Coleman-Weinberg Inflation in light of Planck

We revisit a single field inflationary model based on Coleman-Weinberg potentials. We show that in small field Coleman-Weinberg inflation, the observed amplitude of perturbations needs an extremely small quartic coupling of the inflaton, which might be a signature of radiative origin. However, the spectral index obtained in a standard cosmological scenario turns out to be outside the 2 sigma region of the Planck data. When a non-standard cosmological framework is invoked, such as brane-world cosmology in the Randall-Sundrum model, the spectral index can be made consistent with Planck data within 1 sigma, courtesy of the modification in the evolution of the Hubble parameter in such a scheme. We also show that the required inflaton quartic coupling as well as a phenomenologically viable B-L symmetry breaking together with a natural electroweak symmetry breaking can arise dynamically in a generalized B-L extension of the Standard Model where the full potential is assumed to vanish at a high scale.


Introduction
The Hot Big-Bang Model and General Relativity successfully explain the thermal history of our Universe since its very first nanoseconds of existence up to now. However they left unexplained crucial issues such as the horizon, flatness, and monopole problems. Inflation is the most elegant and simple explanation to address them and therefore no modern cosmological model lacks a prudential stage of inflation. Inflation basically assumes the existence of a period of exponential growth of the scale factor which essentially wipes out any trace of curvature, dilutes unwanted relics and leaves the Universe in a highly symmetric state.
The simplest scenario where this picture can be realized is based on a single scalar field (called inflaton) with a nearly flat potential. The quantum fluctuations of the inflaton would be then responsible for the tiny temperature anisotropies observed in the Cosmic Microwave Background (CMB). Basically an inflationary theory must fulfill two requirements to agree with experimental observations: (i) it has to provide sufficient inflation, i.e., the inflationary potential must drive an increase on the scale factor of 50 -60 e-folds (the precise number depends on details of the particular inflaton model) in order to describe the thermal equilibrium observed in the CMB at least for the scales of interest; (ii) the size of the quantum fluctuations of the inflaton leading to the power spectrum of primordial curvature perturbations, A s , should be at the appropriate level while the spectral index n s must agree with observations. Because of these "mild" requirements most simple inflation models were given a death blow once the Planck mission released its highprecision data [1]. The Planck result, combined with the WMAP large-angle polarization measurements, requires the two observables n s and A s of curvature perturbations to be n s = 0.9603 ± 0.0073 , A s = 2.196 +0.051 −0.060 × 10 −9 (1) at a scale k * = 0.05 Mpc −1 which rules out exact scale-invariance at more than 5σ. Analogously, the tensor-to-scalar ratio r is bounded to be r < 0.11 at 95 % CL, for the same scale k * . These constraints are already powerful enough to rule out or strongly disfavour the most popular and simple inflationary potentials. Consequently, simple and well motivated inflationary potentials are distinctively welcomed. Among these, a special place should be given to Coleman-Weinberg (CW) type of potentials [2] as they not only arise naturally but are unavoidable when loop corrections are taken into account. Inflation with a CW potential has been suggested at an early stage of the inflation theory formulation, and studied extensively, in particular, in association with grand unification theories [3].
In this paper, we revisit the CW inflation accounting for cosmological observations in a scheme where the inflaton potential has a dynamical origin, naturally arising from quantum corrections. As we will see later, the quartic coupling of the CW potential is proportional to the amplitude of primordial perturbations and thus has to be extremely small (∼ 10 −14 ) which might point towards a radiative origin. An attempt to generate a phenomenologically viable scalar potential starting from a vanishing initial condition of the full scalar potential at a cutoff scale has been worked out in the context of the [4]. It has been shown that a Standard Model (SM) Higgs potential consistent with recent LHC data [5] can be generated radiatively once the B − L symmetry is broken appropriately by the usual CW mechansim. One of the distinctive features of this scheme is that the quartic coupling of the B − L scalar arises due to its coupling to right-handed neutrinos in a similar way as the SM scalar quartic coupling arises due to the top Yukawa coupling.
