Comment on "Late Time Behavior of false Vacuum Decay: Possible Implications for Cosmology and Metastable Inflating States"
aa r X i v : . [ a s t r o - ph . C O ] N ov Comment on ”Late Time Behavior of false Vacuum Decay:Possible Implications for Cosmology and Metastable Inflating States”
K. Urbanowski ∗ Institute of Physics, University of Zielona Gora,ul. Prof. Z. Szafrana 4a, 65-516 Zielona Gora, Poland
In the Letter [1] Krauss and Dent analyze late timebehavior of false vacuum decay and discuss its possiblecosmological implications. Their attention is focused onthe possible behavior of the unstable false vacuum at verylate times, where deviations from the exponential decaylaw become to be dominat. In my opinion the discussionpresented in the Letter requires some complements.Let us start from a brief introduction into the prob-lem. If | M i is an initial unstable state then the sur-vival probability, P ( t ), equals P ( t ) = | a ( t ) | , where a ( t ) is the survival amplitude, a ( t ) = h M | M ; t i , and | M ; t i = e − itH | M i , H is the total Hamiltonian of thesystem under considerations. The spectrum, σ ( H ), of H is assumed to be bounded from below, σ ( H ) = [ E min , ∞ )and E min > −∞ . Searching for late time properties ofunstable states one usually uses the integral representa-tion of a ( t ) as the Fourier transform of the energy dis-tribution function, ρ ( E ), (see (2) in [1]), with ρ ( E ) ≥ ρ ( E ) = 0 for E < E min . In the case of quasi–stationary (metastable) states it is convenient to express a ( t ) in the following form [2–5], a ( t ) = a exp ( t ) + a non ( t ),where a exp ( t ) is the exponential part of a ( t ), that is a exp ( t ) = N exp [ − it ( E M − i Γ M )], ( E M is the energyof the system in the state | M i measured at the canonicaldecay times, Γ M is the decay width, N is the normaliza-tion constant), and a non ( t ) is the non–exponential part of a ( t ). From the literature it is known that a non ( t ) exhibitsinverse power–law behavior at the late time region. Thecrossover time T can be found by solving the followingequation, | a exp ( t ) | = | a non ( t ) | .The authors of the Letter find the crossover time T (see (7) in [1]) and then they find a quantitative estima-tion of T , (see (11) in [1]). It should be noticed that thisestimation can not be considered as conclusive becausethe formulae for T depend on the model considered (i.e.on ρ ( E )) in general and they can differ from relation (7)(see, eg. [2–5]). In order to find a proper relation for T one needs a more realistic expression for ρ ( E ) corre-sponding to the unstable false vacuum state. One cannot exclude that T for more realistic ρ ( E ) may be muchshorter than that found in [1], which could change theconclusions contained there. So, result (11) in [1] shouldbe considered only as a very rough approximation.Another, more important remark concerns the energyof the false vacuum state at late times t ≫ T . In [1] it ishypothesized that some false vacuum regions do survivewell up to the time T or later. Let E false be the energyof a state corresponding to the false vacuum measured at the canonical decay time and E true be the energy oftrue vacuum (i.e. the true ground state of the system).The problem is that the energy of those false vacuumregions which survived up to T and much later differsfrom E false . It follows from properties of the instanta-neous energy E M ( t ) of a unstable state | M i , [4, 5]. In theconsidered case, E M ( t ) can be found using the effectiveHamiltonian, h M ( t ), governing the time evolution in asubspace of states spanned by vector | M i , [4, 5]: h M ( t ) = ia ( t ) ∂a ( t ) ∂t ≡ h M | H | M ; t ih M | M ; t i . (1)The instantaneous energy E M ( t ) of the system in thestate | M i is the real part of h M ( t ), E M ( t ) = ℜ ( h M ( t )).There is E M ( t ) = E M at the canonical decay time and, E M ( t ) ≃ E min + c t + c t . . . , (for t ≫ T ) , (2)where c i = c ∗ i , i = 1 , , . . . , and lim t →∞ E M ( t ) = E min .So, if one assumes that E true ≡ E min then one has forthe false vacuum state that at t ≫ TE false ( t ) ≃ E true + c t + c t . . . . (3)The basic physical factor forcing the wave function | M ; t i and thus the amplitude a ( t ) to exhibit inversepower law behavior at t ≫ T is a boundedness frombelow of σ ( H ). This means that if this condition takesplace and R + ∞−∞ ρ ( E ) dE < ∞ , then all properties of a ( t ),including a form of the time–dependence at t ≫ T , arethe mathematical consequence of them both. The sameapplies by (1) to properties of h M ( t ) and concerns theasymptotic form of h M ( t ) and thus of E M ( t ) at t ≫ T .Note that properties of a ( t ) and h M ( t ) discussed above donot take place when σ ( H ) = ( −∞ , + ∞ ). In conclusion,above considerations show that the results concerning thelate time behavior of the false vacuum regions obtainedin [1] should be modified using properties (2), (3). ∗ e–mail: [email protected][1] L. M. Krauss, J. Dent, Phys. Rev. Lett., , 171301(2008).[2] K. M. Sluis, E. A. Gislason, Phys. Rev. A 43 , 4581 (1991).[3] E. Torrontegui, et al