Brane-world inflation

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 [13

, 33

, 100

, 103

, 138

, 139

, 140

, 141

, 158

, 165

, 180

, 195

, 229

, 238

, 276

, 304

, 318

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 [90 , 157 ] or a hot big-bang radiation era, as in the “ekpyrotic” or cyclic scenario [154 , 163 , 193 , 231 , 251 , 301 , 307 ], or in colliding bubble scenarios [29 , 107 , 108 ]. (See also [21 , 69 , 214 ] 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 , 222 , 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 [70 , 145 , 206 , 222 , 226 , 258 , 277 ].

The field satisfies the Klein–Gordon equation

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 (188

), with $m = 0 = Λ = K$ , and Equation (198

), we find that

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

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 (201

) serves to enhance the Hubble rate for a given potential energy, relative to general relativity. Thus there is enhanced Hubble ‘friction’ in Equation (202

), 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 $≈ λ ∕V$ :

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 [70

, 206

, 226

, 258

, 277

]. They also allow for the novel possibility that the inflaton could act as dark matter or quintessence at low energies [4 , 43 , 70

, 206

, 208 , 226

, 239 , 258

, 277

, 285 ].

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

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 (206

) becomes

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 [102 , 122 , 191 , 218 ]. 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 [102 ]. Thus it seems a reasonable approximation in slow-roll to neglect the KK effects carried by $ℰ μν$ when computing the density perturbations.

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 (214

), which at high energies is in excellent agreement with the exact solution, evaluated numerically in slow-roll. (Figure taken from [222

].)

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

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) [317

]. The curvature perturbation on uniform density hypersurfaces is given in terms of the scalar field fluctuations on spatially flat hypersurfaces $δφ$ by

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 (209

), this gives

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

where the slow-roll parameters are given in Equations (203

) and (204

). 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 (206 ) gives the integrated expansion from $φi$ to $φf$ as

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 (200 ) becomes $φ˙2 < 2V 5$ . In the slow-roll approximation (using Equations (201 ) and (202 )) $˙ 3 φ ≈ − M 5∕2πφ$ , and this yields

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 (212

) and $φ = φ f end$ . The second term on the right of Equation (212

) dominates, and we obtain

Imposing the COBE normalization on the curvature perturbations given by Equation (210

) requires

Substituting in the value of $φcobe$ given by Equation (214

) shows that in the limit of strong brane corrections, observations require

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 5

, 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 [203 ]. 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 [203 ], as shown in Figure 6 .

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:

High-energy inflation on the brane also generates a zero-mode (4D graviton mode) of tensor perturbations, and stretches it to super-Hubble scales, as will be discussed below. This zero-mode has the same qualitative features as in general relativity, remaining frozen at constant amplitude while beyond the Hubble horizon. Its amplitude is enhanced at high energies, although the enhancement is much less than for scalar perturbations [192 ]:

Equation (218 ) means that brane-world effects suppress the large-scale tensor contribution to CMB anisotropies. The tensor spectral index at high energy has a smaller magnitude than in general relativity,

but remarkably the same consistency relation as in general relativity holds [145 ]:

This consistency relation persists when $Z2$ symmetry is dropped [146 ] (and in a two-brane model with stabilized radion [118 ]). It holds only to lowest order in slow-roll, as in general relativity, but the reason for this [286 ] and the nature of the corrections [44 ] are not settled.
The massive KK modes of tensor perturbations remain in the vacuum state during slow-roll inflation [121 , 192 ]. The evolution of the super-Hubble zero mode is the same as in general relativity, so that high-energy brane-world effects in the early universe serve only to rescale the amplitude. However, when the zero mode re-enters the Hubble horizon, massive KK modes can be excited.
Vector perturbations in the bulk metric can support vector metric perturbations on the brane, even in the absence of matter perturbations (see the next Section 6). However, there is no normalizable zero mode, and the massive KK modes stay in the vacuum state during brane-world inflation [39 ]. Therefore, as in general relativity, we can neglect vector perturbations in inflationary cosmology.

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 ].