Matter bounce cosmology with a generalized single field: non-Gaussianity and an extended no-go theorem
MMatter bounce cosmology with ageneralized single field:non-Gaussianity and an extendedno-go theorem
Yu-Bin Li, a Jerome Quintin , b Dong-Gang Wang c,d,a and Yi-FuCai a a CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy,University of Science and Technology of China, Chinese Academy of Sciences,Hefei, Anhui 230026, China b Department of Physics, McGill University,3600 rue University, Montréal, QC, H3A 2T8, Canada c Leiden Observatory, Leiden University,2300 RA Leiden, The Netherlands d Lorentz Institute for Theoretical Physics, Leiden University,2333 CA Leiden, The NetherlandsE-mail: [email protected], [email protected],[email protected], [email protected]
Abstract.
We extend the matter bounce scenario to a more general theory in which thebackground dynamics and cosmological perturbations are generated by a k -essence scalarfield with an arbitrary sound speed. When the sound speed is small, the curvature pertur-bation is enhanced, and the tensor-to-scalar ratio, which is excessively large in the originalmodel, can be sufficiently suppressed to be consistent with observational bounds. Then, westudy the primordial three-point correlation function generated during the matter-dominatedcontraction stage and find that it only depends on the sound speed parameter. Similar tothe canonical case, the shape of the bispectrum is mainly dominated by a local form, thoughfor some specific sound speed values a new shape emerges and the scaling behaviour changes.Meanwhile, a small sound speed also results in a large amplitude of non-Gaussianities, whichis disfavored by current observations. As a result, it does not seem possible to suppress thetensor-to-scalar ratio without amplifying the production of non-Gaussianities beyond currentobservational constraints (and vice versa). This suggests an extension of the previously con-jectured no-go theorem in single field nonsingular matter bounce cosmologies, which rulesout a large class of models. However, the non-Gaussianity results remain as a distinguishablesignature of matter bounce cosmology and have the potential to be detected by observationsin the near future. Vanier Canada Graduate Scholar. a r X i v : . [ h e p - t h ] M a r ontents ζ ˙ ζ term 104.2.3 Contribution from the ˙ ζ∂ζ∂χ term 104.2.4 Contribution from the ζ ( ∂ i ∂ j χ ) term 104.2.5 Contribution from the ˙ ζ term 114.2.6 Secondary contributions 114.3 Summary of results 124.3.1 Amplitude 124.3.2 Shape 124.3.3 The squeezed limit 13 λ/ Σ Matter bounce cosmology [1] is a very early universe structure formation scenario alternativeto the paradigm of inflationary cosmology (see, e.g., [2] for a review of inflation, its problemsand its alternatives). The idea is that quantum fluctuations exit the Hubble radius in amatter-dominated contracting phase before the Big Bang, which generates a scale-invariantpower spectrum of curvature perturbations [3, 4]. The contracting phase is then followed bya bounce and the standard phases of hot Big Bang cosmology. This construction solves theusual problems of standard Big Bang cosmology such as the horizon and flatness problems,but in addition, it is free of the trans-Planckian corrections that plague inflationary cosmology[5], and one can naturally avoid reaching a singularity at the time of the Big Bang (contraryto standard inflation [7, 8]) under the assumption that new physics appears at high energyscales [1, 2]. Nonsingular bounces can be constructed in various ways using matter violatingthe Null Energy Condition (NEC), with a modified gravity action, or within a quantum theoryof gravity (see the reviews [1, 9–12] and references therein). The singularity before inflation could be avoided with, for example, bounce inflation (e.g., [6]). – 1 – typical way of constructing a nonsingular matter bounce cosmology is to assume theexistence of a new scalar field. With a canonical Lagrangian, the oscillation of the scalar fieldcan drive a matter-dominated contracting phase when the ratio of the pressure to the energydensity averages zero. As the energy scale of the universe increases, new terms can appear inthe Lagrangian that violate the NEC and drive a nonsingular bounce. For example, using aGalileon scalar field [13] (or equivalently, in Horndeski theory [14]), one can construct a stableNEC violating nonsingular bounce [15–20] that may be free of ghost and gradient instabilities[21, 22] (see, however, the difficulties in doing so as pointed out by [23–26]).To distinguish the matter bounce scenario from inflation observationally, studying pri-mordial non-Gaussianities is a useful tool . In the case of inflation, after the calculation of thebispectra generated in single field slow-roll models [29], there have been many studies in thepast decade trying to extend the simplest result, which largely enriched the phenomenologyof nonlinear perturbations (see [30, 31] for reviews). In particular, one important progresshas been to generalize the canonical inflaton to a k -essence scalar field [32, 33], such as k -inflation [34, 35] and DBI models [36, 37], which are collectively known as general singlefield inflation [38]. In these models, due to the effects of a small sound speed, the amplitudeof the bispectrum is enhanced and interesting shapes emerge [30, 31, 38–41]. In a matter-dominated contracting phase, the calculation of the bispectrum has only been done by [42] forthe original matter bounce model with a canonical scalar field. A natural extension is thus toconsider a k -essence scalar field similarly to what has been done in inflationary cosmology,especially since the appearance of a noncanonical field is quite common in the literature ofnonsingular bouncing cosmology in order