Topological Early Universe Cosmology

Standard Einstein equations (2.3) hold in the non-distributional sense in the whole of ${\cal M}_{{\rm II}}$ including its boundary. That is, all geometric quantities are continuous at $B$ and in particular, the second fundamental form (extrinsic curvature) should also be continuous across $B$. Viewed from phase I, i.e., from ${{\cal M}}_{\rm I}$, the second fundamental form is

$\displaystyle K_{\mu\nu}=\nabla_{\mu}n_{\nu},$

(2.21)

where $n^{\mu}$ is the unit normal vector to the boundary $B$. On the other hand, from the ${\cal M}_{{\rm II}}$ point of view we find

$\displaystyle K_{\mu\nu}=i\nabla_{\mu}n_{\nu},$

(2.22)

since we have to Wick rotate going form ${{\cal M}}_{\rm I}$ to ${\cal M}_{{\rm II}}$. Since the extrinsic curvature is continuous on $B$, the only way both (2.21) and (2.22) to hold is

$\displaystyle K_{\mu\nu}\Big{|}_{B}=0.$

(2.23)

In other words, the common boundary of ${{\cal M}}_{\rm I}$ and ${\cal M}_{{\rm II}}$ has vanishing extrinsic curvature and therefore is a totally geodesic submanifold as in the no-boundary proposal [34].¹¹1A hypersurface like $B$ with vanishing extrinsic curvature is also called “moment of time symmetry” [21]. This also specifies the scalar curvature of $B$. Indeed, from Einstein equations (2.3), the Hamiltonian constraint is

$\displaystyle{}^{3}R+K^{2}-K_{ij}K^{ij}=2M_{\rm p}^{-2}n^{\mu}n^{\nu}T_{\mu\nu}~{}~{}~{}~{}\mbox{in}~{}~{}~{}{\cal M}_{{\rm II}}$

(2.24)

and therefore, due to (2.23), we get that

$\displaystyle{}^{3}R\big{|}_{B}=2M_{\rm p}^{-2}\,\rho\big{|}_{B},$

(2.25)

where $\rho=n^{\mu}n^{\nu}T_{\mu\nu}$ is the energy density. Since $n^{\mu}$ is timelike, the weak energy condition $n^{\mu}n^{\nu}T_{\mu\nu}\geq 0$ gives

$\displaystyle{}^{3}R\big{|}_{B}\geq 0.$

(2.26)

Therefore, the induced metric on $B$ has non-negative scalar curvature. This result should be kept in mind when discussing the flatness problem.

Since (2.25) is nothing else than the Hamiltonian constraint, it provides initial data for Einstein equations. However, these initial data should not be arbitrary but should describe a totally geodesic hypersurface of the Riemannian space ${{\cal M}}_{\rm I}$, where ${{\cal M}}_{\rm I}$ and ${\cal M}_{{\rm II}}$ are glued in an at least $C^{(2)}$ way.

We have seen that geometries with vanishing self dual (or the anti-self dual) part of the Weyl tensor are critical points of Witten’s topological gravity. Therefore, at the boundary $B$ of ${{\cal M}}_{\rm I}$ we will have by continuity

$\displaystyle W_{ABCD}=0~{}~{}~{}~{}~{}\mbox{in}~{}~{}~{}~{}{{\cal M}}_{\rm I},~{}~{}~{}\Longrightarrow~{}~{}~{}W_{ABCD}\big{|}_{B=\partial{{\cal M}}_{\rm I}}=0$

(2.27)

as well. The same condition should hold from the ${\cal M}_{{\rm II}}$ side (phase II) i.e.,

$\displaystyle W_{ABCD}\big{|}_{B=\partial{\cal M}_{{\rm II}}}=0.$

(2.28)

Since we have to Wick rotate from ${{\cal M}}_{\rm I}$ to ${\cal M}_{{\rm II}}$, we have that

	$\displaystyle W^{-}_{\mu\nu\rho\sigma}=\frac{1}{2}\Big{(}W_{\mu\nu\rho\sigma}+{}^{*}W_{\mu\nu\rho\sigma}\Big{)}~{}~{}~{}~{}\mbox{in}~{}~{}~{}~{}{{\cal M}}_{\rm I},$
	$\displaystyle W^{-}_{\mu\nu\rho\sigma}=\frac{1}{2}\Big{(}W_{\mu\nu\rho\sigma}+i{}^{*}W_{\mu\nu\rho\sigma}\Big{)}~{}~{}~{}~{}\mbox{in}~{}~{}~{}~{}{\cal M}_{{\rm II}}.$		(2.29)

