Cosmological Perturbations in the 5D Holographic Big Bang Model
Natacha Altamirano, Elizabeth Gould, Niayesh Afshordi, Robert B. Mann
PPrepared for submission to JCAP
Cosmological Perturbations in the 5DHolographic Big Bang Model
Natacha Altamirano a,b
Elizabeth Gould a,b
Niayesh Afshordi a,b
Robert B. Mann b a Perimeter Institute for Theoretical Physics,31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada b Department of Physics and Astronomy, University of Waterloo,Waterloo, ON, N2L 3G1, CanadaE-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
The 5D Holographic Big Bang is a novel model for the emergence of the early uni-verse out of a 5D collapsing star (an apparent white hole), in the context of Dvali-Gabadadze-Porrati (DGP) cosmology. The model does not have a big bang singularity, and yet can ad-dress cosmological puzzles that are traditionally solved within inflationary cosmology. In thispaper, we compute the exact power spectrum of cosmological curvature perturbations due tothe effect of a thin atmosphere accreting into our 3-brane. The spectrum is scale-invarianton small scales and red on intermediate scales, but becomes blue on scales larger than theheight of the atmosphere. While this behaviour is broadly consistent with the non-parametricmeasurements of the primordial scalar power spectrum, it is marginally disfavoured relativeto a simple power law (at 2.7 σ level). Furthermore, we find that the best fit nucleationtemperature of our 3-brane is at least 3 orders of magnitude larger than the 5D Planck mass,suggesting an origin in a 5D quantum gravity phase. Keywords:
Cosmology, Early Universe, DGP, Scalar Power Spectrum a r X i v : . [ a s t r o - ph . C O ] M a r ontents Modern cosmology continues to experience an astonishing degree of empirical success [1].The agreement with the phenomenological six-parameter ΛCDM paradigm is remarkable, allthe more so as the number of cosmological observations continue to increase at an acceleratedrate. Despite this success, there are still intriguing puzzles left unresolved: the big bang sin-gularity, the horizon and flatness problems (traditionally addressed within the inflationaryparadigm), as well as the nature of dark matter and dark energy. Recently [2], a novel cos-mological model was proposed in which our universe is a 3-brane emergent from the collapseof a 5-dimensional star. Motivated by the desire to see if a more satisfactory (or natural)understanding of these puzzles can emerge from an alternative description of the geometry,this model explains the evolution of our early universe whilst avoiding a big bang singu-larity. Furthermore, the model was shown to have a mechanism via which a homogeneousatmosphere outside the black hole generates a scale invariant power spectrum for primordialcurvature perturbations, (nearly) consistent with current cosmological observations [1].Our 5D holographic origin for the big bang is based on a braneworld theory that includesboth 4 dimensional induced gravity and
5D bulk gravity: the Dvali-Gabadadze-Porrati (DGP)model [3], with action S DGP = 116 πG (cid:90) bulk d x √− gR + 18 πG (cid:90) brane d x √− γK + (cid:90) brane √− γ (cid:18) R πG + L matter (cid:19) , (1.1)where g and γ , G and G , R and R are the metrics, gravitational constants and Ricci– 1 –calars of the bulk and brane respectively, while K is the mean extrinsic curvature of thebrane. Our universe, described by the metric ds = − dτ + a ( τ ) K [ dψ + sin ( ψ )( dθ + sin ( θ ) dφ )] , (1.2)is represented by a hypersurface in a 5 dimensional Schwarzschild black hole spacetime ds = − (cid:16) − µr (cid:17) dt + (cid:16) − µr (cid:17) − dr + r d Ω , (1.3)located at r = a ( τ ) √K . In this context, a pressure singularity is generically found when theenergy density of the holographic fluid ˜ ρ satisfies ˜ ρ = ˜ ρ s = G πG [4]. The authors in [2]showed that the singularity happens at early times in the cosmic history as matter decaysmore slowly than a − . However, under the evolution from smooth initial conditions, thepressure singularity can occur before Big Bang Nucleosynthesis (BBN), and is genericallyinside a white hole horizon. Alternatively, the universe could have emerged from the collapseof a 5D star into a black hole, just before BBN, removing both pressure and big bang/whitehole singularities. As advocated in the fuzzball program [5], the rate of this tunnelling isenhanced due to the large entropy of black hole microstates, which we speculate could matchthose of an expanding 3-brane thermal state. Our universe is represented by the boundary ofa 5D spherically symmetric spacetieme with metric (1.3), in which we impose Z boundaryconditions. This picture will be described in more detail in Section 2.Interestingly, this model not only circumvents the singularity at the origin of time,but can also address other problems of cosmology that are typically solved by inflation.Because the collapsing star could have existed long before its demise, it had enough timeto attain uniform temperature, thereby addressing the Horizon Problem . Furthermore, ifwe assume that the initial Hubble constant was of order of the 5D Planck mass, then thecurvature density − Ω k ∼ ( M r h ) − , where r h is the radius of the black hole. Consequently − Ω k ∼ M /M ∗ could become very small for massive stars, thus solving the Flatness Problem .More generically, the no hair theorem ensures that a 3-brane nucleated just outside the eventhorizon of a massive black hole has a smooth geometry.Yet another feature of the model is that a thermal atmosphere around the brane, com-posed of a gas of massless particles produces scale invariant curvature perturbations. In thiswork, we shall revisit this result, focusing our attention on the mechanism responsible fordeviations from scale-invariance in the primordial curvature power spectrum in the contextof the
