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J Bramante, J Cook, a Delgado, a Martin Low scale inflation with a curvaton Jessica Cook 11/14/17 1608.08625, PRD: J Bramante, J Cook, A Delgado, A Martin The tensor to scalar ratio r < 0.07 φ2 inflation is basically ruled out many of the simplest large field models ruled out large field = Δφ ≥ MP models that are looking good are the strong gravitational coupling models (R2 inflation, Higgs inflation, etc.) and smaller field models, Δφ ≤ MP becomes interesting to ask, what are the best looking smaller field models small field models generally fine tuned. Note we know the amplitude of the scalar power 2 spectrum: 9 H P⇣ =2.2 10− 2 ⇥ ⇡ ✏MP so smaller scalar inflation -> smaller H -> smaller ε ε sets slope of potential inflation requires a pretty flat potential anyway, need a ‘really’ flat potential as you go to smaller scales means couplings of the inflaton must get smaller too general small field models often look like: 2 3 V = V0 + g1φ + g2φ + g3φ + ... often models either hilltop… or inflection point… where derivatives of the potential -> 0 at scales relevant to inflation becomes easier to explain why the potential is so flat at inflation scales Higgs hilltop? V = V µ2φ2 + λφ4 0 − ask where does field have to start, such that you get enough efolds? 60 4M2 · P 2 φ60 = φend e− ⌫ have to start ‘really’ close to the top of the hill! and there is a quantum uncertainty bound, even at temp=0 H 17 φ =4 10− GeV ≥ 2⇡ ⇥ 32 find inflation could only last: N =1.4 10− max ⇥ ideally you could write a model where only two potential terms are really relevant at inflation scales. Like 1 2 2 1 4 V = V0 m φ V = V λφ − 2 0 − 4 And while these models do work in the large field regime, they don’t in the small field regime. or rather they can generate enough inflation in the small field regime, but will generate totally wrong answers for ns. one can write small field models of the form: V = V g φ g φ2 g φ3 + ... 0 − 1 − 2 − 3 but need keep multiple of those g’s and balance them against each other and the tuning of all the parameters will get worse and worse the smaller scale you want to go further the models that work, that give the right As and ns, are so flat, the potential changes so slowly with φ for the V0… end up with leftover dark energy in the end example: V = (1014)4 (5 1010)3φ 103φ3 − ⇥ − imagine adding a higher dimensional term to turn potential around, keep it from being tachyonic φ5 V = (1014)4 (5 1010)3φ 103φ3 + − ⇥ − ⇤ but even for Λ = MP, final term too large, get V ≈ V0 in the end one can get nicer models by adding an extra scalar field φ = inflation, dominates energy density, drives inflation σ = curvaton, responsible for the observed power spectrum some of the simple models for φ then work, because no longer need to fit to As or ns way to get rid of φ’s power spectrum… both fields will generate fluctuations during inflation key is reheating assume that φ decays quickly, quickly becomes radiation. It’s decay particles carry φ’s spectrum of fluctuations, but they redshift like radiation assume σ behaves like matter, decays much more slowly, so σ’s energy density redshifting more quickly. by time σ decays, mostly left with σ and it’s perturbation spectrum don’t care anymore what φ’s spectrum was do require φ’s initial power spectrum small enough to not significantly increase σ’s power spectrum thorough their gravitational coupling so only ask φ to generate enough inflation and only ask σ to produce correct ns and As can place constraints based on reheating. Define end of reheating = when σ has decayed. Time evolution: N inflation reheating BBN inflation ends reheating ends, inflation decays, curvaton decays H Γφ H Γ ⇡ ⇡ σ Will choose a low scale inflaton model… λ φ6 V (φ)=V φ4 + 0 − 4 ⇤2 Will inflate only using the φ4 term. It’s one of the simplest models. Also, many small field models can be expanded into approximately this form, with the inflaton rolling to higher field values. last term prevents the field from becoming tachionic. require that V(φmin) =0 1 2V 3 λ =6 0 φ ⇤4 ✓ ◆ Apply constraint: want to require that there exists a