Quartic Inflation and Radiative Corrections with Non-Minimal Coupling

Journal of Cosmology and Astroparticle Physics

Quartic inflation and radiative corrections with non-minimal coupling

To cite this article: Nilay Bostan and Vedat Nefer enouz JCAP10(2019)028

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This content was downloaded from IP address 129.255.225.189 on 24/12/2019 at 00:03 JCAP10(2019)028 are r quartic 4 hysics λφ P le ic t ar between the inﬂaton and the R 2 1 ξφ and the tensor-to-scalar ratio a, s strop n A https://doi.org/10.1088/1475-7516/2019/10/028 005 is one of the simplest models that agree with the current . 0 & osmology and and osmology ξ ˙ Istanbul 34380, Turkey C and Vedat Nefer S¸eno˘guz a,b 1907.06215 inﬂation, physics of the early universe

It is well known that the non-minimal coupling rnal of rnal

ou An IOP and SISSA journal An IOP and Corresponding author. 1 2019 IOP Publishing Ltd and Sissa Medialab Department of Physics, Mimar No.Silah¸s¨orCad. Sinan 89, Fine Arts University, Department of Physics andIowa Astronomy, University City, Iowa of 52242, Iowa, U.S.A. E-mail: , [email protected] [email protected] b a c

data. After reviewing thethe inflationary radiative predictions corrections of due thisUsing to potential, two we different couplings analyze prescriptions ofcoupling the discussed the parameter effects in values inflaton of for the to which literature, the other we spectral scalar calculate index fields the or range of fermions. these Ricci scalar affects predictionsinflation of potential single with field inflation models. In particular, the in agreement with the data taken by the KeckKeywords: Array/BICEP2 and Planck collaborations. ArXiv ePrint: Nilay Bostan Quartic inflation and radiative corrections with non-minimal coupling J Received July 22, 2019 Accepted September 19, 2019 Published October 8, 2019 Abstract. JCAP10(2019)028 between the Ricci R 2 ξφ quartic potential, providing 4 λφ , for an inflaton potential in the presence r – 1 – , and showing that the model agrees with current data r and s n and the tensor-to-scalar ratio s n 005. We also briefly discuss how and to what extent can the reheating stage affect . 0 & Section4 introduces two prescriptions that can be used to calculate radiative corrections The next two sections, section5 and section6 contain a detailed numerical investigation In this work, we first review in section2 how to calculate the main observables, namely The observational parameter values predicted by different potentials that the inflaton ξ to the inflaton potentiala due conformal to transformation inflaton isfield couplings dependent applied to mass to terms bosons express inframe. or the the one-loop fermions. Whereas action Coleman-Weinberg in potential in In prescription arethe prescription the expressed II, original in I, Einstein the Jordan this field frame; frame. dependent and mass the terms areof taken into how account the in radiative corrections due to inflaton couplings to bosons or fermions modify the of non-minimal coupling.analytical Next, approximations in for section3 wefor review the the values of observables. the spectral index scalar and the inflatoncurved (this space-time is [8–10]), required and by inflationarythe the predictions coefficient are renormalizability of significantly of this altered the term depending scalar on –19]. [11 field theory in 1 Introduction Inflation], [1–4 which isboth an accelerated helps expansion explaining era generalhomogeneity, thought and properties to it occur of leads in the totures the the universe in early primordial such the universe, density as universe. perturbationsof its Up that flatness them to evolve into and now depending the many largebeing struc- on inflationary scale tested models a by have scalar beenpolarization the introduced observations field cosmic that with called microwave have most background become the radiation even inflaton. more temperature sensitive anisotropiesmay in Predictions have and recent were of years calculated [5,6]. these inarticles many models assume articles, are that see forthe the instance other inflaton ref. hand [7]. is the A coupled action vast in to majority general gravitation of also these solely contains through a the coupling term metric. On Contents 1 Introduction1 2 Inflation with non-minimal coupling3 3 Quartic4 potential 4 Radiative7 corrections 5 Radiatively corrected quartic potential:6 prescription I8 Radiatively corrected quartic potential:7 prescription II Conclusion 10 15 JCAP10(2019)028 3 λ [7]. are ρ/ r ≈ /N p 4 Radiative , including ≈ [24, 25] as 3 1 r 2 10 /N . ξ 12 ) and N ≈ (2 -attractor models [41]. r / change due to radiative α 3 r 1 is ref. [29]. In this work − and 1 and ξ ≈ /N s s 2 n n and the tensor-to-scalar ratio − s as discussed in section. 