Ricci Subtraction for Cosmological Coleman-Weinberg Potentials
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Ricci Subtraction for Cosmological Coleman-Weinberg Potentials
S. P. Miao ∗ , S. Park ⋆ and R. P. Woodard † Department of Physics, National Cheng Kung University,No. 1 University Road, Tainan City 70101, TAIWAN CEICO, Institute of Physics of the Czech Academy of Sciences,Na Slovance 2, 18221 Prague 8 CZECH REPUBLIC Department of Physics, University of Florida,Gainesville, FL 32611, UNITED STATES
ABSTRACTWe reconsider the fine-tuning problem of scalar-driven inflation arising fromthe need to couple the inflaton to ordinary matter in order to make reheatingefficient. Quantum fluctuations of this matter induce Coleman-Weinbergcorrections to the inflaton potential, depending (for de Sitter background) ina complex way on the ratio of the inflaton to the Hubble parameter. Thesecorrections are not Planck-suppressed and cannot be completely subtractedbecause they are not even local for a general geometry. A previous studyshowed that it is not satisfactory to subtract a local function of just theinflaton and the initial
Hubble parameter. This paper examines the otherallowed possibility of subtracting a local function of the inflaton and the Ricciscalar. The problem in this case is that the new, scalar degree of freedominduced by the subtraction causes inflation to end almost instantly.PACS numbers: 04.50.Kd, 95.35.+d, 98.62.-g ∗ e-mail: [email protected] ⋆ email: [email protected] † e-mail: [email protected]fl.edu Introduction
Primordial inflation driven by the potential energy of a scalar inflaton, L = R √− g πG − ∂ µ ϕ∂ ν ϕg µν √− g − V ( ϕ ) √− g , (1)suffers from many fine tuning problems [1]. These include the need to makethe potential very flat, the need to choose very special initial conditions tomake inflation start, and the need to keep inflation predictive by avoiding theformation of a multiverse [2]. The implications of the increasingly stringentupper limits on the tensor-to-scalar ratio have caused some of the pioneersof inflation to question its testability [3, 4, 5].This paper is aimed at a different sort of fine tuning problem which mayprove equally serious: the Coleman-Weinberg corrections [6] to the inflatonpotential that are generated when ordinary matter is coupled to the infla-ton to facilitate re-heating. These corrections are too large to be ignoredbecause they are not Planck-suppressed [7]. The usual assumption has beenthat Coleman-Weinberg corrections are local functions of the inflaton whichcould be completely subtracted off, however, it has recently been shown thatcosmological Coleman-Weinberg corrections depend nonlocally on the metric[8], which precludes their complete subtraction.There are two possible local subtraction schemes:1. Subtract a local function of the inflaton which exactly cancels the cos-mological Coleman-Weinberg potential at the beginning of inflation;or2. Subtract a local function of the inflaton and the Ricci scalar whichexactly cancels the cosmological Coleman-Weinberg potential when thefirst slow roll parameter vanishes.A recent study of the first possibility concluded that it is not viable [9]. Whenthe inflaton is coupled to fermions, inflation never ends unless the couplingconstant is chosen so small as to endanger re-heating, and then an initialreduction of the expansion rate still results in de Sitter expansion at a lowerrate. When a charged inflaton is coupled to gauge bosons, inflation endsalmost immediately, again unless the coupling constant is chosen so small asto endanger re-heating. 