5D Holographic origin of the Big Bang . To this end, we consider a thin atmospherethat can be regarded as infalling matter, or the outer envelope of the collapsing 5D star,which resides in the 5D bulk and thus contributes to its energy momentum tensor. In thiscontext, the DGP action Eq. (1.1) is modified to S = S DGP + (cid:90) bulk d x √− g L (1.4)where L accounts for the matter Lagrangian in the bulk. Consistency between cos-mological phenomenology and the DGP model implies that the brane is expanding outwards,and thus eventually encounters this atmosphere . We then compute the resulting power spec-trum of scalar curvature perturbations and the change of the Hubble parameter due to thisencounter. We find that the best fit nucleation temperature of the 3-brane is considerablylarger than the 5D Planck mass, perhaps indicating an origin in a 5D quantum gravity phase.In Section 2, we discuss a possible mechanism for brane nucleation and the differ-ent scales involved in our problem, giving a qualitative description of the different physics– 2 –rocesses. In Section 3, we solve Einstein equations with matter in the bulk, and the con-sequences that it has on the brane. In particular, we solve for the density profile of a 5Dspherically collapsing atmosphere and compute the change on the Hubble parameter as itfalls into our 3-brane. In Section 4, we study cosmological perturbations in the bulk andtheir projection onto the brane, making special emphasis on the curvature perturbation andits power spectrum. Section 5 compares our predictions against Planck data and contrastsit with the power-law power spectrum assumed in the ΛCDM model. We conclude our workwith discussion of the limitations and prospects of our model in Section 6. As described in the introduction, we are working in the context of the 5D HolographicBig Bang model [2] where our universe is modelled as a hypersurface (the brane) in a 5-dimensional Schwarzschild space time according to the embedding r = a ( τ ) √K . This constructionis a solution of the DGP action (1.1) once we impose a Z boundary condition on the brane.As a consequence, via the embedding constraint the brane becomes an outward travellingboundary of the higher dimensional spacetime, an assumption that is necessary if we wantour universe (represented by the brane) to be expanding.From the perspective of an observer in the bulk, this setup is reminiscent of a con-struction proposed by Witten called the ‘bubble of nothing’ [6], in which an interior region ofspace is missing, with space ending smoothly at the surface of this bubble (the brane). Onepossible scenario in the 5D Holographic Big Bang model is that the brane was formed by thequantum tunnelling of a collapsing star in 5 dimensions, with all of the degrees of freedomof the inner part of the collapsing matter becoming degrees of freedom of the brane. This isanalogous to the fuzzball paradigm, a model proposed to solve the information-loss paradox[7], which consists of the explicit construction of black hole microstates with no “datalesshorizon region”. The infalling matter can tunnel to a fuzzball state with amplitude [5] A ∼ e − G (cid:82) R ∼ e − αG M , (2.1)where α = O (1) and we have used the length scale r ∼ G M to estimate the EuclideanEinstein action for tunneling between two configurations that have the length and mass scalesset to those of the black hole. Although this amplitude is very small, the number of fuzzballconfigurations that a black hole can tunnel to depends on the Bekenstein-Hawking entropyas N ∼ e S BH ∼ e G M (2.2)yielding a significant probability to form a fuzzball. In fact, the two exponentials exactlycancel each other [5]. We anticipate a similar principle operating here, in which collapsingmatter at sufficiently high density – just prior to formation of a black hole horiozn– necessarilytunnels to a brane so as to avoid the ensuing quantum paradoxes that follow upon introducingan event horizon.Since it is well established that BBN happened in the formation of our universe, andthat in the 5D Holographic Big Bang (HBB) model [2] the pressure singularity genericallyforms before BBN, we consider a brane that must form before this. This means that thetemperature of nucleation must be at most the temperature of BBN – T BBN ∼ . T nuc ≥ T BBN ∼ . . (2.3)– 3 – igure 1 . Cartoon of the different scales treated in this problem, in the black hole rest frame. Theinner black circular arc represents the brane with radius R and the outer black circular arc representsthe brane with radius 2 R . H − is estimated by tracing light rays on the brane after it has doubledit size; it is small if the brane is traveling near the speed of light. The atmosphere is shown in yellowsitting in between the two red arcs with length ∆ L . Finally, let us mention that the DGP model possesses a scale r c = G G , above which 5-dimensional gravity dominates over 4-dimensional gravity. Constraints on the normal branchof the DGP model [9] give r c (cid:38) H − → M < ( H M / / → M < . (2.4)where M = πG ) / is the reduced 4D Planck mass. In this scenario, we are interested in the effects of a thin atmosphere located just outside thebrane. Although the brane forms a Z boundary, excluding the event horizon in Eq. (1.3),we shall refer to the metric in Eq. (1.3) as the black hole metric.In order to organize the different assumptions, we will review the implied hierarchy ofthe scales present in this problem. If we assume that the Hubble patch of our universe,at/near brane nucleation, is small enough to be insensitive to the curvature of the black holespacetime, we can assume that the atmosphere is just a perturbation around a Minkowskibackground. This limit implies H − (cid:28) R where R is the radius of the black hole, or braneupon nucleation. This assumption is also observationally motivated, since today we measure( HR ) − (cid:28) | Ω k | (cid:28) H − (cid:28) R . This will allow Planck 2015 constraints on the dark energy equation of state roughly imply | w | < .
11 at 95% level, atthe pivot redshift of z (cid:39) .
