The discovery that our universe is undergoing an accelerated expansion   has led to a renewed interest in modified theories of gravity. One of the most important of these, modified gravity, replaces the Ricci scalar in the Einstein-Hilbert action
by a nonlinear function :
(For a review, see Refs.    .)
An alternative to the modified gravity model is the hypothesis that the acceleration is due to a negative pressure dark energy, implying that in the Friedmann equation
(We are using units in which ) In the equation of state , corresponds to the range of values , referred to as quintessence dark energy. The case is equivalent to assuming Einstein’s cosmological constant. It has been forcefully argued by Bousso  that the cosmological constant is the best model for dark energy. In this note, we go a step further and propose that modified gravity implies that is the only allowed value in the equation of state .
2. The Solution
For convenience of notation, we start with the spherically symmetric line element
It was shown by Lobo  that under the assumption that , the Einstein field equations are
where . The curvature scalar is given by
For our purposes, a more convenient form of the line element is
Here the Einstein field equations can be written 
Then if , Lobo’s equations become
Now substituting into the equation of state , we obtain
This equation can be rewritten as follows:
Since we are dealing with a cosmological setting, we may assume the FLRW model, so that :
Observe that we now have
which is independent of time. The significance of the special value in Equation (14) now becomes apparent: the entire equation has become time independent, i.e.,
(For later reference, observe that if , then .) The solution of Equation (19) is
This solution can also be written
where and .
3. Staying Close to Einstein Gravity
In a cosmological setting, modified gravity must remain close to Einstein gravity to be consistent with observation. In this section, we wish to show that it is possible, at least in principle, to choose the arbitrary constants in Equation (21) so that this goal is achieved.
The sinusoidal solution (21) has a large period and a small slope, especially for large . To confirm this statement, observe that the function
for large . So both and approach zero as . As a result, has the approximate form on any interval that is not excessively large, and since the slope is small in absolute value, we have
We can now show that it is possible in principle to choose the arbitrary constants and in such a way that remains close to unity and both and close to zero on one complete period.
Let , so that
First observe that whenever
for all integers . Solving for , we get
Now choose a particular for which on the interval , where
Next, subdivide the interval into subintervals each of which is small enough so that remains in a narrow range. Then on each separate subinterval, construct a tangent line near the midpoint, thereby ensuring that . (See Figure 1) So we may now choose for the arbitrary constant . We then repeat the procedure on the interval , so that on the entire period . Since both and are close to zero [from Equation (22)], the periodicity of guarantees that our modified gravity is close to Einstein gravity for all .
Figure 1. The line segment on the interval (not drawn to scale).
4. The Cosmological Constant
Suppose we return to Equation (14) and substitute Equations (16)-(18). Then we obtain
While we normally assume that , it is noted in Ref.  that , representing a spatially flat universe, is not a dramatic departure from generality when it comes to late-time cosmology.
With , the time-dependent solution is
In the special case , , in agreement with Equation (19) with .
In the previous section, we dealt with a time-independent solution due to the assumption . This allowed our modified model to remain close to Einstein gravity at least in principle. By contrast, solution (27) is time dependent. So if , we are dealing with two possibilities:
(a) if , there is no real solution;
(b) if , then the model is far removed from Einstein gravity, i.e., if increases indefinitely, then the first term in solution (27) goes to zero, while the second term gets large. So cannot remain close to unity.
We conclude that in the equation of state is the only allowed value. Since this note deals with rather reasonable assumptions, the only plausible objection to this conclusion is that the equation of state for dark energy is much more complicated than the perfect-fluid equation of state . This possibility was also raised by Lobo  , who stated that a mixture of various interacting non-ideal fluids may be necessary. This could imply that dark energy is dynamic in nature, thereby forcing us to exclude models with constant , including the cosmological constant.
The starting point in this note is modified gravity in a cosmological setting. We also assume a spatially flat universe to describe late-time cosmology  ; thus in the FLRW model. Our key assumption is the perfect-fluid equation of state to describe the hypothesized dark energy. While is sufficient to yield an accelerated expansion, it is concluded that is the only value which allows our solution to remain close enough to Einstein gravity to be consistent with observation.
Weighing the above assumptions, we conclude that either (1) Einstein’s cosmological constant is the only acceptable model for dark energy, or that (2) the equation of state is far more complicated than the above perfect-fluid equation and may even exclude a constant .
 Nojiri, S. and Odintsov, S.D. (2007) Introduction to Modified Gravity and Gravitational Alternative for Dark Energy. International Journal of Geometric Methods in Modern Physics, 4, 115.
 Rahaman, F., Chakraborty, K., Kuhfittig, P.K.F., Shit, G.C. and Rahman, M. (2014) A New Deterministic Model of Strange Stars. European Physical Journal C, 74, 3126. https://doi.org/10.1140/epjc/s10052-014-3126-6