The present study is motivated by two recent observations. Firstly, the precise measurements of the Higgs boson mass m_H at the Large Hadron Collider (LHC), and of the top quark mass m_t at the Tevatron and LHC, within their 3σ uncertainties, allow the Standard Model to be valid all the way up to the Planck energy scale, without a compelling need for new physics beyond the Standard Model at high energies [1] [2] [3] . And secondly, recent measurements of the dark matter particle comoving root-mean-square velocity v h rms ( 1 ) = a v h rms ( a ) happen to agree with the prediction of the “no freeze-in and no freeze-out” scenario of spin zero dark matter S that decouples early on from the Standard Model sector, e.g. scalar dark matter coupled to the Higgs boson [4] . a ( t ) is the expansion parameter of the universe normalized to a ( t 0 ) = 1 at the present time t 0 . Rotation curves of dwarf galaxies obtain v h rms ( 1 ) = 406 ± 69   m / s [5] . Galaxy rest-frame ultraviolet (UV) luminosity distributions, and stellar mass distributions, at redshifts z = 4 , 6 , 8 and 10, obtain v h rms ( 1 ) = 410 − 120 + 140   m / s [6] . The predictions of the “no freeze-in and no freeze-out” scenario, for dark matter S coupled to the Higgs boson, with the measured dark matter density, are v h rms ( 1 ) = 490 ± 5   m / s ,

842 ± 13 m/s, and 1471 ± 19 m/s, for spin 0, 1 2 , and 1 dark matter particles, respectively [4] . Note that spin 1 2 and spin 1 particles S are disfavored by the

measurements. S has a predicted mass M S = 150 ± 2   eV [4] . If S is real, then S is the dark matter particle.¹ If S is complex, it may decay to two massive gauge bosons V which are the dark matter in this case [10] . Additional observables studied, that obtain consistent results, are the redshift of first galaxies [11] , the ultra-violet luminosity of first galaxies [6] , and the related “reionization optical depth” [6] . For a summary of measurements see [6] .

For these reasons, in the present study we explore inflation options of the Standard Model extended with a scalar warm dark matter field S with a contact coupling − 1 2 λ h S ( ϕ † ϕ ) S 2 to the Higgs boson ϕ .

In Section 2 we take a journey towards the past, based on current observations and on the assumption of the validity of the Standard Model up to high energies. In Sections 3 to 7 we take a journey towards the future, starting from an assumed early stage of inflation of the universe driven by either S or ϕ , and the assumption that quantum fluctuations during inflation are the seeds of galaxies and of the observed large-scale power spectrum of density perturbations. The meeting, or not, of the two journeys provides a consistency test of the preceding assumptions. It is quite amazing, and non-trivial, that current data and the listed assumptions do pass the consistency test with no adjustable parameters! A summary of results is presented in the concluding section.

A summary of the expansion history of the universe is presented in Figure 1. The horizontal axis x ≡ M P r / a is the comoving distance in Planck units, and the vertical axis y = a is the expansion parameter a , both with natural logarithmic scales. M P ≡ 1 / G N ≈ 1.2 × 10 19   GeV is the Planck mass (we adopt the definition of [1] ). The two vertical lines correspond to two particles fixed in space: the vertical axis is the observer, i.e. us, and the red line is a point fixed in space at the reference comoving distance 1 / k = Mpc / 0.05 = exp ( 134.9 ) / M P . The comoving distance to the horizon relative to us, indicated by the blue line, is M P / ( a H ) , where the Hubble parameter H is the relative expansion rate. The

expansion parameter a ( t ) , in a spatially flat and homogeneous universe, evolves according to the Friedmann equation [1] :

H 2 ≡ ( 1 a d a d t ) 2 = 8 π 3 M P 2 ρ ≡ H 0 2 ρ ρ crit . (1)

where ρ crit ≡ 3 H 0 2 / ( 8 π G N ) is the critical density. Today, H ( t 0 ) ≡ H 0 . The density has contributions from radiation, and matter, and we have included the cosmological constant Λ (or vaccuum energy) contribution into ρ: ρ crit Ω Λ = ρ crit Λ / ( 3 H 0 2 ) :

ρ ≡ ρ crit ( Ω r a 4 + Ω m a 3 + Ω Λ ) . (2)

It is handy to have the solutions to (1) when one term in ρ dominates, see Table 1.

