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 [231Jump To The Next Citation Point, 195Jump To The Next Citation Point, 146Jump To The Next Citation Point, 368Jump To The Next Citation Point, 198Jump To The Next Citation Point, 197Jump To The Next Citation Point, 407Jump To The Next Citation Point, 426Jump To The Next Citation Point, 259Jump To The Next Citation Point, 275Jump To The Next Citation Point, 196Jump To The Next Citation Point, 46Jump To The Next Citation Point, 325Jump To The Next Citation Point, 221Jump To The Next Citation Point, 315Jump To The Next Citation Point, 149Jump To The Next Citation Point, 18Jump 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 [134Jump To The Next Citation Point, 220Jump To The Next Citation Point] or a hot big-bang radiation era, as in the “ekpyrotic” or cyclic scenario [229Jump To The Next Citation Point, 215Jump To The Next Citation Point, 339Jump To The Next Citation Point, 403Jump To The Next Citation Point, 273Jump To The Next Citation Point, 317Jump To The Next Citation Point, 412Jump To The Next Citation Point], or in colliding bubble scenarios [40Jump To The Next Citation Point, 156Jump To The Next Citation Point, 157Jump To The Next Citation Point]. (See also [26Jump To The Next Citation Point, 98Jump To The Next Citation Point, 299Jump 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 [287] for a review).

High-energy brane-world modifications to the dynamics of inflation on the brane have been investigated [308Jump To The Next Citation Point, 216, 92, 405, 320, 319, 106, 285Jump To The Next Citation Point, 34, 35, 36, 328, 192, 264, 363, 307]. 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 [308Jump To The Next Citation Point, 99Jump To The Next Citation Point, 312Jump To The Next Citation Point, 369Jump To The Next Citation Point, 346Jump To The Next Citation Point, 286Jump To The Next Citation Point, 205Jump To The Next Citation Point].

The field satisfies the Klein–Gordon equation

′ ϕ¨+ 3H ˙ϕ + V (ϕ) = 0. (204 )
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 (194View Equation), with m = 0 = Λ = K, and Equation (204View Equation), we find that
[ ] 1 1 + 2ρ∕λ ¨a > 0 ⇒ w < − 3- 1-+-ρ-∕λ- , (205 )
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, (206 )
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-- , (207 ) 3 2λ V ′ ˙ϕ ≈ − ---. (208 ) 3H
The brane-world correction term V∕ λ in Equation (207View Equation) serves to enhance the Hubble rate for a given potential energy, relative to general relativity. Thus there is enhanced Hubble ‘friction’ in Equation (208View 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 , (209 ) 2 ( ′′) [ ] η ≡ − -¨ϕ--= M-p- V-- ----1---- . (210 ) 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 , (211 )
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 [99Jump To The Next Citation Point, 312Jump To The Next Citation Point, 369Jump To The Next Citation Point, 346Jump To The Next Citation Point, 286Jump 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 [99Jump To The Next Citation Point, 312Jump To The Next Citation Point, 369Jump To The Next Citation Point, 346Jump To The Next Citation Point, 286Jump To The Next Citation Point, 8, 327, 288, 56, 380].

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

8π ∫ ϕfV [ V ] N ≈ − --2- --′ 1 + --- dϕ. (212 ) 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 (212View Equation) becomes
3∫ ϕf 2 N ≈ − 128π-- V--dϕ. (213 ) 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 [303Jump To The Next Citation Point, 271Jump To The Next Citation Point, 177Jump To The Next Citation Point, 148Jump 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 [148Jump 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.

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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 (220View Equation), which at high energies is in excellent agreement with the exact solution, evaluated numerically in slow-roll. (Figure taken from [308Jump To The Next Citation Point].)