23 (Fig. 5 in [10]), which provides a similar bound on r c , using the DGP Friedmannequation with a cosmological constant. – 4 – igure 2 . Penrose diagram (left), and cartoon of the 5D star collapse (right), followed by thenucleation of a 3-brane (our universe). The star (in yellow) that is collapsing (nearly) forms a blackhole, but the 3-brane (red) will nucleate just prior to the formation of the event horizon, and traversesa thin atmosphere of infalling matter or atmosphere (cross-section shown in the cartoon at right). us to define metric perturbations in Section 4. We will then be interested in finding thebehaviour of the power spectrum of curvature perturbations for modes of wavelength λ , thatare of super-horizon size before BBN, but are now observable in the CMB sky. This restricts R (cid:29) λ (cid:29) H − .Finally, we would like to understand the behaviour of different physical quantities of thebrane, such as the behaviour of the Hubble parameter before and after the encounter withthe atmosphere. Assuming it can be considered to be a thin atmosphere in the 5 dimensionalspace time, the width ∆ L of the atmosphere needs to be smaller than R for the brane to crossthe atmosphere completely in less than a Hubble time. We then get the following hierarchyof different scales H − (cid:28) λ (cid:28) R, and ∆ L (cid:28) R (black hole frame) , (2.5) λ (cid:46) ∆ L (atmosphere frame) . (2.6)As we shall in Section 4, the latter inequality is the key ingredient that leads to a near scale-invariant spectrum of primordial curvature perturbations for large scales. This hierarchy ofscales is illustrated in Fig. 1. We want to study the influence of an atmosphere that is falling into the black hole as shownin in Fig. 3.1. For this, we assume that the brane is moving supersonically (in fact, almostwith the speed of light) into the atmosphere, and thus bulk metric perturbations do not reactto the brane’s presence until it runs into them.– 5 –he Einstein equations on the brane that follow from the action (1.4) are: G µν = 8 πG (cid:16) T µν + ˜ T µν (cid:17) (3.1)where the two different components of the energy-momentum tensor are T µν , the matter livingon the brane, and ˜ T µν the holographic fluid that is induced on the brane via the junctionconditions described below. Due to the symmetry of the (unperturbed) FRW spacetime, T µν has the form of a perfect fluid T µν = ( P + ρ ) u µ u ν + P γ µν , (3.2)where u µ is the 4-velocity of the fluid normalized such that u µ u µ = − T µν = 18 πG ( Kγ µν − K µν ) , (3.3)where K µν ≡ ∇ α n β e αµ e βν is the extrinsic curvature of the brane whose unit normal is n α . Here e αν ≡ ∂ ˆ x α ∂x ν , where we have associated the set of coordinates { ˆ x α } and { x ν } with the bulk andthe brane respectively. In addition to the Einstein equations (3.1), the continuity equationsfor the total matter living on the brane arising from the Bianchi identities are: ∇ µ (cid:16) T µν + ˜ T µν (cid:17) = 0 . (3.4)The Gauss-Codazzi equations constrain the geometric quantities of the brane with the matterpresent in the bulk ∇ µ ( Kg µν − K µν ) = 8 πG T αβ e αν n β , (3.5) R + K µν K µν − K = − πG T αβ n α n β , (3.6)where R = − πG (cid:16) T + ˜ T (cid:17) is the Ricci scalar of the brane. T αβ is the energy momentumtensor of the bulk which satisfies the Einstein’s equations on the bulk G αβ = 8 πG T αβ .Note that the first of the Gauss-Codazzi equations (3.5) reduces to the conservation of theholographic fluid ˜ T µν in the case that T αβ = 0. If the bulk matter flows into the brane, theholographic fluid is not conserved and the effect of the continuity equation (3.4) is to changethe matter on the brane through the holographic fluid in order for the sum of both to beconserved.In the same way, as a result of the symmetries of FRW spacetime, the holographic fluidmust have the form of a perfect fluid. Moreover, the 4-velocity of this fluid must coincidewith the 4-velocity of the normal matter on the brane:˜ T µν = (cid:16) ˜ P + ˜ ρ (cid:17) u µ u ν + ˜ P γ µν . (3.7)Combining Eqs. (3.3) and (3.7) we get: K µν = − πG (cid:20) (cid:16) ˜ P + ˜ ρ (cid:17) u µ u ν + 13 ˜ ρ µν (cid:21) . (3.8)– 6 – .2 Shift in the Hubble The rate of expansion described by the Hubble parameter will change as the brane goesthrough the atmosphere and we can find a general expression for H by studying the Einsteinequations and junction conditions for the DGP brane in the general case. This generalcase treats the bulk as a 5-dimensional Schwarzschild black hole (1.3) and the brane as ahypersurface parametrized by r = a ( τ ) √K , as detailed in the Introduction.From Equations (3.1) and (3.4-3.6) we obtain H + K a = 8 πG ρ + ρ ) (3.9)˙ ρ + ˙˜ ρ + 3 H (cid:16) ρ + ˜ ρ + P + ˜ P (cid:17) = 0 (3.10)˙˜ ρ + 3 H (cid:16) ˜ ρ + ˜ P (cid:17) = T αβ e ατ n β (3.11)0 = T αβ e αi n β (3.12)( ρ + ˜ ρ ) − (cid:16) P + ˜ P (cid:17) + 8 πG G (cid:18)
