Viability of the matter bounce scenario in Loop Quantum Cosmology from BICEP2 last data

JCAP08(2014)025 ournal of Cosmology and Astroparticle Physics An IOP and SISSA journal J Viability of the matter bounce scenario in Loop Quantum Cosmology from BICEP2 last data Jaume de Haro and Jaume Amor´os Departament de Matem`atica Aplicada I, Universitat Polit`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mail: [email protected],[email protected] Received May 14, 2014 Revised July 3, 2014 Accepted July 19, 2014 Published August 12, 2014 Abstract. The CMB map provided by the Planck project constrains the value of the ratio of tensor-to-scalar perturbations, namely r, to be smaller than 0.11 (95 % CL). This bound rules out the simplest models of inflation. However, recent data from BICEP2 is in strong tension with this constrain, as it finds a value r= 0.20+0.07 −0.05 with r= 0 disfavored at 7.0σ, which allows these simplest inflationary models to survive. The remarkable fact is that, even though the BICEP2 experiment was conceived to search for evidence of inflation, its experimental data matches correctly theoretical results coming from the matter bounce scenario (the alternative model to the inflationary paradigm). More precisely, most bouncing cosmologies do not pass Planck’s constrains due to the smallness of the value of the tensor/scalar ratio r≤0.11, but with new BICEP2 data some of them fit well with experimental data. This is the case with the matter bounce scenario in the teleparallel version of Loop Quantum Cosmology. Keywords: alternatives to inflation, cosmological perturbation theory ArXiv ePrint: 1403.6396 c 2014 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1475-7516/2014/08/025 JCAP08(2014)025 Contents 1 Introduction 1 2 Constrains on inflationary models from experimental data 2 3 Calculation of the power spectrum in LQC 4 4 An specific example 7 4.1 Numerical results 9 5 Conclusions 11 1 Introduction The latest Planck temperature data for cosmic inflation constrains the spectral index for scalar perturbations to be ns= 0.9603 ±0.0073, ruling out exact scale invariance with over 5σconfidence, and establishes an upper bound for tensor/scalar ratio given by r≤0.11 (95 % CL) [1]. Such data shrinks the set of allowed simplest inflationary models: power law potentials in chaotic inflation [2], exponential potential models [3], inverse power law potentials [4], are disfavored because they do not provide a good fit to Planck’s data [1,5]. In fact, this data set prefers a subclass of inflationary models with plateau-like inflation potentials (see for example [6]) and R2gravity [7,8]. On the other hand, recent results from the BICEP2 experiment [9], designed to look for the signal of gravitational waves in the B-mode power spectrum, lead to the same constrain for the spectral index, but constrain the ratio of tensor-to-scalar perturbations to be r= 0.20+0.07 −0.05 with r= 0 disfavored at 7.0σ(see figure 13 of [9] to compare Planck’s with BICEP2 data). This higher value of rextends the set of compatible inflationary models, allowing back some of the simplest inflationary models cited above. Dealing with the matter bounce scenario, the alternative to the inflationary paradigm (see [10] for a report about bouncing cosmologies), one encounters a similar problem when one tries to match Planck’s data with theoretical results: theoretical results provide, in general, values of rhigher than 0.11 and, then, to sort out this problem some very complicated mechanism has to be introduced to enhance the power spectrum of scalar perturbations [11], reducing the ratio renough to achieve the bound 0.11. However, in this work we will show that the higher value of rprovided by BICEP2 allows the viability of some bouncing models. This is the main goal of the paper. As a matter of fact, we will deal with the matter bounce scenario in Loop Quantum Cosmology (LQC) which, when one only takes into account holonomy corrections, provides the simplest bounce. More precisely, it is well known that LQC contains two kind of corrections: holonomy corrections and inverse-volume effects. When one deals with the flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) geometry, holonomy corrections always lead to a big bounce (see for instance [13]), however this could not happen when one considers inverse-volume effects. For example, when the universe is filled