The only way to match on their common boundary is the full Weyl tensor to vanish, i.e.,

$\displaystyle W_{\mu\nu\rho\sigma}\big{|}_{B}=0.$

(2.30)

Let us note that the above condition (2.30) can also be written as initial data for the electric and magnetic part of the Weyl tensor, which will be useful below, as follows. The boundary hypersurface $B$ is spacelike with timelike normal $n^{\mu}$. We can decompose the Weyl tensor into its electric and magnetic parts with respect to $n^{\mu}$ as

	$\displaystyle E_{\mu\nu}$	$\displaystyle=$	$\displaystyle W^{\rho}_{\mu\sigma\nu}n_{\rho}n^{\sigma},$		(2.31)
	$\displaystyle B_{\mu\nu}$	$\displaystyle=$	$\displaystyle\frac{1}{2}{\epsilon_{\mu\rho}}^{\kappa\sigma}W^{\lambda}_{\nu\kappa\sigma}n^{\rho}n_{\lambda}.$		(2.32)

In terms of the projection $h_{\mu\nu}$ orthogonal to $n^{\mu}$

$\displaystyle h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu},$

(2.33)

Eqs.(2.31) and (2.32) are expressed as

	$\displaystyle E_{\mu\nu}$	$\displaystyle=$	$\displaystyle-{}^{3}R_{\mu\nu}+KK_{\mu\nu}-K_{\mu\lambda}K^{\lambda}_{\nu}+\frac{1}{2}h^{\rho}_{\mu}h^{\sigma}_{\nu}R_{\rho\sigma}+\left(\frac{1}{2}h^{\rho\sigma}R_{\rho\sigma}-\frac{1}{3}R\right)h_{\mu\nu},$		(2.34)
	$\displaystyle B_{\mu\nu}$	$\displaystyle=$	$\displaystyle D_{\rho}K_{\mu(\sigma}{\epsilon_{\nu)}}^{\sigma\rho\kappa}n_{\kappa},$		(2.35)

where $D_{\rho}$ is the covariant derivative with respect to the induced metric $h_{\mu\nu}$. The vanishing of the Weyl tensor on $B$ then implies

$\displaystyle E_{\mu\nu}\big{|}_{B}=B_{\mu\nu}\big{|}_{B}=0.$

(2.36)

Since the extrinsic curvature of $B$ vanishes according to Eq.(2.23), the magnetic part of the Weyl tensor is automatically zero. On the other hand, the vanishing of the electric part gives that the boundary $B$ is a space of constant curvature [22].

There is a second condition related to the Weyl tensor, namely, the Bach tensor should vanish in ${{\cal M}}_{\rm I}$. Indeed, the variation of (2.4) leads to field equations of conformal gravity

$\displaystyle{\cal B}_{\mu\nu}=\left(\nabla^{\rho}\nabla^{\sigma}+\frac{1}{2}R^{\rho\sigma}\right)W_{\mu\nu\rho\sigma}=0,$

(2.37)

where ${\cal B}_{\mu\nu}$ is the Bach tensor. If the Weyl tensor is self dual or anti-self dual, then the Bach tensor vanishes identically. Indeed, in two-component notation the Bach tensor is written as

	$\displaystyle{\cal B}_{\mu\nu}$	$\displaystyle=$	$\displaystyle 2\Big{(}{\nabla^{C}}_{\dot{A}}{\nabla^{D}}_{\dot{B}}+{\Phi^{CD}}_{\dot{A}\dot{B}}\Big{)}W_{ABCD}$		(2.38)
		$\displaystyle=$	$\displaystyle 2\Big{(}{\nabla^{\dot{C}}}_{A}{\nabla^{\dot{D}}}_{B}+{\Phi^{\dot{C}\dot{D}}}_{AB}\Big{)}W_{\dot{A}\dot{B}\dot{C}\dot{D}},$

where $\Phi_{AB\dot{A}\dot{B}}=-\frac{1}{2}\big{(}R_{\mu\nu}-\frac{1}{4}Rg_{\mu\nu}\big{)}$ is the Ricci spinor. Clearly ${\cal B}_{\mu\nu}=0$ for self dual ($W_{\dot{A}\dot{B}\dot{C}\dot{D}}=0$), or anti-self dual $(W_{ABCD}=0$) Weyl. Therefore, the Bach tensor should vanish in ${{\cal M}}_{\rm I}$ and by continuity, it should vanish also on the boundary $B$, i.e.,

$\displaystyle{\cal B}_{\mu\nu}\big{|}_{B}=0.$

(2.39)

The vanishing of the Bach tensor on the boundary $B$ should also be true for observers in ${\cal M}_{{\rm II}}$.

