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Is the island universe model consistent with observations?

We study the island universe model, in which initially the universe is in a cosmological constant sea, then the local quantum fluctuations violating the null energy condition create the islands of matter, some of which might corresponds to our observable universe. We examine the possibility that the island universe model is regarded as an alternative scenario of the origin of observable universe.

Recent observations of the cosmic microwave background (CMB) have implied that the inflation [1, 2] is a very consistent cosmological scenario. The inflation stage took place at the earlier moments of the universe, which superluminally stretched a tiny patch to become our observable universe today, and ended with a period of reheating that bring the universe back to the usual FRW evolution. The quantum fluctuations in the inflation period are drew out the horizon and, when reentering into the horizon during radiation/matter domination after the end of inflation, are responsible for the formation of cosmological structure.
However, the inflation has still its own problem of fine tuning, in addition it is not also able to address the initial singularity problem, the cosmological constant problem, super Planck fluctuations problem, as was described in Ref. [3]. Thus it seems desirable to search for an alternative to the inflation model. Recently, Dutta and Vachaspatian have proposed a model [4], in which initially the universe is in a cosmological constant (Λ) sea, then the local quantum fluctuations violating the null energy condition (NEC) [5,6] create the islands of matter, some of which might corresponds to our observable universe. The headstream of this model in some sense may be backward to the eternal inflation [7,8], and especially the recycling universe proposed by Garriga and Vilenkin [9], in which the thermalised regions asymptotically approaching dS spacetime may be fluctuated and recycled back to the false vacuum and then the nucleated false vacuum region will serve as a seed for a new (eternally) inflating domain. The difference of island universe is that after the NEC violating fluctuation is over, instead of the inflation followed, the thermalisation occurs instantly and the radiation fills the volume rapidly. Though the island universe model brings us an interesting environment, people are still able to doubt whether the island universe model is consistent with our observations. Whether we can live in "Islands"? We firstly attempt to phenomenally (semiclassically) understand it, following closely Ref. [10]. We set 8π/m 2 p = 1 and work with the parameter ǫ = −ḣ/h 2 , where h ≡ȧ/a is the local Hubble constant. The "local" means here that the quantities, such as a and h, only character the value of the NEC violating region. In the island universe model, the scale of the NEC violating region is required to be larger than that of present horizon, which has been clearly stated by Dutta and Vachaspatian [4], and is consistent with Refs. [11,12], see also Refs. [13,14,15,16] for relevant discussions. This result provides the initial value of local evolution of a and h. The ǫ can be rewritten as ǫ ≃ 1 h∆t ∆h h , thus in some sense ǫ actually describes the change of h in unit of Hubble time.
During the NEC violating fluctuation,ḣ > 0, thus ǫ < 0 can be deduced. The time of this fluctuation can be given by where the subscript i and e denote the initial and end value of the NEC violating fluctuation respectively. In this model h 2 i ≡ h 2 0 ∼ Λ, where the subscript 0 denotes the present value. The reasonable and simplest selection for the local evolution of scale factor a(t), due tȯ h > 0, is [10,17] a(t) ∼ (−t) n , where t is from −∞ to 0 − , and n is a negative constant.
We have h = n/t, and thus ǫ = 1/n. To make T → 0 in which the NEC violating fluctuate can be so strong as to be able to create the islands of our observable universe [4], from (1), |ǫ| → ∞ is required, which results in n → 0 − . Thus though the change of h is large, the expanding proportion of the scale factor is very small. Further in some sense before and after the fluctuation a is nearly unchanged, which can be also seen from following discussions.
In the conformal time dη = dt/a, i.e. −η ∼ (−t) −n+1 , we obtain where the prime denotes the derivative with respect to η. The perturbations leaving the horizon during the NEC violating fluctuation might reenter the horizon during radiation/matter domination after the end of fluctuation, which are responsible for the structure formation of our observable universe. The efolding number which measures the quantity of perturbation leaving the horizon during the NEC violating fluctuation can be defined as From (2) and (3), we obtain Thus for the constant ǫ, we have We can see that for the negative enough ǫ, the change ∆a/a = (a e − a i )/a i ≃ N /(1 − ǫ) of a will be very small. Taking the logarithm in both sides of (5), we obtain We plot Fig.1 to further illustrate the island universe model, in which ln (1/ah) is regarded as the function of ln a. In the NEC violating region, n → 0 − , thus the slop (1−n)/n → −∞, which makes the evolution of ln (1/ah) during the fluctuation correspond to a nearly vertical line in Fig.1. The (7) can be also shown and actually applied for the expansions with arbitrary constant n. Thus during the Λ, radiation and matter domination, we have the slope −1, 1 and 1/2 respectively, which have been reflected in the Fig.1.