We find that in a general CW inflation model, the number of e-folding required for solving the horizon problem cannot be made consistent with the observed spectral index. That is, the spectral index resulting from getting enough number of e-folds to take care of the horizon problem, can only fit within the 3σ range of the Planck data for a symmetry breaking scale larger than 10 15 GeV. In particular, when using a value of the symmetry breaking scale favoured for providing natural electroweak symmetry breaking in a generalized B − L extension of the SM, the resulting spectral index is well outside the 5σ range of the Planck data. This motivates us to consider the CW inflation in a non-standard cosmological scenario, a typical example of which is brane cosmology in a higher dimensional space such as the Randall-Sundrum model [6,7]. In this case, the extra dimension does not evolve in time while the evolution of four-dimensional spacetime is described by a modified Friedman equation, where H 2 ∝ ρ 2 in the large energy density limit. Thus, the effective inflaton energy increases by a factor, V 0 /Λ, with V 0 being the original inflaton energy and Λ being the brane tension, which makes the CW inflation fully compatible with the observations for an appropriate choice of the brane tension. A remarkable feature of the CW inflation on the brane is that the correlations between the observables are kept the same as in the standard CW inflation.
The paper is organized as follows. We begin by revisiting CW inflation in standard cosmology in Section 2. In Section 3 the setup is extended to brane cosmology and the inflationary observables are calculated for the modified Friedman equation. We show in Section 4 how the required quartic coupling and symmetry breaking scale for the CW inflaton potential can be generated in the context of the B − L extension of the SM which is motivated by the explanation of the neutrino masses and mixing. Finally, conclusions are drawn in Section 5.

CW inflation in standard cosmology
A general CW potential evaluated at the renormalization scale Q = φ = v φ takes a rather simple form: which satisfies V (v φ ) = 0 and V (v φ ) = 0. Here A = A(v φ ) is determined by the beta function for the scalar quartic coupling defined at Q = v φ , which is a function of gauge, Figure 1: The CW potential realizing a small field inflation.
Yukawa and other scalar couplings of the inflaton field φ. A detailed form of the beta function and the CW potential applied to the B − L gauge symmetry will be discussed later. Depending on the values of A and v φ , the inflaton can have small or large values compared to the Planck scale during observable inflation. In the latter case inflation mimics chaotic inflation. We will be interested in the scenario where the inflaton starts its journey from small values of the field, φ/v φ < 1, in the flat region of the CW potential as drawn in Fig. 1. There is a recent overview on various inflation models including the CW type inflation [8].
During inflation, the inflaton rolls down towards the minimum of its potential, evolving according toφ + 3Hφ + ∂V /∂φ = 0 where the Hubble rate H is given by Here M P ≈ 2.44 × 10 18 GeV is the reduced Planck mass. If the field is moving along a region where the potential is so flat that the evolution of the field is friction dominated, (what goes in the literature under the name of slow-roll approximation) the equation of motion is basically given by 3Hφ + ∂V /∂φ ≈ 0.
Within this approximation, the number of e-folds of inflation generated since the modes that are entering the observable Universe now left the horizon (at φ * ) till the end of inflation (at φ f ) is given by where V (φ) = ∂V /∂φ and φ f is the value of the field at which inflation stops. This happens when At this time, quantum fluctuations on the scale observed today were produced too and its size is given by while the spectral index of these density perturbations and its dependence on the scale read Here, η and ξ 2 are the second and third slow-roll parameters that can be expressed in terms of the inflaton potential as Besides the observed density (scalar) perturbations, the inflaton generates the yet unobserved gravitational waves or tensor perturbations. Normally, the tensor amplitude is expressed in terms of the tensor/scalar ratio as Expressed in terms of the parameters of the CW potential (2), the above inflation quantities are given explicitly by where Ei stands for the exponential integral.
From the equations above, one can see that for slow-roll inflation to take place, a field value φ * at horizon exit can be always found for any choice of v φ and it lays in the region of φ * /v φ 1 as the slow-roll parameters have a large factor of (4M P /v φ ) 2 . This leads to | | |η| ≈ 3|ξ 2 |/2. As a consequence, small field CW inflation predicts remarkable correlations: independently of the precise values of v φ and φ/v φ . Similarly, the power spectrum of perturbation is which is insensitive to φ/v φ . Considering the central value of n s = 0.96 and P R ≈ 2.2×10 −9 measured by the Planck mission at the pivot scale k * = 0.05 Mpc −1 [1], one gets N ≈ 75, and A ∼ 10 −14 mildly depending on φ * . The tensor-to-scalar ratio given by is severely suppressed by a factor of (v φ /M P ) 4 while being insensitive to φ.