1he purpose of this paper is to study the second possible subtractionscheme. Section 2 details the form of cosmological Coleman-Weinberg po-tentials for fermionic and for bosonic couplings. Section 3 gives the evolutionequations associated with the second subtraction scheme. The effect on thesimple V ( ϕ ) = m ϕ model is worked out for fermion and boson couplingsin section 4. We discuss the results in section 5 The purpose of this section is to explain the cosmological Coleman-Weinbergcorrections from fermions and from gauge bosons, which differ profoundlyfrom the simple ∓ ϕ ln( ϕ ) form that pertains in flat space [6]. The section be-gins by reviewing explicit results from computations in de Sitter background.We then explain our assumption for how to generalize these de Sitter resultsto a general spatially flat, homogeneous and isotropic background geometry, ds = − dt + a ( t ) d~x · d~x = ⇒ H ( t ) ≡ ˙ aa , ǫ ( t ) ≡ − ˙ HH . (2)The section closes with a discussion of the Ricci subtraction scheme. Explicit results have so far only been obtained for de Sitter background,which corresponds to ǫ = 0, with H exactly constant. Suppose the inflaton ϕ is Yukawa-coupled to a massless, Dirac fermion on this background via theinteraction L Yukawa = − λϕψψ √− g . The one loop correction to the inflatoneffective potential on de Sitter was originally derived by Candelas and Rainein 1975 [10]. Of course their result depends slightly on conventions of regu-larization and renormalization. Our more recent computation [11] employeddimensional regularization in D spacetime dimensions with conformal andquartic counterterms,∆ L f = − δξ f ϕ R √− g − δλ f ϕ √− g . (3)To simplify the result we took the dimensional regularization scale µ to beproportional to the constant Hubble parameter of de Sitter, δξ f = 4 λ H D − (4 π ) D ( Γ(1 − D ) D ( D −
1) + (1 − γ )6 + O ( D − ) , (4)2 λ f = 24 λ H D − (4 π ) D ( Γ (cid:16) − D (cid:17) + 2 ζ (3) − γ + O ( D − ) , (5)where γ = 0 . ... is the Euler-Mascheroni constant. These choices resultin a cosmological Coleman-Weinberg potential of the form ∆ V f ( ϕ, H ) = − H π × f ( z ), where z ≡ λϕH , and the function f ( z ) is, f ( z ) = 2 γz − [ ζ (3) − γ ] z + 2 Z z dx ( x + x ) h ψ (1+ ix ) + ψ (1 − ix ) i , (6)and ψ ( x ) ≡ ddx ln[Γ( x )] is the digamma function.We assume that de Sitter results can be extended to general homoge-neous and siotropic geometries (2) by simply replacing the constant de SitterHubble parameter with the time dependent H ( t ) for a general background.However, we must be careful to keep the dimensional regularization scaleconstant, which amounts to a small change of the counterterms (4-5), δξ f = 4 λ µ D − (4 π ) D ( Γ(1 − D ) D ( D −
