The Lambda CDM-model in quantum field theory on curved spacetime and Dark Radiation
aa r X i v : . [ g r- q c ] J un The Λ CDM-model in quantum field theory on curved spacetime and Dark Radiation
Thomas-Paul Hack ∗ Dipartimento di Matematica, Universit`a degli Studi di Genova, I-16146 Genova, Italy. (Dated: June 14, 2013)In the standard model of cosmology, the universe is described by a Robertson-Walker spacetime,while its matter/energy content is modeled by a perfect fluid with three components correspondingto matter/dust, radiation, and a cosmological constant. On the other hand, in particle physicsmatter and radiation are described in terms of quantum field theory on Minkowski spacetime. Weunify these seemingly different theoretical frameworks by analysing the standard model of cosmologyfrom first principles within quantum field theory on curved spacetime: assuming that the universe ishomogeneous and isotropic on large scales, we specify a class of quantum states whose expectationvalue of the energy density is qualitatively and quantitatively of the standard perfect fluid form up topotential corrections. Qualitatively, these corrections depend on new parameters not present in thestandard ΛCDM-model and can account for e.g. the phenomenon of Dark Radiation ( N eff > . PACS numbers: 04.62.+v, 98.80.-k, 95.36.+x, 95.35.+d
I. INTRODUCTION
Quantum field theory on curved spacetime (QFT onCST, see e.g. the monographs and reviews [7, 8, 10,32, 54, 59, 80]) is a framework in which matter is mod-elled as quantum fields propagating on a classical curvedspacetime which is treated according to the principles ofGeneral Relativity. As such, QFT on CST is the sim-plest and natural generalisation of quantum field theoryon Minkowski spacetime which takes into account thatonly in certain regimes a flat Minkowski spacetime is agood description for our universe. Taking the spacetimeas being classical, QFT on CST is presumably only validin situations where the spacetime curvature scale is belowthe Planck scale and thus QFT on CST is the ideal frame-work for describing physics in the regime of medium-sizedspacetime curvature, e.g. in the vicinity of black holes orin the early universe.Indeed, quantum field theory on curved spacetime is animportant ingredient in modern theoretical cosmology, asquantum fluctuations of the scalar field(s) responsible forthe rapid expansion of the universe in the scenario of In-flation are believed to be the seeds of the structures in ourpresent universe, see e.g. [53, 74] and the recent works[22, 63], and the analysis of these fluctuations is donein the framework of QFT on CST. However, other as-pects of theoretical cosmology are usually not dealt withwithin QFT on CST, but simplified and less fundamentaldescriptions are used.According to the Standard Model of Cosmology – theΛCDM-model – our universe contains matter, radiation,and Dark Energy, whose combined energy density de- ∗ Electronic address: [email protected] termines the expansion of the universe, see for exam-ple [21, 53]. In the ΛCDM-model, these three kinds ofmatter-energy are modelled macroscopically as a perfectfluid and are thus completely determined by an energydensity ρ and a pressure p , with different equations ofstate p = p ( ρ ) = wρ , w = 0 , , − II. QUANTIZATION OF THE KLEIN-GORDONFIELD ON GENERAL CURVED SPACETIMES
We first review the quantization of a free neutral scalarfield φ on a general curved spacetime before analysing thecase of cosmological spacetimes in more detail. Here, acurved spacetime is a tuple ( M, g ), where M is a four-dimensional smooth (i.e. infinitely often differentiable)manifold and g is a smooth metric on M with signature(+ , − , − , − ); we shall often abbreviate ( M, g ) by M forsimplicity. The classical neutral scalar field is charac-terised by satisfying the Klein-Gordon equation
P φ := (cid:0) (cid:3) + ξR + m (cid:1) φ = 0 , (1) (cid:3) := ∇ µ ∇ µ , where ∇ is the Levi-Civita covariant derivative associatedto g , R is the corresponding Ricci curvature scalar, ξ denotes the coupling of φ to R and m is the mass ofthe scalar field. For technical reasons one often assumesthat the spacetime ( M, g ) is globally hyperbolic in orderto guarantee that (1) has unique solutions for any initialconditions given on an equal-time surface, see e.g. [7,32] for details; the Robertson-Walker spacetimes used incosmology belong to this class.In Minkowski spacetime, (time) translation invarianceis an important ingredient in the usual construction ofQFTs: first of all it is necessary in order to have awell-defined notion of energy ; given this, one can de-fine a vacuum state Ω and the associated notions of creation/annihilation operators , particles and Fock space by requiring that all excitations of the vacuum have pos-itive energy. General curved spacetimes, e.g. cosmologi-cal spacetimes, are not invariant under time-translationsand thus a unique notion of vacuum does not exist. Onemay be able to find several candidates for generalisedvacuum states, as we will also discuss in the following,but choosing any two of them, say Ω A and Ω B , onetypically finds because of the infinitely many degrees offreedom of a quantum field that Ω A contains infinitelymany particles w.r.t. Ω B , s.t. the two states are notunitarily equivalent. In order to discuss the quantizationof the Klein-Gordon on arbitrary curved spacetimes itthus seems advisable to consider a more fundamental ap-proach to quantization, the so-called algebraic approach [31]. In this framework, one first defines an abstract al-gebra A which encodes all fundamental algebraic rela-tions of the quantum fields, e.g. equations of motion andcanonical (anti)commutation relations, and then analy-ses the possible states, i.e. the possible representationsof A as operators on a Hilbert space.In the case of the neutral scalar field, the algebra A KG encodes the Klein-Gordon equation (1), the reality con-dition φ ( x ) ∗ = φ ( x ) and the canonical commutation re-lations ( ~ = 1) [ φ ( x ) , φ ( y )] = iG ( x, y ) (2)where G ( x, y ) = G − ( x, y ) − G + ( x, y ) is the Pauli-Jordan commutator function constructed from the ad-vanced/retarded Green’s functions G ± ( x, y ) which sat-isfy P x G ± ( x, y ) = δ ( x, y )and vanish for x in the past/future of y . This covariant form of the canonical commutation relations is equivalentto the often used form where commutation relations ofthe field and its canonically conjugate momentum arespecified at equal times, see e.g. [7, 32, 80] for this andfor a more detailed account of the quantization of theKlein-Gordon field on a curved spacetime in the algebraicframework.As already anticipated, many unitarily-inequivalentHilbert space representations of A KG exist and, even ifin general none of them is preferred, it would be nice tohave criteria in order to select those which are of phys-ical relevance. As the vacuum state Ω in Minkowskispacetime is pure and Gaussian (quasi-free), it seems ad-visable to concentrate on these states as well in curvedspacetimes; they are completely characterised in terms oftheir two-point correlation functionΩ( x, y ) := h φ ( x ) φ ( y ) i Ω in particular. Note that the algebraic approach toQFT on CST is certainly powerful enough to treat non-Gaussian states with ease, we just concentrate on theGaussian ones here in order to initially maintain as manyproperties of Ω as possible. To arrive at further physicalconstraints on Ω, we consider the functional propertiesof the vacuum two-point function Ω ( x, y ) in Minkowskispacetime. Indeed Ω ( x, y ) is divergent if x and y areconnected by a light-like geodesic and the exact formof this singularity is important for assuring that all ex-pressions appearing in Wick’s theorem for the productof normal-ordered quantities are well-defined. Using theseminal results in [64, 65], the authors of [11, 12] showedthat the correct generalisation of this singular structureto curved spacetime is the so-called
Hadamard condition [42], which is satisfied by a state Ω if its two-point func-tion is of the formΩ( x, y ) = H ( x, y ) + W Ω ( x, y ) H ( x, y ) := 18 π (cid:18) U ( x, y ) σ ( x, y ) + V ( x, y ) log (cid:18) σ ( x, y ) λ (cid:19)(cid:19) where σ ( x, y ) is the half squared geodesic distance be-tween x and y [81], λ is an arbitrary length scale and U ( x, y ), V ( x, y ) and W Ω ( x, y ) are smooth functions. V ( x, y ) can be expanded w.r.t. to σ ( x, y ), viz. V ( x, y ) = ∞ X n =0 V n ( x, y ) σ n ( x, y ) , and U ( x, y ) and all V n ( x, y ) can be completely specifiedin terms of (derivatives of) the metric g , and the pa-rameters ξ and m appearing in (1) in a recursive man-ner. Thus, the Hadamard singularity H ( x, y ) is universalamong all Hadamard states Ω and only the regular partof their two-point function W Ω ( x, y ) depends on Ω. Ingeneral it is not possible to prove the convergence of theseries V ( x, y ), but while this is dissatisfactory from thethe conceptual point of view, it is not relevant for thecomputation performed in the following, as there in thelimit x → y all but a finite number of terms in the seriesvanish.We have briefly reviewed the algebraic approach toquantization and the notion and physical and mathemat-ical properties of Hadamard states only for the free neu-tral scalar field, but these concepts have been developedfor fields of higher spin, gauge fields and interacting fieldsas well. For this and further details we refer the readerto e.g. [12, 15, 16, 25, 27, 30, 32, 33, 38, 67, 68]. III. THE SEMICLASSICAL EINSTEINEQUATION
The starting point for the understanding of theΛCDM-model within quantum field theory in curvedspacetime is the semiclassical Einstein equation G µν = 8 πG h : T µν : i Ω , (3)where G µν is the Einstein tensor, G is Newton’s constant,: T µν : is the regularised stress-energy tensor of all quan-tum fields in the model, and Ω is a suitable state. Thereare several conceptual issues related to this equation ofwhich we would like to mention a few in the following.First of all in (3) one equates a classical quantity witha probabilistic one, which makes sense only if the fluc-tuations of the latter are small. From the mathematicalproperties of Hadamard states it follows that the fluctu-ations of (a suitably regularised) : T µν : are finite in anyHadamard state Ω [11], whereas a discussion of the actualsize of the fluctuations can be found in [61]. However, asalready mentioned it is not at all clear how to pick apreferred state among the class of all Hadamard states,and in fact writing down the semiclassical Einstein equa-tions implies that we are able to specify a map Ω( M )which assigns to each spacetime M a Hadamard state ina coherent manner. Unfortunately, it has been explicitlyproven in [26] that this is impossible if one wants to dothis for all (globally hyperbolic) spacetimes. However, apossible way out is to restrict the allowed class of space-times. Indeed, as we will review in the next section, it ispossible to assign coherently a Hadamard state to eachRobertson-Walker spacetime.The next conceptual issue we would like to mention isthe actual definition of h : T µν : i Ω or : T µν : respectively. Inusual particle physics experiments we always measure the difference of the expectation value of : T µν : in two states,e.g. the vacuum and a many-particle state. However,gravity is sensitive to the absolute value of h : T µν : i Ω ,thus the unambiguous specification of h : T µν : i Ω corre-sponds to a specification of a ”zero point” in the absoluteenergy scale, but this is impossible within quantum fieldtheory in curved spacetime. In more detail, one couldask the question: what is the most general expression for: T µν : which is compatible with all physical consistencyconditions I can impose? The conditions one could im-pose are: a) correct commutation relations with otherobservables, b) covariant conservation ∇ µ : T µν := 0, c): T µν ( x ) : should be a local object and depend only on x in a suitable sense, but a state is non-local on account ofthe equations of motion; thus, the observable : T µν ( x ) :should be defined in a state-independent manner. In-deed, one can show that there is no unique expressionwhich satisfies all these conditions (and further technicalones) [36, 52, 77]. The most general expression for theexpectation value h : T µν : i Ω turns out to be h : T µν : i Ω = h : T µν : i Ω + α g µν + α G µν + α I µν + α J µν . (4)Here, α and β can be interpreted as a (renormalisa-tion of) the cosmological constant and a renormalisationof Newton’s constant, whereas I µν and J µν are conservedlocal curvature tensors which contain fourth derivativesof the metric [36, 52, 78] and are obtained as func-tional derivatives with respect to the metric of the La-grangeans √− gR and √− gR µν R µν respectively; thesehigher-derivative contributions are usually ruled out inclassical General Relativity but one can show that theycan not be avoided in QFT on CST [78]. Moreover, a”model” h : T µν : i Ω is [52] h : T µν : i Ω := lim x → y (cid:18) D µν − g µν P x (cid:19) (Ω( x, y ) − H ( x, y ))(5) D µν := (1 − ξ ) g ν ′ ν ∇ µ ∇ ν ′ − ξ ∇ µ ∇ ν − ξG µν + g µν (cid:26) ξ (cid:3) x + (cid:18) ξ − (cid:19) g ρ ′ ρ ∇ ρ ∇ ρ ′ + 12 m (cid:27) . Here g ν ′ ν denotes the parallel transport of a vector from x to y along the geodesic connecting x and y , the formof D µν follows directly from the classical stress-energytensor of the scalar field, viz. T µν = (1 − ξ ) ( ∇ ν φ ) ∇ µ φ − ξφ ∇ µ ∇ ν φ − ξG µν φ + g µν (cid:26) ξφ (cid:3) φ + (cid:18) ξ − (cid:19) ( ∇ ρ φ ) ∇ ρ φ + 12 m φ (cid:27) and the modification term − g µν P x is necessary in orderto have a covariantly conserved h : T µν : i Ω [52]. Notethat h : T µν : i Ω is not unambiguously defined itself as itdepends on the length scale λ in H ( x, y ). Indeed, if onechanges the length scale λ in H ( x, y ) as appearing in (5)to a new value λ ′ , then h : T µν : i Ω changes by2 log λ/λ ′ π (cid:18) (6 ξ − m G µν − (6) − m g µν I µν − J µν − (6 ξ − I µν (cid:19) . The parameters α i are free parameters of the theorywhich are independent of the field content and the space-time M and can in principle be fixed by experiment, justlike the mass m . In the following we will take the pointof view that α is not a free parameter because New-ton’s constant has been measured already. In order to dothis, we have to fix a value for the length scale λ in theHadamard singularity H ( x, y ), we do this by confining λ to be a scale in the range in which the strength of grav-ity has been measured. Because of the smallness of thePlanck length, the actual value of λ in this range doesnot matter as changing λ in this interval gives a negli-gible contribution to h : T µν : i Ω . Moreover, in the case ofconformal coupling ξ = , which we shall assume mostof the time, α is independent of λ as one can infer from(6). One could also take a more conservative point ofview and consider α to be a free parameter, in this casecomparison with cosmological data, e.g. from Big BangNucleosynthesis, would presumably constrain α to bevery small once λ is in the discussed range.Thus, we are left with three free parameters in thedefinition of h : T µν : i Ω : one of them corresponds to thecosmological constant which is already a free parameterin classical General Relativity, whereas the other two pa-rameters do not appear there and thus will by themselveslead to an extension of the ΛCDM-model.To close this section, we would like to highlight thepoint of view on the so-called cosmological constant prob-lem taken in this work, as well as in most works on QFTon CST in the algebraic approach and e.g. the review[9]. It is often said that QFT predicts a value for the cos-mological constant which is way too large in comparisonto the one measured. This conclusion is reached by com-puting one or several contributions to the vacuum energyin Minkowski spacetime Λ vac and finding them all to betoo large, such that, at best, a fine-tuned subtraction interms of a negative bare cosmological constant Λ bare isnecessary in order to obtain the small value Λ vac + Λ bare we observe. In this work, we assume as already men-tioned the point of view that it is not possible to providean absolute definition of energy density within QFT onCST, and thus neither Λ vac nor Λ bare have any physicalmeaning by themselves; only Λ vac + Λ bare is physical andmeasureable and any cancellation which happens in thissum is purely mathematical. The fact that the magni-tude of Λ vac depends on the way it is computed, e.g. theloop or perturbation order, cf. e.g. [71], is considered tobe unnatural following the usual intuition from QFT onflat spacetime. However, it seems more convincing to usto accept that Λ vac and Λ bare have no relevance on theirown, which does not lead to any contradiction betweentheory and observations, rather than the opposite.In the recent work [39] it is argued that a partial andunambiguous relevance can be attributed to Λ vac by de-manding Λ bare to be analytic in all coupling constantsand masses of the theory; taking this point of view, onecould give the contribution to Λ vac which is non-analyticin these constants an unambiguous meaning. Indeed theauthors of [39] compute a non-perturbative and hencenon-analytic contribution to Λ vac , which turns out to besmall. In the view of this, one could reformulate ourstatement in the above paragraph and say that contribu-tions to Λ vac and Λ bare which are analytic in masses andcoupling constants have no physical relevance on theirown. IV. STATES OF INTEREST ONCOSMOLOGICAL SPACETIMES
After the review of the quantum theory of a free scalarfield on general curved spacetimes we shall consider onlycosmological spacetimes in the following. To wit, weassume that the spacetime is given by a spatially flatRobertson-Walker spacetime, i.e. a subset of R withthe metric ds = dt − a ( t ) d~x . The translational and rotational invariance of this metricin the spatial coordinates reflects the paradigm that ouruniverse is homogeneous and isotropic on large scales,while we have chosen a Robertson-Walker spacetimewithout spatial curvature in order to simplify computa-tions and because observations are compatible with theassumptions of vanishing spatial curvature [1].As usual, t is cosmological or co-moving time, whereas a ( t ) is the scale factor whose expansion rate is the Hub-ble rate H := ˙ a/a . Further possible time variables arethe conformal time τ , and, if H is strictly positive (ornegative), the scale factor a itself as well as the redshift z := a /a −
1, where a is the scale factor of today, usu-ally set to a = 1 by convention. These time variablesare related by dt = adτ = daaH = − dz (1 + z ) H .
