Perturbations of bounce inflation scenario from f(T) modified gravity revisited
Abstract
In this work, we revisit the perturbations that are generated in the bounce inflation scenario constructed within the framework of f(T) theory. It has been well known that pure f(T) theory cannot give rise to bounce inflation behavior, so aside from the gravity part, we also employ a canonical scalar field for minimal extension. We calculate the perturbations in f(T) theory using the wellestablished ADM formalism, and find various conditions to avoid their pathologies. We find that it is indeed very difficult to obtain a healthy model without those pathologies, however, one may find a way out if a potential requirement, say, to keep every function continuous, is abandoned.
1 introduction
Inflation [1, 2, 3] has been viewed as one of the most successful theory in modern cosmology. Not only is it simple and elegant, it can also simultaneously solve several cosmological puzzles of BigBang, such as horizon, flatness, unwanted relics and so on, as well as predicts nearly scaleinvariant power spectrum, which is verified by the observational data [4]. Nonetheless, inflation cannot avoid the notorious BigBang Singularity, whose existence has been proved by Hawking and Penrose [5, 6, 7]. one of the easiest ways of avoiding the singularity point might be to assume that the universe starts from a contracting phase, and bounce into the expanding one as we observed [8]. Together with the inflation period that follows up, this can be called as “bounce inflation” scenario of the early universe [9, 10, 11].
For the universe to bounce, some conditions must be satisfied, such as the Null Energy Condition violating [12]. In order to do so, one may either introduce exotic matter which can violate the NEC, or modify the classical General Relativity. Recently the studies of bounce cosmology encountered a boost in the literatures and fruitful bounce models are built in both two ways. The first way includes doublescalarfield bounce [13, 14, 15] and higherorder singlescalarfield bounce [16, 17, 18, 19, 20], while the second way includes nonminimal coupling bounce [21, 22], f(R) bounce [23], f(T) bounce [24, 25], Loopquantum bounce [26] and so on.
In Ref. [27] (see also [28, 29, 30]), it is proved that it is indeed very difficult for a single scalar to make a alhealthy bounce (inflation) scenarios, which needs to go even beyond Horndeski theory [31, 32, 33, 34]. However, the conclusion only applies to single scalar models, and for modified gravity driven bounces whether there is such “nogo” theorem is unknown. In this paper, we will focus on an interesting type of bounce inflation scenario, driven by the f(T) modified gravity theory. The f(T) theory is an extension of the socalled “Teleparallel Equivalent General Relativity (TEGR)”. Although TEGR, as a torsion theory, is equivalent to General Relativity, f(T) is no longer equivalent to the extension to GR, namely f(R) theory, but act as a totally new theory, with many interesting properties not shared by GR or f(R) theories. For more information on f(T) theory, see reviews [35, 36].
We will perform a detailed investigation of perturbations generated by f(T) modified gravity theory, and apply it into the bounce inflation scenario. Note that the general formulation of perturbations on f(T) has been investigated in, for instance, [37, 38, 39, 40, 41, 42]. We study on what conditions could the perturbations remain healthy passing through the bounce. The rest of our paper is organized as follows: in Sect. 2 we set up with the basic formulation of f(T) theory, as well as the FRW background analysis which can be applied to the bounce inflation scenario. In Sect. 3, we focus on the perturbations (both tensor and scalar) generated from f(T) theory, respectively, and demonstrate the conditions for a healthy bounce inflation scenario. An explicit example of f(T) bounce inflation scenario satisfying all the conditions are given in Sects. 4, and 5 comes the final remarks.