The blue straight lines in Figure 1 can be understood by noting that the comoving distance to the horizon 1/( a H ) is proportional to a − 1 , a , a 1 / 2 , and a − 1 , for ρ dominated, respectively, by vacuum energy during inflation until a = a f , radiation until a eq , matter until a Λ , and the cosmological constant (extended to the future to be seen on the scale of the figure). (There are intervals of a in the range a f < a < a r with ρ ∝ a − 3 if the inflaton field has a dominating quadratic potential, but this will not be the case in the present scenario.)

Note that a point in space at the reference comoving distance 1 / k exits the horizon at a i , and re-enters the horizon at a r , i.e.

k = a i H i = a r H r . (3)

Radiation decouples from matter at a dec as indicated by the black horizontal line in Figure 1. The Cosmic Microwave Background Radiation that we observe today propagates freely since t_dec from the “CMB” point marked with an X in Figure 1.

The slope of “light” rays is

d y d x = 1 ± ( r H ) − 1 ± 1 , (4)

so d y / d x ≈ 0 if r is inside the horizon, while d y / d x ≈ ± 1 if r is outside the horizon. Segments of light rays are shown with symbols “−” and “×” in Figure 1.

The Einstein equation of General Relativity obtains (1), and also the energy conservation equation

d ρ d t + 3 H ( ρ + p ) = 0, (5)

where p is the pressure:

p = ρ crit ( Ω r 3 a 4 + 0 − Ω Λ ) . (6)

Let us now obtain the temperature in the range m t < T ( a ) < T reh , where T_reh is the reheating temperature when the Standard Model particles and S obtain thermal equilibrium. From conservation of entropy

g reh a 3 T 3 = g γ a 0 3 T 0 3 + g ν a 0 3 T ν 0 3 + g S a 0 3 T S 0 3 , (7)

where T 0 = 2.7255   K is the present photon temperature [1] , and g ≡ N b + 7 8 N f

for each particle species, with N_b or N_f the number of spin states for bosons or fermions respectively. We have neglected gauge singlet neutrinos with both Yukawa and Majorana mass terms (needed to give the observed mass and mixing to neutrinos, and for baryogenesis via leptogenesis [12] ). g γ = 2 for the photon, g ν = 7 / 8 ⋅ 3 ⋅ 2 for three left-handed neutrinos plus right-handed anti-neutrinos, and g S = 1 for the scalar warm dark matter S , T ν 0 = ( 4 / 11 ) 1 / 3 T 0 , and T S 0 = ( 43 / ( 11 ⋅ 95.25 ) ) 1 / 3 T 0 . g_reh is the effective number of degrees of freedom at the end of the reheating phase. For the present model with warm dark matter, g reh = 106.75 + 1 . So,

a a 0 = 0.332 T 0 T . (8)

The corresponding density is

ρ = π 2 30 g reh T 4 . (9)

In this Section we take S to be the field that drives inflation. Our metric for flat space is

d s 2 = g μ ν d x μ d x ν = d t 2 − a ( t ) 2 ( d x 2 + d y 2 + d z 2 ) . (10)

Greek indices go from 0 to 3, while latin indices go from 1 to 3. The Lagrangian, wave equation, energy-momentum tensor, energy density, and pressure of a real scalar field S , are [1]

L = 1 2 ∂ μ S ⋅ ∂ μ S − V ( S ) , (11)

0 = ∂ L ∂ S − ∂ μ ∂ L ∂ ( ∂ μ S ) = − d V ( S ) d S − ∂ μ ∂ μ S , (12)

T μ ν = ∂ L ∂ ( ∂ μ S ) ∂ ν S − g μ ν L = ∂ μ S ⋅ ∂ ν S − g μ ν L (13)

T 0 0 = ρ = 1 2 S ˙ 2 + V ( S ) , (14)