3 1 n 3 ≈ = 1 case of the 10 s α n . [29]. ξ r by – 2 – -attractor-type inflation models due to inflaton plane, however the size of these deviations differ N and α r s n and s n 1 is required [23]. In this limit, the observational parameters ξ to be free parameters, with 1. A related work which includes the case of ξ is ambiguous unless the inflationary part of the Lagrangian is embedded in a specific theory ξ , the inflationary predictions of these models coincide with those of conformal Here, we extend previous works by considering both prescriptions I and II, and ξ and 2 λ → ∞ ξ In this work we take the inflaton to be a SM singlet scalar field, and take the self- The reheating phase of Higgs and Finally, we note that the non-minimal quartic inflation model given by eq. (3.1) is a The effect of radiative corrections to the predictions of the non-minimal quartic potential The value of See also refs. [35–38] for related work. are rather insignificant according to prescription I. However, according to prescription II, 1 2 values that we consider, the average equation of state during reheating is given by s coupling (see e.g. refs. [19, 30]). according to the prescriptionplateau used type for structure the of calculation.n the As Einstein discussed frame inradiative potential corrections refs. remains lead [21, intact to23, and aa],28 linear the scalar the term deviations field in with in thethe a Einstein canonical coefficient frame kinetic potential of term. writtenpredictions this in If move terms term towards the of the is inflaton linearIf is positive, potential dominantly the and predictions coupling inflaton to asleading is bosons this to dominantly coefficient a coupling reduction is in to increased the fermions the values the of inflationary coefficient of thiscorrections term for a is SM negative, we singlet focus inflaton on studying have the beenthe effect studied of case by radiative of corrections refs. for [31–33]. general values Unlike of these works, The relation between these types of models iscouplings elucidated to in additional ref. fields [42]. While has the been observational parameter discussed values in alsogeneral, a depend for number on the the of special details works,ξ of case see the of e.g. reheating the refs. phase non-minimal in [43–48]. as quartic we discuss inflation in model3. section then and to The for a number the good of rangemain e-folds approximation of and effect independent the of of observational the inflaton parameterdue reheat values couplings to temperature. are the to reheating Thus, additional phase ininflation, fields but our which rather on case we due the the focus to observational on the parameters radiative this corrections is work. to not the potential during special case of thelimit universal attractor models discussed in ref. [39]. In the strong coupling in the Starobinskycalled model Starobinsky [26, point27]. in the Radiative corrections lead to deviations from this so the inflaton is assumed toa couple potential to which fermions andto coincides prescription bosons. with II the is used.inflaton potential coupling Ref. discussed to [34] bosons in consideres orcoupling section6 fermions. parameterfor values For each for inflaton case whichin we the coupling calculate spectral agreement the index with regions the incorrections the current in plane data. these of regions. We also display how attractor models [40], which correspond to the has been discussed mostlyfor instance in refs. the [21, context22] of and the standard references model within. (SM) In Higgs this inflation context, since [20], the see self coupling predictions of the non-minimalresults quartic in potential, section7. for each prescription. We summarize our of the inflaton isare known, given in terms of the e-fold number JCAP10(2019)028 ) = GeV φ (2.4) (2.5) (2.8) (2.3) (2.6) (2.7) (2.1) (2.2) ( 18 F field, the 10 exited the × σ ∗ 4 . k 2 with a canonical ≈ σ πG , the tensor-to-scalar 8 s n √ , ), so that the Lagrangian / , 2 ) φ , ) ( ) φ ) φ ( = 1 φ φ ( ( J ( /F P J V F V V µν m g − − , . ) = = ˜ φ φ φ is given by ν , ν σσ ( V , µν V 0 after inflation. 