2 f(T) modified gravity and the bounce inflation ansatz
We are focusing on the bounce inflation solution given by f(T) modified gravity theory. The bounce, by definition, is the scenario where the universe goes from contracting phase (\(H<0\)) to expanding phase (\(H>0\)), therefore there must be a pivot point where \(H=0\), \(\dot{H}>0\) is satisfied, which we call the bounce point. However, in absence of the matter part, namely \(\rho _m=p_m=0\), from Eqs. (12) and (13) one can only get a trivial solution of \(f(T)=T/3+\lambda /\sqrt{T}\) with an integral constant \(\lambda \), and \(\dot{H}=0\) forever, so no bounce will happen. This is a wellknown result [24] and that’s why a matter part will be needed. Moreover, in order to solve the inconsistency problem in usual bounce model with single scalar degree of freedom (namely one cannot both solve the anisotropy problem and get the scaleinvariant power spectrum) (last two references in [16, 17, 18, 19, 20]), we explore the bounce inflation model where the contracting phase has a large equation of state, \(w\ge 1\), or, in terms of the slowvarying parameter \(\epsilon \equiv 3(1+w)/2\), \(\epsilon \ge 3\), while in expanding phase usual slowroll conditions for inflation, \(w\simeq 1\), \(\epsilon \simeq 0\), is imposed. Note that other bounce inflation solutions in f(T) theory has been discussed in Ref. [44].
In principle, one can employ the reconstruction method to obtain the functional form of f(T), which gives the bounce inflation solution, as has been done in [24, 45]. However, there will be several conditions coming from perturbations, namely ghostfree and gradient stable conditions for both scalar and tensor perturbations, violating which will make the model pathologic. So before heading to specific models, we will first analyze the perturbation theory of f(T) in a very general form, to find whether these conditions will impose rigid constraints on f(T) models.
3 Perturbations generated from f(T) modified gravity
3.1 \(3+1\) decomposition
3.2 Perturbations generated from bounce inflation: tensor part
3.3 Perturbations generated from bounce inflation: scalar part
3.4 A “nogo theorem”?
 1.

the gravity theory with pure f(T) cannot give rise to a bounce universe;
 2.

bounce can be realized with the help of exotic matter, e.g., a canonical scalar field. However, if the field doesn’t contribute the perturbations, then in order for the perturbations be stable within all scales, at least at bounce point \(\dot{f}_T\) cannot be a continuous function with respect to t.
Lemma. For any function V(t)which satisfies \(V(t)>0\) for \(t>0\), \(V(t)<0\) for \(t<0\), or vice versa, then at \(t=0\) point, V(t)can either be vanishing, or become discontinuous.
This is easy to prove. if V(t) is continuous acrossing \(t=0\) point, and assume \(V(t=0)=V_*\ne 0\), then we always have a small number \(\varepsilon \), such that \(V(t=0+\varepsilon )\approx V_*+V^\prime _*\varepsilon \), and \(V(t=0\varepsilon )\approx V_*V^\prime _*\varepsilon \), where \(V^\prime _*\varepsilon \) is the time derivative of V(t) at \(t=0\). So \(V(t=0+\varepsilon )\cdot V(t=0\varepsilon )=(V_*+V^\prime _*\varepsilon )(V_*V^\prime _*\varepsilon )\approx V_*^2V^{\prime 2}_*\varepsilon ^2\approx V_*^2>0\) for small enough \(\varepsilon \) and regular \(V^\prime _*\varepsilon \). This violates the condition that V(t) changes its sign before and after \(t=0\) point. Proof completed.
Now we prove the second item of the theorem. Since H must change its sign when crossing the bounce point, in order to have \(\alpha _1>0\), i.e., to eliminate the ghost problem, \(\dot{f}_T3Hf_T\) must change its sign when crossing the bounce point, which we set to be \(t=0\). According to the lemma, \(\dot{f}_T3Hf_T\) can either cross 0 or become discontinuous. If \(\dot{f}_T3Hf_T\) crosses 0 at \(t=0\), it means that \(3af_T^2/(\dot{f}_T3Hf_T)\) gets divergent when t approaches to zero, unless\(f_T\) also goes to zero to compensate the divergence. In that case, as \(3af_T^2/(\dot{f}_T3Hf_T)\) blows up, its time derivative will be at least positive, and \(f_T\) is also positive considering the stability of tensor perturbations, which gives rise to \(\alpha _2<0\), leading to a gradient instability.