A serious shortcoming of the standard big bang cosmology is the horizon problem, which is associated to the puzzling homogeneity of the observed universe across patches which had never been in causal contact since the onset of the cosmological evolution. In the present scenario, all regions of the universe have emerged from a single phase I. As can be seen from Fig. 1, two past light cones, although have never been in contact in ${\cal M}_{{\rm II}}$, have emerged, through the boundary $B$, from the common phase I of the Riemannian ${{\cal M}}_{\rm I}$ space. Therefore, their homogeneity is the result of the same universal initial conditions set from phase I.

This is also related by the fact that the initial data for the cosmological evolution in phase II in ${\cal M}_{{\rm II}}$, as we have seen, is the vanishing of the Weyl tensor and the extrinsic curvature on the common boundary $B$.

Let us recall that it is generally accepted that an initial vanishing Weyl tensor, a hypothesis that is usually referred to as “the Weyl curvature hypothesis”, suffices to explain the isotropy and homogeneity of the universe [8, 9, 10]. In fact, it has been conjectured that an initially zero Weyl tensor necessary implies an FRW cosmology. Although a general proof is lacking, this conjecture has been proved for a universe filled with a perfect fluid with equation of state $P=w\rho$ for $0<w\leq 1$ [23, 24, 25, 26, 22].

Although it seems that the horizon problem and the associated homogeneity and isotropy of the universe can easily be explained withing the present set up, this is not automatically the case for the flatness problem. Indeed, as we have seen above, the conditions Eqs. (2.23) and (2.30) lead to a constant curvature 3D boundary $B$, see equation (2.26). This poses a threat to the solution to the flatness problem as both possibilities $k=0$ and $k=1$ of the spatial geometry of the FRW universe are allowed.

One may hope that a flat $B$ ($k=0$) can be selected by adding an appropriate theory on $B$. Such a boundary theory should be conformal as we want the conformal invariance to be broken only by anomalies. Luckily, such a theory exists and is a 3D conformally invariant version of conventional gravity of Chern-Simons type with action [27, 28, 29]

$\displaystyle{\cal S}_{HW}=\int_{B}\epsilon^{ijk}\left\{\omega_{ia}\big{(}\partial_{j}\omega_{k}^{a}-\partial_{k}\omega_{j}^{a}\big{)}+\frac{2}{3}\epsilon^{abc}\omega_{ia}\omega_{jb}\omega_{kc}\right\},$

(3.1)

where ${\omega_{i}}^{a}$ is the spin connection. As it has been proven in [29], 3D conformal gravity described by (3.1) is classically equivalent to a Chern-Simons theory for the conformal group $SO(3,2)$. Variation of (3.1) gives

$\displaystyle\delta{\cal S}_{HW}=\int_{B}\epsilon^{ijk}{R_{ij}}^{ab}\delta\omega_{kab}.$

(3.2)

In the usual treatment of the gravittional Chern-Simons action (3.1), one trades the variation of the spin connection with that of the vierbeins. This leads to the equation of motion

$\displaystyle C_{ijk}=\nabla_{k}W_{ij}-\nabla_{j}W_{ki},$

(3.3)

where $W_{ij}=R_{ij}-1/4Rg_{ij}$, i.e., to the vanishing of the 3D Cotton tensor $C_{ijk}$. Therefore, one is tempting to conclude that $B$ should be only conformally flat (due to the vanishing of the Cotton tensor), but not necessarily flat ($k=0$). However, in our case, $B$ is the boundary of both ${{\cal M}}_{\rm I}$ and ${\cal M}_{{\rm II}}$ so that variations of the vierbein vanishes

$\displaystyle\delta{e_{i}}^{a}\big{|}_{B}=0.$

(3.4)

This means that we should treat the spin connection $\omega_{iab}$ as independent field as in Palatini formulation, so that its equation of motion is

$\displaystyle{R_{ij}}^{ab}=0.$

(3.5)

Therefore we see that the addition of the conformal invarinat 3D action Eq. (3.1) selects a flat boundary $B$ solving the flatness problem.

