Thermal effects in Ising Cosmology

We consider a real scalar field in de Sitter background and compute its thermal propagators. We propose that in a dS/CFT context, non-trivial thermal effects as seen by an 'out' observer can be encoded in the anomalous dimensions of the $d = 3$ Ising model. One of these anomalous dimensions, the critical exponent $\eta$, fixes completely a number of cosmological observables, which we compute


Introduction
The rapidly expanding phase of the universe can be modelled by de Sitter (dS) space and the simplest form of matter by a real scalar.It is believed that basic effects that left an imprint on the Cosmic Microwave Background (CMB) were of thermal nature.Therefore a simple model that could explain some of the observed features of the CMB is a real scalar field ϕ in the expanding Poincare patch of dS space [1,2,3], formulated in the context of thermal quantum field theory [4].The action is which we will quantize taking into account finite temperature effects.Here m is the mass of the scalar field, ξ is the non-minimal coupling of the field to gravity and R the scalar curvature.
Consider a d + 1 dimensional FRW spacetime with metric with τ the conformal time and a(τ ) the scale factor.The expanding Poincare patch of de Sitter space corresponds to a = − 1 Hτ where H = a ′ a 2 the Hubble constant with a ′ = da dτ .The expanding Poincare patch of dS space is parametrized by τ ∈ (−∞, 0].The scalar field mode in d-dimensional momentum space ϕ |k| = χ |k| a in this background yields the classical Klein-Gordon equation of motion (k = (k 0 , k) is the four-momentum and the prime is derivative with respect to τ ) with ω 2 |k| = |k| 2 + m 2 dS and a time-dependent mass given by m 2 dS = 1 τ 2 (M 2 − d 2 −1 4 ).The dS mass parameter is M 2 = µ 2 H + 12ξ with µ 2 H = m 2 H 2 and H the inverse curvature parameter of dS space, satisfying R = 12H 2 .The solutions to Eq. (1.3) are linear combinations of the Hankel function H ν cl (τ, |k|) and its complex conjugate, of weight ν cl , with and as we will revisit later on, ν cl is a part of the classical scaling dimensions of the bulk and boundary operators.
Focusing on a given vacum at a specific time, we can Taylor expand in the usual way the scalar field and calculate the Hamiltonian [2]: where α k , α † k are the annihilation and creation operators that satisfy the commutation relations: In addition, Ω |k| and Λ |k| are defined as where u |k| , u * |k| are the mode functions that pair with the ladder operators α k , α † |k| .The frequency ω 2 |k| will be defined below and it carries the time dependence.One of the main key points of a QFT in a curved spacetime is that there is not a single choice for a vacuum state, while different choices lead to different ladder operators β k , β † k and mode functions that are connected by the Bogolyubov Transformation (BT) [2]: with c |k| , d |k| the Bogolyubov coefficients.Using the commutation relations defined above, one can show in a straightforward manner that the Hamiltonian (1.5) is hermitian for a given ground state defined at a given time τ . 4  The work presented in this paper follows the related work [5] where a possible connection between a bulk dS theory and a boundary Ising model was examined along with its implications to the value of the cosmological spectral index n S .In particular, experiments [6], [7] find that n S deviates slightly from unity which shows that the CMB is nearly scale invariant.Although, it is true that other endeavors to explain this deviation exist (e.g.[8,9]), the idea of using the critical exponent of a boundary Ising field in order to predict cosmological observables is new.Furthermore, it is important to note that the calculations and results of the current paper are model-independent, meaning that we only take for granted the experimental value of n S and the assumption that the inflation era of the Universe can be explained by the expanding Poincare patch of dS spacetime.
In Section 2 we compute thermal propagators in dS spacetime by generalizing methods first produced in flat spacetime (as firmly discussed in [4]) i.e. the Schwinger-Keldysh (SK) path integral and the Thermofield Dynamics (TFD).To our knowledge, while there are cases upon which the SK path integral has been used in dS before (e.g.[10]), the use of the TFD formalism in a general FRW spacetime is novel.Moreover, a plain but non-trivial connection between the two formalisms is presented which can hold for other spacetime choices other than dS and enables one to avoid ambiguities rising in the usual SK construction.In Section 3, we make use of the dS thermal propagators in order to incorporate the thermal corrections that arise when one proceeds to calculate the scalar spectral index n S .In contrast to previous works [11] where the spectral index to leading order was found to be equal to unity, here we argue that thermal effects actually slightly break scale invariance, resulting in n S ̸ = 1.This leads to a parametric freedom which can be fixed by an RG flow argument that has its origins in the d = 3 Ising model and in such a way, we can match the deviation of n S away from unity to the experimental data.Finally, using the newly formed thermal dS propagator, we can extract more cosmological observables determined by the scalar metric fluctuations, namely the running of the spectral index n (1) S and the non-Gaussianity parameter f N L . 4 An analysis concerning the full time range that spans the entire dS space (instead of only its expanding patch that we consider here) is more delicate and Hermiticity maybe lost.

