Exploring Bouncing Cosmologies with Cosmological Surveys

Yi-Fu Cai1, ∗ 1Department of Physics, McGill University, Montr´eal,QC, H3A 2T8, Canada In light of the recent observational data coming from the sky we have two significant directions in the field of theoretical cosmology recently. First, we are now able to make use of present observations, such as the Planck and BICEP2 data, to examine theoretical predictions from the standard inflationary ΛCDM which were made decades of years ago. Second, we can search for new cosmological signatures as a way to explore physics beyond the standard cosmic paradigm. In particular, a subset of early universe models admit a nonsingular bouncing solution that attempts to address the issue of the big bang singularity. These models have achieved a series of considerable developments in recent years, in particular in their perturbative frameworks, which made brand-new predictions of cosmological signatures that could be visible in current and forthcoming observations. In this article we present two representative paradigms of very early universe physics. The first is the so-called new matter (or matter-ekpyrotic) bounce scenario in which the universe starts with a matter-dominated contraction phase and transitions into an ekpyrotic phase. In the setting of this paradigm, we propose some possible mechanisms of generating a red tilt for primordial curvature perturbations and confront its general predictions with recent cosmological observations. The second is the matter-bounce inflation scenario which can be viewed as an extension of inflationary cosmology with a matter contraction before inflation. We present a class of possible model constructions and review its implications on the current CMB experiments. Last, we review the significant achievements of these paradigms beyond the inflationary ΛCDM model, which we expect to shed new light on the direction of observational cosmology in the next decade years.

PACS numbers: 98.80.Cq, 98.80.Es

I. INTRODUCTION power spectrum of primordial curvature perturbations has been confirmed to high precision by the observations of the CMB temperature anisotropies in recent years [2]. The first evidence for the primordial B-mode polariza- Inflation also predicted a nearly scale-invariant power tion of the cosmic microwave background (CMB) has very spectrum of primordial tensor perturbations [9], which likely been detected by the Background Imaging of Cos- can seed the B-mode polarization as detected by the BI- mic Extragalactic Polarization (BICEP2) telescope at the CEP2 experiment. Provided that all polarization signals South Pole in March of 2014 [1]. Its statistical significance were contributed by inflationary gravitational waves, one is claimed to be above the 5σ confidence level (C.L.). The can apply the recent data to constrain inflation models. simplest and also the most reasonable theoretical inter- pretation of these signals is the primordial gravitational Although a major achievement has been made, one wave which was arisen from the ripples in the spacetime ought to be aware of the fact that inflationary cosmology in the very early universe. Obviously, this important is not the unique paradigm describing the very early uni- measurement, together with another CMB observation – verse. Moreover, general inflation models have difficulty the Planck data released in 2013 [2], have brought major in explaining the Planck and BICEP2 data simultane- information to the field of very early universe physics and ously. Additionally, we will point out in the main text theoretical physics in general. that there exist several conceptual challenges that can- The recent developments in observational cosmology not be resolved in the context of inflationary cosmology. arXiv:1405.1369v2 [hep-th] 18 Jun 2014 has a series of significant implications on inflation models. Having the recent experimental developments and theo- In this picture, cosmologists believe that when the uni- retical challenges in mind, we are motivated to search for verse was very young, namely about 10−35 second after new proposals for the theory of the very early universe the big bang, it has undergone a period of extremely beyond the paradigm of inflationary ΛCDM cosmology. rapid expansion, known as ”inflation”, when its volume In the literature there are indeed several alternative sce- increased by a factor of up to 1080 in less than 10−32 narios to inflation that describe the very early moment second. This paradigm, as proposed in the early 1980s to of the universe. Namely, the “pre big bang” scenario understand the initial condition issue in the hot big bang [10] suggests that the universe may start from an initial cosmology [3–5] (see also [6–8] for early works), has so far state characterized by a very small curvature, based on become the most prevailing in describing the very early the scale-factor duality [11] of string cosmology. More- universe. Its critical prediction of a nearly scale-invariant over, the “ekpyrotic/cyclic universe” configuration [12] suggests that the periodic collision of two membranes in a high-dimensional spacetime could give rise to a cyclic solution. One another interesting paradigm of very early ∗Electronic address: [email protected] universe is the so-called “emergent universe” cosmology 2