In the following we shall always assume
H > t ( τ ) shall be denoted by ˙ ( ′ ). Using a or z as time variables is often convenient because it doesnot require the explicit knowledge of a as a function of t or τ . Moreover, the redshift z is a direct observable incontrast to the other time parameters.The goal of this section is to introduce a class of stateswhich we believe to be a good model for the actual quan-tum states that describe the content of our universe onlarge scales, as will be justified in the next section. Asmost of the energy density in the ΛCDM-model is be-lieved to be of thermal origin, we seek states which can beconsidered as generalised thermal states [82]. However,for this we first need to specify a good class of generalisedvacuum states, which is what we shall do now. To thisavail, we recall that a pure and Gaussian isotropic andhomogeneous state for the Klein-Gordon field on a spa-tially flat Robertson-Walker spacetime is determined bya two-point correlation function of the form [49]Ω( x, y ) = 18 π a ( τ x ) a ( τ y ) Z R d~k χ k ( τ x ) χ k ( τ y ) e i~k ( ~x − ~y ) , where the modes χ k satisfy the ordinary differential equa-tion (cid:18) ∂ τ + k + m a + (cid:18) ξ − (cid:19) Ra (cid:19) χ k ( τ ) = 0 (7)and the normalisation condition χ k ′ χ k − χ k χ ′ k = i . (8)Here, k := | ~k | and · denotes complex conjugation. Choos-ing a solution of the above equation for each k amountsto specifying the state.Distinguished states are the adiabatic states introducedin [57]. They are specified by modes of the form χ k ( τ ) = 1 p ω ( k, τ ) exp (cid:18) − i Z ττ ω ( k, τ ′ ) dτ ′ (cid:19) , (9)where ω ( k, τ ) solves a non-linear differential equation in τ obtained by inserting this ansatz into (7) and finding ω ( k, τ ) = f ( ω ( k, τ ) ′′ , ω ( k, τ ) ′ , ω ( k, τ ) , a ( τ ))for a suitable function f . While this ansatz in principleholds for any state, the adiabatic states are specified bysolving the differential equation for ω ( k, τ ) iteratively as ω n +1 ( k, τ ) := f ( ω n ( k, τ ) ′′ , ω n ( k, τ ) ′ , ω n ( k, τ ) , a ( τ ))starting from ω ( k, τ ) = q k + m a + (cid:0) ξ − (cid:1) Ra .Truncating this iteration after n steps defines the adi-abatic states of order n . Note that, while the result-ing modes satisfy the normalisation condition (8) exactly,they satisfy (7) only up to terms which vanish in the limitof constant a or of infinite k and/or m . Thus they consti-tute only approximate states. This can be cured by usingthe adiabatic modes of order n only for the specificationof the initial conditions for exact solutions of (7), see [49].Regarding the UV properties of such defined ‘proper’ adi-abatic states, it has been shown in [40] (for spacetimeswith compact spatial sections) that they are in generalnot as UV-regular as Hadamard states, but that theyapproach the UV-regularity of Hadamard states in a cer-tain sense in the limit of large n . In the following weshall often use the ‘improper’ adiabatic modes of order0, χ ,k ( τ ) := exp( − i R ττ ω ( k, τ ′ ) dτ ′ )) / p ω ( k, τ ). Adi-abatic states have also been constructed for Dirac fields,see [37, 48], and general curved spacetimes [37, 40].A further class of states of interest in cosmology, and infact our candidates for generalised vacuum states, are the states of low energy (SLE) introduced in [55], motivatedby results of [24]. These states are defined by minimisingthe energy density per mode ρ k ρ k := 116 a π (cid:18) | χ ′ k | + (cid:18) ξ − (cid:19) a ℜ χ ′ k χ k ++ (cid:18) k + m a − (cid:18) ξ − (cid:19) H a (cid:19) | χ k | (cid:19) integrated in (cosmological) time with a sampling func-tion f and thus loosely speaking minimise the energyin the time interval where the sampling function is sup-ported. The minimisation is performed by choosing ar-bitrary basis modes χ k and then determining the Bogoli-ubov coefficients λ ( k ), µ ( k ) with respect to these modes,such that the resulting modes of the state of low energyare χ f,k = λ ( k ) χ k + µ ( k ) χ k with λ ( k ) := e i ( π − arg c ( k )) s c ( k ) p c ( k ) − | c ( k ) | + 12 µ ( k ) := p | λ ( k ) | − c ( k ) := 12 Z dtf ( t ) 1 a (cid:18) | χ ′ k | + (cid:18) ξ − (cid:19) a ℜ χ ′ k χ k ++ (cid:18) k + m a − (cid:18) ξ − (cid:19) H a (cid:19) | χ k | (cid:19) c ( k ) := 12 Z dtf ( t ) 1 a (cid:18) χ ′ k + (cid:18) ξ − (cid:19) aχ ′ k χ k ++ (cid:18) k + m a − (cid:18) ξ − (cid:19) H a (cid:19) χ k (cid:19) . [55] only discusses the case of minimal coupling, i.e. ξ = 0 and proves that the corresponding SLE satisfy theHadamard condition for sampling functions f which aresmooth and of compact support in time. However, we shall use these states for the case of conformal coupling ξ = , and, although we do not prove that they satisfythe Hadamard condition, we shall find them to be at leastregular enough for computing the energy density. More-over, it is not difficult to see that the SLE constructionyields the conformal vacuum χ f,k ( τ ) = 1 √ k e − ikτ , and thus a Hadamard state [60], for all sampling func-tions f in the massless case. This demonstrates both thatthe SLE construction for ξ = yields Hadamard states atleast in special cases and that states of low energy deserveto be considered as generalised vacuum states on curvedspacetimes. The SLE construction has recently been gen-eralised to spacetimes with less symmetry in [75].A conceptual advantage of states of low energy isthe fact that they can be consistently defined an allRobertson-Walker spacetimes at once just by specifyingthe sampling function f once and for all (with respectto e.g. cosmological time and a fixed origin of the timeaxis). Thus, they solve the conceptual problem men-tioned in Section III, namely the necessity to specify astate in way which does not depend on the spacetime inorder for the semiclassical Einstein equation to be well-defined a priori.We now proceed to construct the anticipated gener-alised thermal states on the basis of states of low energy.To this avail, we recall a result of [17]: given a pure,isotropic and homogeneous state, i.e. a set of modes χ k ,one can construct generalised thermal states with a two-point correlation function of the formΩ( x, y ) = 18 π a ( τ x ) a ( τ y ) Z R d~k , e i~k ( ~x − ~y ) ×× χ k ( τ x ) χ k ( τ y )1 − e − βk + χ k ( τ x ) χ k ( τ y ) e βk − , with k := q k + m a F . It has been shown in [17] that for the case of confor-mal coupling, special Robertson-Walker spacetimes andparticular generalised vacuum modes χ k on these space-times, these states satisfy certain generalised thermody-namic laws and the Hadamard condition, and one canshow that they satisfy the Hadamard condition on gen-eral Robertson-Walker spacetimes if the pure state spec-ified by χ k is already a Hadamard state by using (slightgeneralisations of) results of [61].We shall assume in the following that the quantumfields in our model are in a generalised thermal state ofthe form as above, with generalised vacuum modes χ k specified by a state of low energy with suitable samplingfunction f . If m >