Propagators and temperature
Quantization of this system results in the notion of a time-dependent vacuum state and a doubled Hilbert space.Regarding the vacua, we will be concerned with the so called "in" vacuum defined at τ = −∞ and the "out" vacuum defined at the boundary (i.e. the horizon) of the expanding patch, at τ = 0.These are empty vacua from the perspective of corresponding local (in conformal time) observers.The |in⟩ will be chosen to be the maximally symmetric Bunch-Davies (BD) vacuum [12,13].The two vacua are related via the BT ⟨J| Φ I = ⟨I| Φ J where I, J = in, out is a label of the vacuum and Φ I is the field operator with mode function χ I |k| .Note that the field is the same in both vacua, with the mode functions and the creation and annihilation operators inside it being the vacuum dependent quantities.Common notation is χ in |k| = u |k| and The doubled Hilbert space can be understood in the context of the SK path integral as being related to a + (or forward) branch and a − (or backward) branch in conformal time evolution.The field propagator D in such a basis has a 2 × 2 matrix structure and is (T (T * ) denoting time (anti-time) ordering and ⟨0| is a generic vacuum): and where Note that, since the vacuum state |0⟩ is time dependent, we construct the above formulas without choosing a specific vacuum for now.In addition, there is no need for a time ordered product in Eq. (2.1) because the two fields commute since they are defined in different parts of the SK contour.
The above matrix elements satisfy the relation Hidden in these expressions is the iε shift, implementing the projection on the vacuum at τ = −∞.It can be chosen so that in the flat limit the propagator becomes diagonal with . The above construction of the propagator at zero temperature in dS spacetime has been recently studied in [10].
The thermal generalization of the propagator components in Eq. (2.1) and Eq.(2.2) is our next goal.If the Hamiltonian of the system was time-independent, one could just follow the process described in Appendix A and show that the propagator satisfies the KMS condition [14], which ensures that it is a good thermal propagator.Here however we are dealing with a time-dependent Hamiltonian and this is not straightforward.Instead, we will use the method introduced in [15] that takes advantage of the SK contour, by adding an extra,"thermal" leg to it.In particular, if C + is the forward branch where time evolution follows the path τ in → τ out , C − is the backward branch where τ out → τ in , we attach an extra part to the contour C 3 , where τ in → τ in − i β 2 and β = 1/T is the inverse temperature parameter: Furthermore, we introduce the propagators where Φ 3 (τ ) is the field operator living on C 3 and τ 1 , τ 2 ∈ C and we demand that the junction conditions for a ∈ {+, −, 3}: are satisfied at the time instance τ = τ out where the C + and C − contours meet, while the conditions need to be satisfied at τ = τ in where C − and C 3 meet.Finally for the SK analogue of the KMS condition to hold, we need to sew together C + and C 3 which results in the conditions (2.7) that ensure the consistency of the deformed contour and yield a good thermal propagator.
The above conditions will introduce corrections of thermal nature into the propagators Eq. (2.1) and (2.2), which we compute by making two assumptions.Since the chosen contour allows for an imaginary time flow, we assume that there is no inflation in that direction.This means that the mode functions living on the C 3 leg of the contour can be taken to have a plane wave form.In addition, at τ = τ in we assume the BD vacuum so that the mode functions are expressed in terms of the Hankel functions of ν cl = 3/2 order.According to these assumptions, the solution to the conditions results in the in-in thermal propagator components [15]: with n B the Bose-Einstein distribution parameter We can express conveniently this propagator collectively in a matrix notation as: with and the parametrization s(β/2) ≡ sinh θ |k| (β/2) = n B (β/2) and c(β/2) ≡ cosh θ |k| (β/2). 5It is easy to see that this thermal propagator satisfies a condition like Eq. (2.3).
