Bouncing cosmologies while offering a compelling alternative to inflationary models face challenges from the growth of vector perturbations during the contracting phase. While linear vector instabilities can be avoided with specific initial conditions or the absence of vector degrees of freedom we demonstrate the significant role of secondary vector perturbations generated by nonlinear interactions with scalar fluctuations. Our analysis reveals that in a broad class of singlefield matter bounce scenarios these secondary vector perturbations inevitably get unacceptably large amplitudes provided the curvature fluctuations are consistent with cosmic microwave background observations. This finding underscores the crucial importance of scalarinduced vector perturbations in bouncing cosmology and highlights the need for further investigation into their potential impact on the viability of these models.
['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']
Constraining matter bounce scenario from scalarinduced vector perturbations
{'Mian Zhu a Chao Chen b,c, 1': '- a Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30348 Krakow, Poland\n- b Department of Physics, School of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, China\n- c Jockey Club Institute for Advanced Study, The Hong Kong University of Science and Technology, Hong Kong, China  \nE-mail: mian.zhu@uj.edu.pl, chaochen012@gmail.com  \nAbstract. Bouncing cosmologies, while offering a compelling alternative to inflationary models, face challenges from the growth of vector perturbations during the contracting phase. While linear vector instabilities can be avoided with specific initial conditions or the absence of vector degrees of freedom, we demonstrate the significant role of secondary vector perturbations generated by non-linear interactions with scalar fluctuations. Our analysis reveals that in a broad class of single-field matter bounce scenarios, these secondary vector perturbations inevitably get unacceptably large amplitudes, provided the curvature fluctuations are consistent with cosmic microwave background observations. This finding underscores the crucial importance of scalar-induced vector perturbations in bouncing cosmology and highlights the need for further investigation into their potential impact on the viability of these models.', '1 Introduction': 'Inflation [1], the standard paradigm of the early-universe cosmology, provides a natural way to explain the formation of large-scale structures (LSS) and the observation of cosmic microwave background (CMB). Nonetheless, inflationary cosmology may suffer from the initial singularity problem [2-5] and the trans-Planckian problem [6-8]. These challenges motivate us to explore alternative early universe scenarios such as the non-singular bouncing cosmology [9-12], where a contraction phase takes place in prior to the expansion phase. While bouncing cosmology offers an intriguing alternative for the early universe, it faces significant challenges. Conceptual issues [13, 14] and its compatibility with CMB observations [15, 16] remain critical concerns. There are also extensive debate surrounding specific problems of bouncing cosmologies [17-36] and proposed solutions [37-50], a comprehensive review of these challenges is available in the reference [51].  \nIn this paper, we highlight another challenge for bouncing cosmology, the overproduction of vector perturbations, a problem overlooked in the community. Early studies [52, 53] demonstrated that linear vector perturbations scale as S i ( k ) ∝ a -2 , leading to its growth that can break down the perturbation theory. Resolving this issue typically requires specific model constructions or finely-tuned initial conditions for vector perturbations. For instance, a single-field bouncing scenario lacks vector degrees of freedom, preventing primordial vector fluctuations from vacuum fluctuations.  \nHowever, secondary vector perturbations inevitably arise from non-linear interactions with primordial curvature fluctuations ζ . Those fluctuations cannot be arbitrarily fine-tuned, as the power spectrum of curvature fluctuation P ζ is determined by CMB observations. In Ref. [54], scalar-induced vector perturbations (SIVP) are investigated in specific collapsing universes with theoretical considerations. For the first time, we in this paper connect the power spectrum P ζ to CMB observations, establishing a lower bound for the energy density of SIVP. Specifically, we work in matter bounce scenario [55], a simple-yet-significant bouncing scenario where nearly scale-invariant curvature fluctuation is generated in a matterdominated contraction phase (i.e., the effective equation-of-state parameter is zero). Our results demonstrate that the energy density of SIVP becomes comparable to the background  \nenergy density at the end of the matter contraction phase, provided the contraction is driven by a k-essence scalar field. This significant back-reaction poses a serious challenge to the viability of the matter bounce scenario.  \nThis paper is organized as follows: Section 2 introduces the theoretical framework. Section 3 presents the calculation of the energy density ratio of SIVP to the background. We conclude in Section 4. Technical details are provided in the appendices. Throughout the paper, we set the Planck mass M p = 1 . A dot denotes derivative with respect to cosmic time t , and a prime denotes differentiation with respect to τ , unless otherwise specified.', '2 Theoretical setup': "We work in a spatially-flat FLRW universe  \nd s 2 = -d t 2 + a ( t ) 2 d x i d x i = a ( τ ) 2 ( -d τ 2 +d x i d x i ) , (2.1)  \nwhere d τ = d t/a is the conformal time. In matter bounce, the scalar factor scales as a ∝ τ 2 , and can be parameterized as  \na ( τ ) = ( τ/τ 0 ) 2 , τ < τ 0 < 0 , (2.2)  \nwhere τ 0 labels the end of the contraction phase. It will also be useful to define a comoving Hubble parameter H ≡ a ' /a = 2 /τ < 0 . The background energy density is given by the Friedmann's equation, ρ bg ( τ ) = 3 H 2 = 12 τ 4 0 /τ 6 . In the framework of k-essence theory [56],  \nS = ∫ d 4 x √ -g [ R 2 + K ( ϕ, X ) ] , X ≡ -1 2 ∂ µ ϕ∂ µ ϕ , (2.3)  \nthe quadratic action for curvature fluctuation ζ is [57-59]  \nS (2) ζ = ∫ d τ d 3 x z 2 s 2 [ ζ ' 2 -c 2 s ( ∂ i ζ ) 2 ] , z 2 s = 3 a 2 c 2 s , (2.4)  \nwhere we used the fact that the effective slow-roll parameter ϵ ≡ -˙ H/H 2 = 3 / 2 in the matter contraction phase, and we regarded the sound speed for curvature perturbations, c s ≡ K ,X / ( K ,X +2 XK ,XX ) , as a constant for simplicity. Working in the Fourier space with a canonical mode function v k = z s ζ k , the dynamical equation for curvature perturbations becomes  \nv '' k + ( c 2 s k 2 -2 τ 2 ) v k = 0 . (2.5)  \nImposing the vacuum initial condition, we get the expression for curvature fluctuations as  \nζ k ( τ ) ≡ v k ( τ ) z s = e -ikc s τ c s √ 6 c s k ( 1 -i c s kτ ) ( τ 0 τ ) 2 . (2.6)  \nIn contrast to the vanilla slow-roll inflation case, the curvature perturbations grow on superhorizon scales | kτ | ≪ 1 in the matter contraction phase (see e.g., Ref. [60]). Hence, one needs to evaluate the curvature power spectrum at the end of the contraction phase:  \n⟨ ζ ⃗ k ζ ⃗ p ⟩ ( τ = τ 0 ) = (2 π ) 3 δ ( ⃗ k + ⃗ p ) | ζ ⃗ k | 2 = (2 π ) 3 δ ( ⃗ k + ⃗ p ) c s 6 k ( 1 + 1 k 2 c 2 s τ 2 0 ) . (2.7)  \nFrom the definition of the scalar power spectrum,  \n⟨ ζ ⃗ k ζ ⃗ p ⟩ = (2 π ) 3 δ ( ⃗ k + ⃗ p ) 2 π 2 k 3 P ζ ( k ) , (2.8)  \nwe derive,  \nP ζ ( k, τ 0 ) = k 2 c s 12 π 2 ( 1 + 1 k 2 c 2 s τ 2 0 ) ≃ 1 12 π 2 c s τ 2 0 , (2.9)  \nwhich is scale-invariant.  \nIn the FLRW universe, the most general perturbed metric, including only vector perturbation, is given by [61]  \nd s 2 = a 2 ( τ ) [ -d τ 2 -2 G i d τ d x i +( δ ij + F ij )d x i d x j ] , (2.10)  \nwhere F ij satisfies F ij = ∂ i F j + ∂ j F i and ∂ i F i = 0 , and G i is divergent free, ∂ i G i = 0 . Since there is no vector degree of freedom in our setup, F i and G i should be regarded as secondorder fluctuations induced by the first-order perturbations. We work in the Newtonian gauge where F i = 0 , and the metric involving scalar and vector perturbations can be written as  \nd s 2 = a 2 ( τ ) [ -e 2Φ d τ 2 -2 G i d τ d x i + e -2Φ δ ij d x i d x j ] , (2.11)  \nwhere the scalar perturbation Φ is related to the curvature fluctuation via [62, 63]  \nζ = Φ + H H 2 -H ' (Φ ' + H Φ) . (2.12)  \nThe vector power spectrum, defined as  \n⟨ G λ ( ⃗ k ) G s ( ⃗ p ) ⟩ ≡ (2 π ) 3 δ ( ⃗ k + ⃗ p ) δ λs 2 π 2 k 3 P G ( τ, ⃗ k ) . (2.13)  \nAs discussed above, G i can be sourced by Φ (or equivalently ζ via Eq. (2.12)) through their nonlinear coupling, its power spectrum is computed as  \nP G ( τ, k ) = ∫ ∞ 1 √ 2 d t ∫ 1 √ 2 -1 √ 2 d s (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 ×P ζ ( k √ 2 ( t -s ) , τ ) P ζ ( k √ 2 ( t + s ) , τ ) |I ( t, s, z ) | 2 , (2.14)  \nwhere z ≡ kτ . We summarize the computational details in App. A.", '3 Energy density of SIVP': "The energy density of SIVP is given by [64]  \nρ V ( τ, ⃗x ) = 1 4 a 2 ∂ i G j ( τ, ⃗x ) ∂ i G j ( τ, ⃗x ) . (3.1)  \nFrom Eq. (3.1), the energy density of SIVP is related to P G as  \nρ V ( τ ) = 1 2 a 2 ∫ d kk P G ( τ, ⃗ k ) . (3.2)  \nThe back-reaction is represented by the ratio of the energy density of SIVP against the background one at τ = τ 0 :  \nδ V ≡ ρ V ( τ 0 ) ρ bg ( τ 0 ) = 1 24 ∫ ( kτ 0 ) P G ( k, τ 0 )d( kτ 0 ) . (3.3)  \nThe P ζ on super-horizon scales is associated to CMB observations:  \nP ζ ( ⃗ k, τ 0 ) ≃ 1 12 π 2 c s τ 2 0 = A s , (3.4)  \nwhere A s = 2 . 1 × 10 -9 from Planck collaboration [65]. We denote the scales of fluctuations that 'cross the horizon' at the beginning/end of contraction phase to be k min and k max :  \nk min = c -1 s |H ( τ ini ) | = -2( c s τ ini ) -1 , k max = c -1 s |H ( τ 0 ) | = -2( c s τ 0 ) -1 , (3.5)  \nwhere τ ini labels the initial time of matter contraction phase. The modes with k min < k < k max becomes super-horizon during the matter contraction phase, and we adopt a minimal curvature power spectrum for super-horizon perturbations  \nP ζ = { A s , k min < k < k max , 0 , otherwise . (3.6)  \nIt is possible that modes with k < k min or k > k max become super-horizon before or after the matter contraction phase, hence also give a positive contribution to SIVP. Adopting the ansatz in Eq. (3.6) captures the dominant contribution to SIVP from modes entering the horizon during the contraction phase, providing a lower bound sufficient to analyze the instability.  \nThe scale of scale-invariant curvature fluctuation indicated by CMB observations ranges from k C /a today ≃ 10 -4 Mpc -1 to k L /a today ≃ 1 Mpc -1 . To match CMB data, modes with k = k C must be well within the horizon ( k C ≫ k min ), while k L must be super-horizon at τ = τ 0 ( k L ≤ k max ). Introducing a dimensionless scaling factor  \ng ≡ τ ini τ 0 = k max k min , (3.7)  \nand g must be larger than 10 4 .  \nFigure 1 . Left: The SIVP power spectra (2.14) at the end of matter contraction τ 0 as functions of c s with g = 10 4 ; Right: The SIVP power spectra (2.14) as functions of g with c s = 1 .  \n<!-- image -->  \nThe resulting vector power spectrum is determined by the dimensionless parameters c s and g , as τ 0 is fixed by c s through Eq. (3.4). We numerically evaluate P G and present the result in Fig. 1. In App. B, we demonstrate that P G scales as A 2 s c -2 s g 4 log g . Consequently,  \nδ V ∝ A 2 s c 4 s g 4 (log g + O (1)) , (3.8)  \nfrom our numerical results. We further confirm this scaling through analytical estimations presented in App. C. As a comparison, the energy density of linear curvature perturbations scales as g 2 assuming a scale-invariant P ζ . As a result, the back-reaction problem associated with SIVP can be more severe than that of curvature fluctuations due to its g 4 log g scaling.  \nThe value of δ V is presented in Table 1, revealing that δ V is less than unity only when c s ≃ 1 and g ≃ 10 4 . In all other cases, δ V > 1 , indicating either a significant back-reaction on the background evolution or a breakdown of perturbation theory. Considering the consistency relation in the context of k-essence theory during matter contraction, we have r = 24 c s [33], where r is the tensor-to-scalar ratio constrained by r 0 . 