0, the phenomenological interpreta-tion of these states is that they are the quantum stateof a massive field which has been in thermal equilib-rium in the hot early universe and has departed fromthis equilibrium at the ‘freeze-out time’ a = a F . In themassless case, these states are just conformal rescalingsof the thermal equilibrium state with temperature 1 /β inMinkowski spacetime.The generalised thermal states we use here have beendiscussed also for Dirac fields, see [17]. Moreover, wewould like to mention that several definitions of gener-alised thermal states on curved spacetimes have been pro-posed so far, including almost equilibrium states [47] and local thermal equilibrium states [69, 76]. A comparisonof these different proposals in the context of cosmolog-ical applications would certainly be interesting, but isbeyond the scope of this work. V. COMPUTATION OF THE ENERGYDENSITY
We now approach the first main result of this work, ademonstration that the energy density in the ΛCDM-model can be reproduced from first principles withinquantum field theory in curved spacetime. To this avail,we consider the following setup: we model radiation by aconformally coupled massless scalar quantum field, andmatter/dust by a conformally coupled massive scalarquantum field. We choose conformal coupling also forthe massive scalar field because this simplifies analyticalcomputations a lot and we also found numerical compu-tations to be more stable with this value of non-minimalcoupling to the curvature. Moreover, both quantumfields are assumed to be in generalised thermal equilib-rium states as introduced in the previous section, wherethe state and field parameters β (possibly different val-ues for the two quantum fields), m and a F , as well as thesampling functions f determining the generalised vac-uum states of the two fields, are considered to be un-determined for the time being. Let us stress once morethat there is no principal obstruction for formulating thismodel with more realistic quantum fields of higher spin,we just consider scalar quantum fields for simplicity andease of presentation.An exact computation of the energy density of the twoquantum fields in the generalised thermal states wouldrequire to solve the coupled system – the so-called back-reaction problem – consisting of the quantum fields prop-agating on a Robertson-Walker spacetime, which in turnis a solution of the semiclassical Friedmann equation H = 8 πG (cid:0) ρ + ρ m (cid:1) , (10)where ρ m = h : T m : i Ω m , ρ = h : T : i Ω are the en-ergy densities of the two quantum fields in the respec-tive generalised thermal states and the 00-componentof the stress-energy tensor is here taken with respect to cosmological time t . An exact solution of the back-reaction problem is quite involved, as it requires solv-ing simultaneously the mode equation (7) for all k andthe semiclassical Friedmann equation. Notwithstandingthere have been quantitative numerical treatments of thebackreaction problem, see e.g. [2–6], as well as numerousqualitative treatments including [23], where the backre-action problem in Robertson-Walker spacetimes is set upin full generality from the point of view of the algebraicapproach to QFT on CST, [14], where the same pointof view is considered and the coupled system is solvedexactly for conformally coupled massless scalar quan-tum fields and approximately for massive ones, and [60],where a variant of the backreaction problem is solvedexactly for conformally coupled massive scalar quantumfields in the vicinity of the Big Bang on Robertson-Walkerspacetimes with a lightlike Big Bang hypersurface.However, in this work we follow a simplified strategyin order to avoid solving the full backreaction problem,which is justified in view of our aim. We assume that thetwo quantum fields in our model are propagating on aRobertson-Walker spacetime which is an exact solutionof the Friedmann equation in the ΛCDM-model, i.e. H H = ρ ΛCDM ρ = Ω Λ + Ω m a + Ω r a , (11)where H ≃ − eV denotes the Hubble rate of today,the so-called Hubble constant , ρ ≃ − eV is the en-ergy density of today and Ω Λ , Ω m and Ω r denote respec-tively the present-day fraction of the total energy densitycontributed by the cosmological constant, matter/dustand radiation. For definiteness we consider the samplevalues Ω m = 0 .
30, Ω r = 10 − , Ω Λ = 1 − Ω m − Ω r , ratherthan currently measured values from e.g. the Planck col-laboration [1], because the exact values are not essen-tial for our results. Given this background spacetime,we strive to prove that the field and state parameters ofour model, as well as the SLE sampling functions, canbe adjusted in such a way that the energy density ofthe quantum fields in our model matches the one in theΛCDM-model up to negligible corrections for all redshifts z ∈ [0 , ], i.e. ρ + ρ m ρ ≃ Ω Λ + Ω m a + Ω r a = ρ ΛCDM ρ . Once we succeed to obtain this result, we have clearlysolved the full coupled system in an approximative senseto a good accuracy in particular.In order to compute the quantum energy density ρ + ρ m , we start from (4) and (5). The former equa-tion parametrises the freedom in defining the energy den-sity as an observable, whereas the latter gives a pos-sible “model definition”. The renormalisation freedomfor the energy density is readily computed as g = 1, G = 3 H and J = 13 I = 6 ˙ H −