23 ˜ ρ + 2 ˜ P ˜ ρ (cid:19) = − G G T αβ n α n β (3.13)The quantity T αβ is the stress-energy of the atmosphere outside the black hole. This atmo-sphere will have two effects on the brane. It will induce metric and matter perturbationsin our universe, which we shall use to compute the curvature perturbation in Sec. 4 below.However it will also make the Hubble parameter change its value as the brane crosses theatmosphere: the brane will expand more slowly due to an extra source of infalling matter.Combining Eqs. (3.9) and (3.10) and ignoring the curvature term we get∆ H = − πG (cid:90) ( P T + ρ T ) dτ, (3.14)where the integral is performed in the proper time of the brane and P T = P + ˜ P , ρ T = ρ + ˜ ρ .Let’s first look at the behavior of the holographic fluid ˜ P and ˜ ρ . From Equation (3.13)we have ˜ P + ˜ ρ = 13 (cid:0) ˜ ρ ˜ ρ s − (cid:1) (cid:20) − G G T nn − ρ + 2 ˜ ρ ˜ ρ s + T (cid:21) , (3.15)where T = 3 P − ρ , T nn = T αβ n α n β and ˜ ρ s = G πG . In order to avoid the pressure singularitywe require ˜ ρ (cid:29) ˜ ρ s and in this limit the last equation becomes˜ P + ˜ ρ = 13 (cid:20) − G G T nn ˜ ρ s ˜ ρ + 2 ˜ ρ + T ˜ ρ s ˜ ρ (cid:21) . (3.16)Now let’s analyze the behavior for the the matter on the brane P and ρ . CombiningEqs. (3.10) and (3.11) and assuming an equation of state P = wρ the fluid on the branesatisfies ˙ ρ + 3 H ( ρ + P ) = − T αβ e ατ n β ⇒ ddτ ( ρa w +1) ) = − T αβ e ατ n β a w +1) . (3.17)If we now assume that the atmosphere is thin enough so that we can approximate the matterdistribution as a delta function (i.e. H ∆ τ (cid:28)
1, during the impact time ∆ τ ), we see that thelast equation will give a jump in the density (and hence in the pressure) proportional to astep function. In fact, if we consider the system of equations (3.9-3.13), with P = wρ , and adelta function T αβ , the only consistent solution would have a delta function in ˜ P , with step– 7 –unction jumps in other variables. As such, the biggest contribution in Eq. (3.14) is givenby the first term on the right hand side of Eq. (3.16):∆ H = G G (cid:90) T αβ n α n β ˜ ρ dτ [1 + O ( H ∆ τ )] . (3.18)We shall see in Sec. 4 below that the amplitude of curvature perturbations depends on∆ ln H = ∆ HH . To compute this, we note that from Eq. (3.9) we can write H ≈ πG ( ˜ ρ + ρ ) ≈ πG ρ in the regime where ρ (cid:29) ˜ ρ . Then∆ ln H ≈ G G (cid:114) πG √ ρ ˜ ρ (cid:90) T αβ n α n β dτ , (3.19)We can now use the solution in vacuum for ˜ ρ found in [2]˜ ρ = ˜ ρ s (cid:18) (cid:115) − ρ BH − ρ )˜ ρ s (cid:19) , (3.20)where ρ BH = k H r h πG a . In the approximation where ρ (cid:29) ρ BH and ρ (cid:29) ˜ ρ s we find˜ ρ ≈ ρ (cid:115) ρ s ρ (3.21)and then Eq. (3.19) reads ∆ ln H ≈ ρ (cid:90) T αβ n α n β dτ , (3.22)implying that the relative jump in the Hubble parameter due to a thin atmosphere is theratio of the work done by the pressure of the atmosphere to the energy of the brane. So far we have considered a general energy momentum tensor on the bulk responsible ofdynamic features on the brane. Let’s now consider that the bulk is filled with a relativisticspherically symmetric, collapsing 5D radiation atmosphere ( P = ρ ) that represents theatmosphere whose energy momentum tensor is T αβ ( w ) = ρ ( w ) (cid:20)(cid:18) (cid:19) δ α δ β + 14 η αβ (cid:21) . (3.23)In order to study the effect of this atmosphere we need to introduce scalar homogeneousperturbations in the bulk. A generalization of 4D perturbation theory allows us to write theperturbed metric of the bulk in the Newtonian gauge in 5D as ds = − [1 + 2Φ ( x α )] dt + [1 − ( x α )][ dx + dy + dz + dw ] . (3.24)where Φ and Ψ represent the scalar perturbations of the bulk and x α are bulk coordinates.Our universe is represented as a hypersurface in the 5D bulk, whose trajectory is given by theconstraint w = f ( x µ ). In this setup, the brane will inherit three bulk coordinates { t, x, y, z } and will respond to perturbations that are just functions of the bulk time via the relation w = f ( x µ ). Consider that the metric perturbations and the brane position are homogeneous:Φ = (cid:15) Φ ( w ) , Ψ = (cid:15) Ψ ( w ) , f = f ( t ) , (3.25)– 8 –here (cid:15) (cid:28) ∇ Φ = 8 πG ρ , (3.26) ∇ Φ = − ∇ ρ ρ . (3.27)These equations can be solved exactly in Minkowski background: ρ ( w ) = ¯ ρ (cid:26) − tanh (cid:20) ¯ γ (cid:18) w ¯ w − (cid:19)(cid:21)(cid:27) , (3.28)where ¯ ρ ≡ πG ¯ γ ¯ w , while ¯ γ and ¯ w are constants of integration.The relationship between the energy density and the temperature of the atmospherecan be computed by integrating the Bose-Einstein distribution in 4+1 dimensions ρ ( w ) = (cid:90) d k (2 π ) ω exp [ ω/T ( w )] − ζ R (5) π T ( w ) , (3.29)where ω = k α k α . The above expression allows us to write T ( w ) = (cid:18) ρ ( w ) π ζ R (5) (cid:19) = 1 . ρ ( w ) / . (3.30)Note that the characteristic thickness of the atmosphere is given by∆ L = ¯ w √ γ . (3.31)We are now in position to compute Eq. (3.22) and we will do so in the reference frameof the atmosphere. In this frame ρ does not depend on time, but the normal to the brane n α will depend on the relative velocity of the brane and the atmosphere. First consider thefluid velocity u α = (1 , v ) / √ − v , where v is the relative 3 velocity between the brane andthe atmosphere. If we now require n α n α = 1 and n α u α = 0 we have n α = ( v, v /v ) / √ − v .With this we can write T αβ n α n β = ρ ( w )(1 + 4 v ) / − v ) and the RHS of Eq. (3.22) reads∆ ln H = 12 ρ (cid:90) T αβ n α n β dτ = 12 ρ (cid:90) ∞ T αβ n α n β (cid:112) − v dt , = 12 ρ (cid:90) ∞ T αβ n α n β (cid:112) − v dwv , = (1 + 4 v ) v √ − v ¯ ρ ρ ¯ w ¯ γ (cid:20) γ ) (cid:21) . (3.32) As discussed in the last section, if the velocity of the brane is near the speed of light we canassume that the Hubble patch of the universe will be smaller than the curvature radius ofthe black hole spacetime. In this regime it is safe to approximate the bulk as Minkowskispacetime and analyze the perturbations around it. In the last section we have briefly intro-duced the homogeneous scalar perturbations and in Appendix A we present the anisotropic– 9 –erturbations. The curvature perturbation can be written as function of the scalar gaugeinvariant quantities (A.8) ζ = Ψ − H ˙ H (cid:18) H Φ + ˙Ψ (cid:19) . (4.1)Note that in our framework the Hubble constant is of first order in the perturbation (seeEq.