to violate the NEC as explained above. Becausethe perturbations behave differently in matter bounce cosmology compared to inflation, inparticular due to the growth of curvature perturbations on super-Hubble scales during thematter-dominated contracting phase, the canonical matter bounce yields non-Gaussianitieswith negative sign and order one amplitude, which differs from the results in canonical singlefield inflation. It would be interesting to explore how these non-Gaussianity results changewhen one generalizes the original matter bounce scenario to be based on a k -essence scalarfield. Besides non-Gaussianity, another interesting observable for very early universe models isthe tensor-to-scalar ratio r . In the original matter bounce scenario, this ratio is predicted to bevery large [49, 50]. Indeed, the scalar and tensor power spectra share the same amplitude, andaccordingly, the tensor-to-scalar ratio is naturally of order unity [51]. This is well beyond thecurrent observational bound from the Cosmic Microwave Background (CMB), which statesthat r < . at confidence [52].A resolution to this problem is to allow for the growth of curvature perturbations duringthe bounce phase, which suppresses the tensor-to-scalar ratio. However, curvature perturba-tions tend to remain constant through the bounce phase on super-Hubble scales [20, 53]. Infact, amplification can only be achieved under some tuning of the parameters, and the overallgrowth is still limited [51]. Yet, if the scalar modes are amplified, another problem follows Another observable quantity, besides non-Gaussianities, that would allow one to differentiate betweeninflation and the matter bounce scenario is the running of the scalar spectral index (see [27, 28]). This could be easily further generalized to a Galileon field [43], which has also been done for inflation(see, e.g., [44–48]). The studies of Refs. [20, 51, 53] have been carried out for models where the nonsingular bounce is attributedto a noncanonical scalar field. Loop quantum cosmology (LQC) provides an alternative class of nonsingularbouncing models that could suppress r during the bounce. In LQC, the amplitude of the suppression dependson the equation of state during the bounce; if it is close to zero, then the suppression is very strong (see – 2 –n that it leads to the production of large non-Gaussianities [51], a problem that might begeneric to a large class of nonsingular bounces [56, 57]. Again, these large non-Gaussianitiesare excluded by current measurements from the CMB [58]. This leads to conjecture thatsingle field matter bounce cosmology suffers from a no-go theorem [51], which states that onecannot satisfy the bound on r without violating the bounds on non-Gaussianities and viceversa.There is another way to suppress the tensor-to-scalar ratio if the sound speed of theperturbations can be smaller than the speed of light during the matter-dominated contractingphase. For example, in the Λ CDM bounce scenario [54] (and its extension [59]; see the review[28]), if there exists a form of dark matter with a small sound speed that dominates thecontracting phase when the scale-invariant power spectra are generated, then the tensor-to-scalar ratio is already suppressed proportionally to the sound speed. Therefore, this providesanother motivation to study non-Gaussianities when the sound speed is small during thematter-dominated contracting phase. An immediate question is whether the no-go theoremstill holds true in this case or whether it can be circumvented. In this work, we want toexplore this possibility of having a k -essence scalar field that would mimic dust-like matterwith a small sound speed at low energies and that could play the role of the NEC violatingscalar field during the bounce.In this paper, we will evaluate the bispectrum produced by a k -essence scalar field ina matter-dominated contracting universe. This more general setup will yield richer features,which have the potential to be detected by future non-Gaussianity observations. In particular,the shapes, amplitudes, and scaling behaviors will be studied systematically. We will showthat a small sound speed implies a large amplitude associated with the three-point function.Accordingly, we will claim that the no-go theorem is not circumvented but rather extended:in single field matter bounce cosmology, one cannot suppress the tensor-to-scalar ratio, eitherfrom the onset of the initial conditions in the matter contracting phase or from the ampli-fication of the curvature perturbations during the bouncing phase, without producing largenon-Gaussianities.The outline of the paper is as follows. In section 2, we first introduce the backgrounddynamics of the matter bounce scenario and introduce the class of k -essence scalar fieldmodels that we study in this paper. In section 3, we calculate the power spectra of curvatureperturbations and tensor modes and show how a small sound speed coming from the k -essence scalar field allows for the suppression of the tensor-to-scalar ratio. We then considerthe primordial non-Gaussianity in section 4. Using the in-in formalism, we evaluate everycontribution to the three-point function and give a detailed analysis of the size and shapesof the resulting bispectrum. In section 5, we compute the amplitude parameter of non-Gaussianities in different limits and finally combine these results with the bound on thesound speed from section 3 to show that the no-go theorem in matter bounce cosmology isextended. We summarize our results in section 6. Throughout this paper, we use the mostlyplus metric convention, and we define the reduced Planck mass to be M Pl = (8 πG N ) − / ,where G N is Newton’s gravitational constant. The idea of the matter bounce scenario is to begin with a matter-dominated