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Is it possible to show the universe is static (i.e., $a=a_*=\rm{const}$) without assuming $a=\rm{const}$ to begin with, and using a mix of $w=0$ and $w=-1$? Let $\Lambda=0$ in the Friedmann equations, i.e.,

$$\left(\frac{\dot a}{a}\right)^2 = \frac{8 \pi}{3} \rho - \frac{K}{a^2}\tag{1}$$
and $$\frac{\ddot a}{a} = \frac{-4 \pi}{3} (\rho + 3p)\tag{2}$$

where, for simplicity, $G=c=1$.

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind May 6 at 20:28
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I think you are asking "Is it possible to have a static universe without incorporating a positive cosmological constant?". Please correct me if I'm wrong. If this is your question, FYI, a number of great minds has considered your idea before and, e.g., Schrödinger believed this can happen. I think this is a good question.

A static universe implies $a=const.$, no doubt. If you want to get this result ($a=const.$) from the field equations, the r.h.s. of the second equation (in your post) must be zero, which is only possible if $\rho+3p=0$. And, from the first equation, we must have $k=\frac{{8\pi\rho{a^2}}}{3}$, which implies $\dot a=0$. In conclusion, with the aforementioned conditions, it is possible to have a static universe without a cosmological constant (for this claim, see Weinberg [1], page 54). This was first realized by E. Schrödinger in Ref. [2]. Schrödinger proposed that the Einstein’s static universe can be obtained from the Einstein field equations without a positive cosmological constant if a negative-pressure term was added to the stress-energy tensor on the right-hand side of the Einstein field equations! But this is controversial since this proposal is equivalent to introducing a cosmological constant at the level of the action of the theory. After this proposal, Einstein stated that he, in this way, indeed discovered that the cosmological constant is able to account theoretically for the existence of a finite mean density in a static universe.

So, it is mandatory to have a nonzero pressure in a static universe without a cosmological constant. However, assuming that $p=0$ for matter (which is reasonable according to observational data) leads to the inconsistent solution, i.e., $\dot a=0$ but $\ddot a\ne 0$!. This latter result was first understood by Einstein and de Sitter in Ref. [3] (open access version).

For these reasons, you always hear that Einstein's gravitational field equations without a cosmological constant will never appear as a static universe, but, as seen, at least, a number of great minds has considered your idea.

And note that the case of $\omega =-1$ is for the vacuum if the cosmological constant is nonzero ($\Lambda \ne 0$).

For a nice (historical) review with evidence about the Einstein static universe, see this paper (open access). This paper covers my answer in a good manner.

References

[1] S. Weinberg, Cosmology, Oxford university press (2008).

[2] E. Schrödinger, "Über ein Lösungssystem der allgemein kovarianten Gravitationsgleichungen." Physikalische Zeitschrift 19 (1918) 20.

[3] A. Einstein and W. de Sitter, On the Relation between the Expansion and the Mean Density of the Universe, Proceedings of the National Academy of Sciences of the United States of America 18 (1932) 213.

[4] C. O’Raifeartaigh, M. O’Keeffe, W. Nahm, and S. Mitton, Einstein’s 1917 static model of the universe: a centennial review, The European Physical Journal H 42 (2017) 431.

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind May 6 at 20:27

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