2 | R , φ∂ = 16 g / φ∂ σ σ 3 µ ) µ σ = V d → ∂ | ∂ φ V V ,V with a canonical kinetic term and a poten- ( V φ φ µν ) µν π Z d ∗ ˜ g φ φ 3 g σ e η , r 1 ( 1 ) , η 1 2 σ p √ φ F 2 Z 1 or – 3 – 2 ( 1 − + 2 = Z + = = σ → 2 R σ 6 V 2 ∗ V ) 2 ) d ) R from the Planck measurement [5] with the pivot scale ) φ − φ N − φ ( φ ( 9 ∆ ( ˜ ( 0 R 1 2 − F F F 1 2 F 2 1 = 1 10 = 2 3 s = = × n = ˜ g 4 g . E J ) − . The number of e-folds is given by 2 − L 1 φ L √ 1 ( √ − ≈ Z ” denotes quantities when the scale corresponding to 2 R ∗ indicates that the Lagrangian is specified in Jordan frame, and J is the inflaton value at the end of inflation, which can be estimated by 002 Mpc . If we make a field redefinition . ) frame by applying a Weyl rescaling e φ d σ E = 0 are given in the slow-roll approximation by ∗ F/ k r d ): ’s in the subscript denote derivatives. The spectral index φ ≡ . We are using units where the reduced Planck scale σ ( 0 2 J Once the Einstein frame potential is expressed in terms of the canonical The amplitude of the curvature perturbation ∆ For calculating the observational parameters given eq. (2.1), it is convenient to switch to F V ) = 1. ξφ e σ ( we obtain thekinetic Lagrangian term. density for a minimally coupledobservational scalar parameters field can bereview calculated using and the references): slow-roll parameters (see ref. [50] for a where ratio and chosen at where the subscript “ 2 Inflation with non-minimalConsider coupling a non-minimally coupledtial scalar field 1 + and which should satisfy ∆ horizon, and the Einstein ( where the subscript is set equal to unity, so we require density takes the form [49] where JCAP10(2019)028 . ∗ N (3.2) (3.1) (2.9) (2.14) (2.10) (2.11) (2.12) (2.13) . ) is the energy 4 P r As discussed in ∗ ρ m φ 3 ( ln V ≈ 1 4 ∗ . ρ − 4 ) is the energy density at the , r λφ r ω φ 4 1 ρ 2 . 1 , − . r φ e 0 φ 00 ρ ν 2 Z 3(1 + V ) V 0 ln φ p φ∂ . d ) V 4 1 µ = we also need a numerical value of 2 + φ 4 ∂ ) , φ − r 2 4 P | e Z( µν 0 ∗ λφ ρ g 2 ∗ m ξφ ρ V / e φ 4) | 1 2 + sgn( , η 3 φ and / ln ln Z φ 2 Z V − s ) 1 2 (1 ) √ (1 + – 4 – r 0 exited the horizon, n 0 Zη R ω π ) ∗ V V 3 2 k = 1 7 + 1 . ) = √ ξφ φ 2 1 2 64 ( 3(1 + V , η = = sgn(V ≈ = − φ (1 + ∗ ∗ φ R 1 2 4 P N ∗ N Z ∆ ρ m = = g ln J − 1 2 L is the equation of state parameter during reheating. √ r , following the approach in e.g. ref. [51]. Using eq. (2.4), eq. (2.5) can ω 7 + ) is the energy density at the end of inflation, . φ e φ 64 3 is generally a good approximation for the potentials which we investigate. ( / V ≈ ∗ 2) = 1 / N r ω = (3 e ρ The Lagrangian of the non-minimal quartic inflation model is given by It is convenient for numerical calculations to rewrite these slow-roll expressions in terms To calculate the numerical values of For a derivation of eq. (2.13) see e.g. ref. [52]. 3 For this case section3, independent of the reheat temperature. 3 Quartic potential Inflationary predictions ofe.g. non-minimal refs. quartic [14, inflation15 , comment on25, have an29, been analytical53–56].the approximation studied reheating used in Here stage in after on detail, that the work, summarizing see and inflationary the briefly predictions. results discuss following the ref. effect of [55], we Similarly, eqs. (2.7) and2.8) ( can be written as Assuming a standard thermal history after inflation, density when the scale correspondingend to of reheating and In Einstein frame, the potential is of the original field be written as where we defined Here JCAP10(2019)028 (3.7) (3.3) (3.6) (3.8) (3.9) (3.4) (3.5) 1. How- . ξ . 1 is given by eq. (2.14), − ∗ ): , 1 branch of the Lambert . x ! N − ! )] ln 2 e − ψ ξψ , , 3 . ) = 1 as follows: sN . ln( ) e e 4 ) 2 ψ + 4 ψ ψ ψ ξ ) − ) ( 3 2 ψ ξ ln(1 + ξ x ) ψ 1 + 2 3 1 + ξ sN 2 3 1 + ) − ln 4 ψ (1 + 6 ) + ln /s ≈ ln /s ξ ξ 1 e 3 4 ) − 1 + ψ ξψ x N 3 e − 4 , 1, see below for a discussion) to find the value − ) ( 64(1 + (1 + (1 + 6 + 8 ln e 1 3 + 1 + 32 ψ 2 & 2(1 + 6 ) – 5 – s (1 + (1 + 6 ψ − ψ 1 + ψ ξ ξ p 5 3 N/ + W − 2 R N ∆ ψ 1 1 + ( 2 3 1 + sN ≡ s π − 3 4 exp [4 4 0