We are interested now in scalar perturbations in ${{\cal M}}_{\rm I}$. Such perturbations will provide initial conditions for the two-point correlators at the Cauchy surface $B$ and determine therefore the spectrum of curvature perturbations in the ${\cal M}_{{\rm II}}$ universe. As long as the theory in ${{\cal M}}_{\rm I}$ is conformal, conformal transformations of the metric

$\displaystyle\delta g_{\mu\nu}=2hg_{\mu\nu}$

(4.15)

do not change the theory. Therefore, the conformal mode corresponding to the conformal perturbation

$\displaystyle g_{\mu\nu}=\big{(}1+2h\big{)}\bar{g}_{\mu\nu},~{}~{}~{}~{}\tau=\bar{\tau}+h,$

(4.16)

where $\bar{g}_{\mu\nu}$ is the background metric, is not physical. However, when conformal invariance is broken, the conformal mode becomes physical (dilaton). In particular, if conformal invariance is broken due to an anomaly, the trace of the energy-momentum tensor is given by

$\displaystyle T^{\mu}_{\mu}=\frac{c}{1920\pi^{2}}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}-\frac{a}{5760\pi^{2}}E_{4},$

(4.17)

normalized such that a real scalar contribute with $a=c=1$ to the anomaly. The response of the effective action under (4.16) is

$\displaystyle\delta_{h}{\cal S}=\int{\rm d}^{4}x\sqrt{g}\,h\left(\frac{c}{1920\pi^{2}}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}-\frac{a}{5760\pi^{2}}E_{4}\right).$

(4.18)

Then, it is straightforward to verify that

$\displaystyle{\cal S}_{h}=-\frac{a}{720\pi^{2}}\int{\rm d}^{4}x\sqrt{\bar{g}}\bigg{\{}\bar{G}^{\mu\nu}\,\partial_{\mu}h\partial_{\nu}h+\bar{R}^{\mu\nu}h\partial_{\mu}h\partial_{\nu}h+\partial_{\mu}h\partial_{\nu}h\partial^{\mu}h\partial^{\nu}h+\mathcal{O}(h^{5})\bigg{\}},$

(4.19)

to fourth order where $\bar{G}^{\mu\nu}$ ($\bar{R}^{\mu\nu}$) the background Einstein (Ricci) tensor. Therefore, the conformal mode is dynamical now. Its two-point correlator can be found by using the Ward identity for scale invariance. The latter is written as

$\displaystyle\langle T^{\mu}_{\mu}(x){\cal O}_{1}(x_{1})\cdots{\cal O}_{n}(x_{n})\rangle=-\sum_{i=1}^{n}\delta^{(4)}(x-x_{i})\Delta_{i}\langle{\cal O}_{1}(x_{1})\cdots{\cal O}_{i}(x_{i})\cdots{\cal O}_{n}(x_{n})\rangle,$

(4.20)

where $\Delta_{i}$ is the dimension of ${\cal O}_{i}$. In particular, for ${\cal O}_{i}=h$, the Ward identity (4.20) for the two-point correlator is written as

$\displaystyle\langle T^{\mu}_{\mu}(x)h(y)h(0)\rangle=-\Delta\,\delta^{(4)}(x-y)\langle h(y)h(0)\rangle-\Delta\,\delta^{(4)}(x)\langle h(y)h(0)\rangle.$

(4.21)

Therefore, after integration we get that

	$\displaystyle 2\Delta\langle h(y)h(0)\rangle$	$\displaystyle=-\int_{{{\cal M}}_{\rm I}}{\rm d}^{4}x\sqrt{g}\langle T^{\mu}_{\mu}(x)h(y)h(0)\rangle$
		$\displaystyle\approx-\int_{{{\cal M}}_{\rm I}}{\rm d}^{4}x\sqrt{g}\,\langle T_{\mu\nu}(x)\rangle\langle h(y)h(0)\rangle.$		(4.22)

where the approximation $\langle T^{\mu}_{\mu}(x)h(y)h(0)\rangle\approx\langle T^{\mu}_{\mu}(x)\rangle\langle h(y)h(0)\rangle$ has been used. Using the integrated conformal anomaly

$\displaystyle\int_{{{\cal M}}_{\rm I}}{\rm d}^{4}x\sqrt{g}\big{<}T^{\mu}_{\mu}(x)\big{>}=\frac{c}{240}\langle I_{W}\rangle-\frac{a}{180}\chi_{\tiny{R}},$

