will keep expanding. Gravity will slow the expansion down, but it'll never slow to the point where the universe reverses and starts contracting.

• Thank you for the answer. Matter in the universe is not enough now but it was some day in the past to stop the expansion and reverse it when matter was dominant when the universe was far smaller than now. It didn't though Cosmological Constant was probably not enough through the small volume of that baby universe to yield enough repulsive force to prevent this! – Ali Alskeif Dec 11 '17 at 1:17
• Well no - in the expression for the critical density, H (the Hubble Constant) is not actually a constant. Long ago during matter domination, the universe was expanding much faster, and the corresponding critical density was higher. A universe that has average matter density less than the critical density (as currently) will forever and has forever had lower matter density than critical. – Allure Dec 11 '17 at 1:24
• I thought the wider the universe is, the higher the speed of expansion depending on Hubble Constant. I mean, if a galaxy is travelling away from us 75 km/sec/megaparsec, another galaxy which is already twice the distance far away from us will have double the speed. That's why I believe that a younger universe will be expanding slower than the current one (except for the inflation period for sure). – Ali Alskeif Dec 11 '17 at 1:36
• That doesn't preclude the possibility that, several billion years ago, the two galaxies in your example were travelling away from us at 30 km/s & 60 km/s. The Hubble constant increases as one goes back in time. See e.g. physics.stackexchange.com/questions/136056/… – Allure Dec 11 '17 at 1:49

From the point of view of cosmology we measure a very small but non-zero cosmological constant and want it to be this value we measured.

From the point of view of quantum field theory, however, it is very unnatural that the cosmological constant is non-zero and very small. The question is similar to the matter-antimatter asymmetry: In the early universe we had a lot of matter and a lot of antimatter, about the same amount of each but not exactly. Then matter and antimatter annihilated but due to the very small difference in amount some matter remained which is all the matter which we observe today. So we could ask: Why were the amounts of matter and antimatter so similar but still not exactly equal?

Similarly, there are hypotheses that the small value of the cosmological constant emerges due to zero point energies of several fields (which individually are very big, rough estimates give over 120 orders of magnitude compared to the value of the cosmological constant) almost cancelling out, but not exactly, leaving a small difference which is the cosmological constant we measure. This is the so called naturalness problem: It seems like an unnatural coincidence that two or several very big values add up to a very small but non-zero value.

• The OP expressed an elementary misconception. This answer fails to correct that, and it also talk about a lot of things that are only tangentially related to the question. – user4552 Dec 10 '17 at 19:18
• Quantum theory has not very common with this area, as it really is about high energy and gravity. Cosmological constant is a free parameter in General Relativity Theory. Of course it can be computed from Standard Model of elementary particles, but results clearly shows that something very important is missing. Even if we have better agreement between theories, as they are completely not unified, better agreement, in my opinion, would be even more surprising – kakaz Dec 10 '17 at 20:06
• @BenCrowell: OP asked why physicists want the cosmological constant to be zero, the answer imho is the naturalness problem which I explained in my answer. He also asked why it is not exactly zero which I explained referring to cosmological observations. I am not sure which misconception of the OP you refer to. Probably the fact that he wrote "expansion" rather than "accelerated expansion". If so, yep, I missed that. – Photon Dec 10 '17 at 21:01
• Thank you for answering my question. Do you think the universe will really be more life supporting had the cosmological constant been exactly zero? Lawrence Krauss once claimed this without explaining why! Anyway, how would the universe expand without a cosmological constant? what would make it expand without it? – Ali Alskeif Dec 11 '17 at 1:21
• @AliAlskeif: I don't know, what exactly he meant by "life supporting". Without a cosmological constant the universe would expand following some initial momentum of the big bang, but it would decelerate (and start to contract at some point if the density is high enough). What we observe, however, is an accelerated expansion, so we have to conclude that a non-zero cosmological constant is present. The case with a zero cosmological constant is like a rock thrown into the sky, the case with a non-zero cosmological constant like a rocket with an engine which can accelerate it. – Photon Dec 11 '17 at 6:52