s can be obtained from 2 = N ≈ = 24 φ 1 + (1 + 6 − e e = 12 ψ ψ ψ ) values deviate from the numerical solution by at most 2%

→ 2 N λ − r 1 φ ( 128 − ). Here we point out that a slightly more complicated but better s s + 1 . Using eq. (2.11) we obtain = = 1 n 2 sW (4 s N r ). Here − n ξφ ψ/ ξ 3 = ≡ , we can calculate the observational parameters by numerically solving ≈ ψ 1, and ξ ψ (1 + 6 value. / + 1 ) 1 the reheating stage includes a phase where the Einstein frame potential for ξ ξ ξ N & (6 ξ ≡ 3 after inflation], [58 and as a result the number of e-folds s / As ref. [55] point out, one can find reasonable approximations to the numerical solution For the minimally coupled case, quartic potential implies an equation of state parameter . Formally, inverting eq. (3.5) gives a solution in terms of the ∗ = 1 φ r by utilizing approximation can be obtained by using where we defined For any value of eqs. (3.5) and2.14) ( (withof a correction for function: Inserting eq. (3.8)ref. in [55] eqs. only (3.3) with the and modification (3.4), we obtain the eqs. (3.19), (3.20) andComparison (3.22) of in the numerical solutionsis and shown the in two analytical figure1.accurate approximations discussed for As here can be seen from the figure, our analytical approximation is more ω (3%) for any Using eqs. (2.6) and (2.9), we obtain and using eq. (2.12) we obtain where independent of the reheatobservational parameter temperature. values due Thisever, to result, for the which reheat removes temperature, the is uncertainty also in valid for the JCAP10(2019)028 ) for 2 (3.10) (3.12) (3.11) ξ . Thus, 1000 1000 (3 2 R / curves show ∆ r 0 2 = 4 N π 4 e . 2) φ → 1 1 / )) ξ 4 +1) / = (3 ξ ξ 1 N V (3 / 4 P 2 m 0.001 0.001 √ = φ ( is proportional to , V . -6 e -6 4 ρ 10 10 / ln ξ 1 Bezrukov and Gorbunov -5 -8 4 2 -11 − 1 / 12

10 10 √ 1

10 0.050 0.010 0.005 0.100

3 λ r e + 2 R ρ due to the quadratic phase is approximately = ∆ 4 P e ∗ r φ ρ 2 m – 6 – N π ln 2 3 V 0.980 0.980 1 3 ln N' (N+1)-> − 1 12 0.975 0.975 ξ < 4 P ∗ are shown as a function of the non-minimal coupling parameter ρ m λ ln 0.970 0.970 2 1 and s s n n after inflation. In this case r Lambert 0.965 0.965 , 4 7 + s with respect to the value calculated by eq. (2.14). The difference in . 1, the reduction in φ n [60–62]. From eqs. (2.7) and (2.6), we obtain ∗ 64 ∝ [59]. 0.960 0.960 N /ξ ) ≈ ξ ξ φ ∗ ( 0.955 0.955 N V 6) ln / 1, 0.950 0.950 (1 . 1 -6 − . The values of 2 1 we use this expression instead of eq. (2.14) in figure1. The quadratic phase slightly

1000

0.001 0.100 0.050 0.010 0.005

ξ ξφ r & For the non-minimal quartic potential, expanding the action around the vacuum reveals 1. Thus, for ξ between the two expressions is approximately given by ∗ . “Lambert” curves show numerical solutions that are obtained using eq. (3.7), “Bezrukov and requiring Λ to be higher than the energy scale during inflation corresponds to For reduces the value of N Figure 1 the improved analytical expressions(68%) using CL eq. contour (3.9). based on The data pink taken (red) by contour the corresponds Keck Array/BICEP2 tothe and the canonical Planck 95% scalar collaborations [57]. fieldnumber is in quadratic this [43, case as 44]. Following refs.43, [ 59 ], we obtain the e-fold Gorbunov” curves show the approximate analytical expressions in ref.], [55 ( ξ Since ξ a cut-off scale Λ = 1 given by JCAP10(2019)028 r – s n (4.5) (4.4) (4.1) (4.2) (4.3) ) denote φ ( i M . , the effect of the 2 3 χ 2 χ values with the recent m r 1 2 λ , , are slightly more stringent + up to 10 ξ 2 and 2 2 ξ ) χ value. Furthermore, preheating 2 s φ 2 ( φ n ∗ , µ i 2 , h N g M . Ψ 1 2