(4.23)

where $\chi_{\tiny{R}}$ is given by

$\displaystyle\chi_{\tiny{R}}=\frac{1}{32\pi^{2}}\int\limits_{{{\cal M}}_{\rm I}}E_{4},$

(4.24)

and $I_{W}$ denotes the integral

$\displaystyle I_{W}=\frac{1}{8\pi^{2}}\int\limits_{{{\cal M}}_{\rm I}}W_{\mu\nu\rho\sigma}^{2}\,\sqrt{g}{\rm d}^{4}x,$

(4.25)

we find that the scaling dimension of $h$ turns out to be

$\displaystyle\Delta\approx\frac{a}{360}\chi_{\tiny{R}}-\frac{c}{480}\langle I_{W}\rangle.$

(4.26)

Note that $\chi_{\tiny{R}}$ is the Euler number for a compact 4D manifolds. If there are boundaries, then the Euler number gets boundary corrections [31] (and so does the integrated conformal anomaly [32, 33]) so that

$\displaystyle\chi_{\tiny{R}}=\frac{1}{32\pi^{2}}\int\limits_{{{\cal M}}_{\rm I}}E_{4}-\frac{1}{4\pi^{2}}\int\limits_{\partial{{\cal M}}_{\rm I}}X,$

(4.27)

where $X$ is

	$\displaystyle X$	$\displaystyle=K^{\mu\nu}n^{\kappa}n^{\lambda}R_{\kappa\mu\lambda\nu}-K^{\mu\nu}R_{\mu\nu}-KR_{\mu\nu}n^{\mu}n^{\nu}$
		$\displaystyle+\frac{1}{2}KR-\frac{1}{3}K^{3}+KK^{\mu\nu}K_{\mu\nu}+\frac{2}{3}K^{\mu\kappa}K_{\kappa\lambda}K^{\lambda}_{\nu}.$		(4.28)

In the present case, since the extrinsic curvature $K_{\mu\nu}$ vanishes, we have $X=0$ and therefore, there are no boundary corrections to the Euler number. In other words, the Euler number of ${{\cal M}}_{\rm I}$ with boundary the totally geodesic codimension-one space $B$, is given just by (4.24).

An estimate of the term $\langle I_{W}\rangle$ gives

$\displaystyle\langle I_{W}\rangle\sim\alpha_{W}\Lambda_{UV}^{4}L_{I}^{4},$

(4.29)

where $\Lambda_{UV}$ is the UV cutoff in ${{\cal M}}_{\rm I}$, $L_{I}$ is a characteristic scale in ${{\cal M}}_{\rm I}$ (curvature scale) and $a_{W}=g_{W}^{2}/8\pi$. Indeed, based on dimensional grounds, $\langle W_{\mu\nu\rho\sigma}^{2}\rangle\sim\alpha_{W}\Lambda_{UV}^{4}$ and therefore, we expect (4.29) to hold.

Note that we may use the doubling trick to construct a new compact manifold out of ${{\cal M}}_{\rm I}$ if its boundary is only $B$. The rule is the following: since the extrinsic curvature of ${{\cal M}}_{\rm I}$ vanishes, we may construct the manifold $2{{\cal M}}_{\rm I}$ consisting of two copies ${{\cal M}}_{\rm I}^{-}$ and ${{\cal M}}_{\rm I}^{+}$ joined across $B$. Since $K_{ij}=0$ on $B$, the induced metric on $2{{\cal M}}_{\rm I}$ from the metrics on ${{\cal M}}_{\rm I}^{\pm}$ is at least $C^{(1)}$. The manifold $2{{\cal M}}_{\rm I}$ is a compact manifold without boundary, it has $\chi_{R}(2{{\cal M}}_{\rm I})=2\chi_{R}({{\cal M}}_{\rm I})$ and signature $\tau_{R}(2{{\cal M}}_{\rm I})=0$ [34]. In addition, ${{\cal M}}_{\rm I}$ admits a reflection map $\theta$ interchanging ${{\cal M}}_{\rm I}^{+}$ and ${{\cal M}}_{\rm I}^{-}$ while leaving $B$ invariant. In this case, the metric close to the boundary $B$ is of the form

$\displaystyle{\rm d}s_{\rm I}^{2}\approx{\rm d}t_{\rm I}^{2}+g_{ij}(x,t_{\rm I}^{2}){\rm d}x^{i}{\rm d}x^{j},$

(4.30)

where the reflection map is realized by $t_{\rm I}\to-t_{\rm I}$. Note also that this form of the metric close to $B$ allows the Wick rotation $t_{\rm I}\to it_{\rm II}$. Then the boundary $B$ is just the fixed point of the reflection map $\theta$ (acting here as $t_{\rm I}\to-t_{\rm I}$).

