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 + g1 + g2 + g3 + ... 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 µ22 + 4 0 ask where does field have to start, such that you get enough efolds? 60 4M2 · P 2 60 = end 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 =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 V = V 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 g 2 g 3 + ... 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)3 1033 ⇥ imagine adding a higher dimensional term to turn potential around, keep it from being tachyonic

5 V = (1014)4 (5 1010)3 1033 + ⇥ ⇤

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 H ⇡ ⇡ Will choose a low scale inflaton model…

6 V ()=V 4 + 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 =6 0 ⇤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 = ⇢ + ⇢ expand out for perfect fluid in FRW

Nfinal 3 N (1+w)dN ⇢final = ⇢initiale initial R 3Nre(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 () 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() 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: 6 V ()=V 4 + 0 4 ⇤2 6 V ()=V 4 + 0 4 ⇤2

⇤=MP 6 V ()=V 4 + 0 4 ⇤2

⇤= 1015 GeV 6 V ()=V 4 + 0 4 ⇤2

⇤= 105GeV 6 V ()=V 4 + 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… ⇤2 m2 = m2 0 24⇡2 compare to how small we 1 2 2 1 4 mmax (.1) need the mass to be, say: 2 ⇤  4 ⇤ 1/3 m V max =0.72 0 1/3 mnat MP ⇤ pN ⇤

favors larger scale inflation: m max ⇤

V0 ⇤ mnat ! MP pN ! ⇤ 1 ex: V 4 ⇤ 1015 N = 30 0 ! ! ⇤ 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 4 V ()=V0 + 2 take: 4 MP basically same situation if doing φ3 hilltop g 1 V ()=V 3 + 5 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 ⇤ 10 N = 30 ! ! ⇤

mmax 5 =5 10 mnat ⇥ Could also try an inflation model: 1 V ()=V m22 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 6 V ()=V m22 4 + 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 s 3H2 note η of σ, not φ

small field, ε -> 0 2 ησ V 1 n = 2 s 3H2

need curvaton model with V negative so need curvaton with hilltop like shape…

1 V ()=V 4 0 4 so looking at basically same type models for curvaton and inflaton 6 V ()=V 4 + 0 4 ⇤2

adding the higher dimension operator, keeps it from being tachyonic total picture: 6 6 V = V 4 + 4 + 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.

and during inflation each φ and σ set up their own spectrum of perturbations, but it’s only σ’s perturbations which last through reheating and are visible today

solving for the curvaton’s power spectrum: since we said curvaton’s potential much flatter than inflation’s, assume V H2 | |⌧ 2 ¨ ˙ k +3H + 2 =0 a H = 2⇡

(same solution you get for tensor modes, since they also have no potential term, they decouple from the inflaton at 1st order) to get power spectrum: use ⇢ ⇣ = H ⇢˙

depends on ⇢tot but can break into separate pieces: and each will separately be conserved outside the horizon as long as 1. they don’t interact 2. they have canonical kinetic terms

so expand out ⇢ by taking: = 0 + and use: 3 2 k 2 H P = ⇣ = 6 ⇣ 2⇡2 h i h i 2k3 9H P⇣ = 2 2 6 4⇡ ⇤ require that this give the standard 2.2 x 10-9 then can take derivate of power spectrum, get ns:

d ln P⇣ 1 d n 1= ns 1= P⇣ s d ln k HP dt k = aH ⇣ ✓ ◆

2 2 n 1= 2✏ ⇤ s H2 0 so for a given H, say we picked an inflation sector and choose V0/ H then power spectrum and ns give 2 equations for 2 unknowns, and ⇤ 2 ⇡ As 3 3H 1 = (1 ns) = ⇤ 18 ⇤ ⇡pA (1 n ) s s note precisely defined 14 4 10 ⇡ ⇥ 3H 1 = ⇤ ⇤ ⇡pA (1 n ) s s

blue line = prediction for generating correct ns and As

orange line is constraint that σ slowly roll down its potential rather than be dominated by quantum fluctuations H > ⇤ 2⇡ other important curvaton constrains:

1. don’t want curvaton to generate a second wave of inflation. Require curvaton inflation to last lest than an efold after φ stops generating inflation.