φ µ ) m 4 and to a Dirac fermion Ψ: ln h χ 4 values for ln 4 ) 4 ¯ ΨΨ + r φ − ( Ψ is a renormalization scale and i 4 κφ g m µ ( M ±

– 7 – and 005 (0.01) at 95% (68%) confidence level (CL). F 4 2 2 3 approximation hereafter. λ , hφ . s / φ 0 π π 1) 1 n 4 λ ¯ − ΨΨ + 32 64 & ( 1 difference in the 2 = 1 ξ hφ ≡ r i ) = . ω κ X φ + , g ( . With this constraint, taking into account the quadratic 4 3 2 χ V φ ) = m 4 λ φ ( V 2 ∆ φ Ψ) = 2 g . For this reason all numerical results in the following sections will be 5 . 1) for bosons (fermions), φ, χ, 2 ( − values up to 10 ) sign corresponds to the case of the inflaton dominantly coupling to bosons V 10 − ξ . ξ is +1 ( F Generalizing eq. (4.4) to the non-minimally coupled case is subject to ambiguity unless Finally, note that since we compare the numerical First let us consider the potential terms for a minimally coupled inflaton field with a the ultraviolet completion ofin the refs. low-energy [21, effective23, field, 28 theory is64]. specified, as In discussed the literature, typically two prescriptions for the calculation (fermions) and we have defined the radiative correction coupling parameter where the + ( leading to displayed for effects can makeparameter this values due difference to even theFor reheating smaller. instance, stage is in rather Thus, smallquadratic figure1 the for phase non-minimalwhich uncertainty amounts quartic to in shows inflation. curves. the the Therefore, barely observational we visible will hook-like use part the at the bottom end of the compared to earlier works. Namely, non-minimalArray/BICEP2 quartic and inflation is Planck compatible data with for the Keck where Under the assumptions the inflaton potential includingeq. the (4.1) Coleman-Weinberg take (CW) the one-loop form: corrections given by phase after inflation makes only a quartic potential, which couples to another scalar Keck Array/BICEP2 and Planck data [57], our constraints on 4 Radiative corrections Interactions of the inflaton withcorrections other in fields, the required inflaton forfollows efficient potential. [63]: reheating, These lead to corrections radiative can be expressed at leading order as field dependent masses. JCAP10(2019)028 r ξ ξ λ no and r (4.7) (4.8) (4.6) value and ξ values ∗ s r φ n and value. We s , n λ ) and ) sign for the . 2 χ φ values change s ( κ − n m r F , κ = values. Finally, the and 2 χ r is increased holding s ˜ and m n κ ξ , and ) ) s φ φ values match. The ( using eq. (2.11). The e-fold ( n . As ∗ . Ψ F ∗ κ values change as a function of N φ m p r 2 . ξφ ) = φ 1+ and φ µ φ 2 ( s √ solutions for each point used as initial 2 ) Ψ µ ) n 2 using prescription I and prescription II, 2 ln λ ˜ m should also be specified. However, shifting rather than 4 ξφ κ ) = 1, and ξφ , e ln µ ξ 4 κφ φ 4 / ( 3 and ) (1 + ± values. It is clear from the figures that κφ (1 + – 8 – Ψ φ 4 ξ ( ξ φ ± F 4 λ 4 values for a relatively narrow range of φ using 4 λ r e ) = ˜ φ Ψ = φ , ( ) = , using the potential in eq. (4.7), with a + ( ) and V φ κ values, however this last result is subject to some caveats as φ for chosen s ( ( n κ φ κ V F values are compatible with the current data. Figure3 shows how and p r ξ ) point, we start the calculation by assigning an initial = ˜ and φ s , . ξ, κ n 2 ) µ ) φ φ ( . Thus, for fixed values of the coupling parameters ( J λ does not change the forms of eqs. (4.7) and (4.8), corresponding only to a shift V F µ is calculated using eq. (2.12) and compared with eq. (2.14). The initial value of ∗ ) = values change with N φ r ( plane where V For prescription I and inflaton coupling to bosons, figure2 shows the region in the For the numerical calculations, a value for Note that the potentials in eqs. (4.7) and (4.8) are approximations that can beIn obtained the next two sections, we numerically investigate how the κ and plane. For each ( s longer change at evendiscussed larger below. fixed, there is a transition in values depend more sensitively on the value of and n do not depend on 5 Radiatively corrected quarticIn potential: this section prescription we I the numerically coupling investigate parameters howinflaton the dominantly coupling to bosons (fermions). values of their neighbors. the value of obtained this way is pluggedcalculation in is eqs. repeated (2.9) and over) (2.6 the to whole yield grid, the with in the value of is then adjusted and the calculation is repeated until the two number then calculate numerical values of as a function ofrespectively. the The calculation coupling procedure parameters κ is as follows: we form a grid of points in the from the one-loop renormalizationfor group a improved effective discussion actions, of see this for point. instance ref. [29] In prescription II, thethe field dependent Jordan masses frame, inJordan so the frame. that one-loop Therefore CW eq. potential the (4.4) are Einstein corresponds expressed frame in to potential in the this one-loop case corrected is potential given by in the of radiative correctionsone-loop are CW adopted. potential are In expressed in prescription the I, Einstein the frame. field Using dependent the transformations masses in the the one-loop corrected potential is obtained in the Einstein frame as JCAP10(2019)028 -3 -2 -1 ξ=0 ξ=10 ξ=10 ξ=10 ξ=10 as a function r -6 and s -3 -2 -1 n -8 κ=0 ξ=0 ξ=10 ξ=10 ξ=10 ξ=10 -10 κ 10 log -12 0.975 -14 values are within the 95% (68%) CL contours 0.970 -16 r and