Here we are actually interested in the out-out thermal propagator.We will first derive the result using a novel shortcut and then we will show that it indeed yields the correct result.The shortcut uses the TFD formalism, where the doubled Hilbert space is seen as the tensor product of the Hilbert spaces of positive and negative momenta H and H.The fields living in these Hilbert spaces are Φ and Φ correspondingly.The validity of this strategy is based on the fact that the SK structure can be read also as a TFD structure, in which case the passage to finite temperature is via the transformation That this is an allowed operation on dS propagators is supported by the fact that a transformation by the matrix . Hence, we essentially calculate the thermal corrections that the BT has on the 5 The flat limit of this propagator is diagonal and its ++ component is such that the iε shift of the zero temperature propagator denominator becomes iE = iε coth(βω |k| /2) in the thermal state.
propagator via the TFD formalism.The result of the rotation gives the out-out thermal propagator One immediately notices that the two expressions in Eq. (2.10) and Eq.(2.12) disagree in the thermal correction, as the latter has an extra term along sinh θ |k| cosh θ |k| .This might seem troublesome at first, however they both contain the same physical information.Taking advantage of the trivial identity the propagators in Eq. (2.10) and Eq.(2.12) are seen to be equal for β ′ = β.Note that the above identity does hold in the sinh θ |k| and cosh θ |k| parametrization, where it reads s 2 (β)+s(β)c(β) = s 2 (β/2).We have therefore proved that the known form of the dS thermal propagator of [15] can be equivalently obtained via a TFD rotation of the zero temperature SK propagator of the half thermal parameter.The equivalence of the two expressions reflects of course the universal nature of the dS temperature as measured at an arbitrary time instance by the in and out observers.The advantage of the TFD rotation operation is that it is very simple and can be easily generalized to any background.Thus, we will use this point of view in the following.
The result of all allowed thermal transformations of D are correlators of the form The doublet field, now in the language of TFD, is (Φ I ) T = (Φ I , ΦI ) and γ is a thermal index, associated with any combination of thermal transformations of the form Eq. (2.11).The label (not index) I on the field is a reminder of the vacuum state to which the mode functions belong.
The two types of thermal transformations that are relevant to us are the insertion of an explicit density matrix, resulting in a transformation by a unitary operator U , as |I; β⟩ = U |I⟩, where the eigenvalue of U is U β (θ) and the Gibbons-Hawking (GH) effect [17] (for which we will momentarily use the parameter δ to distinguish it from β) that is expressed as |I⟩ = |J; δ⟩ with I ̸ = J.But the only temperature that dS space can sustain is the GH temperature which means that 1/β dS = T dS = H/2π = 1/δ.It is then sufficient to know the form of the thermal dS-scalar propagator for some generic temperature and then set β = β dS .

The spectral index with thermal effects
The propagators in Eq. (2.10) and Eq.(2.12) determine several important observables.At equal space-time points and at the time of horizon exit, defined as |τ |H = 1 and concentrating on horizon exiting modes specified by |kτ | ≲ 1, they determine various cosmological indices derived from the thermal scalar power spectrum [11] (here 1 is the 2 × 2 matrix with unit elements) in terms of a single parameter (when the temperature takes its natural value T = T dS ): where ω |k| is the frequency defined under Eq.(1.3).This parameter can be traded for the weight of the Hankel function, as determined by the Klein-Gordon equation, in Eq. (1.4).Of special importance in d = 3 is the choice M = 0, or κ = i, which is known to generate a scale invariant CMB spectrum.This corresponds to ν cl = 3 2 and decaying modes at the time of exit.A particularly useful point of view [18] is to recognize the system at τ = −∞ as related to a UV Conformal Field Theory (CFT) labeled by the weight ν cl and associated with the Gaussian fixed point of the d = 3 real scalar theory, that flows towards an interacting IR fixed point and the corresponding CFT at τ = 0.It is clear that in the present context, exact scale invariance is realized in the |in⟩ vacuum, with the deviations generated by a spontaneous shift in M that, according to Eq. (2.12), should have a finite temperature origin.Deviations can be encoded in general in a shift of the weight ν cl → ν = ν cl + ν q that can be interpreted as a shift in the scaling dimension of a dS scalar field There is a corresponding shadow partner solution to this with ∆ + = d 2 + ν.In this letter, we will be concerned with (∆ − , ∆ + ) cl = (0, 3).
In order to understand ν q (which will turn out to be a non-trivial zero) we first point out that the |out; β⟩ (β > β dS ) state is a BT of the Bunch-Davies vacuum.The mode functions before and after the transformation solve the same Bessel equation with frequency ω |k| .Upon a time-dependent BT however, the frequency that an observer sees for a time other than his own, is [19]: As a result, the horizon exit parameter is transformed as where we have defined the dimensionless temperature parameter x = πH 2πT , that takes values in [π, ∞].The transformed state in general has a reduced isometry with respect to the Bunch-Davies state.This can be seen by the fact that the BT introduces a non-zero mass term and that the late time equations of motion have no non-trivial solution with H = const.and a non-zero, finite mass term.