plausible reheating scenario want wre, Tre, Nre, and N during inflation to take on physically plausible values Assume perfect fluid during reheating, and integrate conservation of energy equation… µ 0= µT0 r 1 d⇢ = 3H(1 + w) ⇢ dt − ⇢ = ⇢tot = ⇢φ + ⇢σ expand out for perfect fluid in FRW Nfinal 3 N (1+w)dN ⇢final = ⇢initiale− initial R 3∆Nre(1+ wre ) ⇢re = ⇢ende− h i note ⇢ end set at the end of inflation by φ, since φ dominates energy density at the end of inflation 3 ⇢ = V (φ) end 2 end and ⇢ re set by σ, since assumption φ’s energy density has redshifted away by then. ⇡2 ⇢ = g T 4 re 30 re re assume conservation of entropy after reheating… 7 a3 g T 3 = a3(2T 3 +6 T 3 ) re re re 0 0 · 8 ⌫0 putting everything together… have equations relating inflaton parameters to the temperature at the end of reheating first just constrain inflaton… 1 4 4 Vend(φ) Nre = 61.6 ln N (1 3wre) − H − ⇤ − " ⇤ !# 26 N Nre Tre =2.5 10 H e− ⇤ e− ⇥ ⇤ can use to constrain inflaton models, without specifying exact curvaton model example: λ φ6 V (φ)=V φ4 + 0 − 4 ⇤2 λ φ6 V (φ)=V φ4 + 0 − 4 ⇤2 ⇤=MP λ φ6 V (φ)=V φ4 + 0 − 4 ⇤2 ⇤= 1015 GeV λ φ6 V (φ)=V φ4 + 0 − 4 ⇤2 ⇤= 105GeV λ φ6 V (φ)=V φ4 + 0 − 4 ⇤2 one of the problems though with the model: is how come no φ2 term, especially as we say inflation occurs close to top of hill when φ -> 0. Would expect a φ2 term to dominate there if we’re allowing a φ4 term, can’t forbid a φ2 term could ask how large of a φ2 term would you expect based on the φ4 term… λ⇤2 m2 = m2 0 − 24⇡2 compare to how small we 1 2 2 1 4 mmaxφ (.1) λφ need the mass to be, say: 2 ⇤ 4 ⇤ 1/3 m V max =0.72 0φ 1/3 mnat MP ⇤ pN ⇤ favors larger scale inflation: m max ⇤ V0 ⇤ mnat ! MP pN ! ⇤ 1 ex: V 4 ⇤ 1015 N = 30 0 φ ! ! ⇤ mmax 5 =5 10− mnat ⇥ could alleviate the tuning on the mass, if you allow for λ extra tiny, which requires that the higher dimensional term is extra suppressed… 6 λ 4 φ V (φ)=V0 φ + δ 2 take: − 4 MP basically same situation if doing φ3 hilltop g 1 V (φ)=V φ3 + φ5 0 − 3 ⇤ still want V(φmin) = 0 still expect a mass term to be generated, and want it irrelevant at inflation scales… g2⇤2 m2 = m2 0 − 9 24⇡2m2 · 0 if all 3 of those terms comparable… 1 g⇤ mnat = 2r 3⇡ 3 m V 10 max =0.7 0 1/5 mnat ⇤ MP pN ⇤ favors large scale inflation best case, get V ⇤ 0 ! then get the same ratio mmax ⇤ mnat ! MP pN ⇤ 1 4 15 ex: V0 φ ⇤ 10 N = 30 ! ! ⇤ mmax 5 =5 10− mnat ⇥ Could also try an inflation model: 1 V (φ)=V m2φ2 0 − 2 question if want to use a φ4 or φ6 term to make potential turn around expect the φ6 term to be there anyway…. but then find the masses compatible with generating enough inflation and without generating too large a power spectrum are so small, don’t allow for V(φmin) = 0 1 1 φ6 V (φ)=V m2φ2 λφ4 + 0 − 2 − 4 ⇤ have to add a negative φ4 term, even if it’s the φ2 which is dominating during inflation. further, the masses are so small, you even end up still in the regime the λφ4 term should have generated a larger mass then you want Constraints on curvaton: want σ to produce correct As and ns find to get the right ns out of curvaton, assuming slow roll: ns 1= 6✏ +2⌘ − − V n 1= 6✏ +2 σσ s − − 3H2 note η of σ, not φ small field, ε -> 0 2 ησ V 1 n = 2 σσ − s − 3H2 need curvaton model with V σσ negative so need curvaton with hilltop like shape… 1 V (σ)=V λσ4 0 − 4 so looking at basically same type models for curvaton and inflaton λ σ6 V (σ)=V σ4 + 0 − 4 ⇤2 adding the higher dimension operator, keeps it from being tachyonic total picture: λ φ6 λ σ6 V = V φ4 + σ σ4 + 0 − 4 ⇤2 − 4 ⇤2 it’s really V0 which is driving inflation choosing φ’s potential to be more steep. Want φ to determine how long inflation lasts, determine when inflation ends. This way, φ’s potential terms subtract the most from V0, and are what cancel the dark energy V0 couldn’t have done this with only one field. This way we can give σ a very shallow potential, have it evolve very slowly, coupled with a large-ish V0, and we can get the right As and ns. If we gave a field this combo of large-ish V0 and very shallow potential, then we would normally be left with dark energy in the end.
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