− T i j = p δ j i = [ 1 2 S ˙ 2 − V ( S ) ] δ j i . (15)

We have neglected spatial derivatives in these last two equations. ρ and p are the density and pressure of the homogeneous fluid in its rest frame. From (5), (10) and (14) we obtain

S ¨ + ( 3 H + Γ ) S ˙ = − d V d S . (16)

Dots are derivatives with respect to t. In this equation we have included the total decay rate Γ of S to discuss reheating in Sections 6 and 7. It is convenient to define the “slow-roll” parameters:

ε ≡ M P 2 16 π ( d V / d S V ) 2   and   η ≡ M P 2 8 π d 2 V / d S 2 V . (17)

Initial conditions for S ( t 1 ) ≡ S 1 and a ( t 1 ) ≡ a 1 are such that V ( S 1 ) ≫ S ˙ 2 / 2 at t 1 . While S ¨ can be neglected in (16) we are in the “slow-roll” regime. Necessary conditions for the slow-roll approximation to be valid are ε ≪ 1 and η ≪ 1 . In this approximation ε ≈ − H ˙ / H 2 . Note that inflation ends, i.e. a ¨ = 0 , when − H ˙ / H 2 = 1 .

In the slow-roll approximation, H, ρ, and p are approximately constant, and a ( t ) ∝ e H t grows approximately exponentially with time. In the slow-roll approximation, the number of “e-folds” of growth of a ( t ) until ε = 1 , is

N ≡ ln ( a f a i ) = ∫ t i t f     H d t = − 2 π M P ∫ S i S f d S ε . (18)

The dimensionless power spectrum² of density fluctuations at horizon re-entry is [1] [13] [14]

P ζ ( k ) = H i 2 π ε i M P 2 ≈ 1 24 π 2 V i ε i ( 8 π M P 2 ) 2 , (19)

and the tensor gravitational wave power spectrum at horizon re-entry is

P t ( k ) = ( H i 2 π ) 2 64 π M P 2 ≈ 2 3 π 2 V i ( 8 π M P 2 ) 2 . (20)

Note that H i and ε i correspond to a i . The power spectrum P ζ ( k ) at the reference wavenumber k ref = 0.05   Mpc − 1 is measured to be exp ( 3.044 ± 0.014 ) × 10 − 10 ≈ 2.1 × 10 − 9 [1] . The tensor-to-scalar ratio for single-field slow-roll inflation is [1]

r ≡ P t P ζ = 16 ε i (21)

A “pedestrian” derivation of (19) is presented in Appendix A.

Let us now consider the quadratic potential

V ( S ) = 1 2 m ¯ S 2 S 2 . (22)

The slow-roll parameters are

ε = η = 1 4 π ( M P S ) 2 . (23)

The solution of the evolution equations in the slow-roll approximation are

S = S i − m ¯ S M P 12 π ( t − t i ) , (24)

a ( t ) = a i exp ( 4 π 3 m ¯ S M P ∫ t i t     S d t ) . (25)

End of inflation, i.e. ε = 1 , corresponds to S f = M P / 4 π . The number of e-folds of inflation is:

N ≡ ln ( a f a i ) = 2 π S i 2 − S f 2 M P 2 . (26)

The power spectrum of density fluctuations, and the spectral tilt, are

P ζ = 16 π 3 ( m ¯ S M P ) 2 ( S i M P ) 4 , (27)

n s − 1 ≡ d ln P ζ d ln k ≈ − 6 ε i + 2 η i ≈ − 2 N . (28)

From the measured n s ≈ 0.965 , we obtain N ≈ 57.1 and S i ≈ 3.0 M P , and from the measured P ζ ≈ 2.1 × 10 − 9 , we obtain m ¯ S ≈ 1.2 × 10 − 6 M P . In conclusion, a quadratic potential is appropriate for cold dark matter, not for the warm dark matter studied in this article. Also, preheating with a quadratic potential requires M S ≫ m H , i.e. cold dark matter. We therefore turn to a study of inflation with a quartic potential.

