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5.1 Brane-world inflation

In 1-brane RS-type brane-worlds, where the bulk has only a vacuum energy, inflation on the brane must be driven by a 4D scalar field trapped on the brane. In more general brane-worlds, where the bulk contains a 5D scalar field, it is possible that the 5D field induces inflation on the brane via its effective projection [13Jump To The Next Citation Point, 33Jump To The Next Citation Point, 100Jump To The Next Citation Point, 103Jump To The Next Citation Point, 138Jump To The Next Citation Point, 139Jump To The Next Citation Point, 140Jump To The Next Citation Point, 141Jump To The Next Citation Point, 158Jump To The Next Citation Point, 165Jump To The Next Citation Point, 180Jump To The Next Citation Point, 195Jump To The Next Citation Point, 229Jump To The Next Citation Point, 238Jump To The Next Citation Point, 276Jump To The Next Citation Point, 304Jump To The Next Citation Point, 318Jump To The Next Citation Point].

More exotic possibilities arise from the interaction between two branes, including possible collision, which is mediated by a 5D scalar field and which can induce either inflation [90Jump To The Next Citation Point, 157Jump To The Next Citation Point] or a hot big-bang radiation era, as in the “ekpyrotic” or cyclic scenario [154Jump To The Next Citation Point, 163Jump To The Next Citation Point, 193Jump To The Next Citation Point, 231Jump To The Next Citation Point, 251Jump To The Next Citation Point, 301Jump To The Next Citation Point, 307Jump To The Next Citation Point], or in colliding bubble scenarios [29Jump To The Next Citation Point, 107Jump To The Next Citation Point, 108Jump To The Next Citation Point]. (See also [21Jump To The Next Citation Point, 69Jump To The Next Citation Point, 214Jump To The Next Citation Point] for colliding branes in an M theory approach.) Here we discuss the simplest case of a 4D scalar field φ with potential V (φ) (see [207] for a review).

High-energy brane-world modifications to the dynamics of inflation on the brane have been investigated [23, 24, 25, 63, 74, 135, 156, 184, 204, 221, 222Jump To The Next Citation Point, 233, 234, 240, 270, 302]. Essentially, the high-energy corrections provide increased Hubble damping, since ρ ≫ λ implies that H is larger for a given energy than in 4D general relativity. This makes slow-roll inflation possible even for potentials that would be too steep in standard cosmology [70Jump To The Next Citation Point, 145Jump To The Next Citation Point, 206Jump To The Next Citation Point, 222Jump To The Next Citation Point, 226Jump To The Next Citation Point, 258Jump To The Next Citation Point, 277Jump To The Next Citation Point].

The field satisfies the Klein–Gordon equation

′ φ¨+ 3H ˙φ + V (φ) = 0. (198 )
In 4D general relativity, the condition for inflation, ¨a > 0, is φ˙2 < V (φ), i.e., p < − 1ρ 3, where 1˙2 ρ = 2φ + V and 1 ˙2 p = 2φ − V. The modified Friedmann equation leads to a stronger condition for inflation: Using Equation (188View Equation), with m = 0 = Λ = K, and Equation (198View Equation), we find that
[ ] 1 1 + 2ρ∕λ ¨a > 0 ⇒ w < − 3- 1-+-ρ-∕λ- , (199 )
where the square brackets enclose the brane correction to the general relativity result. As ρ∕λ → 0, the 4D result w < − 1 3 is recovered, but for ρ > λ, w must be more negative for inflation. In the very high-energy limit ρ∕λ → ∞, we have w < − 2 3. When the only matter in the universe is a self-interacting scalar field, the condition for inflation becomes
[ ( )] ˙2 -12 ˙φ2 +-V 5-˙2 1- φ − V + λ 4φ − 2V < 0, (200 )
which reduces to φ˙2 < V when ρφ = 1˙φ2 + V ≪ λ 2.