[13], in which our universe emerges from a non-zero mini- scenario of bounce inflation was earlier proposed in [29] mal length scale and experienced a sufficiently long period where the universe is assumed to evolve from a phase of of quasi-Minkowski expansion to then begin the normal fast-roll contraction to the inflationary era with a nonsin- big bang expansion. This scenario was originally achieved gular bounce in between, and hence, the power spectrum in the “string gas” cosmology in terms of a Hagedorn of primordial curvature perturbation becomes strongly phase of a thermal system composed of a number of fun- blue at extremely large scales [30]. Alternatively, if one damental strings [14]. Recently it was also realized by a makes an analogue with the evolution of our universe, it cosmic fermion condensate model [15, 16]. In addition, seems more natural to consider the universe to be domi- from the perspective of phenomenological considerations, nated by regular matter fields in the contracting phase. there is the “matter bounce” scenario [17, 18] that gives As a result, the scenario of matter-bounce inflation was rise to almost scale invariant power spectra of primordial implemented by a double-field quintom model in [21]. perturbations and thus can fit to observations very well. Within this scenario, the power spectra of primordial cur- We refer to [19] for a comprehensive review of various vature perturbations are almost scale-invariant at both proposals of very early universe models. large and small length scales while some local features Amongst all of these competitive configurations, we appear due to the phase transition [31]. A representative focus in this article on two phenomenological scenarios al- phenomenon is that the amplitude of the power spectrum ternative to the inflationary ΛCDM model with one being of primordial curvature perturbations could experience the healthy nonsingular bouncing cosmology [20] and the a step feature at a critical scale, which may be applied other the matter-bounce inflation scenario [21]. The idea to explain the suppression of the low ` CMB multipoles of the bouncing cosmology appeared even much earlier as observed by the WMAP and Planck experiments [32]. than inflationary cosmology, namely, one can trace back Moreover, the tensor spectrum of this scenario can have a to Tolman’s work in the 1930s [22]. A modern version of step feature as well and thus, it allows for more freedom bouncing cosmology suggests that the expansion of the in the parameter space to fit to the CMB observations. In universe is preceded by an initial phase of contraction and particular, this scenario was found to be very powerful in then a non-vanishing bouncing point happened to connect reconciling the tension between the Planck and BICEP2 the contraction and the expansion [23]. It was recently observations when compared with the ΛCDM [33]. Ad- pointed out in Ref. [24] that if such a bounce model ditionally, from a top-down viewpoint this scenario can describes a realistic universe within General Relativity, easily be embedded into some fundamental theories, for then motivated by the study of dark energy physics [25] instance, the loop quantum cosmology [34]. a matter field with a so-called quintom scenario is needed The article is organized as follows. In Section II, we that the equation-of-state parameter has to evolve below briefly list the main predictions made by the standard the cosmological constant boundary w = −1 twice. This inflationary cosmology. While these results are roughly type of bounce model was extensively studied in the liter- consistent with the present observations, we intriguingly ature in recent years (for example see [26] for a related point out that experimental implications as well as con- review of Quintom cosmology and references therein for ceptual issues may hint to physics beyond this paradigm. extended analyses). However, it is not easy to construct Thus, we introduce in Sections III and IV a class of very a model that can realize a nonsingular bouncing solution early universe models which are promising in addressing without theoretical pathologies since the quintom scenario issues existing in inflationary cosmology. In particular, is associated with the violation of the null energy condi- we present two representative paradigms. In Section III tion (NEC) which is accompanied with various quantum we study in detail the new matter bounce scenario in instabilities [27]. Furthermore, a general challenge for which the universe starts with a matter-dominated con- bouncing cosmologies is to ensure that their contracting traction phase and evolves into an ekpyrotic phase and phases are stable against the instability to the growth finally bounces into a thermal expansion. We review the of some unexpected anisotropic stress, whose associated predictions of this model on primordial perturbations and −6 energy density grows as a . This is recognized as the confront them with the latest cosmological observations. famous Belinsky-Khalatnikov-Lifshitz (BKL) instability Afterwards, we point out that the simplest model suffers issue of any cosmological models involving a contract- from the issue of the non-red tilt spectrum, and around ing phase [28]. After years of unremitting efforts, it was this point, we propose possible mechanisms of generating recently proposed in Ref. [20] that an effective field de- a reasonable red tilt for primordial curvature perturba- scription that combined the benefits of matter bounce and tions in this model. The second is the matter-bounce ekpyrotic scenarios can give rise to a nonsingular bouncing inflation scenario as will be investigated in Section IV. cosmology without pathologies through a Galileon-like This paradigm can be viewed as an extension of the reg- Lagrangian as well as explaining the formation of the ular inflationary cosmology with a matter contraction large scale structure (LSS) in our universe. before inflation. In this section we study its phenomeno- The matter-bounce inflation scenario is another promis- logical implications on the present cosmological observa- ing paradigm of the very early universe in which the infla- tions. Particularly, we show explicitly that the models tionary cosmology is generalized by introducing a matter- within this scenario generally have significant advantages like contracting phase before the inflationary phase. The in explaining simultaneously the Planck and BICEP2 ob- 3 servations better than the standard ΛCDM model, and where kpivot is the pivot scale and As is the spectrum are sensitive to be verified by the future CMB polarization amplitude of curvature perturbations. The subscript I measurements. Then, we briefly introduce a class of model denotes that the value of the Hubble parameter is taken constructions that can realize this scenario from effective during the inflationary phase. field approach. Eventually, we conclude our work by ad- The slow roll parameter  is defined by dressing some unsettled issues in Section V. Throughout H˙ the article we take the normalization of natural√ units and  ≡ − 2 , (3) define the reduced Planck mass by Mp = 1/ 8πG. H and hence is determined only by the background dynamics of the inflaton field. In order to realize a sufficiently II. GENERAL DISCUSSION ON INFLATIONARY COSMOLOGY long inflationary phase,  is required to be much less than unity. The sound speed parameter cs characterizes the propagating rate of the inflaton field fluctuations. We begin with a brief discussion on general predictions Theoretically, its value is constrained between 0 and 1 of inflationary perturbations within the framework of Gen- so that the model is free from the gradient instability eral Relativity. Inflationary cosmology has become the and super-luminal propagation. Moreover, the recent most prevailing paradigm for describing the very early no-detection of primordial non-gaussianity by the Planck universe based on two reasons. Firstly, it can resolve data [44] strongly indicates that c cannot be very small. some conceptual issues of the hot big bang cosmology. s The spectral index nS can be derived straightforwardly Secondly, it was the first model to provide a causal mech- from its definition through anism for generating primordial curvature perturbations which can explain the formation of the LSS [35, 36] and to d ln P n − 1 ≡ S = −4 + 2η − s , (4) interpret the existence of primordial gravitational waves S d ln k in the CMB [9, 37]. More specifically, the perturbation theory developed in this paradigm predicted nearly scale- with two other slow roll parameters invariant power spectra of primordial perturbations both ˙ c˙s of scalar and tensor type which are highly Gaussian and η ≡  − , s ≡ , (5) 2H Hcs adiabatic with a slightly red tilt. Most of these predic- tions have been accurately confirmed in a number of being introduced. According to the current CMB ob- cosmological observations in the past decades of years servation, the spectral index ns takes a value which is [1, 2, 38–40]. slightly less than unity and hence, the power spectrum of The inflationary perturbation theory was comprehen- primordial curvature perturbations is red tilted. sively reviewed in [41]. In this picture, both the primor- For primordial tensor fluctuations, the associated re- dial density perturbations and gravitational waves were lations are even simpler if the gravity theory is General originated from quantum fluctuations of the spacetime Relativity. The corresponding power spectrum takes the metric in a nearly exponentially expanding universe at form of high energy scales. During this exponential expansion,  k nT the physical wavelengthes of metric fluctuations can be PT = AT , (6) stretched out of the Hubble radius and then form classical kpivot perturbations as observed by the CMB surveys. Such a with convenient causal mechanism can spectacularly determine 2 the power spectra of primordial perturbations of scalar 2HI AT = 2 2 , (7) and tensor types by a series of simple relations. π Mp Specifically, we would like to start by considering a single field inflation model with a K-essence Lagrangian where the coefficient AT is the amplitude of tensor spec- [42, 43]. The corresponding power spectrum of primordial trum at the pivot scale. It is easy to see that the amplitude curvature perturbations is associated with four inflation- of primordial tensor fluctuations only depends on the infla- tionary Hubble parameter and the corresponding spectral ary parameters, the Hubble rate H, the spectral index ns, index n , the latter of which, by definition, is given by the slow roll parameter , and the sound speed cs (cs = 1 T when the Lagrangian is canonical), via the following form d ln PT of parametrization n ≡ = −2 . (8) T d ln k n −1  k  S The expressions (6) and (8) are generic to any inflation PS = AS , (1) kpivot models which minimally couple to General Relativity. Furthermore, it is convenient to define the tensor-to-scalar with ratio as follows, 2 HI AT AS = 2 2 , (2) r ≡ = 16cs , (9) 8π csMp AS 4 which also characterizes the magnitude of primordial ten- III. TOWARD THE BIG BANG IN BOUNCING sor fluctuations. COSMOLOGY: THE PARADIGM OF MATTER-EKPYROTIC BOUNCE Based on these relations, one can make use of observa- tional data to constrain the parameters introduced above and then further discriminate inflation models. Namely, Non-singular bouncing cosmologies can resolve the ini- according to the Planck observation of the CMB, the tial singularity problem of the inflationary ΛCDM model amplitude of the power spectrum of primordial curvature and hence have attracted a lot of attention in the litera- perturbations is constrained to be [2] ture. They appear in many theoretical settings where ei- ther the gravitational sector is modified as in Ho˘rava grav- ity [56–58], non-relativistic gravitational action[59], tor- 10 +0.024 ln(10 AS) = 3.089−0.027 (1σ CL) , (10) sion gravity [60, 61], Lagrange-multiplier gravity [62, 63], nonlinear massive gravity [64] and non-local gravity [65], or by making use of matter fields with the NEC viola- at the pivot scale k = 0.002 Mpc−1. Moreover, the re- tion such as in the quintom bounce [24, 66], the Lee-Wick cently released BICEP2 result requires that [1] bounce [67, 68], the ghost condensate bounce [69–71], and the S-brane bounce [72] models. A non-singular bounce may also be achieved in a universe with non-flat spatial +0.07 r = 0.20−0.05 (1σ CL) . (11) geometry (see e.g. [73, 74]). We refer to [75, 76] for re- cent reviews of various bouncing cosmologies. However, the BKL instability appears in contracting cosmologies However, from the historical perspective, it was known as the effective energy density contributed by the back- even before inflation that a roughly scale-invariant and reaction of anisotropies increases faster than the dust and almost adiabatic spectrum of cosmological perturbations radiation densities, unless one finely tunes the initial con- could be a reasonable interpretation for the distribution ditions to be nearly perfectly isotropic in order to ensure of galaxies historically [45, 46], which is now known as that anisotropies never dominate. However, this issue is the famous “Harrison-Zel’dovich” (HZ) power spectrum avoided in the ekpyrotic scenario where a scalar field with [47, 48]. Moreover, despite the phenomenological accom- a steep and negative-valued potential always dominates plishment, there exist some critical conceptual issues of over anisotropies in a contracting universe [77] and so it inflationary cosmology that deserve to be treated seriously is justified to neglect anisotropies in the presence of an nowadays. ekpyrotic scalar field. Recently, it has been shown how it is possible to com- For one thing, inflationary cosmology does not resolve bine an era of ekpyrotic contraction with a non-singular the problem of the initial singularity inherited from the bounce by introducing a scalar field with a Horndeski- hot big bang cosmology [49]. On the other hand, it is well type non-standard kinetic term and a negative exponential known that the Planck mass suppressed corrections to the potential [20]. Furthermore, one may include a regular inflaton potential generally lead to the masses of the order dust field and assume the universe began in a state of of the Hubble scale and then spoil the slow roll conditions matter-dominated contraction thus combining the matter rendering a sustained inflationary stage impossible [50]. bounce with the ekpyrotic scenario. It can be explicitly This issue would be even worse if the field variation of the checked that anisotropies remain small throughout the en- inflaton is super-Planckian as indicated by the BICEP2 tire cosmological evolution of the matter-ekpyrotic bounce [51]. Moreover, if we trace backward along the cosmo- model, including at the bounce point [78], and therefore logical perturbations observed today, their length scales this model successfully avoids the BKL instability that could go beyond the Planck length at the onset of infla- arises for a large family of non-singular bounce models, tion [52, 53]. Additionally, in order to study quantum as pointed out in [79, 80]. Among many possible imple- field theory during inflation, it is inevitably necessary mentations of the matter bounce, a concrete realization to systematically study the nonlinear corrections of field of the matter-ekpyrotic bounce was constructed in [81] fluctuations that are on one side not ultraviolet (UV) that involves two matter fields with one being the scalar complete, and on the other side yield observably large field that causes the bounce and the other representing infrared effects that were not detected in experiments the matter field that is dominant at the beginning of the [54, 55]. contracting phase. Note that this effective field theory Keeping the above theoretical remarks and the previous model of a non-singular bounce can be developed into phenomenological motivations in mind, it is worth looking a supersymmetric version [82], or embedded into loop for plausible alternative paradigm that might not only be quantum cosmology (LQC) [83] (also see the LQC real- as successful as inflation in phenomenologically explaining izations of matter bounce in [84], ekpyrotic bounce in the CMB and LSS of our universe, but also can resolve [85], and matter-bounce inflation in [148] separately). In or at least circumvent some conceptual issues mentioned the present section, we shall study the matter-ekpyrotic above. In the following sections, we will present two bounce within the context of effective field approach and interesting paradigms. its cosmological implications to observations. 5