The NEC violating fluctuations in the Λ sea with the observable value of cosmological constant create some thermalised matter islands, which subsequently evolute as the usual FRW universe. The radiation and matter will be diluted with the expansion of island and eventually this part of volume will return to the Λ sea again. The total evolution may be as the efolding number when the perturbation with the present horizon scale leaves the horizon during the NEC violating fluctuation, and from (4), we have Thus for the island universe model, in which h i = h 0 and a 0 ≫ a i ≥ 1/h i , we obtain N ei > N ec . This implies that as long as the islands created are suitable for our existence, the efolding number required to solve the horizon problem of FRW universe may be always enough, independent of the energy scale of thermalisation.
Then we discuss the the primordial perturbations from the island universe model. Dutta and Vachaspatian [4] have shown that the spectrum from the quantum fields other than the NEC violating field can be scale invariant, but the amplitude of perturbation is too small to seed the structure of observable universe. However, they in their calculations regarded the NEC violating region as an instant link between the Λ sea and FRW evolution, which to some extent leads that the effect of NEC violating region on the perturbation spectrums is lose. Thus we here will focus on the NEC violating region.
To simplification, we adopt the work hypothesis in which the NEC violating fluctuation behaves like the phantom energy. The simplest implementation of phantom energy is a scalar field with the reverse sign in its dynamical term, in which the NEC ρ + p = −φ 2 < 0 is violated. To make ρ > 0, the potential V(ϕ) >φ 2 /2 of phantom field is also required.
Thus in this case the local evolution of h can be written as h 2 ∼ −φ 2 /2 + V(ϕ). This in some sense is equal to introducing a creation field, see Ref. [18]. Thus we have Due to V(ϕ) >φ 2 /2, this inequality actually requires the denominator ∼ 0, which gives V(ϕ) ∼φ 2 /2. Further, combiningφ 2 /2 =ḣ = |ǫ|h 2 , we may approximately deducė This operation dose not mean that the phantom field is actually required in the island universe model. The aim that we appeal to the phantom field here is only to phenomenally simulate the NEC violating fluctuation, which may be convenient to calculate the primordial perturbation spectra from the NEC violating region. But it maybe also a possibility that the phantom field only exists instantly for producing the NEC violating fluctuation and after the end of fluctuation, the phantom field thermalises into the radiation, see Ref. [19] for a different scenario.
In the momentum space, the equation of motion of gauge invariant variable u k , which is related to the Bardeen potential Φ by u k ≡ aΦ k /ϕ ′ , is 1 1 In the momentum space, the equation of motion of gauge invariant variable v k , which is related to the We have, from (2), a ′′ /a = (2 − ǫ)/(ǫ − 1) 2 η 2 . In the regime kη → ∞, the mode v k ∼ e −ikη (2k) 1/2 can be taken as the initial condition. In long wave limit, the expansion of the Bessel functions to the leading term of k gives where ν = (2 − ǫ)/(ǫ − 1) 2 + 1/4.
We have, from (2), The general solutions of this equation are the Bessels functions. In the regime kη → ∞, the mode u k are very deep in the horizon of Λ, see Fig.1 and Fig.2. Thus u k ∼ e −ikη (2k) 3/2 can be taken as the initial condition. In the regime kη → 0, the mode u k are far out the horizon, and become unstable and grows. In long wave limit, the expansion of the Bessel functions to the leading term of k gives where ν = ǫ/(ǫ − 1) 2 + 1/4. We can see that if ν ≃ 1/2, the spectrum of Φ k will be nearly scale invariant. This requires Thus for the NEC violating fluctuation with ǫ → −∞ 2 , we will have the nearly scale invariant spectrum [10].