by a field under the action of a non-negative potential (to guarantee a positive energy density), one will obtain a non bouncing universe because the Hubble parameter never vanishes (see equations (5) and (8) – 1 – JCAP08(2014)025 of [14]). That is the reason why, in this paper, we will do not take into account inverse-volume corrections. On the other hand, for the flat FLRW geometry, it has been recently showed in [15,16] that holonomy corrected LQC can be formulated as a particular example of teleparallel F(T) gravity, where Tis the so-called torsion scalar whose value in the flat FLRW spacetime is equal to −6H2. This new formulation of LQC with holonomy corrections has been named teleparallel LQC and only coincides with the standard holonomy corrected LQC in the FLRW geometry. Dealing with cosmological perturbations both formulations lead to different perturbation equations and, of course, to different results. The reason of this difference is that in holonomy corrected LQC, working in the Hamiltonian framework, the corresponding perturbation equations are obtained replacing the Ashtekar connection by a suitable sinus function in the classical Hamiltonian and inserting in it counter-terms to preserve the algebra of constrains [19,20]). In constrast to holonomy corrected LQC, the perturbation equations in teleparallel LQC are directly obtained, in the Lagrangian framework, from the well-known perturbation equations in teleparallel F(T) gravity [21–23]. In fact, it has been shown in [12] that for scalar perturbations both formulations lead to the same kind of results, the difference appears when one deals with tensor perturbations, because in teleparallel LQC the equation of perturbations [12] is a regular equation, but in holonomy corrected LQC the corresponding equation [20] has two singular points (at the beginning and end of the super-inflationary phase). This difference is what leads to completely different results. To show that, we deal with the matter bounce scenario in LQC, where the universe is filled by only a scalar field whose potential is the simplest one leading, at early times, to a matter domination in the contracting phase. In this case, the conservation equation is a second order differential equation (a Klein-Gordon equation). Each orbit, i.e. each solution of this differential equation, depicts a different matter dominated universe at early times. We will see that one of these orbits can be calculated analytically (the orbit that depicts a matter dominated universe for all time), but the other ones have to be calculated numerically. Then, for all of these orbits we will calculate analytically and numerically, the corresponding tensor/scalar ratio for adiabatic perturbations (we only considers one matter field, meaning there are not entropy perturbations) coming from holonomy corrected and teleparallel LQC, and we will check that in the case of teleparallel LQC there are orbits leading to theoretical results that match correctly with BICEP2 data, and there are other orbits that provide theoretical results that fit well with Planck’s data. On the other hand, we will also show numerically that holonomy corrected LQC, provides theoretical results, that only match correctly with Planck’s data. The units used in the paper are ~=c= 8πG = 1. 2 Constrains on inflationary models from experimental data Slow-roll inflation is essentially based in two parameters [24]: =−˙ H H2and η= 2−˙ 2H,(2.1) where ˙ is the derivative with respect to the cosmic time. In the slow-roll phase, i.e., when the dynamics of the system is given by equations H2∼ =V( ¯ϕ) 3and 3H˙ ¯ϕ+V¯ϕ∼ =0,(2.2) – 2 – JCAP08(2014)025 where ¯ϕ(t) is the homogeneous part of the scalar field, are given by ∼ =1 2V¯ϕ V2 and η∼ =V¯ϕ¯ϕ V.