The two-point function turns out to be

$\displaystyle\langle h(x)h(0)\rangle\sim\frac{720\pi^{2}L_{I}^{2}}{a}\frac{1}{|x|^{2\Delta}},$

(4.31)

and therefore, we have at the boundary $B$

$\displaystyle\langle h(x)h(0)\rangle\Big{|}_{B}=\langle h(0,\vec{x})h(0,\vec{0})\rangle\sim\frac{720\pi^{2}L_{I}^{2}}{a}\frac{1}{|\vec{x}|^{2\Delta}}.$

(4.32)

This is the initial condition for the scalar perturbation in ${\cal M}_{{\rm II}}$. The factor $L_{I}^{2}/a$ originates from the coupling of the conformal mode in (4.19). In particular, $h|_{B}=\Psi|_{B}$ and, after Fourier transforming, we get that the initial condition for the two-point correlator in ${\cal M}_{{\rm II}}$ should be

$\displaystyle|\Psi_{k}|^{2}\Big{|}_{B}\sim\frac{L_{I}^{2}M_{\rm p}^{2}}{a}k^{-3+2\Delta},$

(4.33)

since $\Psi$ and $h$ are differently normalized as their kinetic terms are multiplied with $M_{\rm p}^{2}$ and $L_{I}^{-2}$, respectively. This leads to a spectral index

$\displaystyle n_{s}=1+2\Delta=1+\frac{a}{180}\chi_{\tiny{R}}-\frac{c}{240}\,\alpha_{W}\Lambda_{UV}^{4}L_{I}^{4}$

(4.34)

and an amplitude

$\displaystyle A\sim\frac{L_{I}^{2}M_{\rm p}^{2}}{a}.$

(4.35)

We encounter now a problem related to the spectral index in Eq. (4.34). Consider the simplest most symmetric case of an $S^{4}$ of radius $L$ as a candidate for $2{{\cal M}}_{\rm I}$ and the standard metric on $S^{4}$

$\displaystyle{\rm d}s^{2}=L^{2}{\rm d}\tau^{2}+L^{2}\cos^{2}\tau\,{\rm d}\Omega_{3},~{}~{}~{}~{}-\frac{\pi}{2}\leq\tau\leq\frac{\pi}{2},$

(4.36)

where ${\rm d}\Omega_{3}$ is the metric of the unit three-sphere and we have shifted the azimuthial angle $\tau$ by $-\pi/2$. Then, the equatorial $\tau=0$ has vanishing extrinsic curvature $K_{ij}=0$ and therefore, it is the boundary $B$ which will be continued into a Lorentzian FRW ${\cal M}_{{\rm II}}$ with $k=1$. However, we find that in this case, $\chi_{\tiny{R}}=1$, and therefore we will have a ridiculously large blue spectrum as follows from (4.34) and (4.35). Now, in order to get an amplitude $A\sim 10^{-10}$, we should have $a\sim 10^{10}L_{I}^{2}M_{\rm p}^{2}$. In addition, we need

$\displaystyle-\frac{a}{180}\chi_{R}\approx 0.03$

(4.37)

for a red spectrum with $n_{s}\approx 0.97$. However, according to Hopf conjecture, the Euler number of a compact manifold is no-negative, so that

$\displaystyle\chi_{R}({{\cal M}}_{\rm I})=\frac{1}{2}\chi_{R}(2{{\cal M}}_{\rm I})=1-b_{1}+\frac{1}{2}b_{2}\geq 0$

(4.38)

where $b_{1},b_{2}$ are the Betti numbers of $2{{\cal M}}_{\rm I}$ [34]. Therefore, we end up in general in an enormous blue spectrum for scalar perturbations. The only way to avoid this is the manifolds ${{\cal M}}_{\rm I}$ to have vanishing Euler number

$\displaystyle\chi_{R}=0.$

(4.39)

In this case, the spectral index is given by

$\displaystyle n_{s}=1+2\Delta=1-\frac{c}{480}\,\alpha_{W}\Lambda_{UV}^{4}L_{I}^{4},$

(4.40)

which may have the correct value for appropriate values of the parameters [1].