The two limiting values of x are interesting.Its natural value x = π where T = T dS gives Λ = ∞ for κ = i.This is a special case where we recover a dS solution of maximal isometry that corresponds to |out; β dS ⟩.As in the BD vacuum, no modes are seen to exit the horizon, this time due to their ultra-short wavelength.In the limit x → ∞ on the other hand, the out observer sees modes of any wavelength as exiting modes, since in this limit the time of exit approaches the horizon.This means that if he calls his frequencies Ω |k| , then his horizon exit parameter will be forced to Λ 0 ≡ lim τ →0 (Ω |k| τ ) → 0.6 This suggests to construct a trajectory from (Λ, x) ∼ (∞, π) to (0, ∞) along which the value of some yet to be defined thermal effect is kept non-zero and constant, starting from a position a bit shifted away from the scale invariant limit (∞, π).Deviations from exact dS isometry due to finite temperature effects can be encoded in the shift of the spectral index of scalar curvature fluctuations [20] n S,β = 1 + d ln |k| 3 P S,β where P S,β is the thermal scalar power spectrum defined in Eq. (3.1).
In the previous section, we showed that the SK and TFD formalisms result to equivalent propagators.Consequently, from Eq. (3.1) they both determine the same thermal deviation of n S away from unity.Observe that in |out; β dS ⟩ where x = π and Λ = ∞, δn S vanishes and we see a scale invariant spectrum.Moving a bit away from it, x ≳ π,7 the state is |out; β⟩ and δn S becomes a one-parameter expression of Λ.We can fix this freedom by determining the value n S,β by interpreting its deviation from unity as an anomalous dimension in the dual field theory in the spirit of the dS/CFT correspondence.Then we can reach x = ∞ along a trajectory which keeps this value constant for all temperatures.
In [5] it is proposed that within the dual field theory that lives on the horizon, the anomalous dimension that shifts the spectral index is the critical exponent η, whose non-perturbative value is around 0.036.Thus, near the horizon while the experimentally measured value is equal to [6]:

.10)
A known fact about the dual field theory of dS is that it is expected to be non-unitary [8].Thus one could argue that the Ising model (or its large N relatives) which is unitary, is not a good candidate as the dual to dS theory.The suggestion made here is that the field theory dual to the AdS version of the model under discussion -which could be in the universality class of the Ising model (or its large N relatives)-is an analytic continuation of dS.This is complemented 0.01 1600 0.5 14.8 x Z e X S e u s 6 p 9 X a 7 e 1 S v 0 q j 6 M I G y S H 3 B e P w A 4 T p b n < / l a t e x i t > |out; i < l a t e x i t s h a 1 _ b a s e 6 4 = " R q h X i U F j m r n 6 S b 4 R j D 0 s e 3 w 3 i P A = " > A A A B 7 X i c d V D L S g M x F M 3 4 r P V V d e k m W A R X w 0 w 7 r X V X 6 s Z l h b 6 g H U o m z b S x S W Z I M k I Z + g 9 u X C j i 1 v 9 x 5 9 + Y a S u o 6 I E L h 3 P u 5 d 5 7 g p h R p R 3 n w 1 p b 3 9 j c 2 s 7 t 5 H f 3 9 g 8 O C 0 f H H R U l E p M 2 j l g k e w F S h F F B 2 p p q R n q x J I g H j H S D 6 X X m d + + J by the fact that Eq. (3.9) is invariant under an analytic continuation (see [5] for a more detailed justification).This is a constraining statement that leaves no free parameters.In [5] it is shown that the quantity by which ∆ +,cl shifts is the operator anomalous dimension of the trace of the Ising stress energy tensor Θ, which is an exact zero, realized as the cancellation Γ Θ = γ O − 2γ σ = 0, where γ O is the "total" operator anomalous dimension and γ σ is the field anomalous dimension, or the so called wave function renormalization.It is therefore in this sense that ν q is a nontrivial zero, being related to the vanishing anomalous dimension of a special operator that is the energy-momentum tensor.In [5] it is also demonstrated that it is the total anomalous dimension γ O that ends up shifting the spectral index n S .Of course, outside the fixed point where the Ising field is massive, M deviates from zero in the bulk and the solution to Eq. (3.6) is not dS.It is important however to understand that the main effect on n S comes from the critical value η of 2γ σ and the deviation from the critical value is small as long as the system sits in the vicinity of the fixed point.For this reason the leading order results are independent of the source of the breaking.In a sense the only assumption here is that there is a mechanism of spontaneous breaking of scale invariance.From the point of view of the boundary this could be for example justified as some sort of a Coleman-Weinberg mechanism.