To the Standard Model we add a warm dark matter real scalar field S , with Z₂ symmetry S ↔ − S , with a contact coupling λ h S to the Higgs boson field ϕ . (An alternative is complex S with a Lagrangian with U(1) symmetry that is, or is not, broken by the S and ϕ ground state [10] . Complex S only requires minimal changes of notation, e.g. S 2 → S * S .) The action is

S J = ∫ d 4 x − g { L grav + L S + L S ϕ + L S M } , (29)

L grav = − ( M P 2 2 ( 8 π ) + ξ h ϕ † ϕ + ξ S S 2 ) R , (30)

L S = + 1 2 ∂ μ S ⋅ ∂ μ S − 1 2 m ¯ S 2 S 2 − λ S 4 S 4 , (31)

L S ϕ = − 1 2 λ h S ( ϕ † ϕ ) S 2 . (32)

Note that the Higgs field ϕ and dark matter field S have non-minimal couplings ξ h and ξ S to the curvature scalar R. These terms are needed for inflation (see Table 2 and Table 3) and also for renormalization [15] . The Higgs part of the Standard Model Lagrangian L S M , in the unitary gauge with

ϕ = 1 2 ( 0 v h + h ) , (33)

is

L h = + 1 2 ∂ μ h ⋅ ∂ μ h − λ h 4 h 4 − λ h v h h 3 − λ h v h 2 h 2 . (34)

The dark matter particle mass is M S = m ¯ S 2 + 1 2 λ h S v h 2 = 150 ± 2   eV [4] , with m ¯ S 2

assumed positive, while λ h S is negative. The Higgs boson mass is m H = 2 λ h v h ≈ 125   GeV . These masses play no significant role in inflation.

In this section we will assume that the field h drives inflation, and therefore set S = 0 initially, and so neglect the term with ξ S .

The Lagrangian can be brought to the “Einstein form”, without the non-minimal couplings to R, with a transformation of the metric [15]

g ^ μ ν = Ω 2 g μ ν ,   Ω 2 = 1 + ξ h h 2 8 π M P 2 , (35)

and a redefinition of the field. We distinguish two versions of General Relativity: the “metric” version, and the “Palatini” version. In the “Palatini” version of General Relativity the metric g μ ν and the affine connection Γ β γ α are treated as independent variables. The metric and Palatini versions differ if the non-minimal couplings to gravity, ξ h or ξ S , are different from zero. In the metric version, the needed field redefinition and the transformed potential, for χ ≫ 6 / ( 8 π ) M P , are [1] [15] :

h ≈ M P 8 π ξ h exp ( χ 8 π 6 M p ) , (36)

U ( χ ) ≈ λ h 4 ξ h 2 ( M P 2 8 π ) 2 [ 1 + exp ( − 2 χ 8 π 6 M P ) ] − 2 (37)

In the Palatini version, the needed field redefinition and the transformed potential are [16] :

h = M P 8 π ξ h sinh ( 8 π ξ h χ M P ) . (38)

U ( χ ) = λ h 4 ξ h 2 ( M P 2 8 π ) 2 [ tanh ( 8 π ξ h χ M P ) ] 4 . (39)

The slow-roll parameters are defined as

ε ≡ M P 2 16 π ( d U / d χ U ) 2 ≈ − d H / d t H 2 , (40)

η ≡ M P 2 8 π d 2 U / d χ 2 U . (41)

In the slow-roll approximation, while ε ≪ 1 and η ≪ 1 , in the metric [15] or Palatini [16] versions of General Relativity, we obtain

ε ≈ 4 3 ξ h 2 h 4 ( M P 2 8 π ) 2   or   ≈ 8 ξ h h 4 ( M P 2 8 π ) 2 , (42)

η ≈ − 4 3 ξ h h 2 ( M P 2 8 π )   or   ≈ − 8 h 2 M P 2 8 π , (43)

N ≈ 3 4 ξ h ( h i 2 − h f 2 ) ( 8 π M P 2 )   or   ≈ π M P 2 ( h i 2 − h f 2 ) , (44)

n s − 1 ≡ d ln P ζ d ln k ≈ − 2 N   or   ≈ − 2 N , (45)

r ≡ P t ( k ref ) P ζ ( k ref ) ≈ 12 N 2   or   ≈ 2 ξ h N 2 , (46)