A. The model building with two fields 1. The phase of matter contraction

We briefly review how it is possible to obtain a non- Since the second scalar χ is viewed as a regular matter singular bounce and avoid the BKL instability in the field, we may simply take the Lagrangian to be that of a new matter bounce cosmology. We phenomenologically free canonically normalized massive scalar field: consider a background model involving two scalar fields 1 µ 1 2 2 with a Lagrangian of the following type [81]: Lψ = ∂µχ∂ χ − m χ . (17) 2 2 Since χ oscillates around its vacuum state χ = 0, the L = K(φ, X) + G(φ, X)φ + Lχ , (12) time-averaged background equation-of-state parameter is where φ is the bounce field and χ is a regular matter w = 0. At the beginning of the cosmic evolution, if the field. Specifically we choose the operators K and G of universe is dominated by χ, then the scale factor evolves the bounce field to be as  2/3 2 t − t˜E βX a(t) ' aE , (18) K = [1 − g(φ)]X + 4 − V (φ) , (13) tE − t˜E Mp γX where tE denotes the final moment of matter contraction G = 3 , (14) and the beginning of the Ekpyrotic phase, and aE is the Mp value of the scale factor at the time tE. In the above, t˜E is an integration constant which is introduced to match with X being defined as the regular kinetic term X ≡ µν the Hubble parameter continuously at the time tE, i.e., g (∂µφ)(∂ν φ)/2, while β and γ are coupling constants 2 µν t˜E ' tE − . In this phase the time-averaged Hubble and ≡ g ∇ ∇ is the standard d’Alembertian opera- 3HE  µ ν parameter can be approximated by tor. Note that in this model the K operator involves the 2 2 hH(t)i = . (19) term βX which can stabilize the kinetic energy of the 3(t − t˜E) scalar field at high energy scales when β is positive-definite. where the angular brackets stand for averaging over time. A bouncing phase can be triggered by allowing g to be- come larger than unity for a short while which leads to the emergence of a ghost condensate. The value of g is 2. The phase of Ekpyrotic contraction required to be small far away from the bouncing phase so that the kinetic term in the Lagrangian is well ap- We assume a homogeneous scalar field φ which is ini- proximated by the standard kinetic term before and after tially placed in the region φ  −M in the phase of the bounce. Additionally, the potential V (φ) governs the p matter contraction. In this case, the Lagrangian for φ ap- dynamics of φ away from the bounce as well as determines proaches the conventional canonical form. Once φ begins the energy scale the bounce occurs at. In order to dilute to dominate the universe, the equation-of-state parameter the unwanted anisotropies to avoid the BKL instability, then approaches an Ekpyrotic one one takes the potential to have the ekpyrotic form of 2 a negative exponential (at least for φ  −Mp). To be w ' −1 + . (20) specific, following [20, 81] we take the function g(φ) and 3q the potential V (φ) to be During the phase of Ekpyrotic contraction, the scale factor evolves as 2g0 q g(φ) = q q , (15)  ˜  − 2 φ b 2 φ t − tB− e pg Mp + e g pg Mp a(t) ' aB− , (21) tB− − t˜B− 2V0 q V (φ) = − q q , (16) ˜ 2 φ 2 φ where tB− = tB− − H is chosen such that the Hubble − q M bV q M B− e p + e p parameter at the end of the phase of Ekpyrotic contraction matches with the one at the beginning of the bounce where pg, q, bg, bV and g0 ≡ g(0) are dimensionless posi- phase, and aB− is the value of scale factor at the time tive constants and V0 is also positive with dimensions of 4 tB−. Therefore, the Hubble parameter is given by (mass) . We choose g0 to be slightly larger than unity q so that the scalar can form a ghost condensate state H(t) ' . (22) when φ is near zero. The critical value of g that signals t − t˜B− the onset of the non-singular bouncing phase is therefore Additionally, we require the scale factor to evolve g(φ ) = 1. By solving this equation, one finds the approx- ∗ smoothly and continuously at the time tE. This leads to imate values of φ∗ where the non-singular bouncing phase the relation begins and ends, respectively φ∗− ' −Mp ln(2g0/pg) and  q HB− φ∗+ ' Mp ln(2g0/bgpg). Moreover, in order to obtain aE ' aB− . (23) ekpyrotic contraction, we take q < 1. HE 6

3. The nonsingular bouncing phase

1 . 5 x 1 0 - 4 In this model the scalar field evolves monotonically 1 . 5 x 1 0 - 4 from φ  −Mp to φ  Mp. When φ evolves into the 1 . 0 x 1 0 - 4 1 . 0 x 1 0 - 4 regime between φ∗− and φ∗+, the coefficient g(φ) becomes 5 . 0 x 1 0 - 5 0 . 0 larger than unity and thus the universe enters a ghost 5 . 0 x 1 0 - 5 - 5 . 0 x 1 0 - 5 condensate state. As shown in Ref. [20], during this - 1 . 0 x 1 0 - 4 0 . 0 - 4