1) + (1 − γ )6 + O ( D − ) , (7) δλ f = 24 λ µ D − (4 π ) D ( Γ (cid:16) − D (cid:17) + 2 ζ (3) − γ + O ( D − ) . (8)The net effect is to change ∆ V f ( ϕ, H ) to,∆ V f ( ϕ, H ) = − H π ( f ( z ) + z ln (cid:16) H µ (cid:17) + 12 z ln (cid:16) H µ (cid:17)) . (9) The contribution of a gauge boson to a charged inflaton, L vector = − (cid:16) ∂ µ + ieA µ (cid:17) ϕ ∗ (cid:16) ∂ ν − ieA ν (cid:17) ϕg µν √− g , (10)was originally computed on de Sitter background using mode sums by Allenin 1983 [12]. As always, the precise result depends on conventions of reg-ularization and renormalization. Our more recent, dimensionally regulatedcomputation [13] employed the massive photon propagator [14] with confor-mal and quartic counterterms,∆ L b = − δξ b ϕ ∗ ϕR √− g − δλ b ( ϕ ∗ ϕ ) √− g . (11)3e again chose the dimensional regularization mass scale µ to be proportionalto the constant de Sitter Hubble parameter, δξ b = e H D − (4 π ) D ( − D + 12 γ + O ( D − ) , (12) δλ b = D ( D − e H D − (4 π ) D ( − D + γ −
32 + O ( D − ) , (13)These choices result in a cosmological Coleman-Weinberg potential of theform ∆ V b ( ϕ ∗ ϕ, H ) = + H π × b ( z ), where z ≡ e ϕ ∗ ϕH , and b ( z ) is, b ( z ) = (cid:16) − γ (cid:17) z + (cid:16) −
32 + γ (cid:17) z + Z z dx (1+ x ) " ψ (cid:16)
32 + 12 √ − x (cid:17) + ψ (cid:16) − √ − x (cid:17) . (14)When generalizing the constant de Sitter Hubble parameter to the time-dependent one of a general homogeneous and isotropic geometry we mustrevise the counterterms (12-13) to keep the mass scale of dimensional regu-larization strictly constant, δξ b = e µ D − (4 π ) D ( − D + 12 γ + O ( D − ) , (15) δλ b = D ( D − e µ D − (4 π ) D ( − D + γ −
32 + O ( D − ) . (16)The net effect is to change ∆ V b ( ϕ ∗ ϕ, H ) to,∆ V b ( ϕ ∗ ϕ, H ) = + 3 H π ( b ( z ) + z ln (cid:16) H µ (cid:17) + 12 z ln (cid:16) H µ (cid:17)) . (17) Ricci subtraction amounts to subtracting the primitive contribution with thereplacement H ( t ) → R ( t ). In a homogeneous and isotropic geometry thiscan be thought of as an ǫ -dependent Hubble parameter H ( t ), R ( t ) = 12 H ( t ) + 6 ˙ H ( t ) = 12 h − ǫ ( t ) i H ( t ) ≡ H ( t ) . (18)4ecause the quartic terms cancel between the primitive potential and theRicci subtraction, the full fermionic result takes the form ∆ V f ( ϕ, H ) − ∆ V f ( ϕ, H ) where,∆ V f ( ϕ, H ) = − H π ( ∆ f ( z ) + z ln (cid:16) H H (cid:17)) , z ≡ λϕH , (19)∆ f ( z ) = 2 γz − z ln( z ) + 2 Z z dx ( x + x ) h ψ (1+ ix ) + ψ (1 − ix ) i . (20)Note that we have chosen the constant mass scale to be the Hubble parameterat the beginning of inflation, µ = H inf . The full bosonic result takes the form∆ V b ( ϕ ∗ ϕ, H ) − ∆ V b ( ϕ ∗ ϕ, H ) where,∆ V b ( ϕ ∗ ϕ, H ) = + 3 H π ( ∆ b ( z ) + z ln (cid:16) H H (cid:17)) , z ≡ e ϕ ∗ ϕH , (21)∆ b ( z ) = (cid:16) − γ (cid:17) z − z ln(2 z )+ Z z dx (1+ x ) " ψ (cid:16)
32 + 12 √ − x (cid:17) + ψ (cid:16) − √ − x (cid:17) . (22) The purpose of this section is to work out the two Friedmann equations forthe case in which the cosmological Coleman-Weinberg potential depends onthe Hubble parameter, and it is subtracted by a function which depends onthe Ricci scalar. We also change to dimensionless dependent and independentvariables.It is useful to change the evolution variable from co-moving time t to thenumber of e-foldings from the beginning of inflation, n ≡ ln (cid:16) a ( t ) a ( t i ) (cid:17) = ⇒ ddt = H ddn , d dt = H h d dn − ǫ ddn i . (23)It is also useful to make the dependent variables dimensionless, φ ( n ) ≡ √ πG × ϕ ( t ) , χ ( n ) ≡ √ πG × H ( t ) . (24)With these variables the slow roll approximation to the (already dimension-less) scalar power spectrum becomes,∆ R ≃ GH πǫ = 18 π χ ǫ . (25)5inally, it is natural to use a dimensionless potential and mass parameter, U ≡ (8 πG ) × V , k ≡ πG × m . (26)The simplest way of expressing the modified field equations is to imag-ine that the dimensionless form of the classical potential plus the primitiveColeman-Weinberg potential takes the form U ( φ, χ ). The Ricci-subtractiontakes the similar form U sub ( φ, χ ), where we define, χ ≡ s − ǫ χ = s χ + 12 χχ ′ . (27)The two potentials enter the scalar evolution equation the same way, χ h φ ′′ + (3 − ǫ ) φ ′ i + ∂U∂φ + ∂U sub ∂φ = 0 . (28)However, the fact that the subtracted potential U sub depends upon ǫ , inaddition to χ , makes the form of its contributions to the gravitational fieldequations very different. The 1st Friedmann equation becomes,3 χ = 12 χ φ ′ + U − χ ∂U∂χ + U sub −
12 (1 − ǫ ) χ ∂U sub ∂χ + 12 χ ddn ∂U sub ∂χ . (29)The 2nd Friedmann equation is, − (3 − ǫ ) χ = 12 χ φ ′ − U + χ ∂U∂χ + 13 χ ddn ∂U∂χ − U sub + 12 (cid:16) − ǫ (cid:17) χ ∂U sub ∂χ − χ h ddn +2 − ǫ i ddn ∂U sub ∂χ . (30)One consequence of the final term in equation (29) is that the first Fried-mann equation involves second derivatives of χ ( n ). To see this, use the chainrule to exhibit the implicit higher derivatives,12 χ ddn ∂U sub ∂χ = 12 χ ( φ ′ ∂ U sub ∂φ∂χ + h χχ ′ + 12 χ ′ + 12 χχ ′′ i ∂ U sub ∂χ ) . (31)Recalling that ǫ = − χ ′ /χ allows us to express the first Friedmann equation(29) as, ǫ ′ = − ǫ + 2 ǫ + 4 χ ∂ U sub ∂χ ( − χ h − φ ′ i + U + U sub − χ " ∂U∂χ + 12 (1 − ǫ ) ∂U sub ∂χ + 12 χ φ ′ ∂ U sub ∂φ∂χ ) . (32)6he natural initial conditions derive from the slow roll solutions for thepurely classical model ( U = k φ and U sub = 0), φ ( n ) ≃ φ (0) − n , χ ( n ) ≃ k h φ (0) − n i , ǫ ( n ) ≃ φ (0) − n . (33)Hence we obtain a 2-parameter family of initial conditions based on φ (0) ≡ φ and k , φ (0) = φ , φ ′ (0) = − φ , (34) χ (0) = kφ q − ( φ ) , χ ′ (0) = − χ φ . (35)Note that these initial conditions (34-35) exactly satisfy the classical Fried-mann equation 3 χ = χ φ ′ + k φ and also make the first slow roll pa-rameter ǫ = 2 /φ agree with the slow roll approximation (33).Using the slow roll approximations (33) we see that φ = 20 will giveabout 100 total e-foldings of inflation. We can also express the power spec-trum and the scalar spectral index in terms of the evolving first slow rollparameter ǫ ( n ), ∆ R ≃ π χ ǫ −→ π k ǫ , (36)1 − n s ≃ ǫ + ǫ ′ ǫ −→ ǫ . (37)Of course relations (36) and (37) allow us to determine the constant k interms of the measured scalar amplitude A s and spectral index n s [15], k ≃ π (1 − n s ) s A s ≃ . × − . (38) m ϕ Model
The purpose of this section is to numerically simulate the effect of primitiveColeman-Weinberg potentials with Ricci subtraction in the context of theclassical V class = m ϕ model. We begin with the case of fermionic cor-rections, and then discuss bosonic corrections. The generic problem in eachcase is that the χ ′′ ( n ) terms in the first Friedmann eqaution (29) excite anew scalar degree of freedom that causes inflation to end almost immedi-ately when starting from the classical initial