(2.3) Using slow-roll parameters and ηthe spectral index for scalar perturbations and the ratio of tensor-to-scalar perturbations are given by ns∼ =1+2η−6and r∼ =16. (2.4) To compare theoretical results with current observations we need the number of e-folds during inflation, namely N, which in slow-roll approximation is given by N=Zte tb Hdt ∼ =Z¯ϕb ¯ϕe V V¯ϕ d¯ϕ, (2.5) where the sub-index b(resp. e) refers to the beginning (resp. end) of inflation. As a first example to compare theoretical with experimental results, we choose a power law potential V( ¯ϕ) = λ¯ϕ2n. For this potential one has ns∼ =1−4n(n+ 1) ¯ϕ2 b , r ∼ =32n2 ¯ϕ2 b and N∼ =¯ϕ2 b−2n2 4n,(2.6) where we have chosen as the end of inflation the condition = 1, which is equivalent to ¯ϕ2 e= 2n2, and to calculate nsand rwe have evaluated and ηat the beginning of inflation. Removing ¯ϕ2 bin (2.6), i.e., writing nsand rin terms of the number of e-folds, one gets ns∼ =1−2(n+ 1) 2N+n, r ∼ =16n 2N+n=⇒ns∼ =1−n+ 1 8nr. (2.7) In the case of a quadratic potential n= 1, for 60 e-folds, the minimum needed to solve the horizon and flatness problems if inflation starts at GUT energies [25], one gets ns= 0.9669 and r= 0.132. When one increases the number of e-folds, nsincreases and r decreases. Then, for the maximal allowed value of the spectral index ns= 0.9676 one has r= 0.1296, which means that the model with a quadratic potential does not fit well neither with Planck’s nor with BICEP2 data. In the same way, for the maximum value allowed of the spectral index, i.e. for ns= 0.9676 the value of ris minimum and is given by r=8n n+1 ×0.0324. Since rincreases as long as the parameter nincreases, and its minimum value is r= 0.1296 (reached when n= 1), one can conclude that inflationary power law models are disfavored by Planck’s data. However, using BICEP2 data, the model n= 2 with 70 e-folds is acceptable because it satisfies ns= 0.9577 and r= 0.2253. To be more specific, from the third equation of (2.7)r is constrained to belong in the interval 8n n+ 1 ×0.0324,8n n+ 1 ×0.047.(2.8) Then, for n≥1 the interval (2.8) has a non-empty intersection with (0.15,0.27). This means that for all values of n≥1, there exist values of Nsuch that nsand rare allowed from BICEP2 data. However, we need that Nwas greater than 60, which can be checked as follows: first of all, we have – 3 – JCAP08(2014)025 1. For n= 1, the allowed values of rbelong in (0.15,0.188). 2. For n= 2 the allowed values of rbelong in 8n n+1 ×0.0324,8n n+1 ×0.047. 3. For n≥3 the allowed values of rbelong in 8n n+1 ×0.0324,0.27. Finally, from the value of r(the second equation of (2.7)) one has 1. For n= 1, Nbelongs in (42.05,52.8). 2. For n≥2, one has N≥62,82, meaning that for n≥2, the model matches correctly with BICEP2 data. As a second example we consider R2gravity, sometimes called Starobinsky model (see [26] for a detailed description of the model). In R2gravity one has [7,8] ns= 1 −2 N, r =12 N2=⇒ns= 1 −rr 3.(2.9) Using the data ns= 0.9603 ±0.0073 and equation (2.9) one obtains the constrain 0.0031 ≤r≤0.0066, what means that BICEP2 data disregards this model. However, the model matches correctly with Planck’s data. Effectively, for 60 e-folds one has ns= 0.9666 and r= 0,0033 which enters perfectly in the range of values obtained from Planck’s temperature anisotropy mesurements. 3 Calculation of the power spectrum in LQC In this section we will obtain the formulas to calculate the power spectrum for scalar and tensor perturbations, in both holonomy corrected and teleparallel LQC, when one deals with the matter bounce scenario. It is well known that, when one only takes into account holonomy corrections, the modified Friedmann equation in the flat FLRW geometry is given by the following ellipse in the plane (H, ρ) H2=ρ 31−ρ ρc,(3.1) where ρcis the so-called critical density. On the other hand, as we have already explained in the introduction, the equation (3.1) could be obtained as a particular case of teleparallel F(T) gravity. In [15,16] this example has been found to be F±(T) = ±r−Tρc 2arcsin s−2T ρc!+G±(T),(3.2) with G±(T) = ρc 2 1±s1 + 2T ρc!,(3.3) where + correspond to the super-inflationary phase, i.e. to ρ>ρc/2, and −to ρ<ρc/2. – 4 – JCAP08(2014)025 Now, dealing with adiabatic cosmological perturbations in the longitudinal gauge ds2= (1+2Φ)dt2−a2(1−2Φ)dx2where Φ is the Bardeen potential, and assuming that the matter part of the Lagrangian is depicted by only one scalar field ϕ= ¯ϕ+δϕ, where ¯ϕis the homogeneous part of the field, one can show that the Mukhanov-Sasaki (M-K) equations for adiabatic perturbations are given by [12,19,20] v00 S(T);h(t)−c2 s;h(t)∆vS(T);h(t)−z00 S(T);h(t) zS(T);h(t) vS(T);h(t)= 0,(3.4) where 0represents the derivative with respect the conformal time, Smeans scalar perturbations, Ttensor perturbations, hholonomy corrected LQC and tteleparallel LQC, and the square of the velocity of sound in the corresponding approach is given by c2 s,h ≡Ω=1−2ρ ρc ;c2 s,t =|c2 s,h| arcsin 2q3 ρcH 2q3 ρcH .