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

ζ ≡ ℛ − H-δρ, (214 ) 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) [425Jump 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
δϕ ζ = H --. (215 ) ϕ˙
If one makes the assumption that backreaction due to metric perturbations in the bulk can be neglected, 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 [425]. 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 (215View Equation), this gives
( ) [ ]3|| 2 512π--V-3- 2λ +-V- | A s ≈ 75M 6 V′2 2λ || . (216 ) p 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. UpdateJump To The Next Update Information

A crucial assumption is that backreaction due to metric perturbations in the bulk can be neglected. In the extreme slow-roll limit this is necessarily correct because the coupling between inflaton fluctuations and metric perturbations vanishes; however, this is not necessarily the case when slow-roll corrections are included in the calculation. Previous work [250, 253, 255] has shown that such bulk effects can be subtle and interesting (see also [109, 114] for other approaches). In particular, subhorizon inflaton fluctuations on a brane excite an infinite ladder of Kaluza–Klein modes of the bulk metric perturbations at first order in slow-roll parameters, and a naive slow-roll expansion breaks down in the high-energy regime once one takes into account the backreaction of the bulk metric perturbations, as confirmed by direct numerical simulations [200Jump To The Next Citation Point]. However, an order-one correction to the behaviour of inflaton fluctuations on subhorizon scales does not necessarily imply that the amplitude of the inflaton perturbations receives corrections of order one on large scales; one must consistently quantise the coupled brane inflaton fluctuations and bulk metric perturbations. This requires a detailed analysis of the coupled brane-bulk system [70Jump To The Next Citation Point, 252Jump To The Next Citation Point].

It was shown that the coupling to bulk metric perturbations cannot be ignored in the equations of motion. Indeed, there are order-unity differences between the classical solutions without coupling and with slow-roll induced coupling. However, the change in the amplitude of quantum-generated perturbations is at next-to-leading order [252] because there is still no mixing at leading order between positive and negative frequencies when scales observable today crossed the horizon, so the Bogoliubov coefficients receive no corrections at leading order. The amplitude of perturbations generated is also subject to the usual slow-roll corrections on super-horizon scales. The next-order slow-roll corrections from bulk gravitational perturbations are calculated in [254] and they are the same order as the usual Stewart–Lyth correction [404]. These results also show that the ratio of tensor-to-scalar perturbation amplitudes are not influenced by brane-bulk interactions at leading order in slow-roll. It is remarkable that the predictions from inflation theories should be so robust that this result holds in spite of the leading-order change to the solutions of the classical equations of motion.

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

dln A2s ns − 1 ≡ -------≈ − 4𝜖 + 2η, (217 ) dlnk
where the slow-roll parameters are given in Equations (209View Equation) and (210View 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 V = 12m2 ϕ2. Equation (212View Equation) gives the integrated expansion from ϕi to ϕf as

2π-( 2 2) π2m2-( 4 4) N ≈ M 2 ϕ i − ϕ f + 3M 6 ϕi − ϕf . (218 ) p 5
The new high-energy term on the right leads to more inflation for a given initial inflaton value ϕi.

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 (206View Equation) becomes ˙2 2 ϕ < 5V. In the slow-roll approximation (using Equations (207View Equation) and (208View Equation)) ϕ˙≈ − M 35∕2πϕ, and this yields

5 (M5 )2 ϕ4end ≈ --2- ---- M 45. (219 ) 4π m
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 N ≈ 55 cobe in Equation (218View Equation) and ϕf = ϕend. The second term on the right of Equation (218View Equation) dominates, and we obtain
( )2 4 165- M5-- 4 ϕcobe ≈ π2 m M 5. (220 )
Imposing the COBE normalization on the curvature perturbations given by Equation (216View Equation) requires
( 8π2 ) m4 ϕ5 As ≈ ---- ----c6obe-≈ 2 × 10 −5. (221 ) 45 M 5
Substituting in the value of ϕcobe given by Equation (220View Equation) shows that in the limit of strong brane corrections, observations require
−5 2 m ≈ 5 × 10 M5, ϕcobe ≈ 3 × 10 M5. (222 )
Thus for M5 < 1017 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 M4.

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 [284Jump 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 [284Jump 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 [284].)

Other perturbation modes have also been investigated:

Brane-world effects on large-scale isocurvature perturbations in 2-field inflation have also been considered [17]. Brane-world (p)reheating after inflation is discussed in [414, 429, 9, 415, 96].

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