H - 1 . 5 x 1 0 phase, the Hubble parameter evolves approximately as - 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 8 - 5 . 0 x 1 0 - 5 H(t) ' Υt , (24) - 1 . 0 x 1 0 - 4 as well as the background scalar - 1 . 5 x 1 0 - 4 2 2 ˙ ˙ −t /T 3 3 0 3 3 φ(t) ' φBe , (25) - 2 x 1 0 - 1 x 1 0 1 x 1 0 2 x 1 0 t where the coefficient Υ is set by the detailed microphysics of the bounce. The coefficient T can be determined by matching the detailed evolution of the scalar field at the beginning or the end of the bounce phase. Thus, during 6 the bounce the scale factor evolves as 4

2 2 a(t) ' a eΥt /2 , (26) B 0

- 2 6 which is exactly a nonsingular bouncing solution. w 4 - 4 2 0

- 2 - 6 4. The phase of fast-roll expansion - 4 - 8 - 6 - 8

- 1 0 - 1 0 - 4 - 2 0 2 4 After the bounce, the universe enters the expanding phase, where the universe is still dominated by the scalar - 1 0 0 - 5 0 0 5 0 1 0 0 field φ. During this stage, the motion of φ is dominated t by its kinetic term while the potential is negligible. Thus, the background equation-of-state parameter is w ' 1. FIG. 1: Numerical plot of the Hubble parameter H and the This corresponds to a period of fast-roll expansion, where equation-of-state parameter w as a function of cosmic time the scale factor evolves as in the new matter bounce cosmology. The insert shows the  1/3 detailed evolution around the bounce point. Planck units are t − t˜B+ used. From [20]. a(t) ' aB+ , (27) tB+ − t˜B+ where tB+ represents the end of the bounce phase and the beginning of the fast-roll period, and aB+ is the value of the scale factor at that moment. Then one can write down the Hubble parameter in the fast-roll phase One can see from the upper panel of Fig. 1 that the Hub- 1 H(t) ' , (28) ble parameter H evolves smoothly through the bounce 3(t − t˜B+) point with a dependence on cosmic time which is close to linear. In the lower panel of the figure, one can find that and the continuity of the Hubble parameter at tB+ yields w in the contracting phase can be larger than unity which 1 t˜B+ = tB+ − . By virtue of this phase, it is then 3HB+ corresponds to an ekpyrotic phase. If one traces back- natural to connect to the thermal expanding history of ward to an earlier time, then there would be a matter-like the big bang cosmology. contracting phase which is not presented in the plot. In addition, after the bounce, the equation-of-state parame- ter quickly approaches the value w = 1 which corresponds 5. Numerics to the kinetic-driven phase of expansion. During this pe- riod the contribution of the bounce field φ will be diluted In Fig. 1, we numerically plot the evolution of the quickly relative to the contributions of regular matter and Hubble parameter and the equation-of-state in the new radiation. Thus, the universe in this model can connect matter bounce paradigm, respectively. Also shown are smoothly to the thermal expansion of the hot big bang zoomed-in views of the evolution around the bounce point. cosmology. 7

B. Observational constraints tB− and tB+, respectively. Up to leading order, the power spectrum of curvature fluctuations is given by It is known that, both in inflation or matter bounce F 2H2 a2 cosmology, the amplitudes of scalar and tensor fluctua- ζ E E 2 2 2 Pζ ' 2 4 γζ m |Uζ | , (31) tions are originally comparable. For inflation models, the 8π Mp scalar perturbations can achieve an enhancement due to the slow roll parameter. For matter bounce cosmologies, with γζ ' γψ and the tensor-to-scalar ratio is in general predicted to be H 27q of order unity since the slow roll condition is no longer U = −(25 + 49q)i E − , ζ 24m 24 required in this scenario. Thus, at first glance, this issue 3 √ t 2 2 4  t  2 B+  2(2+3ΥT +Υ T ) B+ is disturbing for a large class of matter bounce cosmolo- 2 2+ΥT + √ 3 T 3 2+ΥT 2 T gies [67]. However, it also implies a theoretical upper Fζ ' e . (32) bound for the tensor-to-scalar ratio in the matter bounce Combining Eqs. (29) and (31), one can write down the paradigm. If there is any mechanism that can amplify tensor-to-scalar ratio in this model as follows, scalar perturbations, then one can always easily suppress this ratio to fit to observations. For example, in some 2 2 PT FψMp explicit models this value can be suppressed due to the r ≡ ' 2 . (33) P (k) nontrivial physics of the bouncing phase, namely, the ζ 2 2 2 2 2(2q − 3) Fζ aEm Uζ matter bounce curvaton [87] and the new matter bounce cosmology [81]. In Ref. [88], the observational constraints on the pa- In the new matter bounce model introduced in the rameter space of the new matter bounce scenario have previous subsection, we have introduced two scalar fields been investigated numerically. It is interesting to notice with χ driving a phase of matter contraction and φ being that, in this model there are only three main parameters responsible for the ekpyrotic contraction and the bounce. that are most sensitive to observations, which are, the Correspondingly, there exist primordial curvature per- slope parameter Υ, the Hubble rate at the beginning of turbations and entropy fluctuations during the matter- the ekpyrotic phase HE, and the dimensionless duration dominated contracting phase. The fluctuations seeded parameter tB+/T of the bouncing phase. by φ, which are the entropy modes originally, can be The correlation between Υ and tB+/T is shown in Fig. converted into curvature perturbations when the universe 2. From the figure, one can find that Υ and tB+/T are enters the ekpyrotic phase. Therefore, the power spec- slightly negatively correlated. This implies that one ex- trum of cosmological perturbations in this model is highly pects either a slow bounce with a long duration or a adiabatic after the bounce. In this model, when the uni- fast bounce with a short duration. In addition, the con- verse evolves into the bouncing phase, the kinetic term of straint on the dimensionless duration parameter tB+/T the scalar field that triggers the bounce could vary rapidly. is very tight with a value slightly less than 2. Thus, it This process also effectively leads to a tachyonic-like mass is important to examine whether the model predictions for curvature perturbations, and hence, the amplitude accommodate with observations after choosing a fixed can be amplified. Correspondingly, the tensor-to-scalar value of tB+/T . ratio is suppressed when primordial perturbations pass Afterwards, we analyze the correlation between Υ and through the bouncing phase in the new matter bounce HE by setting tB+/T = 1.86. The allowed parameter cosmology. space is depicted by the intersection of the blue and red The observational constraint of this scenario was per- bands shown in Fig. 3. From the numerical result, we formed in detail in [88]. To be specific, the power spectrum find that the Hubble scale HE and the slope parameter for primordial gravitational waves is expressed as Υ introduced in the new matter bounce cosmology are constrained to be in the following ranges 2 2 2 FψγψHE PT ' , (29) −8 −6 2 2 2 1.9 × 10 HE/Mp 1.9 × 10 , (34) 16π (2q − 3) Mp . . −7 2 −7 4.9 × 10 . Υ/Mp . 8.5 × 10 , (35) with respectively. It is easy to see that, for the new matter 1 γψ ' , bounce cosmology, if we assume that the bounce occurs at 2(1 − 3q) the GUT scale, then the duration of the bouncing phase  √  4 2 3/2 3 is roughly O(10 ) Planck times. Also, according to the Fψ ' exp 2 ΥtB+ + Υ tB+ , (30) 3 constraint the amplitude of the Hubble scale HE has to be lower than the GUT scale. This allows for a sufficiently at leading order. In the above expression, we have applied long ekpyrotic contracting phase that can suppress the the assumption that the values of the scale factor before unwanted primordial anisotropies as addressed in [78]. and after the bounce are comparable. We denote the time From the analysis of the new matter bounce cosmology, at the beginning and the end of the bouncing phase by we can conclude that, a nonsingular bouncing cosmology 8

7 7 10− observations. These interesting results encourage further × studies of bouncing cosmologies following the effective field approach.