s 0.050 0.010 0.005 0.100

– 9 – n s 0.965 r n -6 0.960 -8 0.955 plane for which -10 κ values. The pink (red) contour in the top ﬁgure corresponds to the 95% κ 10 – ξ ξ log

-12

0.100 0.050 0.010 0.005 r -14 -16 . For prescription I and inflaton coupling to bosons, the top figure shows in light green . For prescription I and inflaton coupling to bosons, the change in values in these regions.

0.970 0.965 0.960 0.955 0.975

s n is plotted for selected and κ s of (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [57]. based on data taken byn the Keck Array/BICEP2 and Planck collaborations [57]. Bottom figures show Figure 3 Figure 2 (green) the regions in the JCAP10(2019)028 . . . r ξ ξ ξ λ is ξ ξ κ max ξ ln (5.1) (5.2) κ κ and ln 2) ln κ s κ/ n κ ( . However, − κ ) sign for the = 0 curve as rather than 4 1. If 2 − κ ξ λ/ values for chosen ≡ ξ . The second branch A value. This is also true for max max κ 2 κ ξ max 10 κ . almost exactly equals 2 − values change as a function of λ 10 r and the σ × = 1 for convenience. The potential s 6 κ ξ . µ 4 replaced by and r ln 2 s λ/ n κ values and away from the observationally − s n + 2 = 0. There it was pointed out that there are 2 ξ ξ 2 exp 1 and take 2 R . We label the branch of solutions with larger 2 – 10 – ξ − ∆ 2 1 sN 2 π 3. Finally, using eq. (2.7), we obtain decreases. These solutions cease to exist for 2 72 ξφ A ξ values change with κ ≈ r sN/ ≈ , using the potential in eq. (4.8), with a + ( 4 λ plane. ) κ σ κ ≈ ( and values depend more sensitively on the value of – ) values are compatible with the current data, for the first branch ξ V values are compatible with the current data. Figure8 shows how r s σ and plane as r n 6 r becomes the same order of magnitude as value that is smaller than a maximum ξ r s/ – κ and κ s and and p s n values depending on s n s n n max κ as the first branch, and the other branch of solutions as the second branch. For κ . Furthermore, higher loop corrections will eventually become also important. there is no solution, that is, eqs. (2.11), (2.12) and) (2.14 cannot be simultaneously κ plane where plane where max 6= 0, with For prescription II and inflaton coupling to bosons, figure7 shows the region in the For prescription I and inflaton coupling to fermions, figure4 shows the region in the The case of inflaton having a quartic potential with radiative corrections due to coupling Figure6 shows how Even if we take the potential in eq.4.7) ( at face value, the inflationary solutions for large In contrast to the other cases covered in subsequent sections, we find that eqs.), (2.11 κ κ ξ decreases. As can be seen from the bottom panels, significant change in the values can only be obtained for fine tuned values of the coupling parameters. To show this, and the coupling parameters inflaton dominantly coupling to bosons (fermions). solutions, on the other hand, move towards small as inflation with sufficient duration cannot be obtained. 6 Radiatively corrected quarticIn potential: this section prescription we II numerically investigate how the favored region in the for for a given κ > κ κ values only occur when satisfied. two solutions for every of solutions. Again, to fermions was discussed in ref. [65] taking as mentioned in4, section theization potential group we use improved is effective anvalues action, approximation of and of this the approximation one-loop will renormal- eventually fail forκ large let us write the potentialthen in approximately the takes limit Using the eq. form), (2.4 eq. this potential (3.2), can with be written as much larger than this term, eq. (5.2) can only be satisfied if and The observationally compatible region forAs the second seen branch from of solutions theonly is