Such large N and tiny A do not fit well with the number of e-folding required to solve the horizon problem for scales that have just entered/are entering the current horizon. The condition for such a thing to happen is which translates into where ρ 0(r) , ρ reh , ρ end are (radiation) energy densities at present, the end of reheating and the end of inflation, respectively, and ρ * , V * are energy density and inflaton potential evaluated at horizon exit for the pivot scale k * = 0.05 Mpc −1 . Note that one can take V * V (0) V end as a good approximation in most cases. For the CW picture of inflation to work, the number of e-folds N | φ * (16) derived from the CW potential should be able to reproduce the number N * (23). This condition together with the COBE normalization, P R 2.2 × 10 −9 , and the CW inflation property, n s 1 + 2η, uniquely fixes the values of the three parameters, A, v φ and φ/v φ of the CW potential.
The obtained spectral index n s for a given symmetry breaking scale v φ is shown in Fig. 2. The largest possible value that n s can take is about 0.945 for v φ close to M P which is below the 2σ range favoured by the Planck data. Only a 3σ compatibility can be achieved for v φ 10 15 GeV, disfavoring CW inflation at the 2σ level. The running of the spectral index is presented in the right panel of Fig. 2 which shows a strong correlation between the spectral index and its running as discussed before. The obtained running has to be compared with Planck's result at 68% CL. It is clear that the results have no definite impact on this class of models, but are are bounded to test them in the near future as the precision on both measurements, the spectral index and its running will be significantly improved. In Fig. 3, we present the values of A and φ/v φ for given v φ showing a correlation between A and φ/v φ . One finds that smaller v φ require smaller A and φ/v φ , but one gets A ∼ 10 −14 which is rather insensitive to v φ . As mentioned earlier, we find φ/v φ 1 for all the region of v φ < M P .
The strong tension of the CW inflationary setup with Planck data arises due to the fact that V * cannot be made large enough to enhance N * as a tiny quartic coupling A is also required by the size of perturbations, and even though one can have v φ as large as M P it is not enough to overcome the strong A suppression. The tension is aggravated even further if one wants to have lower symmetry breaking scales in the CW mechanism. This problem may be avoided in a non-standard cosmological scenario where the relation (23) is altered in a radical way. Most of the modifications we can get within the standard framework, like

CW inflation on the brane
For a drastic change in the thermal history of the universe we can immerse ourselves into a brane world scenario where we live in a brane embedded in a higher dimensional Universe. Within this scheme, the stress-energy momentum in the bulk can take different forms, depending on the specifics of the model. By an appropriate choice of the boundary conditions, the non standard behaviour of the Universe on the brane we are looking for, can be easily achieved. Even more, the Hubble rate itself on the brane changes rather drastically taking the form where M B denotes a certain scale below which the cosmological evolution follows the standard form. A useful model to illustrate these effects is the brane world cosmology of the Randall-Sundrum(RS) model [7] in which the main correction is the term quadratic in the density, i.e., where Λ bulk is the bulk cosmological constant and M 5 is the 5-dimensional Planck scale [6].
During the inflationary stage the energy momentum tensor on the brane is dominated by the scalar field, which is confined to it and therefore still evolves as Eq. (3), as on the brane ∇ ν T µν = 0 holds. The condition to sustain a period of inflation is now for ρ Λ and when the energy density is dominated by the scalar potential In this regime the slow-roll parameters become where we have definedṼ ≡ V /M 4 B . Note that all the slow-roll parameters recover their standard forms forṼ = 0, but are suppressed by 1/Ṽ in the limit ofṼ 1. While the spectral index take the same form: n s = 1 − 6 B + 2η B , its running is given by The power spectrum of primordial quantum fluctuations and the tensor-to-scalar ratios are also modified to turn into The number of e-folding during inflation is also changed to The condition for solving the horizon problem now gets an extra term which leads to an additional number of e-folding for the scales of interest (k * = 0.05 Mpc −1 ) depending onṼ : This increase can be easily understood by seeing in Fig. 4  Before continuing the numerical analysis of the CW brane inflation, we briefly discuss the consistency of introducing a huge change withṼ = V /(2Λ) 1 in the Friedman equation. First, we note that the length scale on the brane is determined by On the other hand, the length scale in the bulk is related to the AdS length scale, l ∼ Λ −1/2 M P , due to the relations to the input parameters in the RS model, l = 6M 3 5 /|Λ bulk | and Λ = 6M 3 5 |Λ bulk | with M 2 P ∼ M 3 5 l. Thus, taking V /(2Λ) 1, we are in the regime that L/l ∼ (Λ/V ) 1/2 1 where the continuum of KK modes in the RS II model with a single brane could be easily excited during inflation and affect the slow-roll inflation on the brane. Therefore, we need to stabilize the radius of extra dimension, for instance, by introducing a second brane in the bulk, so that the KK modes become discrete and decoupled during inflation. We don't go to the details on the radius stabilization in our work and we just assume that a radius stabilization mechanism of Goldberger-Wise type [9] is at work. Then, the radius should be stabilized at a small value such that the KK masses are larger than the Hubble parameter during inflation. Moreover, in order for the inflaton potential not to destabilize the radius, the mass of the radion, namely, the excitation of the radius, must also satisfy m 2 r H 2 where H 2 ∼Ṽ * V * /M 2 P . For instance, for V 1/4 * = 10 5 GeV and Λ 1/4 = M B = 1 GeV, we get V * ∼ 10 20 so the radion mass should be m r 100 GeV. Consequently, we can accommodate in our setup a light radion relevant for collider physics and still be compatible with inflationary phenomenology contrary to the high-scale case.