12 ¨ HH −
36 ˙ HH . (12)In order to compute the energy density for each quan-tum field following from (5), one has to first subtractthe Hadamard singularity from the two-point correlationfunction of the given state and then to apply a suitablebidifferential operator followed by taking the coincidingpoint limit. As the states we consider here are given asintegrals over spatial momenta, it seems advisable to tryto re-write the Hadamard singularity also in this form, inorder to perform a mode-by-mode subtraction and mo-mentum space integral afterwards. This is indeed possi-ble, as elaborated in [14, 18, 23, 61, 69]. The details arequite involved, thus we omit them and present directlythe result. To this avail we follow [18], where results of[69] are used. In [18] only the minimally coupled case ξ = 0 is discussed, but it is not difficult to generalise theresults there to arbitrary ξ .Doing this, we find the following result for the totalenergy density of the massless and massive conformallycoupled scalar fields in the generalised thermal states. ρ + ρ m ρ = ρ m gvac + ρ + ρ m gth + ρ ρ (13)+ γ H H + Ω Λ + δ H H + ǫ J H γ := 8 πGH π Ω Λ = 8 πGα H δ := 8 πGα H ǫ := 8 πGH α + α ) . Here Ω λ , δ and ǫ parametrise the freedom in the defini-tion of the energy density as per (4). The number of freeparameters in this equation has been reduced to three,because I µν and J µν are proportional in Robertson-Walker spacetimes, cf. (12). As already discussed inSection III, we omit the freedom parametrised by δ in the following, as it renormalises the Newton constant and weconsider this to be already given as an external input. Fornow we will also neglect the contribution parametrised by ǫ , as it turns out to be negligible for 0 ≤ ǫ ≪
1; we willanalyse the influence of this new term, which does not ap-pear in the ΛCDM-model, separately in the next section.Thus, for the remainder of this section, Ω λ parametrisesthe residual freedom in the definition of the quantumenergy density. The term proportional to γ , which isalso not present in the ΛCDM-model, appears due to theso-called trace anomaly , which is a genuine quantum andmoreover state-independent contribution to the quantumstress-energy tensor, see e.g. [78]. This term is fixed bythe field content, i.e. by the number and spins of thefields in the model and always proportional to H , bar-ring contributions proportional to J which we preferto subsume in the parameter ǫ . We have given here thevalue of γ for two scalar fields, see Table 1 on page 179 of[10] for the values in case of higher spin. As γ ≃ − and H < H z in the ΛCDM-model for large redshifts,this term can be safely neglected for z < . Finally,the remaining terms in (13) denote the genuinely quan-tum state dependent contributions to the energy den-sities of the two quantum fields. We have split thesecontributions into parts which are already present for in-finite inverse temperature parameter β in the generalisedthermal states, and thus could be considered as contribu-tions due to the generalised vacuum states ( ρ m gvac , ρ ),and into the remaining terms, which could be interpretedas purely thermal contributions ( ρ m gth , ρ ). Note that ρ m gvac , ρ are not uniquely defined in this way, but onlyup to the general renormalisation freedom of the quan-tum energy density, i.e. one could “shuffle parts of” Ω Λ , δ and ǫ into e.g. ρ m gth and vice versa, without changing anyphysical interpretation of the total energy density. Withthis in mind, the state-dependent contributions read asfollows, where the massless case is simply obtained byinserting m = 0, and we give here the result for arbitrarycoupling ξ for completeness. ρ m gvac = 12 π ∞ Z dkk (cid:26) a (cid:18) | χ ′ k | + (cid:18) ξ − (cid:19) a ℜ χ ′ k χ k + (cid:18) k + m a − (cid:18) ξ − (cid:19) H a (cid:19) | χ k | (cid:19) − k a + m − H ( ξ − )4 a k + Θ( k − ma ) − m + (cid:0) ξ − (cid:1) (cid:16) − H m ξ − − H ˙ H + 36 ˙ H − H ¨ H (cid:17) k (14) − (cid:18) ξ − (cid:19) H + 72 H ˙ H + 18 ˙ H − H ˙ H ( ξ − ) −
108 ˙ H ( ξ − )96 π − − π m − H m π ρ m gth = 12 π ∞ Z dkk a e βk − (cid:18) | χ ′ k | + (cid:18) ξ − (cid:19) a ℜ χ ′ k χ k + (cid:18) k + m a − (cid:18) ξ − (cid:19) H a (cid:19) | χ k | (cid:19) (15)In the conformally coupled case ξ = one can show by a straightforward computation that ρ m gvac = 12 π ∞ Z dkk (cid:26) a (cid:0) | χ ′ k | + (cid:0) k + m a (cid:1) | χ k | (cid:1) − (cid:18) k a + m a k − Θ( k − ma ) m k (cid:19)(cid:27) − − π m − H m π (16)= 4 π (2 π ) ∞ Z dkk a (cid:8)(cid:0) | χ ′ k | + (cid:0) k + m a (cid:1) | χ k | (cid:1) − (cid:0) | χ ′ ,k | + (cid:0) k + m a (cid:1) | χ ,k | (cid:1)(cid:9) , where χ ,k are the adiabatic modes of order 0, cf. the pre-vious section. This implies that the so-called Hadamardpoint-splitting regularisation of the energy density coin-cides with the so-called adiabatic regularisation of or-der zero up to the trace anomaly term and terms whichcan be subsumed under the regularisation freedom. Inthe following we analyse the individual state-dependentterms in the energy density.
1. Computation of ρ m gvac Following our general strategy in this section, wefirst aim to show that in states of low energy on theRobertson-Walker spacetime specified by (11) defined bya sampling function of sufficiently large support in time, ρ m gvac is for all z ∈ [0 , ] negligible in comparison tothe total energy density in the ΛCDM-model. Results inthis direction have been reported in [18] for the simpli-fied situation of a de Sitter spacetime background (cor-responding to Ω m = Ω r = 0), here we generalise theseresults to ΛCDM-backgrounds. One can easily see that ρ m gvac = 0 in the case of m = 0. For masses in the rangeof the Hubble constant m ≃ H and states of low energywe have performed numerical computations and found ρ m gvac /ρ ΛCDM < − , see Figures 1, 2. To achievethis result, we have rewritten all expressions in terms ofthe redshift z as a time variable and solved the equation(7) with initial conditions at z = 0 given by the valueand derivative of the adiabatic modes of order zero χ k, there. Note that a state of low energy does not dependon the choice of a mode basis, but the choice we madeseemed to be numerically