ρ i = U i = λ h 4 ξ h 2 ( M P 2 8 π ) 2   or   = λ h 4 ξ h 2 ( M P 2 8 π ) 2 , (47)

ρ f ≈ U f ≈ 0.134 U i   or   ≈ [ tanh ( 8 ξ h ) ] 4 U i , (48)

P ζ ( k ref ) = H i 2 π ε i M P 2 ≈ λ h 2 ( h i M P ) 4   or   ≈ λ h 12 ξ h ( h i M P ) 4 . (49)

Let us consider the metric version of General Relativity. Note that the measured n s obtains N, r, ξ h h i , ε i , and η i , see Table 2. The measured P ζ ( k ) obtains λ h 1 / 4 h i . Together they determine λ h / ξ h 2 , ρ i , ρ f , H i , and H_f. Finally, H_i and k obtains a i , and, with N, obtains a f . The resulting a i and N, with their uncertainties, obtains the green closure box in Figure 1. Note that a different measured P ζ ( k ref ) would shift the green box. Results for the Palatini version of General Relativity are listed in Table 3.

The preceding results, with the interchange S ↔ h , are also valid for inflation driven by S instead of ϕ . We note that v h and M S have no significant effect on inflation.

The Higgs quartic coupling λ h runs from λ h ( m t ) ≈ 0.126 at m t to

λ h ( M P ) = − 0.0143 − 0.0066 ( m t GeV − 173.34 )       + 0.0018 α 3 ( M Z ) − 0.1184 0.0007 + 0.0029 ( m H GeV − 125.15 ) , (50)

at M P [17] . Note that, within 3σ uncertainties, λ h ( M P ) may be negative, zero or positive! The relation λ h ( M P ) ≈ 0 is not a prediction of the Standard Model since it depends on input parameters to the model, e.g. λ h , v h and y t . Either λ h ( M P ) ≈ 0 is a coincidence, or some unknown high energy extension of the Standard Model relates these parameters. Since λ h ( M P ) may be positive up to M P , the possible need for new physics is pushed to very high energies. Note that the Higgs mass m H = 2 λ h v h has relatively little running [17] [18] , so v h 2 running compensates λ h running. β λ also vanishes around the Planck mass. “Near-criticality holds for both the Higgs quartic and the top Yukawa coupling” [17] .

From [17] and [19] we conclude that Electro Weak Symmetry Breaking (EWSB) at T EWSB = 159.5 ± 1.5   GeV [19] is not due to v h 2 running to zero at this temperature, but to a fraction of the Higgs bosons excited to above the potential

barrier ( 1 4 λ h ) 1 / 4 v h = m H / ( 16 λ h ) 1 / 4 . We note that this potential barrier grows

with the renormalization energy scale μ slower than the temperature T = μ , which allows the restoration of the electroweak symmetry U ( 1 ) Y × S U ( 2 ) L at T > T EWSB .

The renormalization energy scale at which λ h crosses zero depends critically on the Yukawa coupling y t , on the SU(3) gauge coupling g 3 , and, to a lesser extent, on g 2 . During inflation the Higgs boson does not have time to become “dressed”, so we should use λ h ( M P ) (not λ h ( m t ) ) which depends critically on m t , g 3 , and m H , and may even be zero or negative, and so, Higgs inflation may break down.

The renormalization group equations for the Standard Model extended with a scalar field coupled to the Higgs boson, have been studied in [20] [21] and [22] . Our conclusions, for the warm dark matter extension of the Standard Model studied in this article, with the expected λ h S of order 10⁻⁸ to 10⁻⁷, is that the running of Standard Model parameters is not significantly changed, while λ S ( μ ) increases with μ , and so does not become zero. For example, if we set λ S ( M Z ) = 0.126 , we obtain λ S ( M P ) ≈ 0.5 .