(3.5) Moreover the M-K variables zS(T);h(t)and vS(T);h(t)are defined as follows: zS;h=a˙ ¯ϕ H, zT;h=a cs;h , zS;t=a|cs;h|˙ ¯ϕ cs;tH, zT;t=acs;t |cs;h|,(3.6) and vS(T);h(t)=ζS(T);h(t)zS(T);h(t), where ζS;h(t)≡Φ + H ˙ ¯ϕδϕ is the curvature fluctuation in co-moving coordinates and ζT;h(t)is the amplitude of tensor perturbations. Remark 3.1 From the definitions of the M-S variables we can see that for scalar perturbations, the equations in holonomy corrected and teleparallel LQC are essentially the same. They are singular at the bouncing point (when Hvanishes), and differ with the value of square of the velocity of sound, which in the case of holonomy corrected LQC becomes negative in the super-inflationary phase (ρ>ρc/2), but as we will see, to calculate the power spectrum of perturbations the term containing the Laplacian could be disregarded. In constrast, for tensor perturbations the equations are completely different. In the case of holonomy corrected LQC it contains two singular points, at the beginning and end of the super-inflationary phase, i.e., when ρ=ρc/2. This does not happen in the teleparallel version where the corresponding M-S equation is always regular. We will see that due to this difference the ratio of tensor to scalar perturbations is completly different depending on the approach used. Once we have the perturbation equations, we can deal with the matter bounce scenario. In this scenario, in order to have a scale invariant spectrum, the universe has to be matter dominated, at early times, in the contracting phase. This is due to the duality, pointed out in [17], between matter domination in the contracting phase and de Sitter regime in the expanding one. Then, since at early times the holonomy effects can be disregarded because ρρc(the universe is in the bottom of the ellipse (3.1)), and the universe is matter dominated at this epoch, one will obtain zS;h=zS;t=√3a, zT;h=zT;t=a, (3.7) where a(t) = 3 4ρct2+ 11/3∼ =3 4ρc1/3t2/3=ρc 12η2, being tthe cosmic time and ηthe conformal time [12]. – 5 – JCAP08(2014)025 As a consequence, at early times, the M-S equations, in Fourier space, will becomes v00 S(T);h(t)+k2−a00 avS(T);h(t)= 0 ⇐⇒ v00 S(T);h(t)+k2−2 η2vS(T);h(t)= 0,(3.8) whose solutions are the mode functions vS(T);h(t)=e−ikη √2k1−i kη ,(3.9) that depict the Bunch-Davies (adiabatic) vacuum when η→ −∞. On the other hand, at early times, modes well outside the Hubble radius satisfy the long wavelength condition k2η21, and thus, the M-S equations (3.4) can be approximated by v00 S(T);h(t)−z00 S(T);h(t) zS(T);h(t) vS(T);h(t)= 0,(3.10) which solution is the so-called long wavelength approximation vS(T);h(t)(η) = AS(T)(k)zS(T);h(t)(η) + BS(T)(k)zS(T);h(t)(η)Zη −∞ d¯η z2 S(T);h(t)(¯η).(3.11) The long wavelength approximation can be explicitely calculated at early times using (3.7), yielding vS;h(t)(η)∼ =AS(k) 4√3ρcη2−4BS(k) √3ρc 1 η, vT;h(t)(η)∼ =AT(k) 12 ρcη2−4BT(k) ρc 1 η.(3.12) To obtain the value of these coefficients on has to match, in the long wavelength regime k2η21, the approximate solutions (3.12) with the exact modes (3.9), giving as a result [12, 28] AS(k) = AT(k) √3=−r8 3 k3/2 ρc , BS(k) = √3BT(k) = ir3 8 ρc 2k3/2.(3.13) Once we have calculated these coefficients we will use the long wavelength approximation (3.11) to calculate, at late times (η→ ∞), the curvature fluctuation in co-moving coordinates ζS,h(t)and the amplitude for tensor perturbations ζT,h(t), obtaining [12] ζS(T),h(t)=vS(T);h(t)(η) zS(T);h(t)(η)=AS(T)(k) + BS(T)(k)RS(T);h(t)∼ =BS(T)(k)RS(T);h(t),(3.14) where RS(T),h(t)∼ =R∞ −∞ d¯η z2 S(T),h(t)(¯η)=R∞ −∞ d¯ t a(t)z2 S(T),h(t)(¯ t). From this result we can calculate the power spectrum of scalar and tensor perturbation, in both approaches, as follows: PS(T);h(t)(k)≡k3 2π2|ζS(T);h(t)|2=3ρ2 c ρpl R2 S(T);h(t),(3.15) where ρpl is the Planck’s energy density, which in our units equals to 64π2. And also the tensor/scalar ratio of perturbations rh(t)≡PT;h(t)(k) PS;h(t)(k)=R2 T,h(t) R2 S,h(t) .(3.16) To end this section, two important final remarks are in order: – 6 – JCAP08(2014)025 1. The formulas (3.15) and (3.16) are essential to perform numerical and analytic calculation in the matter bounce scenario. It is also important to note that, in order to obtain them, only a matter dominated universe at early times in the contracting phase has been required. Indeed, in next section we will provide the simplest example that satisfies this requeriment and allows us to perform, with all the details, all the numerical and analytic calculations. 