favoured. To fix the state oflow energy, we chose a sampling function which was asymmetric bump function in z supported in the interval z ∈ (10 − , − +10 − ) for definiteness. In order to makethe numerical computations feasible, we chose a loga-rithmic sampling of k with 10 sampling points, wherethe boundaries of the sampling region have been chosensuch that the integrand of ρ m gvac , cf. (16), was vanishingin k -space to a large numerical accuracy outside of thesampling region for all z ∈ [0 , ]. We have computedthe mode coefficients c i ( k ) in the mode basis chosen ateach sampling point by a numerical integration in z andfinally the energy density by means of a sum over the FIG. 1: λρ m gvac /ρ ΛCDM for z < m (rescaled for ease of presentation). The dotted line corre-sponds to m = 100 H and λ = 10 − , the dashed line to m = 10 H and λ = 1 and the solid line to m = H and λ = 10 . One sees nicely how the energy density is minimalin the support of the sampling function at around z = 10 − . sampling points in k -space. Thus we have approximatedthe integral in (16) by a Riemann sum with logarithmicsampling. As our main aim here is to demonstrate that ρ m gvac /ρ ΛCDM ≪ ρ m gvac on the widthof the sampling function, but we have observed that themaximum amplitude of ρ m gvac /ρ ΛCDM seems to be mono-tonically growing with shrinking width of the samplingfunction, in accordance with the computations of [18] indeSitter spacetime.Unfortunately, we have not been been able to compute ρ m gvac /ρ ΛCDM for m > H in the way outlined abovebecause for large masses the modes oscillate heavily, andthus it costs a lot of computer power to solve the modeequation for such a large z -interval we are interested inand to the numerical accuracy which is necessary to ob-tain reliable results for the coefficients of the state of lowenergy and ρ m gvac /ρ ΛCDM . However, realistic field massesin the GeV regime are rather of the order of 10 H .In the numerical computations outlined above we haveobserved that ρ m gvac /ρ ΛCDM seemed to grow quadrati-0
FIG. 2: ρ m gvac /ρ ΛCDM for z > m . Theupper line corresponds to m = 100 H , the middle lines to m = 10 H and the lower lines to m = H ; solid lines (dashedlines) indicate results obtained with exact modes (zeroth orderadiabatic modes). ρ m gvac /ρ ΛCDM becomes constant for large z because there both energy densities scale like a − , c.f. (18)and the related discussion. cally with m , see Figure 2, but looking at the resultsof [18] in de Sitter spacetime one could maybe expectthat ρ m gvac /ρ ΛCDM decreases for large masses. Moreover,even if a potential quadratic growth of ρ m gvac /ρ ΛCDM with m would still imply ρ m gvac /ρ ΛCDM ≪ ρ m gvac /ρ ΛCDM ∼ − for m = H , it wouldbe better to have a more firm understanding of the largemass regime.In view of the numerical problems for large masses wehad to resort to an approximation in order to be ableto compute ρ m gvac /ρ ΛCDM . In fact, we have taken theadiabatic modes of order zero as basis modes for com-puting the state of low energy. Of course these modesare not exact solutions of the mode equations, but thefailure of these modes to satisfy the exact mode equationis decreasing with increasing mass and thus one can ex-pect that the error in all quantities derived from thesemodes rather than exact modes is also decreasing withincreasing mass. We have checked numerically that theenergy density computed with adiabatic modes ratherthan exact modes matched the ’exact’ result quite wellalready for masses in the regime m ≃ H , see Figures 2,1. For more details regarding error estimates for adia-batic modes we refer the reader to [56].Inserting the adiabatic modes χ ,k we obtain the fol-lowing expressions for the coefficients c i ( k ) of the statesof low energy. c ( k ) = Z dz f ( z ) (cid:26) m H ω ( k ) (1 + z ) + ω ( k )(1 + z ) H (cid:27) c ( k ) = Z dz f ( z ) (cid:26) m H ω ( k ) (1 + z ) − i m ω ( k ) (cid:19) × × exp (cid:18) − i Z zz ω ( k ) H dz ′ (cid:27) We now perform another approximation. We take as asampling function a Gaussian with mean z and variance σ ≪ f ( z ) = 1 √ πσ exp (cid:18) − ( z − z ) σ (cid:19) and take the zeroth term of the Taylor expansion of boththe expressions in the curly brackets in the integrands of c i ( k ) and of the integrand appearing in the exponent ofthe exponential in c ( k ). Without performing a detailederror analysis we note that this is justified for σ ≪ ∂ z H/H | z = z or H ( z ) /m , both of which are ei-ther smaller than or of order one under the assumptionof large masses and a ΛCDM-background. We can nowperform the z -integrals, which corresponds to consider-ing the Fourier-transform of f in the case of c . Using H /m ≪ H ( z ) /m ≪ c ( k ) > | c ( k ) | < exp (cid:18) − k σ H ( z ) (cid:19) exp (cid:18) − m σ H ( z ) (1 + z ) (cid:19) . For H ( z )(1 + z ) / ( mσ ) ≪ | c ( k ) | ≪ λ ( k ) and µ ( k )as µ ( k ) ≃ | c ( k ) | c ( k ) , λ ( k ) ≃ ρ m gvac as | ρ m gvac | < a ∞ Z dkk ( µ + µ | λ | ) (cid:0) | χ ′ ,k | + ω | χ ,k | (cid:1) < a ∞ Z dkk µ | λ | ω < a H ( z ) mσ exp (cid:18) − m σ H ( z ) (1 + z ) (cid:19) such that, barring our approximations, we indeed get aresult which shows that the energy density decreases –exponentially – for large masses. Note that for not toosmall σ the bound we found is in general small comparedto ρ ΛCDM even if we forget about the exponential because H m is much smaller than the the square of the Planckmass, i.e. 1 /G . We also see that the bound grows withgrowing z , i.e. if we ’prepare’ the state of low energyfurther in the past, and that it diverges if the width ofthe sampling function goes to zero; this is in accord withthe results of [18] in deSitter spacetime. Note that wecould have chosen any rapidly decreasing or even com-pactly supported sampling function in order to obtaina bound which is rapidly decreasing in m/H thus onecould say that the result does not depend on the shape1of the sampling function as long as its width is not toosmall. Finally, one could of course directly take the pointof view that for large masses the adiabatic modes χ ,k de-fine ’good states’ themselves and conclude that in thesestates ρ m gvac = 0 on account of (16).