002 < 0 . 044 [66]. This constraint implies c s < 0 . 02 , inevitably leading to an excessively large δ V . Even if we artificially set c s = 1 , the parameter g is significantly larger than 10 4 in reality because k min ≫ k C . Combining these arguments, we conclude that cosmological models where nearly scale-invariant curvature fluctuations on CMB and LSS scales originate from a matter contraction phase governed by a minimally coupled k-essence field are constrained by the overproduction of SIVP, rendering such models invalid.  \nTable 1 . The values of δ V in Eq. (3.3) for various values of c s and g .", '4 Conclusion': 'Vector fluctuations play a crucial role in bouncing cosmologies, particularly during the contraction phase. This study, for the first time, combines the concept of secondary vector fluctuations induced by scalar fluctuations with observational constraints on curvature perturbations, revealing an overproduction of these modes in a matter contraction phase governed by a k-essence scalar field. This finding highlights the importance of vector modes in bouncing cosmologies and motivates further investigation into their impact on various bouncing scenarios, including matter bounces with more complex actions [47], Ekpyrotic scenarios [37], and scenarios where the bouncing phase significantly influences the evolution of curvature fluctuations [67].  \nOur findings motivate further exploration of perturbation theory within the context of bouncing cosmology. Additionally, our results could be revisited by replacing the cut-off of  \nthe curvature power spectrum with regularized primordial fluctuations. This approach could, in principle, yield more accurate results. However, this area currently lacks sufficient research. Additionally, the growth of anisotropic shear during the contraction phase, whose energy density scales as ρ ∝ a -6 , warrants investigation into scalar-induced shear, which could provide additional theoretical constraints on bouncing models. The secondary vector fluctuations in bouncing cosmology in the presence of vector field(s) deserves future investigation due to its potential connection with topics such as primordial magnetogenesis [68].', 'Acknowledgement': 'M.Z. thanks Yi Wang for discussions which inspired this work, and Xian Gao, Chunshan Lin and George Zahariade for their critical comments. M.Z. also thanks the correspondence with Yi-Fu Cai and Sabino Matarrese for the discussions on several technical problems. We are grateful to Atsuhisa Ota for his contributions during the initial stage of this work. M.Z. was supported by grant No. UMO 2021/42/E/ST9/00260 from the National Science Centre, Poland. C.C. thanks the support from the Jockey Club Institute for Advanced Study at The Hong Kong University of Science and Technology. C.C. is supported by NSFC (Grants No. 12433002).', 'A Scalar-induced vector perturbations': "This appendix derives the second-order vector fluctuations by solving the second-order Einstein equations, following a similar approach used in the study of scalar-induced gravitational waves (SIGW) (see, for example, the review [69]). Throughout this section, superscripts will be used to denote the order of perturbation. For instance, ρ (2) represents the perturbed energy density at second order. The second-order vector G (2) i is determined by the ij components of the Einstein equations,  \n̸  \nG (2) j i = T (2) j i , i = j . (A.1)  \nAlthough it is possible to derive the secondary vector fluctuations using momentum constraints (as there are no vector degrees of freedom in our specific scenario), the method of computing with Einstein equations will prove useful in future studies involving vector fields. Examples include primordial magnetogenesis [68] and baryon asymmetry [70]. For detailed explanations of both methods and their equivalence, see Refs. [71, 72].  \nThe computation of geometric quantities based on the metric perturbations in Eq. (2.11) is straightforward. We present some useful expressions below (utilizing the identity H ' = -H 2 / 2 , applicable in the matter bounce scenario, and the notation ∂ 2 ≡ ∂ i ∂ i ),  \nR = -2 e 2Φ a 2 [ ( ∂ Φ) 2 -2 ∂ 2 Φ ] , G (0)0 0 = -3 H 2 a 2 , (A.2)  \nG (2) j i = 1 2 a 4 d d τ [ a 2 ( ∂ i G j + ∂ j G i )] -1 a 2 [ 2 ∂ i Φ ∂ j Φ + δ j i (4ΦΦ '' +5Φ ' 2 -( ∂ l Φ) 2 +12 H ΦΦ ' ) ] , (A.3)  \nwhere R is the intrinsic curvature defined for a constant-time hypersurfece associated with the metric (2.11).  \nFor the matter sector, we obtain  \nT ν µ = ( ρ + P ) u µ u ν + Pδ ν µ +Σ ν µ , (A.4)  \n̸  \nwhere ρ and P are the energy density and the pressure, respectively; u µ is the four-velocity of the observer; Σ µν is the anisotropic stress subject to the conditions Σ 00 = Σ 0 i = 0 , Σ ij = Σ ji and δ ij Σ ij = 0 . For simplicity, we will set the anisotropic stress to be zero, and leave the study of Σ ij = 0 case in the future work. From (A.4) we get  \nT (2) j i = ( ρ (0) + P (0) ) u (1) i u (1) j + P (2) δ j i . (A.5)  \nThe four-velocity by definition is normalized according to u µ u ν g µν = -1 . Along with the definition u µ = g µν u ν , one arrive  \nu (0)0 = ( -g (0) 00 ) -1 / 2 , u (1)0 = 1 2 ( -g (0) 00 ) -3 / 2 g (1) 00 , (A.6)  \nand accordingly  \nu (1) i = -2 a 3 H 2 ( ∂ i Φ ' + H ∂ i Φ) , u (1) i = -2 3 a H 2 ( ∂ i Φ ' + H ∂ i Φ) . (A.7)  \nThe rest quantities are to be determined by the perturbed Einstein equations G ( n ) µν = T ( n ) µν . For instance,  \nρ (0) = -T (0)0 0 = -G (0)0 0 = 3 H 2 a 2 , (A.8)  \nP (2) = 1 3 T (2) i i -1 3 ( ρ (0) + P (0) ) u (1) i u (1) i = 1 3 G (2) i i -1 3 ( ρ (0) + P (0) ) u (1) i u (1) i . (A.9)  \nWe organize some useful expressions as below,  \nρ (0) = 3 H 2 a 2 , P (0) = 0 , (A.10)  \nP (2) = -1 9 a 2 H 2 [ 4( ∂ i Φ ' ) 2 + H 2 ( ∂ i Φ) 2 +8 H ∂ i Φ ∂ i Φ ' +45 H 2 Φ ' 2 ] . (A.11)  \nFrom Eqs. (A.5), (A.3), (A.10), and (A.11), we obtain the following equation,  \n∂ i G j ' +2 H ∂ i G j -4 ∂ i Φ ' ∂ j Φ ' +8 H ∂ i Φ ' ∂ j Φ+10 H 2 ∂ i Φ ∂ j Φ 3 H 2 +( i ←→ j ) = 0 . (A.12)  \nNow we are about to derive the equation for SIVP. The curvature fluctuation is related to the Φ through the relation (2.12). While directly converting Φ to ζ using Eq. (2.12) is challenging, we can leverage the fact that curvature fluctuations grow on super-horizon scales. By adopting the ansatz in Eq. (3.6), we focus on the contributions from modes that are super-horizon at the end of the contraction phase. This allows us to utilize the simplified relationship Φ = 3 2 ζ , derived by combining Eqs. (2.12) and (2.6) in the super-horizon regime. We will use this relationship to convert Φ to the curvature fluctuation ζ in the following calculations.  \nIn order to extract the transverse vector modes, we define a projection vector [73],  \nV kl i ≡ 1 ∇ 2 ∂ l T k i ≡ 1 ∇ 2 ∂ l ( δ k i -∂ k ∂ i ∇ 2 ) , (A.13)  \nwhich is able to project a term S kl into a transverse vector G i such that  \nG i = V kl i S kl , (A.14)  \nand it is helpful to list the following relationships,  \nV kl i δ kl = 0 , V kl i ∂ k ∂ l Φ = 0 , V kl i ∂ k G l = 0 , V kl i ∂ l G k = G i . (A.15)  \nWe then get the equation of G i as  \nG ' i +2 H G i = V kl i S kl , (A.16)  \nwhere the source term is  \nS kl ≡ 15 ∂ k ζ∂ l ζ + 6( ∂ k ζ ' ∂ l ζ + ∂ k ζ∂ l ζ ' ) H + 6 ∂ k ζ ' ∂ l ζ ' H 2 . (A.17)  \nAs a final step, we move to the Fourier space. We choose a pair of polarization vector { e ( ˆ k ) , ¯ e ( ˆ k ) } , which are orthogonal to each other and ⃗ k , satisfying:  \ne λ i ( ˆ k ) e σ,i ( ˆ k ) = δ λσ , e λ i ( ˆ k ) k i = 0 , ∑ λ e λ,i ( ˆ k ) e λ,j ( ˆ k ) = δ ij -k i k j k 2 . (A.18)  \nThe vector perturbation becomes  \nG i ( τ, ⃗x ) = ∑ λ ∫ d 3 ⃗ k (2 π ) 3 e i ⃗ k · ⃗x G λ ( τ, ⃗ k ) e λ i ( ˆ k ) , (A.19)  \nand we have  \nV ab i S ab ( τ, ⃗x ) = ∫ d 3 ⃗ k (2 π ) 3 ik b k 2 ( δ a i -k a k i k 2 ) e i ⃗ k · ⃗x S ab ( τ, ⃗ k ) , = ∑ λ ∫ d 3 ⃗ k (2 π ) 3 ik b k 2 e λ i ( ˆ k ) e λ,a ( ˆ k ) e i ⃗ k · ⃗x S ab ( τ, ⃗ k ) , (A.20)  \nwhere S ab ( τ, ⃗ k ) is the Fourier transform of S ab ( τ, ⃗x ) . Hence the equation for vector mode becomes  \nG λ ' ( τ, ⃗ k ) + 2 H G λ ( τ, ⃗ k ) = S λ ( τ, ⃗ k ) , (A.21)  \nwhere  \nS λ ( τ, ⃗ k ) = ik