The only loophole is to have \(f_T\) also goes to zero at \(t=0\), as mentioned before. However, it is also impossible. Since \(f_T\) is constrained to be positive either before or after the bounce, \(f_T\rightarrow 0\) means that \(f_T\) decreases (\(\dot{f}_T<0\)) before the bounce, while increases (\(\dot{f}_T>0\)) after the bounce, contradicting with the requirement of \(\alpha _3>0\). Therefore, the only way to have all the three \(\alpha \)’s be positive all the time is to have \(\dot{f}_T3Hf_T\) discontinuous, at least at \(t=0\).
Actually, the implication of discontinuous function in cosmology solutions is not rare at all in the literature. For example, in Refs. [51, 52, 53] people explore interesting observational effects brought by steplike functions in inflation model buildings. In next section, we will give an example of a bounce inflation scenario, which is modeled by f(T) theory with discontinuous \(\dot{f}_T\) at the bounce point.
4 An example
Figure 1 shows the time evolution of the parameters \(\alpha _1\), \(\alpha _2\), \(\alpha _3\) as well as \(f_T\) [derived from Eq. (74)] which describes the stabilities of tensor and scalar perturbations. One can see that although \(f_T\) appears continuous, it forms a sharp peak around the bounce point, demonstrating a discontinuity of its further derivative, \(\dot{f}_T\) (In our numerical calculation, \(\dot{f}_T(0_)\approx 0.13M_p^3\) while \(\dot{f}_T(0_+)\approx 4.39\times 10^{4}M_p^3\)). \(\alpha _i\)’s may also be discontinuous as they contain \(\dot{f}_T\), nonetheless, all the parameters remain positive, leading to a totally stable bounce inflation solution. For parameter choices, we choose \(a_1=10^{100}\), \(a_2=\root 1/3 \of {3}/10\), \(t_1=10^{3}/3M_p^{1}\), \(t_2=0.1M_p^{1}\), \(p_1=1/3\), \(p_2=100\), \(\rho _{m1}=\root 5/2 \of {3}\times 10^{5}M_p^4\), \(\rho _{m2}=1M_p^4\), \(\lambda _1=\lambda _2=0\), and \(r_1=r_2=5/4\), which means \(w_{m1}=3/2\) for contracting phase with \(w_1=1\), and \(w_{m2}\approx 1\) for expanding phase with \(w_2\approx 1\). These choices can ensure the continuity of a(t), T(t), \(\rho _m(t)\) and f(T), however, as all the degrees of freedom are thus used up, one has to abandon the continuities of further derivatives, namely \(\dot{f}_T\).
5 Conclusions
In this paper, we investigated the properties of perturbations generated from f(T) modified gravity theory applied to bounce inflation scenario. We calculated the perturbation action of f(T) theory plus a scalar field, and found conditions for obtaining a stable bounce inflation solution. We found it is actually very difficult to satisfy all the conditions, and one way out is to give up the continuity of derivative function of f(T), say, \(\dot{f}_T\). An example of such a solution is also presented.
As a member of the big modified gravity family, f(T) theory has many interesting features that are deserved further investigation. For example, as it breaks the local Lorentz symmetry [55], an interesting idea is to extend the current study to a more general torsion theory which restored the symmetry. One example of such a torsion theory, namely the Cartan theory, has been explored in Ref. [57]. Moreover, since we only consider perturbations from the gravity part. If the matter part also takes the role, the perturbation analysis will be more complicated since isotropic modes of perturbation also involved in. A higher order perturbations (NonGaussianities) of the system might also be interesting especially for future observational data. Another smokinggun of the work in this paper is that we have chosen the gauge \(\beta =0\) so that the tetrad (17) is made diagonal. However, since one has freedom to choose other gauges, which may make the tetrad nondiagonal, whether the conclusion will be affected is still unknown. We leave all these topics for upcoming works.
Footnotes
 1.
We thank the anonymous referee for pointing this to us.
Notes
Acknowledgements
We thank YiFu Cai, Jun Chen, Wenjie Hou, Keisuke Izumi, Ze Luan, Jiaming Shi, Taishi Katsuragawa, Emmanuel N. Saridakis and YunLong Zheng for useful discussions. This work is supported by NSFC under Grant nos. 11405069 and 11653002.
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