contractingphase. At the background level, this corresponds to having a scale factor as a function of [50, 54, 55] and references therein for a discussion of LQC effects in nonsingular bouncing cosmology). – 3 –hysical time given by a ( t ) = a B (cid:18) t − ˜ t B t B − ˜ t B (cid:19) / , (2.1)and the Hubble parameter follows, H ( t ) = 23( t − ˜ t B ) , (2.2)where t B corresponds to the time of the beginning of the bounce phase and ˜ t B correspondsto the time at which the singularity would occur if no new physics appeared at high energyscales. Accordingly, a B is the value of the scale factor at t B . In terms of the conformal time τ defined by d τ = a − d t , the scale factor is given by a ( τ ) = a B (cid:18) τ − ˜ τ B τ B − ˜ τ B (cid:19) , (2.3)where τ B and ˜ τ B are the conformal times corresponding to t B and ˜ t B . Throughout the restof this paper, the scale factor is normalized such that a B = 1 .One can define the usual “slow-roll” parameters of inflation by (cid:15) ≡ − ˙ HH = 32 (1 + w ) , η ≡ ˙ (cid:15)H(cid:15) , (2.4)where a dot denotes a derivative with respect to physical time, and w ≡ p/ρ is the equation ofstate parameter with p and ρ denoting pressure and energy density, respectively. In the caseof the matter bounce, the matter contracting phase implies that pressure vanishes, which isto say that w = 0 , (cid:15) = 32 , η = 0 . (2.5)If the pressure does not vanish exactly but is still very small, i.e. | p/ρ | (cid:28) , then the valuesfor w , (cid:15) , and η in equation (2.5) are only valid as leading order approximations, and theywill be time dependent rather than constant. In this paper, we will work in the limit whereequation (2.5) is valid.In the usual matter bounce scenario, one would introduce a canonical scalar field todrive the matter-dominated contracting phase and describe the cosmological fluctuations. Inthis paper, we aim for more generality and assume that the perturbations are introduced bya k -essence scale field φ with Lagrangian density of the form L φ = P ( X, φ ) , (2.6)where X ≡ − ∂ µ φ∂ µ φ/ , and we assume that the scalar field is minimally coupled to gravity.The energy density and pressure of this scalar field are then given by ρ = 2 XP ,X − P , p = P , (2.7)where a comma denotes a partial derivative, e.g. P ,X ≡ ∂P/∂X . Thus, the Friedmannequations read M H = 2 XP ,X − P , M ˙ H = − XP ,X . (2.8) For an introduction to such a Lagrangian in early universe cosmology with the derivation of the backgroundequations of motion and the definition of the different parameters, see, e.g., [34, 35, 38, 39]. – 4 –ince we want a matter-dominated contracting phase, the pressure of the scalar field shouldvanish (at least in average), and ρ = 2 XP ,X ∝ a − .It is helpful to have one specific example where a k -essence field drives the mattercontraction. Let us consider the following Lagrangian density: L φ = K ( X ) = 18 ( X − c ) . (2.9)This type of Lagrangian belongs to a subclass of k -essence models P ( X, φ ) where the kineticterms K ( X ) are separate from the potential terms V ( φ ) , i.e. P ( X, φ ) = K ( X ) − V ( φ ) .Moreover, the above Lagrangian has vanishing potential. Then, the ghost condensate solutionis given by X = c and φ ( t ) = ct + π ( t ) , with ˙ π ( t ) (cid:28) c . In this case, the background equationsyield p (cid:39) and ρ ∼ ˙ π ∝ a − , which exactly corresponds to a matter-dominated universe.More details about this model can be found in [60]. We note that there should be also otherforms of P ( X, φ ) that can drive a matter contraction, and remarkably, the analysis thatfollows in this paper is done in a model-independent way and does not rely on the specificmodel of equation (2.9).The sound speed and another “slow-roll” parameter are defined by c ≡ ∂p∂ρ = P ,X P ,X + 2 XP ,XX , s ≡ ˙ c s c s H . (2.10)Calculations will be done for a general sound speed, but as we will argue, we will be interestedin the small sound speed limit, which can be realized with the appropriate form for P ( X, φ ) .For instance, the explicit example given by equation (2.9) yields c s (cid:39) ˙ π/c (cid:28) . Furthermore,we will generally assume later that the sound speed remains nearly constant, which is to saythat | s | (cid:28) . We also define two other variables for later convenience, Σ ≡ XP ,X + 2 X P ,XX = M H (cid:15)c , (2.11)and λ ≡ X P ,XX + 23 X P ,XXX = X ,X −
13 Σ . (2.12)The ratio λ/ Σ will be of particular interest in the following sections. For inflation, it dependson the specific realization of the general single field, such as DBI and k -inflation models. Forthe matter bounce scenario, it can be obtained in an approximately model-independent way.The detailed calculation is in Appendix A, where we find that the ratio λ/ Σ can be expressedin terms of the sound speed, as shown by equation (A.20). We begin with an action of the form S = (cid:90) d x √− g (cid:18) M R + L φ (cid:19) , (3.1)where g is the determinant of the metric tensor and R is the Ricci scalar. Importantly, weassume that the matter Lagrangian L φ has the general form of equation (2.6), but we do We assume that the cosmological perturbations will remain adiabatic throughout the matter-dominatedcontracting phase. – 5 –ot restrict our attention to any specific model. By perturbing up to second order the aboveaction, one finds S (2) = (cid:90) d τ d (cid:126)x z (cid:104) ζ (cid:48) − c ( (cid:126) ∇ ζ ) (cid:105) , (3.2)where ζ ( τ, (cid:126)x ) denotes the curvature perturbation in the comoving gauge, i.e. on slices wherefluctuations of the scalar field vanish ( δφ = 0 ). Also, a prime represents a derivative withrespect to conformal time, (cid:126) ∇ = ∂ i is the spatial gradient, and we define z ≡ (cid:15)a M /c .Transforming to Fourier space, the second-order perturbed action becomes S (2) = (cid:90) d τ (cid:90) d (cid:126)k (2 π ) z (cid:104) ζ (cid:48) ( (cid:126)k ) ζ (cid:48) ( − (cid:126)k ) − c k ζ ( (cid:126)k ) ζ ( − (cid:126)k ) (cid:105) , (3.3)where k ≡ (cid:126)k · (cid:126)k = | (cid:126)k | . Upon quantization of the curvature perturbation, one has ˆ ζ ( τ, (cid:126)k ) = ˆ a † (cid:126)k u k ( τ ) + ˆ a − (cid:126)k u ∗ k ( τ ) , (3.4)where the annihilation and creation operators satisfy the usual commutation relation [ˆ a (cid:126)k , ˆ a † (cid:126)k (cid:48) ] =(2 π ) δ (3) ( (cid:126)k − (cid:126)k (cid:48) ) . The equation of motion of the mode function is then given by v (cid:48)(cid:48) k + (cid:18) c k − z (cid:48)(cid:48) z (cid:19) v k = 0 , (3.5)where the mode function is rescaled as v k = zu k ( v k is called the Mukhanov-Sasaki variable).Together with the commutation relation [ˆ ζ ( (cid:126)k ) , ˆ ζ (cid:48) ( (cid:126)k )] = (2 π ) δ (3) ( (cid:126)k + (cid:126)k ) , one finds (see,e.g., [42]) u k ( τ ) = iA [1 − ic s k ( τ − ˜ τ B )]2 √ (cid:15)c s k ( τ − ˜ τ B ) e ic s k ( τ − ˜ τ B ) (3.6) u (cid:48) k ( τ ) = iA √ (cid:15)c s k (cid:18) − − ic s k ( τ − ˜ τ B )]( τ − ˜ τ B ) + c k ( τ − ˜ τ B ) (cid:19) e ic s k ( τ − ˜ τ B ) (3.7)to be the solution to the equation of motion (3.5) in the context of a matter-dominatedcontracting universe as described in the previous section. Here, A is a normalization con-stant that is determined by the quantum vacuum condition at Hubble radius crossing in thecontracting phase, which is given by A = ( τ B − ˜ τ B ) /M Pl .The general two-point correlation functions are given by (cid:104) ˆ ζ ( τ , (cid:126)k )ˆ ζ ( τ , (cid:126)k ) (cid:105) = (2 π ) δ ( (cid:126)k + (cid:126)k ) u ∗ k ( τ ) u k ( τ ) , (3.8) (cid:104) ˆ ζ ( τ , (cid:126)k )ˆ ζ (cid:48) ( τ , (cid:126)k ) (cid:105) = (2 π ) δ ( (cid:126)k + (cid:126)k ) u ∗ k ( τ ) u (cid:48) k ( τ ) , (3.9)and in particular, the power spectrum, evaluated at the bounce point τ B (well after Hubbleradius exit), is given by (cid:104) ˆ ζ ( τ B , (cid:126)k )ˆ ζ ( τ B , (cid:126)k (cid:48) ) (cid:105) = (2 π ) δ (3) ( (cid:126)k + (cid:126)k (cid:48) ) 2 π k P ζ ( τ B , k ) , (3.10) Again, see, e.g., [30, 35, 38, 39] for a derivation of the perturbation equations in k -essence early universecosmology. – 6 –here P ζ ( τ B , k ) = A π (cid:15)c s ( τ B − ˜ τ B ) = 112 π c s M ( τ B − ˜ τ B ) . (3.11)The scale invariance of the power spectrum in matter bounce cosmology is thus explicit fromthe above.The above focused only on the scalar perturbations, but as mentioned in the introduc-tion, the matter bounce scenario also generates a scale-invariant power spectrum of tensorperturbations. Considering the transverse and traceless perturbations to the spatial metric, δg ij = a h ij , which can be decomposed as h ij ( τ, (cid:126)x ) = h + ( τ, (cid:126)x ) e + ij + h × ( τ, (cid:126)x ) e × ij (3.12)with two fixed polarization tensors e + ij and e × ij , the second-order perturbed action has contri-butions of the form S (2) ⊃ M (cid:90) d τ d (cid:126)x a (cid:104) h (cid:48) − ( (cid:126) ∇ h ) (cid:105) (3.13)for each polarization state h + and h × . By normalizing each state as µ = aM Pl h/ , thesecond-order perturbed action is of canonical form ( µ is the Mukhanov-Sasaki variable), andthe resulting equation of motion for each state is µ (cid:48)(cid:48) k + (cid:18) k − a (cid:48)(cid:48) a (cid:19) µ k = 0 , (3.14)where the equation is written in Fourier space. Since a ∼ τ in a matter-dominated contract-ing phase, one has a (cid:48)(cid:48) /a = 2 /τ , and so, one expects a scale-invariant power spectrum just asin de Sitter space. The tensor power spectrum is given by P t = 2 P h = 2 (cid:18) aM Pl (cid:19) k π | µ k | , (3.15)where the first factor of 2 accounts for the two polarizations + and × , and the factor [2 / ( aM Pl )] comes from the normalization of µ . Upon matching with quantum vacuum ini-tial conditions at Hubble radius crossing similar to the above treatment for scalar modes, onefinds the power spectrum of tensor modes at the bounce point to be given by P t ( τ B , k ) = 2 π M ( τ B − ˜ τ B ) , (3.16)which is indeed independent of scale.The tensor-to-scalar ratio is then defined to be r ≡ P t P ζ . (3.17)It follows from equations (3.11) and (3.16) that r = 24 c s (3.18)in the context of matter bounce cosmology with a general k -essence scalar field . On onehand, this highlights the problem of standard matter bounce cosmology, which is driven by a Of course, this assumes that the perturbations remain constant on super-Hubble scales after the mattercontraction phase, in particular through the bounce and until the beginning of the radiation-dominatedexpanding phase of standard Big Bang cosmology. – 7 –anonical scalar field with c s = 1 , in which case r = 24 . On the other hand, the above resultprovides a natural mechanism to suppress the tensor-to-scalar ratio provided the k -essencescalar field has an appropriately small sound speed. For example, satisfying the observationalbound [52] r < . at confidence imposes a bound on the sound speed of the order of c s (cid:46) . . (3.19) The previous section showed that a k -essence scalar field could yield a small tensor-to-scalarratio in the context of the matter bounce scenario. This is done at the expense of having asmall sound speed. In what follows, the goal is to compute the bispectrum and see how asmall sound speed affects the results. To evaluate the three-point correlation function, we must expand the action (3.1) up to thirdorder. Let us recall the result of [38], the third-order interaction action of a general singlescalar field , S (3) = (cid:90) d t d (cid:126)x (cid:110) − a (cid:104) Σ (cid:16) − c (cid:17) + 2 λ (cid:105) ˙ ζ H + a (cid:15)c ( (cid:15) − c ) ζ ˙ ζ + a(cid:15)c ( (cid:15) − s + 1 − c ) ζ ( ∂ζ ) − a (cid:15)c ˙ ζ ( ∂ζ )( ∂χ ) + a (cid:15) c dd t (cid:18) ηc (cid:19) ζ ˙ ζ + (cid:15) a ( ∂ζ )( ∂χ ) ∂ χ + (cid:15) a ( ∂ ζ )( ∂χ ) + 2 f ( ζ ) δLδζ (cid:12)(cid:12)(cid:12)(cid:12) (cid:111) , (4.1)where it is understood that ( ∂ζ ) = ∂ i ζ∂ i ζ , ( ∂ζ )( ∂χ ) = ∂ i ζ∂ i χ , ∂ ζ = ∂ i ∂ i ζ , and where wedefine χ such that ∂ χ = a (cid:15) ˙ ζ . Also, we have δLδζ (cid:12)(cid:12)(cid:12)(cid:12) = a (cid:18) d ∂ χ d t + H∂ χ − (cid:15)∂ ζ (cid:19) , (4.2) f ( ζ ) = η c ζ + 1 c H ζ ˙ ζ + 14 a H {− ( ∂ζ )( ∂ζ ) + ∂ − [ ∂ i ∂ j ( ∂ i ζ∂ j ζ )] } + 12 a H { ( ∂ζ )( ∂χ ) − ∂ − [ ∂ i ∂ j ( ∂ i ζ∂ j χ )] } , (4.3)where ∂ − is the inverse Laplacian.The first and second terms in the last line of equation (4.1) can be reexpressed as (cid:15) a ( ∂ζ )( ∂χ ) ∂ χ + (cid:15) a ( ∂ ζ )( ∂χ ) = − a (cid:15) ζ ˙ ζ + (cid:15) a ζ ( ∂ i ∂ j χ )( ∂ i ∂ j χ ) + K , (4.4)where the boundary term is given by K = ∂ i (cid:20) ζ ( ∂ i χ )( ∂ χ ) + 12 ( ∂ i ζ )( ∂χ ) − ζ ( ∂ i ∂ j χ )( ∂ j χ ) (cid:21) . (4.5) From here on, we take M Pl = 1 for convenience. – 8 –ince the ∂ i [ ... ] term above does not contribute to the three-point function, the third-orderaction, equation (4.1), is equivalent to S (3) = (cid:90) d t d (cid:126)x (cid:110) − a (cid:104) Σ (cid:16) − c (cid:17) + 2 λ (cid:105) ˙ ζ H + a (cid:15)c ( (cid:15) − c ) ζ ˙ ζ + a(cid:15)c ( (cid:15) − s + 1 − c ) ζ ( ∂ζ ) − a (cid:15)c ˙ ζ ( ∂ζ )( ∂χ ) + a (cid:15) c dd t (cid:18) ηc (cid:19) ζ ˙ ζ − a (cid:15) ζ ˙ ζ + (cid:15) a ζ ( ∂ i ∂ j χ )( ∂ i ∂ j χ ) + 2 f ( ζ ) δLδζ (cid:12)(cid:12)(cid:12)(cid:12) (cid:111) . (4.6)In the case of a canonical field with c s = 1 , this action returns to equation (15) of [42].Meanwhile, as usual the last term in this action is removed by performing the field redefinition ζ → ˜ ζ + f (˜ ζ ) , (4.7)where ˜ ζ denotes the field after redefinition. In this section, we calculate the three-point correlation function using the in-in formalism (toleading order in perturbation theory; see, e.g., [29–31] for the methodology), (cid:104) O ( t ) (cid:105) = − (cid:90) t −∞ d¯ t (cid:104) | O ( t ) L int (¯ t ) | (cid:105) , (4.8)where O represents a set of operators of the form ˆ ζ in our case of interest. Then, the shapefunction, A , is defined such that (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) = (2 π ) δ (3) (cid:16) (cid:88) i (cid:126)k i (cid:17) P ζ (cid:81) i k i A ( (cid:126)k , (cid:126)k , (cid:126)k ) . (4.9)In what follows, we list all the contributions to the shape function coming from the fieldredefinition and the interaction action (4.6). It is easy to check that, when taking the limit c s = 1 , one recovers the results of [42] for the matter bounce with a canonical scalar field asexpected. In momentum space, the field redefinition can be written as ζ (cid:126)k → ˜ ζ (cid:126)k + (cid:90) d (cid:126)k (2 π ) (cid:34) − c − (cid:15) (cid:32) (cid:126)k · ( (cid:126)k − (cid:126)k ) k − ( (cid:126)k · (cid:126)k )[ (cid:126)k · ( (cid:126)k − (cid:126)k )] k k (cid:33)(cid:35) ˜ ζ (cid:126)k ˜ ζ (cid:126)k − (cid:126)k . (4.10)This redefinition has the following contribution to the three-point correlation function, (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) redef = (cid:90) d (cid:126)k (cid:48) (2 π ) (cid:34) − c − (cid:15) (cid:32) (cid:126)k (cid:48) · ( (cid:126)k − (cid:126)k (cid:48) ) k (cid:48) − ( (cid:126)k · (cid:126)k (cid:48) )( (cid:126)k · [ (cid:126)k − (cid:126)k (cid:48) )] k k (cid:48) (cid:33)(cid:35) × (cid:16) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:48) ζ (cid:126)k − (cid:126)k (cid:48) (cid:17) + (2 permutations) , (4.11) We use ζ (cid:126)k i to refer to ˆ ζ ( τ, (cid:126)k i ) to simplify the notation from here on. – 9 –nd accordingly, the contribution to the shape function is A redef = (cid:16) (cid:15) − c (cid:17) (cid:88) i k i + 3 (cid:15) (cid:88) i (cid:54) = j k i k j − (cid:15) (cid:81) i k i (cid:16) (cid:88) i (cid:54) = j k i k j + (cid:88) i (cid:54) = j k i k j − (cid:88) i (cid:54) = j k i k j (cid:17) . (4.12)When c (cid:28) , this contribution is enhanced compared to the canonical case. ζ ˙ ζ term The term ζ ˙ ζ in equation (4.6) yields the following contribution to the bispectrum (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) ζ ˙ ζ = − × (cid:90) τ B −∞ d¯ τ (2 π ) δ (cid:16) (cid:88) i (cid:126)k i (cid:17) a (cid:104) (cid:15)c ( (cid:15) − c ) − (cid:15) (cid:105) × u ∗ k ( τ B ) u k (¯ τ ) u ∗ k ( τ B ) u (cid:48) k (¯ τ ) u ∗ k ( τ B ) u (cid:48) k (¯ τ ) + (2 permutations) . (4.13)To leading order in c s k i ( τ B − ˜ τ B ) (cid:28) , i.e. on scales larger than the sound Hubble radius ,and recalling the solutions for u k and u (cid:48) k [equations (3.6) and (3.7)], we get the followingcontribution to the shape function, A ζ ˙ ζ = − c (cid:20) c ( (cid:15) − c ) − (cid:15) (cid:21) (cid:88) i k i . (4.14)Again, when c (cid:28) , this contribution is enhanced compared to the canonical case. ˙ ζ∂ζ∂χ term A similar computation for this term yields the following contribution to the shape function A ˙ ζ∂ζ∂χ = − (cid:15) (cid:88) i k i + (cid:15) (cid:81) i k i (cid:16) (cid:88) i (cid:54) = j k i k j − (cid:88) i (cid:54) = j k i k j (cid:17) . (4.15)We note that this contribution is independent of c s . ζ ( ∂ i ∂ j χ ) term For this term, the contribution to the shape function is given by A ζ ( ∂ i ∂ j χ ) = − c (cid:15) (cid:88) i k i + c (cid:15) (cid:88) i (cid:54) = j k i k j + c (cid:15) (cid:81) i k i (cid:16) (cid:88) i k i − (cid:88) i (cid:54) = j k i k j + 3 (cid:88) i (cid:54) = j k i k j − (cid:88) i (cid:54) = j k i k j (cid:17) . (4.16)When c (cid:28) , this contribution is suppressed compared to the canonical case. This is also called the Jeans radius; see [54, 61] for an explicit definition of this scale