7 6 10− × C. Mechanisms for generating a red tilt for 2 p primordial perturbations /M

Υ 7 5 10− × In the paradigm of the new matter bounce cosmology, a free scalar field can yield a scale-invariant power spectrum. For example, in the model considered above, the bounce field φ gives rise to a scale-invariant power spectrum for 7 entropy perturbations, which during the ekpyrotic phase 4 10− × 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 are able to be converted into adiabatic modes. However, it tB+/T is interesting to note that this spectrum is too close to the HZ spectrum rather than a red tilt spectrum as observed FIG. 2: Constraints on the dimensionless duration parameter by WMAP and Planck. Therefore, it is important to look tB+/T and the slope parameter Υ of the new matter bounce for suitable mechanisms capable of generating a red tilt for cosmology from the combined Planck and BICEP2 data. The primordial perturbations in matter bounce cosmologies. value of the Hubble parameter at the beginning of the ekpyrotic −7 Before going to the detailed mechanisms, we would phase is fixed to be HE /Mp = 10 . The blue bands show the like to briefly explain how the perturbations form a scale- 1σ and 2σ confidence intervals of the tensor-to-scalar ratio and invariant power spectrum in the very early universe. A the red bands show the confidence intervals of the amplitude of the power spectrum of curvature perturbations. From [88]. generic analysis on the conditions for generating scale- invariant primordial spectra in various cosmological evo- lutions was performed in [89]. Based on that work, we 6 10− study in this subsection the corresponding cosmological 7 implications. 9 10− × To be specific, we investigate the possible ways of gener- 7 8 10− ating a power spectrum of primordial perturbations with × a modified dispersion relation. To seed the fluctuations, 7 10 7 2 − p × a test scalar field ϕ is introduced which does not affect the background evolution in primordial stages, but yields /M 7 6 10− Υ × a new degree of freedom for scalar perturbations δϕ. In Fourier space, the equation of motion for its fluctuation 7 5 10− δϕ is given by × 00 00 2 a vk + (ν − )vk = 0 , (36) 7 a 4 10− 7 6 × 10− 10− where the variable vk is the Mukhanov-Sasaki variable HE/Mp defined by vk ≡ aδϕk [90, 91], and the prime denotes the derivative with respect to the comoving time η ≡ R dt/a. FIG. 3: Constraints on the Hubble parameter HE and the For a constant equation-of-state w, there is slope parameter Υ of the new matter bounce cosmology from the combined Planck and BICEP2 data. The dimensionless a00 p(2p − 1) 1 = , (37) bounce time duration is fixed to be tB+/T = 1.86. As in Fig. a (1 − p)2 η2 2, the blue and red bands show the confidence intervals of r and Pζ , respectively. From [88]. where p = 2/3(1 + w) has been introduced. Moreover, ν(k) is introduced as a generalized frequency for the per- turbation mode. Since the dispersion relation may be has to experience the bouncing phase smoothly enough to modified at high energy scales, and if this happens, one be consistent with the latest CMB observations. For exam- could phenomenologically consider a generalized disper- ple, the Hubble parameter cannot grow very fast during sion relation the bounce since the constraints that we find favor a small value of Υ. In the model we considered, the observed am- ν = k f(kph) , (38) plitude of the spectra of CMB fluctuations is determined with by the Hubble parameter at the moment when primordial  k perturbations were frozen at super-Hubble scales, i.e., H ph α E  ( M ) , kph > M at the onset of the ekpyrotic phase, which is required to f(kph) = (39) be much lower than the GUT scale in order to agree with  1 , kph ≤ M 9 where kph = k/a is a physical wave number and α is • (iii) α = 2 in the first case, which can be obtained a positive-definite parameter. The energy scale M is in the model of Hoˇrva-Lifshitz cosmology[56, 57]; assumed to be associated with new physics beyond which 4+α the standard dispersion relation is modified. • (iv) p = 6(1+α) in the second case. From Eq. (36), it is obvious that only two terms would Among these four conditions, the last one can be applied affect the solution, which are ν2 and a00/a, respectively. a00 to bouncing cosmology, in particular, to the matter bounce Firstly, let us neglect the term a and focus on the asymp- scenario. In the following we will study the spectrum totic solution under the limit |νη|  1. This asymptotic of primordial perturbations in this case. Specifically, solution oscillates like a sine function. This feature co- 0 2 when the universe is in a matter-like contracting phase, incides with an adiabatic condition |ν /ν |  1, which one can fix the background equation-of-state parameter corresponds to the case in which the effective physical as w = 0 and hence get p = 2/3. In this case, the wavelength is deep inside the Hubble radius. Therefore, contribution of a00/a is also fixed. In this type of models, the modes can be regarded as adiabatic when they are cosmological perturbations with a standard dispersion staying in the sub-Hubble regime with |νη|  1, and we relation were analyzed in detail as shown in Refs. [31, 67], may impose a suitable initial condition by virtue of a and primordial non-Gaussianities were studied by Refs. Wentzel-Kramers-Brillouin (WKB) approximation [94, 95]. It turns out that the power spectrum of the IR modes is given by, i 1 −i R η ν(k,η˜)dη˜ vk ' p e . (40) 2ν(k, η)  2 IR HB Pδφ = , (44) a00 4π Secondly, we look at the effective mass term a closely. Note that the generation of primordial fluctu- where H is associated with the maximal Hubble scale ations strongly depends on the background evolution. In B around the bounce point. suitable environments, the variable |νη| could decrease along with the cosmic evolution. Once |νη|  1, the modes would exit the Hubble radius, and the equation of 1. Varying ν2 motion yields another asymptotic solution of which the leading term takes the form of Now we deal with the UV modes of which the comoving   1 − 1 | 1−3p | wave numbers achieve a modification on their dispersion l 2 2 1−p(1+α) vk ∼ η c(k)η . (41) relation. For these modes, we can solve the perturbation equation and obtain the following solution: Afterwards, we match the above two asymptotic solu- 1 1+α tions (40) and (41) at the moment of Hubble crossing UV − 1−2α vk ' √ (νη) . (45) |νη| ∼ 1 and then determine the form of vk at super- 2ν Hubble scales as follows, From this solution, one can read that the amplitude of 1 − 1 | 1−3p | 1   2 2 1−p(1+α) the super-Hubble modes keeps growing until the bounce vk(η) ' √ ν(k, η)η . (42) takes place, and thus we can calculate the power spectrum 2ν around the bounce point, which is given by Applying the solution (42) directly, we obtain the primor- 9α 2(1+α)  − 1−2α dial spectrum as follows: 1 3α k UV 1−2α 1−2α Pδφ = 2(2−α) HB M . (46) 3 1−2α 2 aB k vk(η) 2 2 π Pδφ = | | 2π2 a The corresponding spectral index can be derived as fol-  2−α 1−p−αp lows:  k , (1 − 3p)[1 − p(1 + α)] ≥ 0 ∼ (43) d ln P 9α  4+α−6p(1+α) n ≡ 1 + δφ = 1 − , (47)  k 1−p−αp , (1 − 3p)[1 − p(1 + α)] < 0 . φ d ln k 1 − 2α Finally, we obtain the power spectrum of primordial which is red-tilted in the UV regime. In Fig. 4 we perturbations without considering any specific models. numerically plot the spectral index of the primordial per- From the result (43), we find that there are four sufficient turbations described by the above mechanism. One may conditions for the spectrum to be scale-invariant (i.e., notice that, in order to fit the cosmological observations, 0 Pδφ ∼ k ). They are: e.g. the Planck data [2], the model parameter α cannot • (i) p → +∞ in the first case, which corresponds to deviate from zero too much, which provides a strong con- inflationary cosmology [3]; straint on models of modified gravity. However, this result successfully illustrates that a slightly red spectrum may • (ii) p → −∞ in the first case, which can be achieved be obtained by making a very small modification to the in island cosmology[92, 93]; dispersion relation in a matter bounce model. 10

1.0 1.0

IR regime

0.8

0.8

0.6

0.6

n

n 0.4

0.4

0.2

0.2

0.0

0.00 0.02 0.04 0.06 0.08 0.10

0.0

IR UV 0.67 0.68 0.69 0.70 0.71

FIG. 4: The spectral index of the primordial perturbations p with a modified dispersion relation as a function of the param- FIG. 5: The spectral index of the primordial perturbations eter α in the UV regime. From [89]. with a standard dispersion relation as a function of the pa- rameter p in the IR regime.