Let us now examine how successful inflationary solutions satisfying the conditions, P R,B (k * ) = 2.2 × 10 −9 and N B = N B, * , can arise in the limit ofṼ 1. For given v φ and M B , these conditions can be solved uniquely by appropriate values of A and φ/v φ as shown in Figs. 5 (varying M B ) and 6 (varying v φ ) for high and low v φ (M B ), respectively.
When M B is not so small that the φ-independent prefactor of the modified slow-roll parameters (e.g., η B ≈ η/Ṽ ) remains much larger than one: as in the standard cosmology. Thus, the simple correlations in Eqs. (18, 19, 20) still hold. This behaviour can be seen from the solution lines corresponding to the large region of M B and the small region of v φ in each panel of Figs. 5 and 6, respectively. As M B (v φ ) becomes smaller (larger) starting from the right (left) end in Fig. 5 (6), the CW parameters, φ/v φ and A, as well as the inflationary observables, n s and dn s /dk, increase monotonically, and n s reaches its local maximum at around 0.946 corresponding to φ/v φ ∼ 0.01. This is the exactly same pattern as in Figs. 2 and 3 where the four quantities increase as v φ gets close to M P , and n s (φ/v φ ) approaches 0.945 (0.01) maximally allowed in the region of v φ < M P . One can also check that the relation (19) holds intact in this small φ/v φ region. Note that the spectral can be enhanced a lot even for low v φ (Fig. 6)  is still limited below the Planck's 2σ lowest value in the region of φ/v φ 0.01 as in the standard cosmology.
When M B (v φ ) becomes even smaller (larger) in Fig. 5 (6), one finds solutions with φ/v φ 0.01 eventually approaching the one for which the standard relations (18, 19, 20) are invalidated, but new correlations appear. The spectral index (and also its running) starts to rise rapidly at φ/v φ ∼ 0.1, but it reaches its maximum value ∼ 0.966 (−0.005) and then decreases slowly to approach 0.96 (−0.006). In this large φ/v φ region, the spectral index (and its running) is dominated by the B contribution. One can see that there appears a new correlation between these two observables, which is different from the standard one (19). Note that A becomes much smaller than the typical value ∼ 10 −14 in the asymptotic region preventing the prefactor (4M P /v φ ) 2 (4M 4 B /Av 4 φ ) becoming too small and thus allowing solutions with φ/v φ close to one.
Let us remark that the four lines in each panel of Figs. 5 and 6 show the same behavior as functions of φ/v φ , that is, one can always find an appropriate range of M B and v φ reproducing the same values of n s and dn s /dk, as well as the same correlation between them. Of course, M B cannot be taken to be smaller than O(1) MeV for which the standard big-bang nucleosynthesis prediction is spoiled. We find that a spectral index within 1σ range of the Planck data can be obtained for v φ 3 × 10 7 GeV with the restriction of M B > 10 MeV. As in the standard cosmology, there is also a remarkable correlation between the running of the spectral index and the spectral index itself. Taking the 1σ range of the Planck data for the spectral index, n s = (0.9540, 0.9686), the CW inflation on the brane predicts It is worth noticing that this kind of correlation is unique to CW potentials and cannot be avoided and therefore provides a crucial test on the model. The ratio of tensor to scalar perturbations is not shown because although it is different from zero and negative, its actual value is so tiny that it is effectively zero from an experimental point of view. Again this feature cannot be circumvented in the CW inflation, and therefore the measurement of gravity waves will completely rule out small-field CW inflation. The allowed ranges of the symmetry breaking scale, v φ ∼ 10 8 GeV, and the quartic coupling, A ∼ 10 −14 , shown in Fig. 6, are of our special interest as such small values of A can have a dynamical origin in a model where the full scalar potential, including both the inflaton and Higgs sectors, can be generated purely from radiative corrections.