Line of constant physics and other observables
A line of constant physics (LCP) is a set of points on the phase space upon which the value of a physical quantity remains fixed.What we will demonstrate now is that in the bulk, there is a LCP, labelled by the fixed value δn S = −η, along which the system is heated up from zero temperature where Λ 0 = 0 and x = ∞, up to the dS temperature.A few points on this line and a picture of the LCP can be found in Fig. 1.We stress that for a given x the corresponding value of Λ is fixed by the label of the LCP.Thus near the endpoint of the LCP where x ≃ π, the value Λ π ≃ 1.5117 is a fixed output.It is important to emphasize that the LCP is really meaningful up to just outside its two limiting points.Up to around x ≃ π it is characterized by a non-zero δn S which however at exactly x = π becomes equal to zero, since the trace of the boundary stress-energy tensor to which the bulk scalar couples, vanishes.Analogously, the interpretation of each point on it as a dS space of the same T dS is possible everywhere except at x = ∞, where the intrinsic temperature must become abruptly unobservable.
Since there are no free parameters, several other observables that are determined by P S,β are expected to be also fixed.Define for example the moment n and let us compute it using that n S = 0.The result, evaluated under the same conditions as n S,β , is which, substituting x ≃ π and Λ = Λ π ≃ 1.5117, gives n S,β = 0.0186 (4.3) for the running of the index.The constraints given in [6] are: n S,exp = 0.013 ± 0.012.
Finally, the universal contribution to the non-Gaussianity parameter [21], can be expressed in terms of N = t f t i dtH and its derivatives in the in-vacuum, as [22] f un N L = 5 6 For x ≃ π and Λ = Λ π ≃ 1.5117 this gives while one of the experimental results for f un N L in a certain analysis is [7]:

Conclusion
We considered a thermal scalar in de Sitter background.Starting from the Bunch-Davies |in⟩ vacuum, a Bogolyubov Transformation placed us in the interior of the finite temperature phase diagram in a thermal state |out; β⟩.This state can be connected through holography to the vicinity of an interacting IR fixed point, in the universality class of the 3d Ising model.The system in this state is rather special, in the sense that the boundary operator that couples to the scalar curvature perturbations in the bulk has a classical scaling dimension.The critical exponent η is the order parameter of the breaking of the scale invariant spectrum of curvature fluctuations and a simple argument from the dS/CFT correspondence fixes the parametric freedom in the dS scalar theory, yielding the prediction n S = 0.964.We also computed in the same context additional cosmological observables such as the first moment of the scalar spectral index and the non-Gaussianity bispectrum parameter f N L and evaluated them numerically.Our predicted values of n S , n S,β and f N L are well within current experimental bounds [6,7].A thermal propagator has to satisfy a variant of the KMS condition.The KMS condition originates from the definition ⟨ϕ(t 1 , x 1 )ϕ(t 2 , x 2 )⟩ β = Tr {ϕ(t 1 , x 1 )ϕ(t 2 , x 2 )ρ} Tr{ρ} (A.4) that leads, in principle, to the thermally corrected dS propagator.However for time dependent Hamiltonians the direct computation of the trace is not obvious.
The condition takes a simple form though near τ → −∞, which we can show explicitly.In Thermofield Dynamics, the form of the KMS condition depends on a gauge parametrized by a real number, say α.It is a well known fact that TFD propagators satisfy such a condition in any of these α-gauges [23,24].The condition holds due to the relation [16]  the KMS condition in the α = 1 gauge.Note that this is a relation where the usual form of the KMS condition of thermal field theory can be recognised.These two gauges however can be readily seen to correspond to equivalent Wightman functions, as they can be related by a shift in the imaginary time, by τ 1,2 −→ τ 1,2 − i β 2 .By this shift freedom, one can also see that the diagonal elements of the propagator do not satisfy any non-trivial constraint.In conclusion, to the extent that Eq. (A.4) applies to dS space and the trace is computable, the thermal propagator it defines satisfies a KMS condition.

Figure 1 :
Figure 1: Left: A few points of the nearly conformal LCP defined by δn S = −η.Right: The Bogolyubov Transformation |in⟩ → |out; β⟩ and the LCP, on the complex plane where κ = Λ + i Imκ.