conditions (34-35).7 .1 Fermionic Corrections The Ricci subtraction scheme for fermionic corrections is defined by the po-tentials, U f ( φ, χ ) = 12 k φ − χ π " ∆ f (cid:16) λφχ (cid:17) + (cid:16) λφχ (cid:17) ln (cid:16) χ χ (0) (cid:17) , (39) U f sub ( φ, χ ) = + χ π " ∆ f (cid:16) λφχ (cid:17) + (cid:16) λφχ (cid:17) ln (cid:16) χ χ (0) (cid:17) , (40)where ∆ f ( z ) was defined in (20) and χ ≡ q − ǫ χ . Figure 1 displays theclassical evolution (in blue) versus the quantum-corrected model (in red dots)for a moderate coupling of λ = 5 . × − . × -7 × (cid:0)6 × (cid:1)(cid:2) × (cid:3)(cid:4) n0510152025 (cid:5) [ n ] × - × - × - × - n0.000000.000010.00002 χ [ n ] × - (cid:7) (cid:8)(cid:9) × (cid:10)(cid:11) - (cid:12)(cid:13)(cid:14) × (cid:15)(cid:16) - × (cid:17)(cid:18) - n0.00.5 (cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24) ϵ [ n ] Figure 1: Plots of the dimensionless scalar φ ( n ) (on the left), the dimension-less Hubble parameter χ ( n ) (middle) and the first slow roll parameter ǫ ( n )(on the right) for classical model (in blue) and the quantum-corrected model(in red dots) with Yukawa coupling λ = 5 . × − .While the initial evolution of the scalar and the Hubble parameter is notvisibly affected by the quantum correction, the first slow roll parameter risesabove the inflationary threshold of ǫ = 1 almost immediately.To understand why Ricci subtraction engenders immediate deviations for λ = 5 . × − , first note that the initial conditions of the classical model(34-35) force the initial value of the parameter z ≡ λφχ to be much largerthan one, z = λφ χ = λk s − ( 2 φ ) ≃ . (41)8his means it is valid to use the large z expansion of (20) [8],∆ f ( z ) = − z + z ln( z ) − (cid:16) − γ (cid:17) z + 1160 ln( z ) + O (1) . (42)Substituting (42) in expressions (39-40) implies, U f ( φ, χ ) = 12 k φ − χ π ( − (cid:16) λφχ (cid:17) + h ln (cid:16) λ φ χ (cid:17) −
56 +2 γ i(cid:16) λφχ (cid:17) + 1160 ln (cid:16) λ φ χ (cid:17) + . . . ) , (43) U f sub ( φ, χ ) = χ π ( − (cid:16) λφχ (cid:17) + h ln (cid:16) λ φ χ (cid:17) −
56 +2 γ i(cid:16) λφχ (cid:17) + 1160 ln (cid:16) λ φ χ (cid:17) + . . . ) . (44)The ( λφ ) terms cancel out between (43) and (44) so that the leading con-tribution usually comes from the ( λφ ) term, U f + U f sub − χ " ∂U f ∂χ + 12 (1 − ǫ ) ∂U f sub ∂χ = 12 k φ + χ π × z " ln (cid:16) λ φ χ (cid:17) −
56 +2 γ + . . . (45)12 χ φ ′ ∂ U f sub ∂φ∂χ = χ π × φ ′ φ z " ln (cid:16) λ φ χ (cid:17) + 16 +2 γ + . . . (46)However, the ( λφ ) term makes no contribution to the denominator, so onemust go one order higher, χ ∂ U f sub ∂χ = χ π × " (cid:16) λ φ χ (cid:17) − + . . . (47)Taking account of the fact that the classical terms initially cancel, and that φ ′ /φ = − ǫ , the large z form of (32) is, ǫ ′ ≃ − ǫ + 2 ǫ + 6 z [ln( z ) − +2 γ ] − ǫ z [ln( z )+ +2 γ ] [2 ln( z ) − − ǫ ) − ≃ . × . (48)9his compares with the slow roll result of ǫ ′ = 2 ǫ = 5 × − , and explainswhy the subtraction term brings inflation to such an abrupt end.Because the large fraction in (48) scales like λ one might expect thatdecreasing λ reduces ǫ ′ (0). This is indeed true for as long as the large z regime pertains, but ǫ ′ (0) approaches a constant value of about 6 in thesmall z regime, as can be seen from Figure 