(A.9)) as the brane crosses the atmosphere, and thus the term H Φ can be neglected.With this we have ζ ≈ Ψ − ∆Ψ ∆ ln( H ) , (4.2)where ∆Ψ = Ψ f − Ψ i . Here Ψ i ( f ) stands for the metric perturbation in 4D right before(after) the brane crossed the atmosphere, and we assume that ζ evolves continuously. We areinterested in the value of the curvature perturbation after the brane has passed through theatmosphere where the metric perturbation Ψ f = 0, which makes Ψ f = 0. In this regimethe curvature perturbation becomes ζ = ζ f ≈ Ψ i ∆ ln( H ) . (4.3)We are now ready to analyze the behaviour of the curvature perturbation power spec-trum P ζ ( k ) = (cid:90) d x e i k · x (cid:104) ζ ( x ) ζ (0) (cid:105) . (4.4)If the atmosphere is not in thermal equilibrium, we can relate the 2-point correlation functionof the thermal fluctuations in 5D energy density to the temperature profile of the atmosphere: (cid:104) ρ ( y ) ρ ( y ) (cid:105) = α ( T ( y )) δ ( y − y ) . (4.5)To proceed, we notice that in [2] it was found that the 5D energy density correlation functiondue to a thermal gas is (cid:104) ρ ( y ) ρ ( y ) (cid:105) (cid:39) (cid:20) (cid:90) d k (2 π ) (cid:20) ω/T ) − (cid:21) ω exp[ ik a ( y a − y a )] (cid:21) . (4.6)This expression can be approximated as a delta function (cid:104) ρ ( y ) ρ ( y ) (cid:105) (cid:39) α ( T ( y )) δ ( y − y ), where α = (cid:20) π (945 ζ R (5) − π ) (cid:21) , while we have dropped the power-law UV divergenceand ζ R is the Riemann zeta function. Note, that (4.6) can be approximated by a 4 dimensionaldelta function on length scales larger than the thermal wavelength T − .With the use of the Poisson equation in 5D ∇ Ψ ( y ) = 8 πG ρ ( y ) , (4.7)we can find the power spectrum for the curvature perturbation to be (see Appendix B fordetails) P ( k ) = k π P ζ ( k ) = β k (cid:90) ∞ dw e − | w | k ( T ( w )) (4.8)= ∆ k (cid:90) ∞ dw e − | w | k (cid:8) − tanh [¯ γ ( w/ ¯ w − (cid:9) / (4.9)where ∆ = β [ ¯ T ] , β = α (cid:18) G H π (cid:19) and we have used Eqs.(3.29) and (3.30) of Sec. 3.3to write the temperature of the brane. The power spectrum predicted by our model ischaracterized by 3 free parameters ¯ γ, ¯ w, ∆ which we are going to fit to Planck data in thenext section. – 10 – able 1 . Planck 2015 and BAO best fit parameters and 68% ranges for HBB and ΛCDM models.The last row indicates the χ for each of the models. Note that ¯ w c corresponds to the comoving value ofthe position of the centre of the atmosphere and its related to the physical ¯ w via ¯ w = ¯ w c . × − MeV T nuc . HBB Λ CDM Λ CDM with runningbest fit range best fit range best fit range Ω b h . . ± . . . ± . . . ± . c h . . ± . . . ± . . . ± . θ . . ± . . . ± . . . ± . τ .
081 0 . ± .
014 0 .
067 0 . ± .
013 0 .
069 0 . ± . ∆ .
79 0 . ± .
021 5 .
798 5 . ± .
019 5 .
82 5 . ± . n s . . ± . . . ± . α s − . − . ± . γ .
513 0 . ± . w c [Mpc] 275 297 +39 − χ . . . The standard model of cosmology, ΛCDM, is described by 6 parameters (Ω b h , Ω c h , θ, τ, ∆ , n s ),the baryon density, dark matter density, angular size of the sound horizon at recombination,the optical depth to reionization, amplitude of the scalar power spectrum and its tilt re-spectively. This model characterizes early universe cosmology via the power spectrum of thecurvature perturbations P ( k ) = ∆ (cid:18) kk ∗ (cid:19) n s − , (5.1)where k ∗ = 0 . / Mpc is the comoving pivot scale. This form of the power spectrum, expectedin slow-roll inflationary models, best fits the CMB data with parameter values [1]∆ = (2 . ± . × − n s = 0 . ± . . (5.2)We would like to compare this model with the HBB model (Eq. 4.9) that strictly is repre-sented by the seven parameters (Ω b h , Ω c h , θ, τ, ¯ γ, ¯ w, ∆ ). In order to compare models withthe same number of parameters, will also include ΛCDM with running α s = dn s /d ln q .We have performed the comparison by running the CosmoMC code [11–17] with Planck2015 data, Barionic Acoustic Oscillations (BAO) [18–25] as well as lensing data [26–30].Finally, to determine the best fit parameters and the likelihood, we run the minimizer ex-pressing our results Table 1. Comparing best-fit χ of HBB and ΛCDM (with running), wesee that HBB is disfavoured at roughly 2.7 2.8 σ (2.8 σ ). The Planck angular TT spectrumtogether with the best fit curves and residuals for HBB and ΛCDM are shown in Fig. 3.The best fit primordial scalar power spectrum in both models are also contrasted with anon-parametric reconstruction from Planck 2015 data [31]. We note that the difference be-tween the two models (mostly) lies within the 68% region and the largest disagreement ofthe models is at low l’s or k’s.We are now going to analyze the physical implications of the best fit parameters { ¯ γ, ¯ w c , ∆ } . In particular, we are interested in the temperature and position of the atmo-sphere and the change on Hubble constant of the brane when crossing the atmosphere. All ofthese physical quantities are related to the fitted parameters, and also to the temperature of– 11 – ( l + ) C l / π [ µ K ] Planck