a figure, shown narrow in the region figure5. second in branch the solutions arevalues. compatible The with first observations for branch solutions move from the red points towards the (2.12) and (2.14) can be simultaneously satisfied for arbitrarily large values of Using eq. (2.8), exp(2 The first term in the right hand side is approximately 5 JCAP10(2019)028 values are within the 95% values are within the 95% r r and and s s n n plane for which plane for which κ κ – – – 11 – ξ ξ values in these regions. values in these regions. r r and and s s n n . For prescription I, inflaton coupling to fermions and first branch solutions, the top figure . For prescription I, inflaton coupling to fermions and second branch solutions, the top figure (68%) CL contours basedBottom on figures data show taken by the Keck Array/BICEP2 and Planck collaborations [57]. Figure 5 shows in light green (green) the regions in the (68%) CL contours basedBottom on figures data show taken by the Keck Array/BICEP2 and Planck collaborations [57]. shows in light green (green) the regions in the Figure 4 JCAP10(2019)028 and s -3 -2 -1 n ξ=0 ξ=10 ξ=10 ξ=10 ξ=10 as a function r values only occur and -8 s r n decreases, whereas the and -10 κ s values are compatible with n κ r . This result is not surprising 10 values, that is, the maximum -12 log values. Again, the first branch max . Figure 10 shows how and max κ κ ξ is increased. This approach to the κ s values approach the linear potential -14 κ n or . Similarly to the prescription I case, r ξ = 0 curve as max κ -16 κ and s values. n approaches s n

values. The 0.010 0.005 0.100 0.050 r – 12 – plane where ξ as , the 2 κ 1 during inflation, in which case the Einstein frame values for chosen − /N 4 10 -8 and max ≈ 2 & ξ κ r ξ ξφ for chosen -10 κ values where the two branch of solutions meet. These values are also κ ) and values. The pink (red) contour in the top figure corresponds to the 95% 10 κ values N and the ξ -12 log ξ (2 κ / 3 − -14 1 ≈ s -16 n values change with . For prescription I and inflaton coupling to fermions, the change in becomes the same order of magnitude as r

0.965 0.960 0.955 0.970

κ s n Finally we note that our results for prescription II and inflaton coupling to fermions is plotted for selected and κ values that allow a simultaneous solution of eqs. (2.11), (2.12) and2.14) ( are also shown. values change with s solutions move fromsecond the branch red solutions points move towards towards small the overlap and agree with those of ref. [29]. r the current data, forcompatible the with first the branch current of solutions. data at The any second value branch of of solutions are not since for large enough potential written in termslinear of term the which canonical eventually dominates scalar as field the using value eqs. of (2.3) and (2.4) contains a there are also twoFigure9 branchshows of the solutions region for in prescription the II and inflaton coupling to fermions. κ From the figures we seepredictions that for linear potential predictions wascase also for noted in inflaton refs. couplingwhen [7, to34, fermions,35]. significant Similarly change to in the the prescription I of written in the figure. The bottom figures only shown the first branch solutions. Figure 6 (68%) CL contour basedThe on solid data (dotted) taken portionspoints by of show the the the Keck Array/BICEP2 maximum curves and correspond Planck to collaborations first [57]. (second) branch of solutions. The red JCAP10(2019)028 -3 -2 -1 ξ=0 ξ=10 ξ=10 ξ=10 ξ=10 as a function r and -8 s n -3 -2 -1 , are also written in the ξ ξ=10 ξ=10 ξ=10 ξ=10 κ=0 ξ=0 -10 κ 10 -12 log -8 0.985 -14 0.980 ×10 κ=5.1 values are within the 95% (68%) CL contours -13 -12 -16 r 0.975 and s