2. (cid:25)(cid:26) - (cid:27)(cid:28) (cid:29)(cid:30) - (cid:31) ! - λ " ϵ ′ [ ] Figure 2: Log-log plot of the final term in relation (32) for ǫ ′ (0) as a functionof λ in the range 10 − < λ < − .The asymptotic limit of ǫ ′ (0) ≃ z limit of ǫ ′ (0), note first that the small z ex-pansion of ∆ f ( z ) is [8],∆ f ( z ) = − z ln( z ) + h ζ (3) − γ i z + 23 h ζ (3) − ζ (5) i z + O ( z ) . (49)Comparison with expressions (39) and (40) implies that the small λ limitingforms derive from the conformal renormalization, U f ( φ, χ ) − k φ −→ − λ φ χ π ln (cid:16) χ χ (cid:17) , (50) U f sub ( φ, χ ) −→ + λ φ χ π ln (cid:16) χ χ (cid:17) . (51)Substituting (50-51) in the final term of (32) gives, ǫ ′ −→ − ǫ + 2 ǫ + 6 − ǫ +2(1 − ǫ ) ln(1 − ǫ )(1 − ǫ ) − . (52)10 striking feature of Figure 2 and expression (52) is that the limit λ → λ = 0 for which there is no change to classicalinflation. This seems contradictory but is in fact the standard signature ofa perturbation that changes the number of derivatives. A simple example isthe higher derivative extension of the simple harmonic oscillator consideredin section 2.2 of [16]. The oscillator’s position is x ( t ) and its Lagrangian is, L = − ǫm ω ¨ x + m x − mω x . (53)When ǫ = 0 this system reduces to the simple harmonic oscillator which hastwo pieces of initial value data and whose energy is bounded below. However,for any nonzero value of ǫ the system has four pieces of initial value data, andits energy is unbounded below. Because the effect of the higher derivativeperturbation (in our inflation model) never becomes small, no matter howsmall the coupling constant, it follows that perturbation theory breaks down. Making the inflaton complex causes a few small changes in the key equationsof section 3. Because the two potentials U b ( φ ∗ φ, χ ) and U b sub ( φ ∗ φ, χ ) dependon the norm-squared of the scalar, the evolution equation for the inflatonbecomes, χ h φ ′′ + (3 − ǫ ) φ ′ i + φ " ∂U b ( φ ∗ φ, χ ) ∂φ ∗ φ + ∂U b sub ( φ ∗ φ, χ ) ∂φ ∗ φ = 0 . (54)The first Friedmann equation takes the form,3 χ = χ φ ′∗ φ ′ + U b + U b sub − χ " ∂U b ∂χ + 12 (1 − ǫ ) ∂U b sub ∂χ + 12 χ ddn ∂U b sub ∂χ . (55)This gives an evolution equation for the first slow roll parameter analogousto (32), ǫ ′ = − ǫ + 2 ǫ + 4 χ ∂ U b sub ∂χ ( − χ h − φ ′∗ φ ′ i + U b + U b sub − χ " ∂U b ∂χ + 12 (1 − ǫ ) ∂U b sub ∂χ + 12 χ ( φ ∗ φ ) ′ ∂ U b sub ∂φ ∗ φ∂χ ) . (56)11ven though we do not use it, the second Friedmann equation is, − (3 − ǫ ) χ = χ φ ′∗ φ ′ − U b − U b sub + χ " ∂U b ∂χ + 12 (cid:16) − ǫ (cid:17) ∂U b sub ∂χ + 13 χ h ddn − ǫ i ∂U b ∂χ − χ h ddn +2 − ǫ i ddn ∂U b sub ∂χ . (57)And the initial values (assuming φ is real) become, φ (0) = φ , φ ′ (0) = − φ , (58) χ (0) = kφ q − ( φ ) , χ ′ (0) = − χ φ . (59)We continue to use φ = 20, with the value of k given in (38).The Ricci subtraction scheme for bosons is defined by these potentials, U b ( φ ∗ φ, χ ) = k φ ∗ φ + 3 χ π " ∆ b (cid:16) e φ ∗ φχ (cid:17) + e φ ∗ φχ ln (cid:16) χ χ (0) (cid:17) , (60) U b sub ( φ ∗ φ, χ ) = − χ π " ∆ b (cid:16) e φ ∗ φχ (cid:17) + e φ ∗ φχ ln (cid:16) χ χ (0) (cid:17) , (61)where ∆ b ( z ) was defined in (22) and χ ≡ q − ǫ χ . Figure 3 compares theclassical evolution (in blue) with the quantum-corrected (in red dots) for acharge e ≃ . × − which is three hundred million times weaker thanelectromagnetism. 