HBB Λ CDM ● ●● ** - - ● * [ C a ( l ) - C P l a n c k ( l )] / C a v e ( l ) ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ************************************************************************************ l l Λ CDMBest fit HBB: γ = w c = × - × - [ Mpc - ] × ( k ) Figure 3 . Left-Top: angular power spectrum of CMB temperature anisotropies, comparing Planck2015 data (black dots) with best HBB model (solid/red) for all l. Left-Inset: angular power spectrumof CMB temperature anisotropies, comparing Planck 2015 data with ΛCDM (dotted/blue) and HBB(solid/red) for l <
40. Left-Bottom: relative residuals and difference between ΛCDM and HBB (blacksolid) where the green shaded region indicates the 68% region of Planck 2015 data. Right: Best fitof the primordial power spectrum as predicted by HBB (dashed-red) in comparison with the best fitof ΛCDM model (blue). The grey regions are the ± σ and ± σ constraints from a non-parametricreconstruction using Planck 2015 data [31]. the brane at nucleation T nuc and the Planck mass in the bulk M . Using ¯ w = ¯ w c . × − MeV T nuc and Eqs. (3.28), (3.29) and (3.31) we can find the values for the amplitude of the energydensity, the temperature and the width of the atmosphere respectively. We have summarizedour results in Table 2.The model still allows freedom for the parameters { T nuc , M , v } . These can be con-strained using observational bounds for DGP model and by requiring consistency of ourapproximations. In particular, we have modelled the atmosphere as being thin, which isequivalent to the requirement that the time it takes the brane to cross it is less than aHubble time:∆ L √ − v v ≤ H − = (cid:18) πG ρ (cid:19) / ⇒ T nuc ≤ v √ − v . × − MeV , (5.3)employing the fact that the energy density on the brane at nucleation time is ρ = π g ∗ T ,for g ∗ effective relativistic degrees of freedom. The above bound is consistent with the BBNconstraint ( T nuc ≥ T BBN ) for velocities near the speed of light. On the other hand, asdiscussed in Section 3.2 when the brane encounters the atmosphere its Hubble parameterwill change as predicted by Eq. (3.32). Since this quantity enters in the amplitude of thepower spectrum Eq. (4.9) we can write∆ = β ¯ T , β = (cid:18) G π ∆ ln H (cid:19) (5.4)and thus ∆ ln H = 4 . × − (cid:18) T nuc M (cid:19) / . (5.5)– 12 – able 2 . Physical characteristics of the HBB model using the best fit parameters presented inTable 1. The first column shows the relevant physical parameters and their definitions in terms of thebest fit parameters and related quantities.The second column shows the numerical values and scalingwith M and T nuc . Finally, in the last column we show the limits necessary for the thin atmospherecondition (5.10). The 5th and 6th rows show the shift in the Hubble constant when crossing theatmosphere computed using perturbations and bulk background information, respectively. The 7throw presents a function constraining the velocity of the atmosphere in the bulk that can be computedby equating the results of rows 5 and 6. Note that T nuc ≥ . Thin atmosphere boundQuantity Value M ≤ . × − ( MeV T nuc ) / T nuc (5.10) Position of ¯ w = ¯ w c . × − MeV T nuc . ×
27 1 T nuc . ×
27 1 T nuc atmosphereDensity of ¯ ρ = πG ¯ γ ¯ w . × − M T ≤ . × − ( MeV T nuc ) / T atmosphereTemperature ¯ T = 1 .
26 ¯ ρ / . × − ( M T ) / ≤ . × − ( MeV T nuc ) / T nuc of atmosphereWidth of ∆ L = √ w ¯ γ . ×
28 1 T nuc . ×
28 1 T nuc atmosphereChange ofHubble from ∆ ln H = G T (0) π ∆ ( α ) . × − ( T nuc M ) / ≥ . × − ( T nuc MeV ) / perturbations Eq. (5.5)
Change of ∆ ln H = Hubble from f ( v ) ¯ ρ ρ ¯ w ¯ γ [1 + tanh(¯ γ )] 1 . × − ( M T nuc ) f ( v ) ≥ . × − ( T nuc MeV ) / backgroud Eqs. (5.6) and (3.32)
Velocity f ( v ) = (1+4 v ) v √ − v . × − ( T nuc M ) / ≥ . × T nuc MeV constraint
We also get a constraint on ∆ ln H from the bulk atmosphere using Eq. (3.32)∆ ln H = 1 . × − (1 + 4 v ) v √ − v (cid:18) M T nuc (cid:19) , (5.6)Using Eqs. (5.5) and (5.6) we can constrain the velocity and the speed of sound of the brane f ( v ) = (1 + 4 v ) v √ − v = 2 . × − (cid:18) T nuc M (cid:19) / . (5.7)From the above equation we notice that in order to have a real brane velocity we need tosatisfy T nuc ≥ M . (5.8)– 13 –e can obtain a constraint for { T nuc , M } by combining expressions (5.7) and (5.3) andnoting that in the large velocity limit v √ − v ≈ √ − v ≈ f ( v )5 T nuc ≤ . × − (cid:18) T nuc M (cid:19) / MeV . (5.9)Inverting this we find M ≤ . × − (cid:18) MeV T nuc (cid:19) / T nuc . (5.10)This bound represents the maximum allowed value of M in order for the thin atmospherecondition to be satisfied. In the third column of Table 2 we list the values for the physicalquantities allowing M to saturate the above bound. This constraint must be combined withthe physical constraint (2.3) T nuc > T BBN ∼ . M < , (5.12)on the normal branch of DGP in order to get the allowed region in parameter space for { T nuc , M } – depicted in Fig.4 (blue shaded region).It is interesting to compare the thermal entropy of our brane to the holographic boundexpected from its surface area in 5D. The entropy for the 5D black hole is S BH = A G , whilethe entropy density in a universe dominated by relativistic particles s ( T ) = π g ∗ T (cid:39) . × T , for g ∗ = 10 .
75 effective relativistic degrees of freedom, prior to electron/positronannihilation [32]. This puts a lower bound M > . (cid:18) T nuc . . (cid:19) MeV (holographic bound) (5.13)on the 5D Planck mass M = πG ) / , where T nuc . is the nucleation temperature of thebrane. We show the Holographic bound allowed region in parameter space in Fig. 4 (orangeshaded region). Eqs. (5.13)-(2.4) constrain the 5D Planck mass to be within 1.5 decades inenergy: 0 .
