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Pressure cause attractive gravity so we think that negative pressure should be repulsive. And I've read that negative pressure means constant energy density in an expanding universe which mean energy can be created out of nothing and total energy increases.
Is it certain that accelerated space expansion ( ie. Inflation,Dark Energy) is caused by negative pressure? Or could there be something else other than negative pressure that we haven't discovered yet which cause accelerated space expansion?
Clarification: Inflation is a (hypothetical) phase of accelerated expansion in the very early universe. While equations for cosmological evolution are (mostly) the same, the energy density and negative pressure (approximately equal to minus the energy density) then were many orders of magnitude higher than the average energy density today, and so was the rate of expansion. Unlike the measurements of current cosmic acceleration, evidence for the inflation is far from conclusive and there are other competing theories for that epoch.
Clarification 2: 'I've read that negative pressure means constant energy density' this is incorrect. Only when negative pressure is equal to minus the energy density does energy density stays constant. And yes, in general relativity energy is not generally conserved. You could think of first law of thermodynamics ($dE= - p d V+\dots$): if space is expanding ($dV>0$) and pressure is negative then the energy increases.
The rate of cosmic acceleration is governed by the second of Friedmann equations which are derived from Einstein field equations for the homogeneous and isotropic universe: $$ \frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+3p\right)$$ where $a=a(t)$ is a cosmological scale factor, $\ddot a$ is a cosmic acceleration rate, $\rho$ and $p$ are energy density and pressure (in units where $c=1$). We see that if $3 p< -\rho $ then acceleration would be positive.
No direct observations of this negative pressure currently exist. And since the cosmic acceleration is a reasonably well established using different methods (supernovae observations, baryon acoustic oscillations, galaxy counting) there are three main alternatives:
(1) has the most support among cosmologists. Dark energy would be the carrier of this negative pressure and is assumed to be distributed more of less evenly across the universe. There are many different explanations for what is dark energy: it could be a cosmological constant, a new dynamical field, a vacuum energy of known fields etc.
(2) challenges the assumption of spatial homogeneity and isotropy. This approach postulates that there are no negative pressure and it is spatially matter inhomogeneities accross the universe that produce locally observable cosmic acceleration. The main evidence against this theory is cosmic microwave background (CMB) which is a strong support for homogeneity of the universe. While it does not completely rule out this explanation, it does require a lot of fine-tuning to work.
(3) assumes that Einstein's equation require modifications when dealing with large distances. While general relativity has a lot of experimental support, most of it is for much shorter distances than cosmological scales. So, in principle, it is possible that some modification of GR is unobservable, say, in a solar system or even across the galaxy, yet has some measurable effect at cosmological distances. Some of the modifications for the GR that could account for the cosmic acceleration without the need for negative pressures include (but are not limited to):
In most of such models corrections to the ordinary GR equations could be grouped together to produce 'effective' of 'phantom' negative pressure.