12 × - × - × - × - n0510152025 ϕ [ n ] × - × - × - × - n0.000000.000010.000020.000030.000040.000050.00006 χ [ n ] × - × - × - × - n0.00520.00540.0056 ϵ [ n ] Figure 3: Plots of the dimensionless scalar φ ( n ) (on the left), the dimension-less Hubble parameter χ ( n ) (middle) and the first slow roll parameter ǫ ( n )(on the right) for classical model (in blue) and the quantum-corrected model(in red) with the charge-squared e = π × − . ≃ . × − .The rapid onset of deviations from classical evolution evident in Figure3 has the same explanation for bosons as for fermions. Even for the smallcoupling e ≃ . × − the initial value of z = e φ ∗ φχ is larger than one, z ≡ e φ χ = e k h − φ i ≃ . . (62)Just as for fermions, this means we can simplify relation (56) using the large z expansion of ∆ b ( z ) [8],∆ b ( z ) = − z + z ln(2 z ) − (cid:16) − γ (cid:17) z + 1960 ln( z ) + O (1) . (63)The corresponding large argument expansions of the potentials are, U b ( φ ∗ φ, χ ) = k φ ∗ φ + 3 χ π ( − (cid:16) e φ ∗ φχ (cid:17) + h ln (cid:16) e φ ∗ φχ (cid:17) −
53 +2 γ i e φ ∗ φχ + 1960 ln (cid:16) e φ ∗ φχ (cid:17) + . . . ) , (64) U b sub ( φ ∗ φ, χ ) = − χ π ( − (cid:16) e φ ∗ φχ (cid:17) + h ln (cid:16) e φ ∗ φχ (cid:17) −
53 +2 γ i e φ ∗ φχ + 1960 ln (cid:16) e φ ∗ φχ (cid:17) + . . . ) . (65)13ust as for fermions, the order e φ ∗ φ terms in (64-65) make the dominantcontributions to the numerator of expression (56), but the denominator is acrucial order weaker, ǫ ′ ≃ − ǫ + 2 ǫ + 6 z [ln(2 z ) − +2 γ ] − ǫ z [ln(2 z ) − +2 γ ] [2 ln( z ) − − ǫ ) − ≃ . (66)Just as we found for fermions, ǫ ′ (0) can be decreased by reducing thecoupling constant, but it eventually approaches a value that is still much toolarge. This can be seen from Figure 4. - - - - - - e ϵ ′ [ ] Figure 4: Plot of the 3rd term in expression (56) for ǫ ′ (0) as a function of e .The analytic derivation follows from the small e limiting forms of the quan-tum part of U b and U b sub , U b ( φ ∗ φ, χ ) − k φ ∗ φ −→ e φ ∗ φχ π ln (cid:16) χ χ (cid:17) , (67) U b sub ( φ ∗ φ, χ ) −→ − e φ ∗ φχ π ln (cid:16) χ χ (cid:17) . (68)The analysis, and even the result, is the same as for fermions. Note that thelimit e →
0, for which there is always an instability, again fails to agree with e = 0 model, which is classical inflation. Cosmological Coleman-Weinberg potentials are induced when the inflaton iscoupled to ordinary matter, typically to facilitate re-heating. Without sub-traction, these potentials are disastrous to inflation because they are far too14teep and not Planck-suppressed. If they depended only on the inflaton itwould be straightforward to subtract them but they also involve the metricin a deep and profound way. Explicit computations on de Sitter background,for fermions [10, 11] and for vector bosons [12, 13], reveal complicated func-tions of the dimensionless ratio of the coupling constant times the inflaton,all divided by the Hubble parameter. Indirect arguments show that the con-stant Hubble parameter of de Sitter in this ratio cannot be constant for ageneral geometry, nor can it even be local [8]. That poses a major obstacleto subtracting away cosmological Coleman-Weinberg potentials because onlylocal functions