23 MeV < M < , (5.14)a range that will inevitably shrink with future observations that better constrain BBN,and late-time cosmic expansion history. As we see in Fig.4, the best-fit value for T nuc . from cosmological observations (assuming the thin atmosphere condition) does violate theholographic bound (5.13) by at least 2.5 orders of magnitude, which would decrease the lowerlimit on M in Eq. (5.14) by the same factor .Is this a “show-stopper”? While the holographic bound on entropy remains a very well-motivated conjecture, it is not clear how firm it might be as objects that get close to crossingit are already in the quantum gravity regime where the classical description of spacetimephysics fails. One may argue that since the degrees from responsible for thermal entropy ofour brane are on scales much smaller than the 5D Planck length, they are not accessible bya 5D bulk observer, and thus are not limited by the 5D holographic bound. Note that the exact saturation of the holographic bound predicts a brane velocity that is not real. – 14 –
GP bound T h i n a t m o s p h e r e c o n d i t i o n H o l og r aph i c bound BBN constraint R ea l v e l o c i t yc ond i t i on - T nuc [ MeV ] M [ M e v ] Figure 4 . Theoretical and empirical bounds for the Holographic Big Bang model. The DGP boundEq.(5.12) (thick-black) and the holographic bound Eq.(5.13) (black, thin dashed), together with theBBN bound Eq.(5.11) (vertical black) constitute the theoretical bounds of the model. The top shadedarea (orange) is the allowed region for these three bounds to be satisfied. The real velocity boundEq.(5.8) (thick, grey) and the thin atmosphere condition (5.9) (black, thick dashed) constitute theempirical bounds that HBB must satisfy. The bottom shaded area (blue) is the allowed region of { M , T nuc } parameter space satisfying the empirical bounds, and the arrows indicate the directionsin which the different bounds apply. It is clear that the empirical bounds violate the holographicbound for all possible allowed pairs { M , T nuc } by at least 2.5 orders of magnitude. The least severeviolation of the holographic bound is for parameters at the bottom left of the plot: for T nuc beingthe minimum allowed value by BBN and M the maximum allowed by the thin atmosphere condition(third column of Table 2). The 5D Holographic Big Bang (HBB) is a novel proposal for a holographic origin of ouruniverse as a 3-brane with induced gravity, out of the collapse of 5D star that can addressthe traditional problems of big bang cosmology. The main goal of this study was to providedetailed and concrete predictions for this proposal, and to see whether it can serve as apossible competitor to slow-roll inflationary models to explain cosmological observations.We first focused our attention on a possible mechanism for the nucleation of our 3-branein which the quantum degrees of freedom of the bulk tunnel into a fuzzball configurationreminiscent to the bubble of nothing model. This mechanism not just provides a possiblescenario of brane nucleation but also constrains the Planck mass in the bulk.Previous work has shown that the presence of uniform thermal gas in the 5D bulk leads ascale-invariant primordial power spectrum for cosmological scalar perturbations. To formalizethis result and search for mechanisms that could potentially explain deviations from scale-invariance (observed in the CMB data), we studied cosmological perturbations induced by athin infalling atmosphere. This atmosphere is composed of a spherically symmetric thermalrelativistic gas that the brane encounters after nucleation. We showed that this atmosphereinduces a change in the Hubble parameter and also scalar cosmological perturbations on thebrane. The power spectrum is scale invariant for large k ’s and scales as k for small k ’s.We then tested this prediction for power spectrum against the cosmological observations.The transition is characterized by a decay of 1 /k for scales where the power spectrum ishighly constrained by data [31] as shown in Fig. 3 (right). We found that our model is– 15 –roadly consistent with non-parametric reconstruction of primordial power spectrum, butis disfavoured compared to a pure power-law at 2.7 σ level. We finally outlined varioustheoretical constraints on the nucleation temperature and 5D Planck mass in the HBB model,and found that the best fit nucleation temperature of the 3-brane was at least 3 orders ofmagnitude larger than the 5D Planck mass.This first attempt to understand the detailed consequences of the HBB model for cos-mology relies on several simplifying assumptions that can be relaxed in future work. Someof the issues that remain to be tackled are:1. Perhaps our most perplexing finding was that our best-fit model violated the holo-graphic entropy bound by 8 orders of magnitude. It is not yet clear whether this is afeature or a bug!2. It would be interesting to study how brane cosmological perturbations will be affectedby the large-scale curvature of the bulk (in a 5D Schwarzschild of Kerr spacetime).3. Other observables that remain to be computed are the amplitude of tensor modes andthe non-gaussianity, although we do not expect them to be significant.4. Given that the speed of sound for a relativistic 5D atmosphere is c s = c/
2, one expect O (0 .
2) relativistic corrections to the atmosphere profile, which we have ignored. Thiscould affect the functional shape of the power spectrum at a similar level, potentiallyimproving (or worsening) the fit to the data. A related issue is whether the hydrostaticequilibrium profile for the relativistic thin atmosphere is stable.To conclude, while we believe the 5D holographic big bang remains an intriguing possi-bility for the origin of our universe, there remain empirical and theoretical challenges to itsstatus amongst various scenarios for the early universe cosmology that should be addressedin future work.
Acknowledgements
This work has been partially supported by the National Science and Engineering ResearchCouncil (NSERC), University of Waterloo, and Perimeter Institute for Theoretical Physics(PI). Research at PI is supported by the Government of Canada through the Departmentof Innovation, Science and Economic Development Canada and by the Province of Ontariothrough the Ministry of Research, Innovation and Science.