In the slow-roll approximation, we get

2 [ ] H2 ≈ κ--V 1 + V-- , (201 ) 3 2λ V ′ ˙φ ≈ − ---. (202 ) 3H
The brane-world correction term V∕ λ in Equation (201View Equation) serves to enhance the Hubble rate for a given potential energy, relative to general relativity. Thus there is enhanced Hubble ‘friction’ in Equation (202View Equation), and brane-world effects will reinforce slow-roll at the same potential energy. We can see this by defining slow-roll parameters that reduce to the standard parameters in the low-energy limit:
2( )2 [ ] H˙- M--p V-′ --1-+-V-∕λ-- ε ≡ − H2 = 16π V (1 + V ∕2λ)2 , (203 ) 2 ( ′′) [ ] η ≡ − -¨φ--= M-p- V-- ----1---- . (204 ) H ˙φ 8π V 1 + V∕2 λ
Self-consistency of the slow-roll approximation then requires ε,|η | ≪ 1. At low energies, V ≪ λ, the slow-roll parameters reduce to the standard form. However at high energies, V ≫ λ, the extra contribution to the Hubble expansion helps damp the rolling of the scalar field, and the new factors in square brackets become ≈ λ ∕V:
[ ] [ ] 4λ- 2λ- ε ≈ εgr V , η ≈ ηgr V , (205 )
where εgr,ηgr are the standard general relativity slow-roll parameters. In particular, this means that steep potentials which do not give inflation in general relativity, can inflate the brane-world at high energy and then naturally stop inflating when V drops below λ. These models can be constrained because they typically end inflation in a kinetic-dominated regime and thus generate a blue spectrum of gravitational waves, which can disturb nucleosynthesis [70Jump To The Next Citation Point, 206Jump To The Next Citation Point, 226Jump To The Next Citation Point, 258Jump To The Next Citation Point, 277Jump To The Next Citation Point]. They also allow for the novel possibility that the inflaton could act as dark matter or quintessence at low energies [4, 43, 70Jump To The Next Citation Point, 206Jump To The Next Citation Point, 208, 226Jump To The Next Citation Point, 239, 258Jump To The Next Citation Point, 277Jump To The Next Citation Point, 285].

The number of e-folds during inflation, N = ∫ Hdt, is, in the slow-roll approximation,

8π ∫ φfV [ V ] N ≈ − --2- --′ 1 + --- dφ. (206 ) M p φi V 2λ
Brane-world effects at high energies increase the Hubble rate by a factor V∕2 λ, yielding more inflation between any two values of φ for a given potential. Thus we can obtain a given number of e-folds for a smaller initial inflaton value φi. For V ≫ λ, Equation (206View Equation) becomes
3∫ φf 2 N ≈ − 128π-- V--dφ. (207 ) 3M 65 φi V ′

The key test of any modified gravity theory during inflation will be the spectrum of perturbations produced due to quantum fluctuations of the fields about their homogeneous background values. We will discuss brane-world cosmological perturbations in the next Section 6. In general, perturbations on the brane are coupled to bulk metric perturbations, and the problem is very complicated. However, on large scales on the brane, the density perturbations decouple from the bulk metric perturbations [102Jump To The Next Citation Point, 122Jump To The Next Citation Point, 191Jump To The Next Citation Point, 218Jump To The Next Citation Point]. For 1-brane RS-type models, there is no scalar zero-mode of the bulk graviton, and in the extreme slow-roll (de Sitter) limit, the massive scalar modes are heavy and stay in their vacuum state during inflation [102Jump To The Next Citation Point]. Thus it seems a reasonable approximation in slow-roll to neglect the KK effects carried by ℰ μν when computing the density perturbations.

View Image

Figure 5: The relation between the inflaton mass m ∕M 4 (M ≡ M 4 p) and the brane tension 4 1∕4 (λ∕M 4) necessary to satisfy the COBE constraints. The straight line is the approximation used in Equation (214View Equation), which at high energies is in excellent agreement with the exact solution, evaluated numerically in slow-roll. (Figure taken from [222Jump To The Next Citation Point].)

To quantify the amplitude of scalar (density) perturbations we evaluate the usual gauge-invariant quantity

H ζ ≡ ℛ − --δρ, (208 ) ρ˙
which reduces to the curvature perturbation ℛ on uniform density hypersurfaces (δρ = 0). This is conserved on large scales for purely adiabatic perturbations as a consequence of energy conservation (independently of the field equations) [317Jump To The Next Citation Point]. The curvature perturbation on uniform density hypersurfaces is given in terms of the scalar field fluctuations on spatially flat hypersurfaces δφ by
ζ = H δφ. (209 ) φ˙
The field fluctuations at Hubble crossing (k = aH) in the slow-roll limit are given by 2 2 ⟨δφ ⟩ ≈ (H ∕2π), a result for a massless field in de Sitter space that is also independent of the gravity theory [317]. For a single scalar field the perturbations are adiabatic and hence the curvature perturbation ζ can be related to the density perturbations when modes re-enter the Hubble scale during the matter dominated era which is given by A2 = 4⟨ζ2⟩∕25 s. Using the slow-roll equations and Equation (209View Equation), this gives
( ) [ ] | 2 512 π V3 2λ + V 3|| A s ≈ -----6--′2 ------- ||. (210 ) 75M p V 2 λ k=aH
Thus the amplitude of scalar perturbations is increased relative to the standard result at a fixed value of φ for a given potential.