m k 2 e λ,n ( ˆ k ) S nm ( τ, ⃗ k ) = -∫ d 3 ⃗ p (2 π ) 3 ik m k 2 e λ,n ( ˆ k ) p n p m f ( ⃗ p, ⃗ k, τ ) ζ ⃗ p ( τ 0 ) ζ ⃗ k -⃗ p ( τ 0 ) , (A.22)  \nwith the evolution kernel,  \nf ( ⃗ p, ⃗ k, τ ) = 15 T ( pτ ) T ( | ⃗ k -⃗ p | τ ) + 6 H 2 T ' ( pτ ) T ' ( | ⃗ k -⃗ p | τ ) + 6 H [ T ' ( pτ ) T ( | ⃗ k -⃗ p | τ ) + T ( pτ ) T ' ( | ⃗ k -⃗ p | τ )] . (A.23)  \nNote that the prime in (A.23) denotes derivative with respect to the argument instead of τ . Here, T is the transfer function defined as  \nζ ⃗ k ( τ ) = T ( kτ ) ζ ⃗ k ( τ 0 ) , τ < τ 0 < 0 . (A.24)  \nWith the help of Eq. (2.6), we have  \nT ( kτ ) = τ 3 0 τ 3 e ic s k ( τ 0 -τ ) i -c s kτ i -c s kτ 0 . (A.25)  \nNow we are about to evaluate the two-point correlation function of G λ , defined as  \n⟨ G λ ( ⃗ k ) G s ( ⃗ p ) ⟩ ≡ (2 π ) 3 δ ( ⃗ k + ⃗ p ) δ λs 2 π 2 k 3 P G ( τ, ⃗ k ) . (A.26)  \nThe general solution of G i is given by  \nG λ ( τ, ⃗ k ) = 1 a ( τ ) 2 ∫ τ a (˜ τ ) 2 S λ (˜ τ, ⃗ k )d˜ τ . (A.27)  \nApplying the bouncing background (2.2) and specifying the integration range, we have  \nG λ ( τ, ⃗ k ) = ∫ τ τ ini d˜ τ ( ˜ τ τ ) 4 S λ (˜ τ, ⃗ k ) , (A.28)  \nwhere τ ini is the conformal time at the beginning of the contraction phase. The two-point correlation function of G λ becomes  \n⟨ G λ ( τ, ⃗ k ) G s ( τ, ⃗ k ' ) ⟩ = ∫ τ τ ini d˜ τ 1 ∫ τ τ ini d ˜ τ 2 ( ˜ τ 1 τ ) 4 ( ˜ τ 2 τ ) 4 ⟨ S λ (˜ τ 1 , ⃗ k ) S s (˜ τ 2 , ⃗ k ' ) ⟩ = -1 k 2 k ' 2 ∫ τ τ ini d˜ τ 1 ∫ τ τ ini d˜ τ 2 ( ˜ τ 1 τ ) 4 ( ˜ τ 2 τ ) 4 ∫ d 3 ⃗ p (2 π ) 3 ⟨ ζ ⃗ p ζ ⃗ k -⃗ p ζ ⃗ q ζ ⃗ k ' -⃗ q ⟩ × k m p m p n k ' i q i q j e λ,n ( ˆ k ) e s,j ( ˆ k ' ) f ∗ ( ⃗ p, ⃗ k, ˜ τ 1 ) f ( ⃗q, ⃗ k ' , ˜ τ 2 ) . (A.29)  \nAssuming a Gaussian distribution of curvature fluctuation and with the help of Eq. (2.8), the contraction of the four-point correlator is decomposed as  \n⟨ ζ ⃗ p ζ ⃗ k -⃗ p ζ ⃗ q ζ ⃗ k ' -⃗ q ⟩ = ⟨ ζ ⃗ p ζ ⃗ q ⟩⟨ ζ ⃗ k -⃗ p ζ ⃗ k ' -⃗ q ⟩ + ⟨ ζ ⃗ p ζ ⃗ k ' -⃗ q ⟩⟨ ζ ⃗ k -⃗ p ζ ⃗ q ⟩ = (2 π ) 6 2 π 2 p 3 2 π 2 | ⃗ k -⃗ p | 3 δ ( ⃗ k + ⃗ k ' ) [ δ ( ⃗ p + ⃗q ) + δ ( ⃗q + ⃗ k -⃗ p ) ] P ζ ( ⃗ p ) P ζ ( ⃗ k -⃗ p ) . (A.30)  \nAdopting the coordinates of two polarization vectors and ⃗ k as  \ne ( ˆ k ) = (1 , 0 , 0) , ¯ e ( ˆ k ) = (0 , 1 , 0) , ⃗ k = (0 , 0 , k ) , ⃗ p = p (sin θ cos ψ, sin θ sin ψ, cos θ ) , (A.31) one can simplify the expressions as  \n⟨ G λ ( τ, ⃗ k ) G s ( τ, ⃗ k ' ) ⟩ = 8 π 5 k 3 δ λs δ ( ⃗ k + ⃗ k ' ) ∫ ∞ 0 d y ∫ 1+ y | 1 -y | d x y 2 x 2 [ 1 -( 1 + y 2 -x 2 2 y ) 2 ] × ( 1 + y 2 -x 2 2 y ) 2 P ζ ( ky ) P ζ ( kx ) ∣ ∣ ∣ ∣ ∫ z z ini d˜ z ˜ z 4 z 4 f ( x, y, ˜ z ) ∣ ∣ ∣ ∣ 2 , (A.32)  \nwith the introduction of auxiliary variables  \nx ≡ | ⃗ k -⃗ p | k , y ≡ p k ; z ≡ kτ < 0 , z 0 ≡ kτ 0 < 0 . (A.33)  \nFollowing the convention in the study of induced gravitational waves, we further define  \ns = y -x √ 2 , t = y + x √ 2 , (A.34)  \nand the total vector power spectrum is calculated as  \nP G ( τ, ⃗ k ) = ∫ ∞ 1 √ 2 d t ∫ 1 √ 2 -1 √ 2 d s (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 ×P ζ ( k √ 2 ( t -s ) , τ ) P ζ ( k √ 2 ( t + s ) , τ ) |I ( s, t, z ) | 2 , (A.35)  \nwhere  \nI ( s, t, z ) = ∫ z z ini d˜ z ˜ z 4 z 4 f ( s, t, ˜ z ) , (A.36)  \nis the time integral, z ini corresponds to the value of z at far past, i.e., z ini = kτ ini . Explicitly, we have  \nf ( s, t, ˜ z ) ≡ 3 z 6 0 e -ic s ( √ 2 t ˜ z -2 z 0 ) ( t 2 -s 2 ) 3 ˜ z 6 ( c s z 0 -i ) 2 [ 2 √ 2 ic s t ˜ z (14 -c 2 s ˜ z 2 ( t 2 -s 2 )) +28 -2 c 2 s (9 t 2 -5 s 2 )˜ z 2 + c 4 s ˜ z 4 ( t 2 -s 2 ) 2 ] . (A.37)  \nThe time integral (A.36) can be integrated out analytically 1 ,  \nF (˜ z ) ≡ ∫ d˜ z ˜ z 4 z 4 f ( s, t, ˜ z ) = 3 z 6 0 e -i ( √ 2 c s t ˜ z -2 z 0 ) √ 2˜ zt 3 z 4 ( t 2 -s 2 ) 3 (1 + ic s z 0 ) 2 [ -i ˜ z 3 c 3 s t 2 ( t 2 -s 2 ) 2 + ic s ˜ z ( s 4 -14 s 2 t 2 +21 t 4 ) -√ 2 c 2 s t ˜ z 2 ( s 4 -4 s 2 t 2 +3 t 4 ) + 28 √ 2 t 3 ] . (A.38)  \nNotice that the specific time scale τ 0 appears as we set a reference time scale τ 0 in the definition of transfer function (A.24). Introducing g ≡ z ini /z 0 to label the duration of contraction phase and an auxiliary variable u ≡ c s z 0 , we have  \n|I ( z 