If decays of the inflaton field can be neglected during inflation, then T 3 a 3 is constant, and the temperature is exponentially small, or zero, at the end of this “cold” inflation. A “reheating” period is needed after the end of cold inflation at a f in order to start the radiation-dominated hot big-bang cosmology. After inflation “the field oscillates about a potential minimum and decays. How this occurs is unknown in any detail.” [23] Steven Weinberg makes a similar comment [14] . This oscillation and decay is known as “preheating”. For a potential V ( h ) ∝ h n , the equation of state is w ≡ p / ρ = ( n − 2 ) / ( n + 2 ) [24] . So, while the oscillation is dominated by the quartic potential, the equation of state is w = 1 / 3 , as for radiation. If the potential becomes quadratic, w = 0 , and the inflaton field expands as matter (but this will not be the case in the present scenario).

Assume S is the inflaton field. During preheating S oscillates anharmonically with angular frequency ω ( t ) [25] (defined as 2π divided by the period), and

ρ = H 2 3 M P 2 8 π = 1 ( 2 t ) 2 3 M P 2 8 π = ρ f ( a f a ) 4 ≈ ω 2 S peak 2 2 , (51)

ρ ≡ π 2 30 g reh T 4 ≈ π 2 30 107.75 ( 0.332 T 0 a ) 4 , (52)

where T is a pseudo-temperature since thermal equilibrium has not yet been reached. Note that in the metric version of General Relativity a f and ρ f are obtained from measurements of n s and P ζ ( k ) , and so all parameters are known during preheating. Somehow we need to create particle pairs with opposite momenta ω and − ω . To this end we need density and/or curvature perturbations that couple to the oscillating inflaton field.

Local thermal and chemical equilibrium needs to be achieved before the temperature T drops below T BBN ≈ 1 MeV to not upset the predictions of Big Bang Nucleosynthesis (BBN) [1] . Thermal and chemical equilibrium between S and h needs to be achieved before T drops below m_H, so that dark matter agrees with observations [4] .

For more general presentations see [27] and [26] . Here we will study two channels that need to bring warm dark matter S into thermal and diffusive equilibrium with the Higgs boson before the temperature drops below m_H. This is a requirement of cosmological observations [4] .

We have extended the Standard Model with a scalar warm dark matter field S that couples to the Higgs boson ϕ . This scenario is in agreement with recent cosmological observations if S and ϕ come into thermal and diffusive equilibrium before the temperature drops below m_H [4] . The extended model has two scalar fields, S and ϕ . Either of these fields may drive inflation. We study several inflation options, and arrive at the following conclusions:

1) S -inflation with quadratic potential: If the field that drives inflation is a

scalar S with a quadratic potential 1 2 m ¯ S 2 S 2 , we can satisfy the observed inflation

constraints n s and P ζ ( k ) with a mass m ¯ S ≈ 1.2 × 10 − 6 M P . This mass corresponds to cold dark matter, and so S would not be the warm dark matter favored by observations.

2) S -inflation with quartic potential: If the field that drives inflation is a scalar

S with a dominating quartic potential 1 4 λ S S 4 , and a non-minimal coupling

− ξ S S 2 R to the curvature scalar R , we can satisfy the observed inflation constraints n s and P ζ ( k ) if λ S / ξ S 2 = ( 4.6 ± 1.1 ) × 10 − 10 for the metric version of General Relativity (see Table 2), or λ S / ξ S = ( 7.6 ± 1.8 ) × 10 − 11 for the Palatini version (see Table 3). Thermal and diffusive equilibrium between S and h is achieved at T = m H if | λ h S | ≳ 10 − 7 . S -inflation does not suffer from critical λ S running at Planck energies. For the metric version of General Relativity, the predicted tensor-to-scalar ratio is r = 0.0037 ± 0.0009 , while it is unconstrained in the Palatini version. In summary, this scenario passes current observational tests.

3) Complex field S : An alternative to the real dark matter field S with Z₂ symmetry, is a complex S with a Lagrangian with U ( 1 ) S symmetry, with the ground state of S and ϕ breaking the U ( 1 ) S and U ( 1 ) Y × S U ( 2 ) L symmetries [10] . In this scenario, S decays to two massive gauge bosons V that are the dark matter. The present analysis is also valid for this complex S field with minor notational modifications, e.g. S 2 → S * S .