and its role in matterbounce cosmology when c s (cid:54) = 1 . – 10 – .2.5 Contribution from the ˙ ζ term The ˙ ζ term is a new element in the Lagrangian caused by the nontrivial sound speed, whichdoes not show up in the cubic action of canonical fields. Its contribution to the bispectrum is (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) ˙ ζ = − × (cid:90) τ B −∞ d¯ τ (2 π ) δ (3) (cid:16) (cid:88) i (cid:126)k i (cid:17)(cid:16) − aM (cid:15)Hc (cid:17)(cid:16) − c + 2 λ Σ (cid:17) × u ∗ k ( τ B ) u (cid:48) k (¯ τ ) u ∗ k ( τ B ) u (cid:48) k (¯ τ ) u ∗ k ( τ B ) u (cid:48) k (¯ τ ) , (4.17)where we have used the expression for Σ , equation (2.11). Then the contribution to the shapefunction is expressed as A ˙ ζ = − (cid:18) − c + 2 λ Σ (cid:19) (cid:88) i k i . (4.18)Since this is a new contribution compared to the canonical case, it vanishes for c = 1 . Indeed,when c = 1 , λ/ Σ (cid:39) (1 − c ) / (6 c ) = 0 (see equation (A.20) in Appendix A) and − /c = 0 .We note though that when c (cid:28) , this contribution is large. The contribution from the term a (cid:15) c dd t (cid:16) ηc (cid:17) ζ ˙ ζ in equation (4.6) is exactly zero since η = 0 during the matter contraction. We can alsoneglect the contribution from the term a(cid:15)c ( (cid:15) − s + 1 − c ) ζ ( ∂ζ ) since the leading order term of the resulting bispectrum is proportional to c k i ( τ B − ˜ τ B ) ,which means that it is suppressed outside the sound Hubble radius.The above results differ from the ones of general single field inflation. As pointed outin [42], two main reasons account for the different non-Gaussianities between matter bouncecosmology and inflation. First, here the “slow-roll” parameter (cid:15) is of order one rather thanbeing close to zero, so the amplitudes are larger and the higher-order terms in (cid:15) are not sup-pressed. Second, curvature perturbations grow on super-Hubble scales in a matter-dominatedcontracting universe, and this behaviour manifests itself in the integral of equation (4.8), whilefor inflation, ζ usually remains constant after horizon-exit, so there is no such contribution.In what follows, we summarize the above results and give a detailed analysis of the bis-pectrum. In particular, the differences with the canonical single field matter bounce scenarioare discussed. – 11 – .3 Summary of results One can gather all the contributions above and get the total shape function, A tot = (cid:18) − c + 9 c (cid:19) (cid:88) i k i + 3256 (3 c + 6) (cid:88) i (cid:54) = j k i k j + 3256 (cid:81) i k i × c (cid:88) i k i + (10 − c ) (cid:88) i (cid:54) = j k i k j − (3 c + 6) (cid:88) i (cid:54) = j k i k j + (9 c − (cid:88) i (cid:54) = j k i k j , (4.19)where we have used (cid:15) = 3 / and λ/ Σ = (1 − c ) / c for the matter contraction stage. Nowthe only free parameter in the total shape function is the sound speed c s . In what follows, weshall discuss several interesting aspects of this result. The size of non-Gaussianity is depicted by the dimensionless amplitude parameter f NL ( (cid:126)k , (cid:126)k , (cid:126)k ) = 103 A tot ( (cid:126)k , (cid:126)k , (cid:126)k ) (cid:80) i k i . (4.20)As one can see in equation (4.19), for most values of c s ∈ (0 , , the first term dominates thetotal shape function, and roughly, f NL becomes f NL (cid:39) − c + 15 c , (4.21)which yields f NL < for . (cid:46) c s ≤ and f NL > for c s (cid:46) . . Thus, besides the negativeamplitude in the canonical case [42], a small sound speed in matter bounce cosmology canproduce a positive f NL . In the next section, we shall further discuss its behaviour in differentlimits to confront observations. The shape of non-Gaussianity is described by the dimensionless shape function F ( k /k , k /k ) = A tot k k k . (4.22)Then, the first term in equation (4.19) gives exactly the form of the local shape. Thus, whenthe prefactor of the first term is nonvanishing ( c s (cid:54)≈ . ), the shape function is dominatedby the local form, while the remaining terms just give some corrections. The total shape ofnon-Gaussianity is shown in the left panel of Figure 1, which looks very similar to the plotsin [42] for the canonical matter bounce except that the amplitude is much larger here with c s small.At the same time, this result differs from the one of general single field inflation, wherethe equilateral form dominates the shape of non-Gaussianity for c s (cid:28) [38]. This is mainlycaused by the different generation mechanisms of non-Gaussianity in these two scenarios. Forthe matter bounce scenario, the growth of curvature perturbations after Hubble radius exitmakes a significant contribution to the final bispectrum. Meanwhile, the local form is usually– 12 – igure 1 . The shape of F ( k /k , k /k ) for c s = 0 . (left panel) and c s = 0 . (right panel). thought to be generated on super-Hubble scales since “local” means that the non-Gaussianityat one place is disconnected with the one at other places. For general single field inflation, thedominant contribution is due to the enhanced interaction at horizon-crossing. Thus, thesetwo scenarios behave quite differently with a small sound speed.It is also interesting to note that for c s ≈ . , the first term in equation (4.19) vanishes,so the shape function is dominated by the remaining terms. The shape of non-Gaussianity isplotted in the right panel of Figure 1 for this case, which is a new form different from the localone. To the best of our knowledge, no other scenario can give rise to such a kind of shape,thus it can be seen as a distinguishable signature of matter bounce cosmology for probes ofnon-Gaussianity. Usually people are interested in the squeezed limit of the bispectrum ( k (cid:28) k = k = k ),since its scaling behaviour is helpful for clarifying the shapes of non-Gaussianity analytically.Here in the squeezed limit ( k /k → ), the dimensionless shape function can be expanded as F ( k /k , k /k ) (cid:39) (cid:18) −