0 ) | 2 = 9 u 2 (1 + u 2 ) -2 2 c 2 s t 6 ( t 2 -s 2 ) 6 [ I 1 + I 2 cos ( √ 2(1 -g ) tu ) + I 3 sin ( √ 2(1 -g ) tu )] , (A.39)  \nwhere  \nI 1 = ( g 4 +1 ) ( t 2 -s 2 ) 4 t 4 u 6 +8 ( g 2 +1 ) ( 2 s 2 -3 t 2 ) ( s 2 -t 2 ) 2 t 4 u 4 +2 ( s 8 -28 s 6 t 2 +126 s 4 t 4 -140 s 2 t 6 +105 t 8 ) u 2 +1568 ( 1 + g -2 ) t 6 , (A.40)  \nI 2 = 2 g 2 t 2 u 4 ( s 2 -t 2 ) 2 [ s 4 -t 2 u 2 ( s 2 -t 2 ) 2 -14 s 2 t 2 +21 t 4 ] -4 gt 2 u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] +2 u 2 ( s 4 -14 s 2 t 2 +21 t 4 ) [ -s 4 + t 2 u 2 ( t 2 -s 2 ) 2 +14 s 2 t 2 -21 t 4 ] +112 g -1 t 4 [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] , (A.41)  \nI 3 = 2 √ 2 g 2 t 3 u 3 ( s 2 -t 2 ) 2 [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] -2 √ 2 gtu 3 ( s 4 -4 s 2 t 2 +3 t 4 ) [ -s 4 + t 2 u 2 ( t 2 -s 2 ) 2 +14 s 2 t 2 -21 t 4 ] -2 √ 2 tu ( s 4 -14 s 2 t 2 +21 t 4 ) [ u 2 ( s 4 -4 s 2 t 2 +3 t 4 ) -28 t 2 ] +56 √ 2 g -1 tu [ t 4 u 2 ( s 2 -t 2 ) 2 -t 2 ( s 4 -14 s 2 t 2 +21 t 4 ) ] . (A.42)", 'B Vector Power Spectrum from Numerical Evaluation': "We're interested in the energy density of the induced vector perturbation,  \nρ V ( ⃗x, τ ) = 1 4 a 2 ∂ i G j ( ⃗x, τ ) ∂ i G j ( ⃗x, τ ) , (B.1)  \nwhich is related to the vector power spectrum as  \nρ V ( τ ) = 2 × 1 4 a 2 ∫ k P G ( τ, ⃗ k )d k = 1 2 a 2 ∫ k P G ( τ, ⃗ k )d k . (B.2)  \nThe factor of 2 arises from the two polarizations of vector perturbations. The vector energy density, ρ V , is directly determined by the vector power spectrum, P G . Therefore, we will focus on evaluating the vector power spectrum numerically in the following.  \nFirst, Fig. 2 demonstrates that the oscillatory terms in Eq. (A.39), namely the I 2 and I 3 terms, contribute significantly on larger scales. Conversely, on scales k ≃ k max , the value of Eq. (A.39) is primarily determined by the non-oscillatory term I 1 . In both cases, the power spectrum rapidly diminishes on scales k ≥ k max due to the cutoff of the curvature power spectrum in Equation (3.4). On scales k < k max , P G exhibits a strong blue tilt, which can be parameterized as  \nP G ( τ 0 , k ) = A G ( k k max ) n G , k < k max , (B.3)  \nfrom which we obtain the ratio of ρ V versus the background energy density value ρ bg ,  \nδ V ≡ ρ V ( τ 0 ) ρ bg ( τ 0 ) = 1 24 ∫ ( kτ 0 ) P G ( k, τ 0 )d( kτ 0 ) , (B.4)  \nFigure 2 . Comparison of the vector power spectrum P G calculated using Eq. (A.39) (including oscillatory terms) with P G evaluated using only the non-oscillatory term I 1 . The parameters used are c s = 1 and g = 10 4 .  \n<!-- image -->  \nas  \nδ V ( τ 0 ) = 1 24 ∫ ( kτ 0 ) P G ( k, τ 0 )d( kτ 0 ) < 1 6 c 2 s ∫ 1 0 A G ( k k max ) n G +1 d ( k k max ) = A G 6 c 2 s ( n G +2) . (B.5)  \nAs Fig. 2 illustrates, the non-oscillatory terms primarily determine the amplitude A G of the vector power spectrum, while the oscillatory terms are responsible for its spectral index n G . However, neglecting the oscillatory terms leads to a difference in the calculated n G by a factor of order unity. For example, with c s = 1 and g = 10 4 , we find n G = 8 . 2 when excluding oscillatory terms and n G = 6 . 0 when using the full expression in Equation (A.39). This discrepancy can be understood as follows: When k ≃ k max , we have u ≃ -2 and g ≫ 1 , making I 1 the dominant term since I 1 = O ( g 4 ) while I 2 and I 3 are of order O ( g 2 ) . Conversely, on larger scales where k ≃ k min , we have u ≃ -2 g -1 , resulting in I 1 = O ( g 0 ) and I 2 , I 3 = O ( g -1 ) . In this case, the oscillatory terms become significant. For simplicity, we will proceed with the numerical calculations by evaluating only the contributions from the non-oscillatory term. As we will demonstrate later, the resulting δ V is significantly larger than unity. Therefore, any potential loss of a factor of order unity in this numerical approximation will not significantly impact our conclusions.  \nFigure 1 displays the resulting vector power spectrum P G ( τ 0 , k ) for various values of c s and g . The figure reveals that the amplitude A G is proportional to c -2 s g 4 . More precisely, as we will demonstrate below, A G exhibits the following behavior for g ≫ 1 :  \nA G ≃ A 2 s c 2 s g 4 ( C 1 log g + C 2 ) + O ( g 3 ) . (B.6)  \nThe value of A G is shown in Fig. 3. A direct fit of the numerical result gives C 1 = 15 . 45 and C 2 = -14 . 20 . App. C provides a semi-analytical explanation for Eq. (B.6).  \nNow we conclude the vector power spectrum for different model parameters in Table 2. The log g dependence in Eq. (B.6) is implicitly reflected in the running of the spectral indices.  \nFigure 3 . The value of A G as a function of g with c s = 1 fixed.  \n<!-- image -->  \nTable 2 . Vector power spectra (B.3) and the values of δ V for various values of c s and g .  \nAs argued in Sec. 3, g ≫ 10 4 , making the log g term in Eq. (B.6) dominant. Therefore, we arrive at the scaling,  \nA G ∝ A 2 s c 2 s g 4 log g , δ V ∝ A 2 s c 4 s ( n G +1) g 4 log g . (B.7)  \nAs a comparison, the energy density of linear curvature perturbations is estimated as [64]  \nρ ζ ≡ -1 2 R→ ρ ζ ∝ ∫ k max k min dln k k 2 a 2 P ζ , (B.8)  \nusing Eq. (A.2). Notice that ρ ζ scales as g 2 assuming a scale-invariant curvature power spectrum P ζ . Thus, the energy density of SIVP exhibits a faster growth rate with respect to g , leading to a more severe back-reaction problem during the contraction phase.", 'C Analysis on the SIVP Power Spectrum': 'In this appendix, we conduct a semi-analytical study on the vector power spectrum. Using the identity  \n| a 2 | -2 | ab | + | b 2 | ≤ | a -b | 2 ≤ | a 2 | +2 | ab | + | b 2 | , (C.1)  \nwe get  \nJ 1 -J 2 + J 3 ≤ |F ( z 0 ) -F ( gz 0 ) | 2 ≤ J 1 + J 2 + J 3 , (C.2)  \nFigure 4 . The integration range of momentum integrals. We take g = 4 here for illustrative purposes.  \n<!-- image -->  \nwith  \nJ 1 ≡ |F ( z 0 ) 2 | , J 2 ≡ 2 |F ( z 0 ) F ( gz 0 ) | , J 3 ≡ |F ( gz 0 ) 2 | . (C.3)  \nOur primary interest lies in determining the value of A G , which corresponds to the value of P G ( k, τ 0 ) at k = k max . For the remainder of this discussion, we will fix k to be equal to k max . The integration range relevant to the power spectrum is defined by the following identity:  \nk min ≤ k 1 , k 2 ≤ k max ; | k 1 -k 2 | ≤ k max , k 1 + k 2 ≥ k max , (C.4)  \nwhich we depict in Fig. 4. In terms of variable s and t , it becomes  \n∫ √ 2 g +1 √ 2 g d t ∫ √ 2 -t t -√ 2 d s + ∫ g +1 √ 2 g 1 √ 2 d t ∫ t -√ 2 g √ 2 g -t d s . (C.5)  \nOne can straightforwardly calculate the integrations. However, the resulting formulae is tediously long and we will use a simpler strategy. We observe that contributions to the integrals mainly comes from the boundary near the points t = 1 √ 2 , s = ± 1 √ 2 , where we can expand the integrand as a series of ( t -1 √ 2 ) , which greatly simplify the calculations. Further, we will expand the results around g →∞ , which gives  \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 J 1 ≃ 27 g 4 z 2 0 (49 + c 2 s z 2 0 ) 56(1 + c 2 s z 2 0 ) , (C.6)  \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 J 3 ≃ 3 g 4 c 2 s z 4 0 (45 -84 c 2 s z 2 0 +140 c 4 s z 4 0 ln g ) 280(1 + c 2 s z 2 0 ) 2 , (C.7)  \nup to O ( g 3 ) contributions. We are interested in the value of P G at k = k max , that is z 0 c s = -2 and  \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 P 2 ζ J 1 ( k = k max ) ≃ 1431 g 4 A 2 s 350 c 2 s , (C.8)  \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 P 2 ζ J 3 ( k = k max ) ≃ 6 g 4 A 2 s 875 c 2 s (2240 ln g -291) . (C.9)  \nThe momentum integral for J 2 is a bit tricky, since it contains absolute values. For our purpose, we notice that near the pole, the real part of F ( z 0 ) is always no less than its imaginary part. Thus we take |F ( z 0 ) | ≤ √ 2 ℜ ( F ( z 0 )) . On the other hand, we have F ( gz 0 ) to be governed by its g 2 term in the g →∞ limit. Thus we arrive at  \n∫ d s d t (1 -2 s 2 )(2 t 2 -1)(2 st +1) 2 4( t 2 -s 2 ) 2 P 2 ζ J 2 ( k = k max ) ≃ 4032 g 4 A 2 s 125 c 2 s . (C.10)  \nCombining all pieces, we get  \n15 . 36 log g -30 . 16 ≤ A G c 2 s A 2 s g 4 ≤ 15 . 36 log g +34 . 35 . 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