2. Varying a00/a D. A brief summary After having analyzed the possible effect brought by the effective frequency, now we turn to the study of how the 00 So far we have reviewed the new matter bounce effective mass term a /a affects the spectral index of the paradigm from both the theoretical setup and observa- scalar spectrum. For convenience, we fix the frequency tional implications. As an alternative to inflationary term by taking the regular dispersion relation ν = k, and cosmology, this paradigm is competitively successful in hence, α = 0. In this case, we solve the perturbation explaining the CMB observations. Moreover, this scenario equation and obtain the super-Hubble solution: has the following advantages that cannot be satisfied by inflation models. First of all, in the new matter bounce 1 1−2p IR 1−p vk ' √ (kη) . (48) paradigm the big bang singularity is replaced by a non- 2k singular bouncing phase and thus the initial singularity problem is resolved. Secondly, since the power spectrum of Substituting this solution into the expression of the scalar cosmological perturbation was generated in the contract- spectrum (43), one can derive ing phase during which the physical wavelengthes of these

4−6p observable modes do not change dramatically, there is no 1 p 2−4p 4p−2  k  1−p Trans-Planckian problem for generic nonsingular bouncing IR 1−p 1−p Pδφ = 2 ( ) HB , (49) 4π 1 − p aB cosmologies if the bounce scale is below the Planck scale. This profound property of cosmological perturbations is around the bounce point. It is easy to write down the also in agreement with the theoretical starting point that spectral index as one can describe the nonsingular bouncing cosmologies by the effective field approach without modifying Einstein’s d ln P 4 − 6p General Relativity. n ≡ 1 + δφ = 1 + . (50) φ d ln k 1 − p Moreover, as shown in previous subsections, the (new) matter bounce paradigm usually predicts a large ampli- correspondingly. From this result, we can easily find tude of primordial gravitational waves and hence the that a red tilt can also be obtained if we choose p to be corresponding theoretical upper bound of the tensor-to- slightly larger than 2/3. In Fig. 5 we numerically plot scalar ratio is of order unity. Accordingly, to apply the the spectral index of the primordial perturbations under recent CMB primordial polarization measurement, we the effect of the a00/a term. From the figure, one can see are able to constrain the parameter space of this type of that the spectrum can also be red tilted when p is above models. Our analysis indicates that a reasonable matter- 2/3. This implies that, when the universe experiences the ekpyrotic bounce is expected to occur at an energy scale contracting phase, one expects the background equation- slightly less than the GUT scale and the duration of the of-state parameter to be slightly less than w = 0 so that bouncing phase is about O(104) Planck times. the primordial perturbations can form a power spectrum At last, we discussed two possible mechanisms for gen- with a red tilt as observed by current CMB experiments. erating a red tilt for the power spectrum of primordial It is interesting to study what specific mechanisms can perturbations in matter bounce cosmologies. One is to implement such a scenario in the future research. consider the UV modifications to the dispersion rela- 11 tion of primordial perturbation modes, and the other Hubble radius at some primordial era, and then exit the is to deform the background evolution by requiring the Hubble radius to become classical perturbations and even- equation-of-state parameter to be slightly less than zero. tually re-enter at late times. This mechanism is robust Both mechanisms can successfully lead to a red tilted to both the curvature perturbations and the relic gravita- power spectrum. The difference is that the first one is tional waves. The dynamics of cosmological perturbations only associated with the UV modes while the latter is are convenient to be analyzed by tracking a Fourier mode generally associated with the IR modes. Although we vk along with the background evolution. In the context of have pointed out the possibility of realizing a red tilt General Relativity, if the background equation-of-state pa- in matter bounce cosmologies, it would be interesting rameter is constant, then one can write down the equation to study the explicit realizations from theoretical model of motion for the Fourier mode as follows, buildings. Furthermore, as has been noticed in [87], there a00 could exist different approaches for realizing a red tilted v00 + (k2 − )v = 0 , (51) power spectrum if one takes into account the interac- k a k tion terms among the matter fields in primordial epochs. where a is the scale factor of the universe and the prime These interactions are also associated with the reheating denotes the derivative with respect to the comoving time. process of a universe in bouncing cosmologies [96]. These Specifically, the scale factor often evolves as a(t) = topics are expected to inspire more studies in the future. 1/ aB(t/tB) , where the subscript “B” denotes any refer- ence time which in our case can be referred as the bounce point. Note that the parameter  is defined by Eq. (3) IV. TOWARD THE BIG BANG IN in Section II. Its physical meaning is associated with the INFLATIONARY COSMOLOGY: THE dynamics of background universe. For example, if the MATTER-BOUNCE INFLATION SCENARIO universe is in the inflationary stage as discussed in Sec- tion II,  corresponds to the slow roll parameter; however, In the previous section we studied the paradigm of the if the universe is in some other phase under a constant new matter bounce cosmology which can explain the CMB equation-of-state parameter w, then one has observations without inflation. However, there exists another interesting scenario that can solve the initial 3  = (1 + w) . (52) singularity problem within the context of inflationary 2 cosmology, which suggests that the inflationary period is preceded by a nonsingular bounce and before that the Particularly,  = 3/2 for a dust matter dominated universe universe has experienced a sufficiently long contracting with w = 0. Then, to transform into the comoving time, phase with a pressureless dust matter being dominant. It one gets is interesting to notice that the study of this paradigm  1/(−1) is of observational interest as will be explained in the η a(η) = aB , (53) following subsection. Therefore, in this section, we would ηB like to first review the latest observational constraint upon this scenario, and then go to the discussion of the model and hence, the comoving Hubble parameter is given by building. a0 1 H ≡ = . (54) a ( − 1)η

A. Observational constraints and cosmological From the above expression, one can immediately learn implications that, for inflation with   1 there is |H| ' |1/η|. More- over, for a dust matter dominated universe with  = 3/2, One interesting prediction of the matter-bounce infla- then |H| ' |2/η|. Additionally, there is a very useful tion paradigm is that the amplitudes of the power spectra relation for the effective mass term given by, of primordial perturbations both of scalar and tensor types may undergo a jump feature at a critical scale. It a00 (µ2 − 1/4) ( − 3) = , with µ = ± , (55) was first pointed out in [32], and recently generalized a η2 (2 − 2) in [33], that a series of parameterizations of primordial power spectra can be applied to confront with the latest which can be derived from Eq. (37). cosmological data. In particular, we can perform a global Following the method developed in [89] and applied fit and utilize the corresponding step feature to derive a in Sec. III C, we study the asymptotic solutions to the novel method for constraining the bounce parameters. perturbation equation in the sub-Planckian and super- To begin with, we briefly discuss the generation of pri- Planckian regimes, respectively. First, we assume that mordial perturbations in the framework of a flat FRW all primordial perturbations originate from vacuum fluc- Universe. The standard causal process suggests that, tuations inside the Hubble radius and hence, the WKB√ i R η these observed large scale perturbations should initial approximation suggests that vk ' exp (−i kdη˜)/ 2k, emerge from very tiny spacetime fluctuations inside a when |kη|  1. This is in agreement with the asymptotic 12