332 + 13 c (cid:19) kk + 364 (cid:0) c (cid:1) k k + O (cid:32)(cid:18) k k (cid:19) (cid:33) . (4.23)The leading order term gives the scaling F ∼ k/k and (cid:104) ζ (cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:105) squeezed ∼ k , (4.24)which is consistent with the dominant local form. The only exception is when the coefficientof the first term vanishes ( c s = (cid:112) / ) and another scaling, F ∼ k /k , follows from thenext-to-leading order term. There are three forms of the amplitude parameter f NL that are of particular interest forcosmological observations. They are called the “local form”, the “equilateral form”, and the– 13 –folded form”. The local form requires that one of the three momentum modes exits theHubble radius much earlier than the other two, e.g., k (cid:28) k = k . In this limit, one cansimplify the total shape function, equation (4.19), to find f localNL (cid:39) − c . (5.1)The equilateral form requires that the three momenta form an equilateral triangle, i.e. k = k = k . In this case, we obtain f equilNL (cid:39) − c + 45 c . (5.2)The folded form has k = 2 k = 2 k , hence f foldedNL (cid:39) −
374 + 658 c . (5.3)As a result, in the limit where c (cid:28) , we find that f localNL ≈ f equilNL ≈ f foldedNL ≈ c (cid:29) . (5.4)Let us recall from section 3 that in order to satisfy the observational bound on the tensor-to-scalar ratio, we must impose c s (cid:46) . . This immediately implies f localNL ≈ f equilNL ≈ f foldedNL (cid:38) . × (cid:29) . (5.5)This amplitude of primordial non-Gaussianity is clearly ruled out according to the observa-tions [58], f localNL = 0 . ± . , f equilNL = − ± , f orthoNL = − ± , (5.6)thus ruling out the viability of the class of models studied here.Alternatively, if one requires that, e.g., − . (cid:46) f localNL (cid:46) . (i.e., imposing f localNL to bewithin the measured σ error bars), then one would need c s (cid:38) . . However, this lowerbound on the sound speed yields a tensor-to-scalar ratio r (cid:38) . , which is again clearlyruled out by observations [52].In summary, there is no region of parameter space where c s can give a good, smalltensor-to-scalar ratio (i.e., of order . at most) and good, small non-Gaussianities (i.e., oforder 10 at most). Therefore, independent of what happens during the bounce, we extendthe no-go theorem conjectured in [51] to the following one: No-Go Theorem.
For quantum fluctuations generated during a matter-dominated contract-ing phase, an upper bound on the tensor-to-scalar ratio ( r ) is equivalent to a lower bound onthe amount of primordial non-Gaussianities ( f NL ). Furthermore, if • the matter contraction phase is due to a single (not necessarily canonical) scalar field, • the same single scalar field allows for the violation of the NEC to produce a nonsingularbounce, • and General Relativity holds at all energy scales,then satisfying the current observational upper bound on the tensor-to-scalar ratio cannot bedone without contradicting the current observational upper bounds on f NL (and vice versa). Note that this constraint does not exclude c s ≈ . , for which the new shape of non-Gaussianity in theright panel of Figure 1 emerges. – 14 – Conclusions and discussion
In this paper, we computed the two- and three-point correlation functions produced by ageneric k -essence scalar field in a matter-dominated contracting universe. Comparing thepower spectra of scalar and tensor modes, we found that the tensor-to-scalar ratio can beappropriately suppressed if the sound speed associated with the k -essence scalar field issufficiently small. In turn, we showed that the amplitude of the bispectrum is amplifiedby the smallness of the sound speed . As a result, it seems incompatible to suppress thetensor-to-scalar ratio below current observational bounds without producing excessive non-Gaussianities. This leads us to extend the conjecture of the no-go theorem, which effectivelyrules out a large class of nonsingular matter bounce models.Although this seriously constrains nonsingular matter bounce cosmology as a viablealternative scenario to inflation, there remain several classes of models that are not affectedby this no-go theorem. Indeed, one could still evade the no-go theorem assuming certainmodified gravity models as stated in [51] (see references therein) or with the introductionof one or several new fields. For example, in the matter bounce curvaton scenario [64] (seealso [15, 65, 66] for other nonsingular bouncing models using the curvaton mechanism) andin the two-field matter bounce scenario [67], entropy modes are generated by the presenceof an additional scalar field, which are then converted to curvature perturbations. In bothmodels near the bounce, the kinetic term of the entropy field varies rapidly, which acts asa tachyonic-like mass that amplifies (in a controlled way) the entropy fluctuations while notaffecting the tensor modes. As a result, the tensor-to-scalar ratio is suppressed (see [10, 50]for reviews of this process). Furthermore, the production of non-Gaussianities in the matterbounce curvaton scenario has been estimated in [64], and it indicated that sizable, negativenon-Gaussianities appeared, yet still in agreement with current observations. Accordingly,such a curvaton scenario does not appear to suffer from a no-go theorem. However, there stillremains to do an appropriate extensive analysis of the production of non-Gaussianities whengeneral multifields are included in the matter bounce scenario.A similar curvaton mechanism is used in the new Ekpyrotic model [68, 69] (extended in[70–73]), which generates a nearly scale-invariant power spectrum