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If we consider the metric (3.24) and homogeneous perturbations of the form (3.25) the inducedmetric on the brane is γ µν = g αβ e αµ e βν that to first order in (cid:15) reads: ds = [( f (cid:48) − − (cid:15) (cid:16) ˆˆΨ ( f ) f (cid:48) + ˆΦ ( f ) (cid:17) ] dt + [1 − (cid:15) ˆΨ ( f )][ dx + dy + dz ] , (A.1)where f (cid:48) = d f d t . The induced metric on the brane has to be able to describe a Friedmannuniverse for which we make the following identifications − dτ = [( f (cid:48) − − (cid:15) (cid:16) ˆΨ f (cid:48) + ˆΦ (cid:17) ] dt , a = [1 − (cid:15) ˆΨ ] , (A.2)where τ is the proper time of the brane and a is the scale factor. On top of the homogeneousperturbations we are now going to consider anisotropiesΦ = (cid:15) Φ ( w ) + (cid:15) Φ ( x α ) , Ψ = (cid:15) Ψ ( w ) + (cid:15) Ψ ( x α ) , f = f ( t ) + (cid:15) f ( x α ) , (A.3)where (cid:15) (cid:28) (cid:15)(cid:15) (cid:28) (cid:15), (cid:15) . Theinduced metric can be written as ds = − [1+ 2 (cid:15) f (cid:48) − + ˆΨ f − f (cid:48) f (cid:48) )] dτ +2 a(cid:15) f (cid:48) f ,i (cid:112) f (cid:48) − dx i dτ + a (1 − (cid:15) ) dx , (A.4)where ˆΨ = Ψ ( w = f ( x µ )) are the metric functions projected to the brane. The generalform of a 4D metric including scalar and vector cosmological perturbation in 4D are thatcontains all the terms in Eq. (A.4) ds = − (1 + 2 φ ) dτ − aB i dτ dx i + a (1 − ψ ) dx , (A.5)– 18 –nd thus we identify φ = (cid:15) f (cid:48) − + ˆΨ f − f (cid:48) f (cid:48) ) , ψ = ˆΨ (cid:15) , B i = (cid:15) f (cid:48) f ,i (cid:112) f (cid:48) − . (A.6)Note that the Newtonian gauge on the bulk does not translate into a Newtonian gauge onthe brane and that some perturbations that are scalars in 5D are projected as a vectorialperturbation component in 4D. The scalar gauge invariant quantities can be constructedfrom Eq.(A.4) as Φ = φ − ∂ τ ( aB ) , (A.7)Ψ = ψ + HaB , (A.8)where, H is the Hubble constant and B is the scalar part of the vector metric perturbation B i . From our construction the Hubble constant is H = ˙ aa = − (cid:15) ˆΨ a , (A.9)and thus Eq.(A.8) reduces to Ψ = ψ to first order in (cid:15), (cid:15) . B Derivation of the power spectrum
With use of the Poisson equation in 4D ∇ Ψ ( y ) = 8 πG ρ ( y ) , (B.1)and the expression for the energy density correlation function (cid:104) ρ ( y ) ρ ( y ) (cid:105) (cid:39) α ( T ) δ ( y − y ) , (B.2)we can write the 2-point correlation function of Ψ using the Green’s function for the Lapla-cian operator (cid:104) Ψ ( x )Ψ ( x ) (cid:105) = α (cid:18) πG (cid:19) (cid:18) π (cid:19) (cid:90) d y ( T ( y )) | y − x | | y − x | . (B.3)The junction condition between the 5D and 4D metrics (3.24) and (A.1) allow us to compute (cid:104) Ψ ( x )Ψ ( x ) (cid:105)(cid:104) Ψ ( x )Ψ ( x ) (cid:105) = (cid:104) Ψ ( x , x w = 0)Ψ ( x , x w = 0) (cid:105) = α (cid:18) G π (cid:19) (cid:90) d y dy w ( T ( y w )) ( | x − y | + | y w | )( | x − y | + | y w | ) (B.4)where we have decomposed the bulk coordinate as y = ( y , y w ) and we have set the temper-ature to be just a function of the w direction of the bulk. Combining expressions (4.3), (4.4),(B.4) and setting x = 0 we have P ζ ( k ) = α (cid:18) G π ∆ln H (cid:19) (cid:90) d y dy w ( T ( y w )) | y | + | y w | (cid:90) d x e i k · x | x − y | + | y w | , = α (cid:18) G π ∆ln H (cid:19) (cid:90) d y dy w ( T ( y w )) | y | + | y w | (cid:90) d x (cid:48) e i k · x (cid:48) e i k · y | x (cid:48) | + | y w | , = α (cid:18) G π ∆ln H (cid:19) (cid:90) dy w ( T ( y w )) (cid:18) (cid:90) d x (cid:48) e i k · x (cid:48) | x (cid:48) | + | y w | (cid:19) , = α (cid:18) G π ∆ln H (cid:19) πk ) (cid:90) dy w e − | y w | k ( T ( y w )) , (B.5)– 19 –here we have performed the coordinate transformation x (cid:48) = x − y , and use the result (cid:82) d x e i k · x | x | + | m | = e − k | m | πk .With this we are able to write P ( k ) = k π P ζ ( k ) = β k (cid:90) ∞ dw e − | w | k ( T ( w )) , (B.6)where β = α (cid:18) G H π (cid:19) . Note that we are in integrating between [0 , ∞ ) because of the Z symmetry. Because the temperature profile is analytic it admits a Taylor expansion of theform ( T ( w )) = ∞ (cid:88) n =0 T ( n ) (0 , ¯ γ, ¯ w ) n ! w n , (B.7)and thus the power spectrum Eq.(B.6) admits a series decomposition of the form P ( k ) = β ∞ (cid:88) n =0 T ( n ) (0 , ¯ γ, ¯ w ) k n − − n . (B.8)This last expression implies that for large k the correction to a scale invariant power spectrumgoes as 1 /k . If the integral in Eq.(B.6) is performed in the rage ( −∞ , ∞ ) the the powerspectrum series would be P ( k ) = β ∞ (cid:88) n =0 T (2 n ) (0 , ¯ γ, ¯ w ) k n − − n , (B.9)giving a correction from scale invariant that goes as 1 /k for large k . This correction rendersmodel disfavourable in comparison with the symmetric case, where we have a 1 /k/k