2. As we have already remarked, the M-S equations (3.4) contain singular points, which means that there are infinitely many ways to match solutions at these points, and thus, one has infinitely many mode solutions that lead to infinitely many different power spectrums. However, if one assumes that ζS(T);h(t)(η) has to be an analytic function for all time η, then there is only one solution that satisfies this requirement: the one given by (3.11). That is the reason why we use the long wavelength approximation (3.11) to calculate the power spectrum of scalar and tensor perturbations in both approximations. 4 An specific example In this section we will find a potential that leads to an analytic solution that depicts, all time, a matter dominated universe. For this potential we also find numerically all the other solutions and, from formula (3.16), we will calculate, for each solution, their corresponding tensor/scalar ratio. To find this potential, first of all, we will solve the holonomy corrected Friedmann equation and the conservation equation for a matter dominated universe (see for instance [13]) H2=ρ 31−ρ ρc; ˙ρ=−3Hρ, (4.1) obtaining the following quantities [12] a(t) = 3 4ρct2+ 11/3 and ρ(t) = ρc 3 4ρct2+ 1.(4.2) To find such potential, one can impose that the pressure vanishes, i.e., P≡˙ ¯ϕ2 2−V( ¯ϕ) = 0, which leads to the equation ˙ ¯ϕ2(t) = ρ(t)⇐⇒ ˙ ¯ϕ2(t) = ρc 3 4ρct2+ 1,(4.3) where we have used the second equation of (4.2). This equation has the particular solution ¯ϕ(t) = 2 √3ln r3 4ρct+r3 4ρct2+ 1!,(4.4) which leads to the potential V( ¯ϕ) = 2ρc e−√3 ¯ϕ 1 + e−√3 ¯ϕ2.(4.5) – 7 – JCAP08(2014)025 It is important to realize that the analytic solution (4.4) is special in the sense that it satisfies for all time ˙ ¯ϕ2(t)/2 = V( ¯ϕ(t)), that is, if the universe is described by this solution, it will be matter dominated all the time. However, all the other solutions, that is, the other solutions that only can be obtained numerically from the conservation equation ˙ρ=−3H±(ρ+P)⇐⇒ ¨ ¯ϕ+ 3H±˙ ¯ϕ+V¯ϕ= 0,(4.6) where the Hubble parameter is equal to H−=−qρ 3(1 −ρ ρc) in the contracting phase and H+=qρ 3(1 −ρ ρc) in the expanding one, do not lead to a matter-dominated universe all the time. Only at early and late times the universe is matter dominated because the solution (4.4) is a global repeller at early times and a global attractor at late ones (see [18] for a demonstration). Once we have introduced the simplest potential for the matter bounce scenario in LQC, we deal with scalar perturbations. In the case of holonomy corrected LQC for the analytic solution (4.4) one has zS;h=2a5/2(t) √ρct[28], and which leads, after using formula (3.15), to PS;h(k) = π2 9 ρc ρpl .(4.7) On the other hand, in teleparallel LQC, whose perturbation equations, as we have explained in the introduction, are the ones of F(T) gravity [21–23] applied to a model (see eq. (2.12) and (2.23) of [12]) whose teleparallel Friedmann equation coincides with the holonomy corrected one (3.1), for the particular solution (4.4) one has zS;t(t) = 2 3 ρc1/4a(t)|t|1/2 trarcsin √3ρc|t| a3(t) ,(4.8) giving as a power spectrum PS;t(k) = 16 9 ρc ρpl C2,(4.9) where C∼ =0.9159 is Catalan’s constant. This result has to be compared with the seven-year data of WMAP [29], which constrains the value of the power spectrum for scalar perturbations to be P(k)∼ =2×10−9, which means that, in both cases (holonomy corrected and teleparallel LQC), when one considers the solution (4.4), the value of the critical density has to be of the order ρc∼10−9ρpl. Dealing with the tensor/scalar ratio of perturbation, for the analytical solution (4.4), in holonomy corrected LQC, after using formula (3.16) one has rh= 0 which is an abnormally small value, and in teleparallel LQC we have obtained the following very high value rt= 3Si(π/2) C2∼ =6.7187, where Si(x)≡Rx 0 sin y ydy is the Sine integral function. However, these results do not mean that the matter bounce model depicted by the potential (4.5) has to be disregarded. What they mean is that, for orbits (solutions of (4.6)) near the solution (4.4), the theoretical results given by holonomy corrected and teleparallel LQC do not match with the current experimental data. But, as we will see numerically, in the case of teleparallel LQC, there are other orbits whose theoretical results fit well with data – 8 –