00 solution of Eq. (51) when the last term a /a is negligible. nS. Moreover, in order to determine the amplitude of Second, another asymptotic solution to Eq. (51) can be primordial tensor modes, the standard process is to intro- 1/2 −|µ| obtained at super-Hubble scales as vk ∼ η [c(k)η ], duce the tensor-to-scalar ratio r ≡ AT /AS as introduced in the limit of |kη|  1. By matching these two solutions in (9). around the moment of Hubble crossing |kη| ∼ 1, we can Similar to this analysis, we would like to define a bounce- obtain the super-Planckian perturbation mode as to-inflation ratio of the power spectrum as

1 1 −|µ| Pm vk(η) ' √ (kη) 2 . (56) rB ≡ , (60) 2k AS The above solution is robust to primordial perturbations so that the amplitude of the scalar spectrum before the of both scalar and tensor types. However, the definitions bounce can be characterized. As we will show later, this of the power spectra of these two modes are different. newly introduced parameter is very convenient to be Specifically, for primordial tensor modes, the power spec- constrained in numerical computation. In addition, from trum is defined as the parameterizations of primordial power spectra (59), 3 one can see that there exist two more parameters, kB and 4k vk 2 P ≡ | | . (57) TB. Physically, kB is associated with the occurrence scale T π2 a of the step feature in the spectra, and TB is determined by Moreover, for the scalar modes, the spectrum is expressed the duration of the bouncing phase and hence can depict as the slope of this jump. As a result, the matter-bounce inflation scenario involves three more parameters and 3 k vk from their physical properties, it is apparent that they P ≡ | |2 , (58) S 2π2 z are highly correlated. For the convenience of numerical √ analyses, we can fix the value of TB as the best fit value with z = a 2 being introduced. derived from the combined Planck, WMAP and BICEP2 Substituting the solution (56) into these spectra, one data, and then perform detailed computations to constrain can find that they could be nearly scale invariant either in the remaining two parameters, kB and rB. the matter-dominated contracting phase or during infla- From the expression in (59), one easily observes that, tionary stage. Accordingly, the power spectrum of primor- m when PT = Pm = 0 and in the limit where kB → 0, dial perturbations is overall roughly scale invariant in the the spectra reduces to the regular result in standard matter-bounce inflation paradigm. However, as has been inflationary paradigm. To test the potentially observable discussed previously, the comoving Hubble parameter signals of the bounce in CMB measurements, we adopt evolves as |H| ' |2/η| during the matter contraction but −1 kpivot = 0.05 Mpc . Then we demonstrate the bounce becomes |H| ' |1/η| during inflation. If we require that effect on the CMB temperature and polarization spectra the background evolution is smooth around the bounce with different models in Figs. 6 and 7. In particular, we point along the comoving time, then the comoving Hub- consider the inflationary ΛCDM model and three matter- ble radius RH ≡ 1/|H| would experience a rapid increase bounce inflation models with different parameter values from |ηB/2| to |ηB|. As a result, for the matter-bounce as depicted in the figures. We can explicitly see that there inflation model, the corresponding amplitude of primor- is a suppression on both spectra at large angular scales. dial perturbations could achieve a jump feature around As it is already known, the BICEP2 data strongly the scale kB comparable to the bounce scale. A much favor a nonzero primordial tensor spectrum, namely r = 2 2 more detailed calculation yields that PT = H /2π when 0.2+0.07 in the intermediate ` regime (50 < ` < 100) 2 2 −0.05 k < kB while PT = 2H /π when k ≥ kB for the model [1]. This significantly large value of r, if it still exists of matter-bounce inflation. in the low ` regime, can obviously lead to extra power Following the method developed in [33], we adopt the in the CMB temperature spectrum and make it difficult phenomenological parameterizations of primordial power to fit to the low ` Planck data. In the matter-bounce spectra as follows, inflation models, however, the amplitudes of both scalar   and tensor spectra are able to be suppressed in the low Pζ,i − Pm  k  PS = Pm + 1 + tanh TB log10 ` regime and therefore greatly relax this observational 2 kB tension between the Planck and BICEP2 data sets. In i m   m PT − PT  k  this regard, we can claim that the matter-bounce inflation PT = PT + 1 + tanh TB log10 (59) 2 kB paradigm can explain the combined CMB observations very well compared to the ΛCDM model. m 2 2 i where we have introduced PT ≡ H /2π and PT ≡ Since the BICEP2 experiment can only measure the 2 2 2H /π . In addition, Pζ,i is the regular power spectrum B-mode power spectrum at scales ` > 30 where the sup- during inflation which takes the form of (1), and Pm is pression effect is small, the median value of the tensor- the scalar spectrum before the bounce which has to be to-scalar ratio r in the matter-bounce inflation paradigm less than Pζ,i. Correspondingly, Pζ,i is characterized by is close to that obtained in the standard ΛCDM model. two parameters, the amplitude AS and the spectral index Moreover, the suppression effect becomes very signifi- 13

10000 string inspired inflation models [97].

LCDM, r=0 Low-l

LCDM, r=0.162 High-l

Bounce, r=0

Bounce, r=0.183 B. A toy model by means of the Horndeski and ] 2 ghost-condensate operators K [

/2

TT Qualitatively speaking, a matter-dominated contracting l universe is very similar to what our universe has experi- enced since the moment of radiation-matter equality until l(l+1)C 1000 dark energy dominates over, except that the sign of their Hubble parameters is opposite. However, it is not trivial to achieve such an extended inflationary model that can 10 100 1000 stably realize a bouncing phase before inflation. One Multipole l ought to be aware of the following theoretical constraints. FIG. 6: Signatures of the bounce effect on the CMB tem- First of all, it is not trivial to build a matter-bounce in- perature power spectra for four early universe models. We flation model that is stable enough against any ghost mode compare the ΛCDM model and the matter-bounce inflation since this scenario involves a violation of NEC which is models with and without using the BICEP2 data. From [33]. required by the bouncing phase. Secondly, one should be aware of potentially dangerous gradient instabilities that might bring an unexpected growth to primordial perturba- 0.1 tions. Any specific model that realizes the matter-bounce LCDM, r=0.162 inflationary scenario must have a canonical Lagrangian Bounce, r=0.183

Bounce, r=0.183 (No Step) with all higher order operators being suppressed in the IR

BICEP2

] regime and therefore, the model is free from the graceful- 2

K exit problem.

0.01 [ To stably violate the weak energy condition is of theo-

/2 BB l retical interest in various models of very early universe physics. One plausible mechanism of achieving such a sce-

l(l+1)C nario is to make use of a ghost condensate field in which the kinetic term for the inflaton takes a non-vanishing

1E-3 expectation value in the infrared regime [98, 99]. However, this type of model often suffers from gradient instabilities