The scale-dependence of the perturbations is described by the spectral tilt

2 n − 1 ≡ dln-As-≈ − 4ε + 2η, (211 ) s dlnk
where the slow-roll parameters are given in Equations (203View Equation) and (204View Equation). Because these slow-roll parameters are both suppressed by an extra factor λ∕V at high energies, we see that the spectral index is driven towards the Harrison–Zel’dovich spectrum, ns → 1, as V ∕λ → ∞; however, as explained below, this does not necessarily mean that the brane-world case is closer to scale-invariance than the general relativity case.

As an example, consider the simplest chaotic inflation model 1 2 2 V = 2m φ. Equation (206View Equation) gives the integrated expansion from φi to φf as

2π ( 2 2) π2m2 ( 4 4) N ≈ --2-φ i − φ f + ----6- φi − φf . (212 ) M p 3M 5
The new high-energy term on the right leads to more inflation for a given initial inflaton value φ i.

The standard chaotic inflation scenario requires an inflaton mass 13 m ∼ 10 GeV to match the observed level of anisotropies in the cosmic microwave background (see below). This corresponds to an energy scale ∼ 1016 GeV when the relevant scales left the Hubble scale during inflation, and also to an inflaton field value of order 3Mp. Chaotic inflation has been criticised for requiring super-Planckian field values, since these can lead to nonlinear quantum corrections in the potential.

If the brane tension λ is much below 1016 GeV, corresponding to M5 < 1017 GeV, then the terms quadratic in the energy density dominate the modified Friedmann equation. In particular the condition for the end of inflation given in Equation (200View Equation) becomes φ˙2 < 2V 5. In the slow-roll approximation (using Equations (201View Equation) and (202View Equation)) ˙ 3 φ ≈ − M 5∕2πφ, and this yields

( ) 4 -5-- M5-- 2 4 φ end ≈ 4π2 m M 5. (213 )
In order to estimate the value of φ when scales corresponding to large-angle anisotropies on the microwave background sky left the Hubble scale during inflation, we take Ncobe ≈ 55 in Equation (212View Equation) and φ = φ f end. The second term on the right of Equation (212View Equation) dominates, and we obtain
165 ( M5 )2 φ4cobe ≈ --2- ---- M 45. (214 ) π m
Imposing the COBE normalization on the curvature perturbations given by Equation (210View Equation) requires
( 2 ) 4 5 As ≈ 8π-- m--φcobe-≈ 2 × 10 −5. (215 ) 45 M 65
Substituting in the value of φcobe given by Equation (214View Equation) shows that in the limit of strong brane corrections, observations require
m ≈ 5 × 10 −5M5, φcobe ≈ 3 × 102 M5. (216 )
Thus for 17 M5 < 10 GeV, chaotic inflation can occur for field values below the 4D Planck scale, φcobe < Mp, although still above the 5D scale M5. The relation determined by COBE constraints for arbitrary brane tension is shown in Figure 5View Image, together with the high-energy approximation used above, which provides an excellent fit at low brane tension relative to M 4.

It must be emphasized that in comparing the high-energy brane-world case to the standard 4D case, we implicitly require the same potential energy. However, precisely because of the high-energy effects, large-scale perturbations will be generated at different values of V than in the standard case, specifically at lower values of V, closer to the reheating minimum. Thus there are two competing effects, and it turns out that the shape of the potential determines which is the dominant effect [203Jump To The Next Citation Point]. For the quadratic potential, the lower location on V dominates, and the spectral tilt is slightly further from scale invariance than in the standard case. The same holds for the quartic potential. Data from WMAP and 2dF can be used to constrain inflationary models via their deviation from scale invariance, and the high-energy brane-world versions of the quadratic and quartic potentials are thus under more pressure from data than their standard counterparts [203Jump To The Next Citation Point], as shown in Figure 6View Image.

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Figure 6: Constraints from WMAP data on inflation models with quadratic and quartic potentials, where R is the ratio of tensor to scalar amplitudes and n is the scalar spectral index. The high energy (H.E.) and low energy (L.E.) limits are shown, with intermediate energies in between, and the 1-σ and 2-σ contours are also shown. (Figure taken from [203].)

Other perturbation modes have also been investigated:

Brane-world effects on large-scale isocurvature perturbations in 2-field inflation have also been considered [12]. Brane-world (p)reheating after inflation is discussed in [5, 67, 309, 310, 321].


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