×10 κ=2.2 ×10 κ=8.7 0.010 0.005 0.100 0.050

n s 0.970 r – 13 – n -14 0.965 -8 ×10 κ=2.4 -14 values. These values, increasing with 0.960 κ -10 ×10 κ=1.4 0.955 plane for which κ values. The pink (red) contour in the top ﬁgure corresponds to the 95% κ 10 – ξ -12 ξ log

0.100 0.050 0.010 0.005 r -14 -16 . For prescription II and inflaton coupling to bosons, the top figure shows in light green . For prescription II and inflaton coupling to bosons, the change in values in these regions.

0.98 0.97 0.96 0.95 0.99

s n is plotted for selected and κ s of (68%) CL contour basedThe on red data points taken show by the the maximum Keck Array/BICEP2 and Planck collaborations [57]. Figure 8 based on data taken byn the Keck Array/BICEP2 and Planck collaborations [57]. Bottom ﬁgures show ﬁgure. Figure 7 (green) the regions in the JCAP10(2019)028 -3 -2 -1 ξ=0 ξ=10 ξ=10 ξ=10 ξ=10 as a function r and s n -10 values are within the 95% r -11 κ and 10 -12 s log n -13 -14 plane for which

κ 0.010 0.005 0.001 0.100 0.050 – r – 14 – ξ -10 -11 values in these regions. values where the two branch of solutions meet. These values are also κ r values. The pink (red) contour in the top figure corresponds to the 95% 10 κ ξ -12 log and s n -13 -14 . For prescription II and inflaton coupling to fermions, the change in . For prescription II, inflaton coupling to fermions and first branch solutions, the top figure

0.965 0.960 0.955 0.950 0.970

s n is plotted for selected κ (68%) CL contours basedBottom on figures data show taken by the Keck Array/BICEP2 and Planck collaborations [57]. Figure 10 of written in the figure. The bottom figures only show the first branch solutions. (68%) CL contour basedThe on solid data (dotted) taken portionspoints by of show the the the Keck Array/BICEP2 maximum curves and correspond Planck to collaborations first [57]. (second) branch of solutions. The red Figure 9 shows in light green (green) the regions in the JCAP10(2019)028 ) κ x ln value (1982) − κ ln( is the same − B 108 x κ ln ≈ arXiv:1807.06209 ) , arXiv:1807.06211 x , ]. − ]. Phys. Lett. ( due to inflaton coupling 1 , r − W SPIRE of the coupling parameter SPIRE IN and IN s n max ). For prescription I, the plateau κ N (8 ) using prescription I (prescription II). / 1 10 values significantly unless values. For example, if inflaton couples to (1983) 177[ (1982) 1220[ − r ]. 10 48 max ξ × κ and B 129 9 SPIRE . – 15 – s IN n (1 8 − 10 Cosmology for Grand Unified Theories with Radiatively × between the Ricci scalar and the inflaton, first reviewing the Phys. Lett. ˙ ITAK (The Scientific and Technological Research Council of , Phys. Rev. Lett. 2 . However, as explained in section5, we regard this result as . R (1981) 347[ , 2 κ dependent maximum value ¨ UB . For the prescription I and coupling to bosons case, in contrast is 2 ξφ ξ Planck 2018 results. VI. Cosmological parameters Planck 2018 results. X. Constraints on inflation 1, corresponding to D 23 max max κ κ 2 ξφ ]. Chaotic Inflation A New Inflationary Universe Scenario: A Possible Solution of the Horizon, The Inflationary Universe: A Possible Solution to the Horizon and Flatness = 10, Phys. Rev. ξ collaboration, collaboration, , ]. ]. SPIRE IN SPIRE SPIRE IN IN The two prescriptions for the radiative corrections lead to significantly different poten- Two prescriptions used in the literature to take into account the radiative corrections to Generally, we observed that while the radiative corrections prevent inflation with a [ Planck [ Planck Induced Symmetry Breaking 389[ Flatness, Homogeneity, Isotropy and Primordial Monopole Problems Problems [4] A.D. Linde, [5] [6] [3] A. Albrecht and P.J. Steinhardt, [2] A.D. Linde, [1] A.H. Guth, fermions and Such differences suggest theprescriptions neeed used for in the further literature work to on calculate the the theoretical observational parameters. motivationsAcknowledgments of these This work is supportedTurkey) by project T number 116F385. 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