4) Higgs inflation: Higgs inflation is similar to S -inflation with quartic potential. The main difference is an uncertain quartic self-coupling λ h ( M P ) at the Planck energy scale, that, within 3σ uncertainties, may even be zero or negative, in which case Higgs inflation breaks down!

5) The mass M_S of S , and the Higgs vacuum expectation value v_h do not play a significant role in inflation.

The closure test presented in Figure 1, obtained from observations of n_s and P ζ ( k ) with no free parameters, is truly amazing!

The author declares no conflicts of interest regarding the publication of this paper.

Hoeneisen, B. (2023) Exploring Inflation Options for Warm Dark Matter Coupled to the Higgs Boson. International Journal of Astronomy and Astrophysics, 13, 217-235. https://doi.org/10.4236/ijaa.2023.133013

The purpose of this appendix is to try to understand the quantum origin of the density perturbations that are observed in the Cosmic Microwave Background radiation, which are the seeds of galaxy formation and of the large-scale structure of the universe. This derivation is outlined in [23] . For rigorous derivations see [1] [13] , or [14] .

Consider the comoving length scale 1 / k that exits the horizon at a i , and re-enters the horizon at a r : k = a i H i = a r H r . See Figure 1. We assume that during inflation, and well within the horizon, the field S has a homogeneous component, plus random quantum fluctuations δ S . We apply periodic boundary conditions to a cube of volume L 3 , and consider the lowest non-zero mode with ω = 2 π / L , and assume that the energy in the fluctuating field in this mode

is 1 2 ω , as in the quantum harmonic oscillator. We neglect masses, so take ω = k .

This energy is L 3 〈 S ˙ 2 〉 k = L 3 ω 2 〈 δ S 2 〉 k , so 〈 δ S 2 〉 k = 1 / ( 4 π L 2 ) . As the universe expands, the box expands as L ∝ a , ω = k ∝ a − 1 and 〈 δ S 2 〉 ∝ a − 2 . When the length scale 1 / k exits the horizon, L = 2 π / k , and H i ≈ 1 / L , so, within a factor of O(1), we obtain the result of [1] for the power spectrum of vacuum field fluctuations at horizon exit a i :

〈 δ S i 2 〉 k = ( H i 2 π ) k 2 . (64)

We now consider a “small” volume well within the horizon. The density and field evolve as (5) and (16), which neglect spatial derivatives. Inflation ends at ε f = 1 , so the density ρ f is independent of δ S . The fluctuation 〈 δ S i 2 〉 k , at a i during slow-roll inflation, causes a spread 〈 δ t f 2 〉 k of the time t f of the end of inflation:

〈 δ t f 2 〉 k = 〈 δ S i 2 〉 k ( d S / d t ) i 2 = 〈 δ S i 2 〉 k ( 3 H i − d V / d S ) 2 (65)

= ( H i 2 π ) k 2 ( 3 H i d V / d S ) 2 = 1 π ε i M P 2 . (66)

(Drawing a graph V ( t ) helps.)

We must now pass from slow roll inflation to expansion with ρ ∝ t − 2 characteristic of radiation (or matter), or of oscillations of S in a quartic potential at the end of inflation. The time fluctuation 〈 δ t f 2 〉 k causes a density perturbation at a fixed time t f ′ :

δ ρ f ′ ρ f ′ = d ρ f ′ / d t ρ f ′ 〈 δ t f 2 〉 k = − 2 t f ′ 〈 δ t f 2 〉 k = − 4 H f 〈 δ t f 2 〉 k . (67)

For ρ ∝ t − 2 , and δ ρ ∝ t − 2 , δ ρ / ρ is independent of t, so δ ρ f / ρ f = δ ρ r / ρ r . Also 3 H f ≈ H i , see Table 2, so, within a factor O(1), we obtain the result of [1] :

P ζ ( k ) ≡ ( 〈 δ ρ r 2 〉 ρ r 2 ) k = ( H i 2 π ε i M P 2 ) k . (68)