of curvature perturbations.In this case, however, the smallness of the observed tensor-to-scalar ratio must be attributedto the fact that the tensor modes have a blue power spectrum when they exit the Hubbleradius in a contracting phase with w (cid:29) . The new Ekpyrotic model originally predicted largenon-Gaussianities [74–78] (see also the reviews [79, 80]), but some more recent extensions canresolve this issue [81–85]. Thus, here as well, it appears that these types of models do notsuffer from a similar no-go theorem .We note that one might be able to prove the no-go conjecture of this paper borrowingsimilar techniques to the effective field theory of inflation [40], i.e. by constructing an effectivefield theory of nonsingular bouncing cosmology (e.g., see the recent work of [25, 26]). Incomplete generality, this could allow us to find the exact and explicit relation between the With a small sound speed, one may also reach the strong coupling regime where the perturbative analysisbreaks down. This is known as the strong coupling problem [62, 63], which affects many non-inflationaryscenarios (see in particular Appendix C of [62], which focuses on non-attractor models). It represents a generalindependent theoretical constraint, but in the context of the matter bounce scenario, our no-go theorem ismore constraining due to current observational bounds. Furthermore, Ekpyrotic models are robust against the growth of anisotropies in a contracting universe.This is another challenge with the matter bounce scenario (see [18, 86]) that will have to be overcome to havea viable theory. – 15 –ensor-to-scalar ratio (which involves the power spectra of curvature and tensor modes) andthe bispectrum. In fact, the goal would be to find a consistency relation for the three-pointfunction in single field nonsingular bouncing cosmology similar to what has been done ininflation [29, 87, 88]. This will be explored in a follow-up study.Finally, we would like to emphasize that, for matter bounce cosmology, although thesimplest k -essence model is ruled out by the no-go theorem, the bispectrum with c s (cid:54) = 1 (asan independent result of this paper) remains to be a probable target for future probes of non-Gaussianity. This possibility relies on the aforementioned bouncing models that can evadethe no-go theorem with other mechanisms. In those cases, a nontrivial sound speed may stilllead to the same behaviour of non-Gaussianities found in this paper, which potentially can bedetected by future observations. Particularly, we predict a new shape with an amplitude stillconsistent with current observational limits, which can serve as the distinctive signature ofmatter bounce cosmology and help us distinguish it from other very early universe theories. Acknowledgments
We are grateful to Robert Brandenberger, Ziwei Wang, and Edward Wilson-Ewing for valu-able comments and helpful discussions. YFC, YBL, and DGW are supported in part bythe Chinese National Youth Thousand Talents Program (No. KJ2030220006), by the USTCstart-up funding (No. KY2030000049), by the National Natural Science Foundation of China(NSFC) (Nos. 11421303, 11653002), and by the Fund for Fostering Talents in Basic Scienceof the NSFC (No. J1310021). JQ acknowledges financial support from the Walter C. SumnerMemorial Fellowship and from the Vanier Canada Graduate Scholarship administered by theNatural Sciences and Engineering Research Council of Canada (NSERC). JQ also wishes tothank USTC for hospitality when this work was initiated. DGW is also supported by a deSitter Fellowship of the Netherlands Organization for Scientific Research (NWO). Part of thenumerical computations were done on the computer cluster LINDA in the particle cosmologygroup at USTC.
A The ratio λ/ Σ Let us recall the definition of Σ and λ in equations (2.11) and (2.12). Their ratio is thus givenby λ Σ = 13 (cid:18) X Σ ,X Σ − (cid:19) . (A.1)Recalling the definition of c in equation (2.10), we note that Σ = X ( P ,X + 2 XP ,XX ) = X P ,X c . (A.2)Also, recalling the expression for ρ and p in equation (2.7), we find that XP ,X = ρ + p , andso, the above expression for Σ becomes Σ = ρ + p c . (A.3)Consequently, X Σ ,X Σ =
X ρ ,X + p ,X ρ + p − X c s ,X c s . (A.4)– 16 –orking in the limit where p = 0 , we note that ρ = 2 XP ,X , and so, p ,X = P ,X = ρ/ (2 X ) ,which implies that p ,X /ρ = 1 / (2 X ) . Also, ρ ,X = p ,X /c from the definition of the soundspeed, and thus, ρ ,X ρ = p ,X ρc = 12 c X . (A.5)Therefore, equation (A.4) in the limit where p = 0 becomes X Σ ,X Σ = 12 c + 12 − X c s ,X c s . (A.6)Alternatively, one can evaluate the ratio λ/ Σ as λ Σ = 13 (cid:18) Σ ,X Σ X − (cid:19) = 13 (cid:32) ˙ΣΣ X ˙ X − (cid:33) . (A.7)Since we can write Σ = H M (cid:15)/c and recalling the definition of the slow-roll parameters insection 2, we get ˙Σ H Σ = − (cid:15) + η − s . (A.8)Now, we note that we can write η = ˙ (cid:15)H(cid:15) = ¨ HH ˙ H − HH = ¨ HH ˙ H + 2 (cid:15) . (A.9)Also, the Friedmann equation M ˙ H = − XP ,X implies that ¨ HH ˙ H = 1 H (cid:32) ˙ XX + ˙ P ,X P ,X (cid:33) , (A.10)and so, ˙ XHX = η − (cid:15) − ˙ P ,X P ,X . (A.11)Therefore, combining equation (A.8) and the above yields ˙ΣΣ X ˙ X = − (cid:15) + η − s − (cid:15) + η − ˙ P ,X P ,X . (A.12)In the limit where p = 0 , we recall that (cid:15) = 3 / and η = 0 , and as a result, ˙ΣΣ X ˙ X = 3 + 2 s ˙ P ,X P ,X . (A.13)Comparing the above with equation (A.6), since ( ˙Σ / Σ)( X/ ˙ X ) = X Σ ,X / Σ , we find s ˙ P ,X P ,X = 12 c + 12 − X c s ,X c s , (A.14)– 17 –ut − X c s ,X c s = − X ˙ X ˙ c s c s = − s HX ˙ X = − sη − (cid:15) − ˙ P ,X P ,X , (A.15)where the last equality follows from equation (A.11). Thus, equation (A.14), with (cid:15) = 3 / and η = 0 , leaves us with
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