2. Require the curvaton comprise at least 99.1% of the energy density of the universe by the end of reheating to avoid Plank’s iscocurvature constraints. Constrains the length of reheating. then can talk about observed mass:

mass at the end, when each field just oscillating about it’s vev 1/3 4/3 V0 m ¯ =2 p3 ⇤1/3 for inflation, depends on Λ and V0:

strongest curvaton constraint here comes from requiring that σ not generate a second wave of inflation also note that the inflaton’s mass predicts the scale of inflation to within an order of magnitude.

ex: mφ = 1 Gev -> V01/4 = 0.3 - 10 TeV for curvaton, since we know λσ, mσ only depends on Λ

⇤ m¯ = p3 both inflaton and curvaton must decay. Curvaton must couple to SM such that its decay products are the universe we see.

suppose simple interaction: Higgs portal, add

V (, )= 22 h| |

keeping the hilltop potential for φ and using the known Higgs potential… find a mass mixing matrix between the neutral Higgs and φ standard to define mass eigenstates through the mixing angle:

S1 = h cos ✓ + sin ✓

S2 = h sin ✓ + cos ✓ interested in weak coupling where have mostly inflaton mass eigenstate and mostly higgs mass eigenstate. They don’t have a large effect on altering each others vev’s or masses.

v v v v tan(2✓)= h h we use: tan(2✓)= h h m2 m2 m2 m2 h | h| want to explore allowed parameter space for θ vs. mφ

first, we want to use φ’s hilltop potential to inflate

however, λφ4 gets corrected due to Higgs coupling:

4⇡ h  2 h p ⇡ 16⇡2 14 6 10 h 10 ⇡ ⇡

but vφ may be quite large

v 104 109 GeV ⇠

so even though Λφh is really small… still have significant parameter space where the mixing angle θ can be sizable inflaton could be detected through its Higgs coupling even if it’s Higgs coupling isn’t spoiling the flatness of the inflaton’s potential have a bound from BBN need φ (and σ) to finish decaying before the temp drops to TBBN = 4.7 MeV

This bound depends strongly on the inflaton mass

only decays into photons

decays to e+ e- decays to muons, mesons, baryons…. requiring Γ large enough that reheating ends when T > TBBN= 4.7 MeV can also place bounds based on the fact that we haven’t seen signs of such a new scalar yet

bound on total Higgs decay rate cos2 ✓ h ⇡ h,SM if mostly Higgs mass eignestate would have a reduced coupling strength compared to the SM higgs. at really high masses, best constrains from the electroweak precision parameter Δr, bound from the observed muon lifetime at slightly lower masses, the best constraint comes from LEP which could have seen a light Higgs like state e+e- -> hZ there are also many bounds coming from decays of different mesons

+ B Kµ µ + ! B µ µ Babar, Υ decays s !

B0 K0 +inv !

B + ! K + ! looking to the future…

SHiP - possible future Search for Hidden, Particles at CERN. and if we found the inflaton, and knew its mass this would infer information regarding the scale of inflation the requirement that the inflaton not produce too large of perturbations, and the requirement that the curvton produce the right As and ns restrict V0 as a function of mφ.

obtain a narrow perdition for V0 and Λ even if the curvaton part of the model was incorrect, but we found a low mass inflaton… we can find a mapping between the inflaton and curvaton

V0 ⇤ m = p12 2 m = ⇤ p3 ✓ ◆

assuming the cutoffs Λ are the same, finding one field predicts parameter space for the other

Conclusions

Using a curvaton can make small field inflation more attractive as always with small field inflation there are still lots of tuning issues

Moving to small field inflation, loose the possibility of finding tensor modes, but maybe if you get really lucky, the mass of the curvaton or inflaton could be small enough that they could be seen directly. particle colliders and meson decay experiments could probe inflation models with V1/4 < 1015 GeV