of the inflaton and the Ricci scalar can be employed [17], andneither can completely subtract the potentials.A previous study explored the possibility of subtracting a function of justthe inflaton, chosen to completely cancel the cosmological Coleman-Weinbergpotential at the onset of inflation [9]. What we found for moderate couplingconstants is that inflation never ends for the corrections due to fermions,and it ends too soon for the corrections due to vector bosons. Making theYukawa coupling very small results in a nearly classical evolution until latetimes, at which point the universe approaches de Sitter with a much smallerHubble parameter. An acceptable evolution can only be obtained by makingthe vector boson coupling very small, and this degrades the efficiency ofre-heating.This paper studied the other possibility: subtracting a function of theinflaton and the Ricci scalar. One might think (as we originally hoped) thatcorrections for this type of subtraction would be suppressed by the smallnessof the first slow roll parameter. However, the higher time derivatives in thesubtraction change the first Friedmann equation (29) from being algebraicin the Hubble parameter to containing second derivatives of it, and the par-ticular way (32) this change manifests is fatal for inflation. We were able toconstruct an analytic proof (52) — supported by explicit numerical analysisin Figures 2 and 4 — that the initial value of ǫ ′ can never be less than about6. That compares with its initial value of 5 × − in the classical model,and it means that inflation cannot last more than a single e-folding. So theRicci subtraction scheme is much worse than the initial time subtraction, butneither method is satisfactory.Before closing we should make a few comments. First, the problem with ǫ ′ (0) is almost completely independent of the classical model of inflation.So one should not expect that a different model would lead to a differentresult. Second, we need better control of the ǫ dependence of cosmological15oleman-Weinberg potentials. The present study was carried out by assum-ing that the constant Hubble parameter of de Sitter background becomesthe instantaneous Hubble parameter of an evolving geometry. In reality, thecosmological Coleman-Weinberg potential should depend as well on ǫ ( n ) [18].Accounting for this dependence will tighten the argument, and we expect itto extend the ǫ ′ (0) problem even to the initial time subtraction. Finally, itshould be possible to extend these studies to the case in which derivatives ofthe inflaton are coupled to matter. Such a coupling would not induce a cos-mological Coleman-Weinberg potential but would change the kinetic term.It would be very interesting to work out the consequences for evolution andthe generation of perturbations. Acknowledgements
This work was partially supported by Taiwan MOST grants 103-2112-M-006-001-MY3 and 107-2119-M-006-014; by the European Research Councilunder the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant No. 617656, “Theories and Models of the Dark Sector:Dark Matter, Dark Energy and Gravity”; by NSF grants PHY-1806218 andPHY-1912484; and by the Institute for Fundamental Theory at the Universityof Florida.
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