10 100 when the universe exits from the inflationary phase to the Multipole l normal thermal expansion phase. FIG. 7: Signatures of the bounce effect on the CMB B Another approach to realize the weak energy condition polarization power spectra for four early universe models. We violation is to make use of a Galileon type field [100–102]. compare the ΛCDM model and the matter-bounce inflation The key feature of this model is that it contains higher or- models when using the BICEP2 data. From [33]. der derivative terms in the Lagrangian while the equation of motion remains second order, and thus does not neces- sarily lead to the appearance of ghost modes. Recently cant at very large scales ` < 20 which is expected to the Galileon-type field has been applied to interpret the be sensitive to the forthcoming CMB polarization data late time acceleration of the universe in [103]. It was also as will be released soon by the Planck team. Therefore, applied to drive the inflationary phase at early times, for it is very promising to further constrain or even probe instance, see [104, 105], and see [106] for the discussion the observational signals of a nonsingular bounce in near of the stability issue. The Galileon model might realize a future. nonsingular bounce as well, but was found to suffer from In Ref. [33], we also performed a global fit on var- a severe graceful-exit problem from the bouncing phase ious model parameters in this paradigm based on a to the phase of regular thermal expansion [107, 108]. generalized CosmoMC code package. We refer to that Inspired by the success of the model building in the paper for details, and briefly summarize the main re- new matter bounce paradigm [20], we suggest in this sults as follows. According to the global fitting analy- section that the matter-bounce inflation scenario could sis, the numerical constraint on the model parameters be achieved by a single scalar field possessing the desir- from the combined Planck, WMAP and BICEP2 data able features of both the ghost-condensate and Galileon- are found to be the following: the bounce parameter inspired models. In particular, we phenomenologically −1 scale log10(kB/Mpc ) = −2.4 ± 0.2 and the bounce-to- propose a Lagrangian of the following form inflation ratio rB ≡ Pm/As = 0.71 ± 0.09, at 68% C.L., respectively. Note that, the suppression effect of the CMB 1 γ˜(φ) 1 2 2 2 L = X − V (φ) − g˜(φ)Mp (2X) + (2X) φ ,(61) temperature anisotropies may also be achieved in some Mp 14 which is similar to the form introduced in Eq. (12). The strongly indicate that there exists a fundamental minimal main differences are as follows. For one thing, the co- length scale in the universe. Based on this profound quan- efficient of the Horndeski operator is generalized as a tum property, the nonsingular bouncing cosmology was function of φ; for another, we have modified the ghost- proposed as a promising paradigm attempting to avoid the condensate-like operator by introducing a X1/2 term. initial singularity. This cosmological paradigm suggests Note that, in the expression of (61), the first two terms that there was a contracting phase preceding the regular are the canonical kinetic term and the potential for the thermal expansion of the universe, and in the contraction inflaton field and thus are of canonical forms. The third the universe was dominated by a pressureless matter com- term is introduced to realize a stable ghost condensate ponent, the corresponding power spectra of primordial in the simplest way. Note that this term was earlier ap- perturbations can be almost scale-invariant similar to plied to explain the late time acceleration of the universe the inflationary predictions. Accordingly, this so-called and was dubbed as the “cuscuton” term [109, 110]. In matter bounce scenario could explain the CMB observa- particular, when g > 0 the ghost condensate state would tions and the formation of the LSS just as successfully as form and in the opposite case the scalar field remains the inflationary cosmology, but was found to suffer from the canonical form. The Horndeski operator G is introduced BKL instability against the backreaction of primordial to stabilize the propagation of cosmological perturbations anisotropies. when the bouncing phase occurs. Its effect is automati- Along this line, a so-called new matter bounce (or cally suppressed at low energy scales when X becomes matter-ekpyrotic bounce) cosmology was recently pro- small. Consequently, the model under consideration could posed to combine the advantages of the matter bounce satisfy the theoretical limits pointed out at the beginning scenario and the ekpyrotic cosmology which can dilute of this section. The research along this line is particularly unwanted primordial anisotropies. Moreover, this new interesting from a phenomenological perspective and we paradigm can be described within the context of General would like to leave it for future studies. Relativity by using techniques of effective field approach which can efficiently approximate the unknown underlying theory. This approach is widely applied in general topics V. CONCLUSION of physics. By virtue of this approach, we are able to con- struct the models of the new matter bounce paradigm in Since the 1990s, we witnessed the birth of the era of which the universe starts its evolution from a contracting precision cosmology. Starting from the COBE satellite, phase in the far past and evolves without pathologies. In we have been able to measure the CMB map with high Sec. III we studied in detail the specific model building precision by virtue of accumulated observational data of this paradigm by including the ghost condensate and from the space telescopes such as the WMAP and Planck Horndeski operators into the field Lagrangian. Moreover, spacecrafts; meanwhile, the ground observatories like the we have confronted this type of model with the latest SPT and BICEP2 collaborations have made significant cosmological observations and shown that a new mat- developments on signaling the CMB at specific angular ter bounce model with a bounce scale being lower than scales. With these great improvements of cosmological the GUT scale and a sufficiently long bouncing phase is observations, we have arrived at the urgent status of favored by the present data. addressing the theoretical question of how the current Furthermore, in light of the nonsingular bouncing cos- early universe picture can arise from fundamental theories. mology, we discussed a generalized bounce inflationary In particular, the recent announcement of detection of non- cosmology with a matter-dominated contraction preced- vanishing primordial B-mode polarization by the BICEP2 ing an inflationary phase. This scenario has been obtained collaboration has led the interest of cosmologists into the by the effective field approach and also in loop quantum accurate examinations of very early universe models and cosmology. In particular, this paradigm is of significant ob- inspired a number of extensive studies, for instance, see servational interest due to its predictions on the featured [33, 88, 111–134] and references therein. power spectra of both scalar and tensor perturbations. So far the standard inflationary ΛCDM paradigm has Due to this notable phenomenological feature, we ana- made a number of initial achievements in the past decade, lyzed in Sec. IV A the latest observational constraint on in particular addressing theoretical issues of the hot big the parameter space of this paradigm from the combined bang cosmology. Its prediction of nearly scale invariant Planck and BICEP2 data. This feature is very promising primordial power spectra has been gradually verified by to be confirmed (or ruled out) by the Planck polarization several CMB experiments as reviewed in the article. This data which will be released in the near future. Afterwards, scenario, however, suffers from a criticism that at the we attempted to explore the possibility of realizing this moment of the big bang, there is a singularity of zero scenario by a well-behaved theoretical model. Though it volume and infinite energy. The corresponding physics is is yet to be finalized, we can explicitly see that the asso- beyond all knowledge we have today, and thus we expect ciated model building deserves a comprehensive analysis to find a resolution to this problem. in the future. Most of the recent theoretical developments in quantum Concerning the theoretical implications of nonsingular gravity, such as string theory and loop quantum gravity, bouncing cosmologies, an important lesson is to break a 15 series of singularity theorems proven by Penrose [135] and tinguish them from other early universe paradigms Hawking [136] about half a century ago and later devel- needs to be addressed. oped into a cosmological framework by Borde, Vilenkin and Guth [137]. Such an issue is often accompanied with • Lastly, if the universe did experience a bounce, this the violation of NEC if we restrict our study within the may necessitate another bounce in the future. It framework of classical General Relativity. As a result, the may be possible that it is cyclical nature. phenomenologies of a nonsingular bouncing cosmology in the very early universe could be associated with the For a very long time, physicists have been continuously quintom scenario as inspired by the dark energy study of making attempts to address the above questions. How- the late time acceleration as reviewed in [26]. Along this ever, we believe that the final resolutions will eventually line, we expect that there might be a bridge connecting be discovered under the combined efforts of both observa- the physics of very early universe with that at late times. tional and theoretical cosmology. If a nonsingular bounce Eventually, we would like to end this article by com- is verified, the knowledge about our universe will usher menting on some unsettled issues of bouncing cosmologies. in a new era. Although the topic of nonsingular bouncing cosmologies has become exciting and attracted more and more cos- mologists’ attention in recent years (for example, see [138– 147, 149–153] for recent interesting studies), the detailed Acknowledgments paradigm is still far from being specified. Namely, in the present article we have analyzed two plausible bouncing I am indebted to Robert Brandenberger and all col- cosmologies that can satisfy cosmological observations laborators for long-term collaborations in the field of equally well. In the following we present the main ques- theoretical cosmology. I am also grateful to Damien tions, which in our opinion are crucial to the development Easson, Hong Li, Jerome Quintin, Emmanuel Saridakis, of bouncing cosmologies in the future. Edward Wilson-Ewing, Jun-Qing Xia for kind permis- • Firstly, the fundamental theory that gives rise to sions to include figures from previous works. In addition, a nonsingular bounce is still not understood. The I acknowledge to Prof. Yipeng Jing and Prof. Xinmin connection to quantum physics is unresolved. Zhang for the invitation to contribute to the special issue of “Science China: Physics, Mechanics & Astronomy”. • Secondly, the potentially observable signatures of This work is supported in part by NSERC and by the a nonsingular bounce need to be verified. To dis- Department of Physics at McGill.

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