The scalar potential at tree-level is written as where H denotes the SM Higgs field. As noted in Ref. [4], all of the above Higgs potential parameters, consistent with the recent SM Higgs data [5], can be generated radiatively  Figure 7: The values of v φ vs. λ Φ satisfying the CW minimization condition [4]. The upper (blue), and lower (red) lines correspond to the UV scale M X = 2 × 10 11 GeV, and 10 18 GeV, respectively. assuming a vanishing initial condition at a certain high scale Λ U V . All the scales are generated by dimensional transmutation of the CW mechanism applied to the B − L gauge symmetry breaking. In this setup, there are two free parameters, the extra gauge coupling g X and the right-handed neutrino Yukawa coupling y N , from which the B − L breaking scale v φ as well as the B −L Higgs quartic coupling λ Φ are generated dynamically. Extending the analysis of Ref. [4] to higher v φ scale, we will examine whether there exists an appropriate CW minimization point which is consistent with the observed cosmological quantities derived in the previous section.
Let us now consider the one-loop Coleman-Weinberg potential [2] for the B − L sector. Taking Φ = φ/ √ 2 in the unitary gauge and the normalization condition of V (0) = 0 and V (Q) = 6λ Φ , the one-loop corrected B − L potential is given by [2] where we took one Yukawa coupling for the right-handed neutrinos y N , and added a constant term V 0 normalizing the potential: V X = 0 at the global minimum. Taking the renormalization scale at Q = φ ≡ v φ to avoid the large-log uncertainty in the one-loop approximation [2], one can evaluate the minimization condition of the potential (40) and obtain This relation fixes the B − L breaking scale v φ in terms of input values of λ Φ , g X and y N which evolve from the high scale Λ to v φ by renormalization group. Putting back (41) into (40), one has which is nothing but the general CW potential given in Eq.
(2) with λ Φ ≡ 22A/3. In Figure 7 we present the values of v φ and λ Φ satisfying the CW minimization condition as well as the correct electroweak symmetry breaking for appropriate values of g X . As noted in Ref. [4], one can see that larger v φ requires smaller g X . From Figure 6, one can see that the spectral index n s falls into the 1σ range of the Planck data for v φ 5 × 10 6 GeV which requires λ Φ ≈ 22A/3 2 × 10 −13 . Furthermore, the required values of λ Φ drop rapidly for higher v Φ . Thus one finds viable parameter points around v φ ∼ 10 8 GeV and M B ∼ 1 MeV in which the inflaton and Higgs potentials are generated simultaneously purely from radiative corrections. However, we should remark that the solutions to the minimization condition, λ Φ ∼ 11/π 2 (g 4 X − y 4 N /6), are found for fine-tuned choices of g 4 X y 4 N /6 resulting in highly suppressed values of λ Φ . It is expected that much less fine-tuned solutions would be found in some other U (1) models which have different beta-function coefficients.

Conclusions
We have analyzed the plausibility of a (small field) CW potential naturally arising from quantum corrections to address inflation in and beyond the standard cosmological scenario. We have shown that although it is not possible in the standard scenario to solve the horizon problem within 1σ consistency with the Plank measurement of the spectral index, brane-world scenarios ease the requirements on the numbers of e-folding needed to solve the horizon problem, while preserving the correlations among the other observables, allowing CW potentials to fulfill all the inflation requirements and remain an attractive and natural explanation for inflation. Besides, CW potentials have an inherent prediction on the relation between the running of the spectral index and the spectral index itself, which can be tested in the near future. Tensor modes are essentially absent in CW inflation models and thus, if found by next generation experiments, can rule out CW motivated inflationary potentials altogether.
The inflation observables can easily accommodate a rather small symmetry breaking scale ∼ 10 8 GeV but do require a tiny quartic coupling ∼ 10 −18 . While the CW mechanism generates small scales through dimensional transmutation, such a tiny coupling may be indicative of a radiative origin. We have illustrated how dynamical generation of the inflaton as well as the SM Higgs potential can work consistently with the Planck results as well as the Higgs mass measurement at the LHC in the context of the SM extended with the generalized B − L gauge symmetry to explain the neutrino masses and mixing.