In kη → 0, the Bardeen potential can also be rewritten as [32] Φ = C + hD/a, where C and D are constants dependent on the mode k. In the inflation model in which ǫ ≃ 0, h is nearly unchanged while a increases exponentially, thus D is a decay mode, and Φ is dominated by the constant mode C, and thus is nearly constant in superhorizon scale. This is the reason why people can briefly obtain the amplitude of perturbation reentering into the horizon by calculating the amplitude of perturbation leaving the horizon during the inflation. But for ǫ → −∞, the case is just inverse. a is nearly unchanged while h increases rapidly, thus instead of being regarded as the decay mode, D is a growing mode, which can be seen from (18), and compared with the constant mode C, it will dominate the Bardeen potential Φ, which means that the amplitude of perturbation in the superhorizon scale will be still increasing during the NEC violating fluctuation, while during the radiation/matter domination after the thermalisation, D will not increase any more but decrease, and thus the 2 We notice from (17) that for ǫ ≃ 0 − in which the scale factor expands exponentially, the spectrum is also nearly scale invariant. This is the so called phantom inflation proposed and discussed in detail in Ref. [20], see also Refs. [21,22,23,24]. This result is actually also a direct reflection of the dualities of primordial perturbation spectra, in which the spectrum index of Φ is invariant under the change ǫ → 1/ǫ, which has been pointed out in Refs. [25,26] and further studied in Refs. [17,27,28], and see also Refs. [29,30,31] for other discussions on the dualities. time has been pointed out [33,35] 3 . We will tentatively take this optimistic matching for the following calculations and leave behind some further comments in the final conclusion.
Because in superhorizon scale the Bardeen potential is increased during the NEC violating fluctuation and constant after the end of fluctuation, it may be reasonable to take the value at the end of fluctuation to calculate the amplitude of perturbation, where (11) has been used.
The observations of CMB constrain the amplitude of perturbation is k 3/2 ζ k ∼ k 3/2 Φ k ∼ 10 −5 , which require V e ∼ 10 −10 , which corresponds to h 2 e ∼ 10 −10 /|ǫ|. We assume the thermalisation is almost instantaneous, thus h 2 e ∼ ρ r , where ρ r is the energy density of radiation after the thermalisation. For example, for ρ r ∼ m 4 ew ∼ 10 −60 , where m ew is the electroweak scale, which may be regarded as a loose lower limit of reheating temperature though the lowest limit might be in nucleosynthesis scale, we obtain |ǫ| ∼ 10 50 . The change of a is far smaller than that of h, thus from (4), we have N ≃ ln (h e /h i ) ∼ ln (V 1/2 e /|ǫ| 1/2 Λ 1/2 ) ∼ 127 − ln |ǫ|/2 ≃ 69, which is enough to solve the horizon problem of our observable universe [42]. These results are also implicitly reflected in the depiction of causal structure of island universe of Fig.2. . How do we distinguish the island universe from the inflation model by the observations?
For ǫ → −∞, from (13), we have ν ≃ 1/2, and thus the index of tensor perturbation is 2, which means that the island universe model will produce a more blue gravitational wave spectrum than the inflation model. This character is also actually a reflection of rapid change of background during the NEC violating fluctuation, compared with the nearly unchanged background during the inflation. This result generally leads to the intense suppression of gravitational wave amplitude on large scale. Thus it seems for the island universe model to be hardly possible to search for the imprint of gravitational wave in the CMB, while a stochastic gravitational wave will be consistent with the inflation model.
In conclusion, it seems to be possible that we can live in "Islands", but slightly not optimistic, since the loophole how the perturbations propagate through the thermalisation surface remains, which may be fatal to the island universe model. However, it should be mentioned that the NEC violating fluctuation is actually a quantum phenomenon, and thus the innocence about the thermalisation and matching hypersurface may to some extent reflect the lack of our understanding on the quantum characters of gravity. We think that (at least) at present it will be premature to either confirm or deny the island universe model.
The deep insights into the quantum gravity in the future may show us the final answer.
However, whatever the outcome, it appears that the viability of the island universe model hangs in the balance and the different outcomes will either hold out or end its competing as an alternative scenario of the origin of observable universe. We will back to the relevant issues in the future.
Acknowledgments The author would like to thank Sourish Dutta, and especially Tanmay Vachaspatian for many valuable comments of earlier draft and discussions. The author also very thank Alexander Vilenkin for encouraging comments and Bo Feng, Xinmin Zhang for helpful discussions. This work is supported in part by K.C. Wang Postdoc Foundation, also in part by NNSFC under Grant No: 10405029.