Question

Scalar fields (eg. the Higgs boson is an example of a scalar field) are posited to solve a number of cosmological challenges, namely that of dark energy and inflation. In this shorter problem we are going to determinp the dynamics of scalar fields. A scalar field ф is fully determined by its potential (the function that describes its potential energy): V(ø). You can think of this as a ball 'rolling down a hill', where the shape of the hill is the potential The energy density of the field is given as a sum of the kinetic energy and the potential energy of the field and its pressure is given by 2 V(ф) 1. Using Eqs. (1) and (2) above, write down the Friedmann and Raychaudhuri equations for a universe dominated by this scalar field. 2. By considering the acceleration (Raychaudhuri) equation that you got from the first part and requiring that the universe undergoes acceleration, what can you say about the balance between the kinetic energy and potential energy of such a field? 3, Compute the equation-of-state for such a scalar field 4. Using the acceleration requirement you derived in part two, what can you say about the equation-of- state of a scalar field that is responsible for the acceleration of the universe? How does this compare to the equation-of-state of components we have seen before?

Answer 1

1) ans -

The tachyon of string theory may be described by effective field theory describing some sort of tachyon condensate which in flat space has a Lagrangian density

L = −V (T) p 1 + η µν∂µT ∂νT. (1)

where T is the tachyon field, V (T) is the tachyon potential and ηµν = diag(−, +, +, +) is the metric of Minkowski spacetime. The tachyon potential V (T) has a positive maximum at the origin and has a minimum at T = T0 where the potential vanishes. In Minkowski spacetime the rolling down towards its minimum value is described by a spatially homogeneous but time-dependent solution obtained from the Lagrangian density

L = −V p 1 − T˙ 2 . (2)

During the rolling the Hamiltonian density

ρH = V (T) p 1 − T˙ 2 (3)

has a constant value E. Thus

T˙ = r 1 − V 2 E2 . (4)

As T increases V (T) decreases and T˙ increases to attain its maximum value of 1 in infinite time as T tends to infinity. Note that as explained in the tachyon field behaves like a fluid of positive energy density

ρ = V (T) p 1 − T˙ 2 (5)

and negative pressure

p = −V (T) p 1 − T˙ 2 . (6)

Thus

w = p ρ = −(1 − T˙ 2 ) (7)  

and therefore, −1 ≤ w ≤ 0. Note that both the Weak Energy Condition, ρ > 0 and Dominant Energy Condition, ρ ≥ |p| hold. However because

ρ + 3p = − −2V (T) p 1 − T˙ 2 1 − 3 2 T˙ 2 , (8)

the Strong Energy Condition fails to hold for small |T˙ | but does hold for large |T˙ |. The discussion above has neglected the gravitational field generated by the tachyon condensate. To take it into account we use the Lagrangian density

L = √ −g R 16πG − V (T) p 1 + g µν∂µT ∂νT . (9)

In this case the expressions (5) and (6) for the density and pressure remain valid and thus the Friedman equations are

a˙ 2 a 2 + k a 2 = 8πG 3 V (T) p 1 − T˙ 2 ;    (10)

a¨ a = 8πG 3 V (T) p 1 − T˙ 2 1 − 3 2 T˙ 2 .   (11)

It follows from (4) that as T increases V (T) increases too, but T˙ could decrease since E decreases. In any case T˙ remains positive and so T increases monotonically to its maximum value of 1. It follows from (10) that if k ≤ 0, then ˙a will always be positive. This is because the Weak Energy Condition holds, ρ > 0. From the Raychaudhuri equation (11) it follows that if |T˙ | < 2/3 the scale factor initially accelerates (¨a > 0), but then, when T˙ exceeds p 2/3, the acceleration will stop and deceleration will start. If the Universe is flat (k = 0), then a(t) → constant. In the hyperbolic Universe (k = −1) the scale factor increases linearly with time (a → t). In both cases the final state of the Universe is flat, the case k = −1 being the Milne model. In the spherical case (k = 1) the Universe will re-collapse. The possibility of cosmic acceleration arises from the positive potential V (T).