2. Computation of ρ gtherm, m We now proceed to analyse the thermal parts of thestate-dependent contributions to the total energy density.Inserting ξ = in (15), we find ρ m gth = 12 π a ∞ Z dkk e βk − × (17) × (cid:0) | χ ′ k | + (cid:0) k + m a (cid:1) | χ k | (cid:1) with k = p k + a F m .Before performing actual computations, we would liketo mention a general result about the scaling behaviourof the energy density w.r.t. a [62]. To wit, usingthe equation of motion (7) and the assumption that H > Q k := | χ ′ k | + (cid:0) k + m a (cid:1) | χ k | with respect to a and obtain the fol-lowing inequalities k + a m k + m Q k ( a = 1) a ≤ Q k ( a ) a ≤ Q k ( a = 1) a . (18)From these one can already deduce that ρ m gth has a scal-ing behaviour w.r.t. a which lies between a − and a − and approaches a − in the limit of vanishing a , in factthis still holds if we replace the Bose-Einstein factors inthe generalised thermal states by arbitrary functions of k . Moreover (18) also implies that ρ m gvac can not scalewith a power of a lower than − a on ΛCDMbackgrounds, c.f. (16).Proceeding with actual computations we find that inthe massless case ρ m gth can be computed exactly and an-alytically and the result is ρ = π
30 1 β a . (19)As in the massless case the state of low energy is theconformal vacuum and the associated generalised ther-mal state is the conformal temperature state with tem-perature parameter β = 1 /T , this result in fact holds forfields of all spin, i.e. the generalised thermal energy den-sity in this case is always the one in Minkowski spacetimerescaled by a − . Thus a computation with e.g. photonsor massless neutrinos yields the same result (19) up tonumerical factors due to the number of degrees of free-dom and the difference between Bosons and Fermions.In the massive case it is not possible to compute ρ m gth analytically and exactly, but we have to resort to approx-imations once more. We recall that the massive scalar field in our model should represent baryonic matter andDark Matter in a simplified way. Thus we take typicalvalues of β , a F and m from Chapter 5.2 in [45] computedby means of effective Boltzmann equations. A popularcandidate for Dark Matter is a weakly interacting mas-sive particle (WIMP), e.g. a heavy neutrino, for which[45] computes x f = βa F m ≃
15 + 3 log( m/ GeV) , (20) a F ≃ − ( m/ GeV) − . We shall take these numbers as sample values althoughworking with a scalar field, because for large masses m ≫ H , the “thermal energy densities” ρ m gth in gen-eralised thermal states for free fields of spin 0 and canbe shown to approximately coincide up to constant nu-merical factors on the basis of the results of [17] and [32,Section IV.5].Considering m > ρ m gth approx-imatively as follows. We recall from the computation of ρ m gvac that for large masses m ≫ H one can consider theadiabatic modes of order zero χ ,k as approximative ba-sis modes for the computation of the state of low energyand that with respect to this basis one finds for the co-efficients of the state of low energy λ ≃ µ ≃
0, thuswe can insert those modes in (19) instead of the modesof the state of low energy. Using m ≫ H once more, wehave | χ ′ ,k | + (cid:0) k + m a (cid:1) | χ ,k | ≃ √ k + m a and us-ing x f >
15 we can approximate the Bose-Einstein factorin(19) as 1 / ( e βk − ≃ e − βk . Finally we can rewritethe integral in (19) in terms of the variable y = k/ ( a F m )and compute, using a/a F ≫ z ∈ [0 , ] we are interested in, ρ m gth ≃ π a F m a ∞ Z dyy e − x f √ y +1 . This already gives the desired result ρ m gth ∝ a − . The re-maining integral can be computed numerically, however,for x f ≫ y ≪ p y + 1 ≃ y / ρ m gth ≃ π ) / mβ a x f e − x f , which for a = a F = 1 (unsurprisingly) coincides withthe thermal energy density for massive scalar fields inMinkowski spacetime.
3. The total energy density
Collecting the results of this section, we find for thetotal energy density of our model ρ + ρ m ρ ≃ Ω Λ + 1(2 π ) / mβ a ρ x f e − x f a + π β ρ a , β , β in order to emphasise that thegeneralised thermal states for the massive and masslessconformally coupled scalar fields can have different tem-perature parameters β . We recall that the thermal con-tribution of the massless scalar field has been computedexactly, while the one of the massive scalar field is anapproximative result. The above result shows that weindeed succeeded in modelling radiation by a masslessscalar field and matter/dust by a massive scalar field insuitable generalised thermal states. Obviously, we canchoose the free parameters m , β i , x f in such a waythat the prefactors of the matter and radiation termshave their correct ΛCDM-values Ω m and Ω r , e.g. forthe former we could choose the sample values (20) with m ≃ /β ≃ VI. A NATURAL EXPLANATION FOR DARKRADIATION
Our analysis in the previous section implies that thereexist quantum states in which the total energy density inquantum field theory on curved spacetimes differs fromthe one in the ΛCDM-model only by the higher derivativeterm ǫJ and terms which are generally negligible or be-come important only at redshifts z ≫ . The prefactor ǫ of J is not determined by the theory but a free param-eter so far and it seems advisable to study its impact onthe cosmological expansion. Indeed, as our second mainresult we demonstrate in this section that such term canprovide a natural explanation of Dark Radiation.To start with, we briefly review the notion of DarkRadiation and the related observations. The fraction Ω r of the radiation energy density in the ΛCDM-model iscomputed asΩ r = Ω γ (cid:18) (cid:19) / N eff ! (21)where Ω γ ≃ × − is the fraction due to electromag-netic radiation, which can be computed by inserting into(19) the CMB temperature T CMB ≃ . ρ = 3 H / (8 πG ) ≃ . × − eV (and multiplying by two for the two degrees offreedom of the photon). Moreover, N eff is the number ofneutrino families and the factor 7 / / / = 0 . N eff is not 3 as onewould expect, but rather N eff = 3 .
046 because the value7 / / / in (21) is computed assuming e.g. instan-taneous decoupling of the neutrinos and corrections haveto be taken into account in a more detailed analysis [50];it is customary to take these corrections into accountby considering N = 3 .
046 as the standard value of the‘neutrino family number’ rather than changing the fac-tor 7 / / / in this formula, hence the nomenclature N eff . Consequently, it is convenient to parametrise anycontribution to Ω r which is not due to electromagneticradiation and the three neutrino families in the standardmodel of particle physics by ∆ N eff := N eff − . r and thus N eff is the primordial fraction of light ele-ments in the early universe as resulting from the so-calledBig Bang Nucleosynthesis (BBN), which has occurred ataround z ≃ and thus in the radiation–dominated era,because the nucleosynthesis processes which happenedat that time depend sensitively on the expansion rate H ≃ H √ Ω r /a , see e.g. [21, 44, 45]. The other mainobservational source for the determination of N eff is thecosmic microwave background radiation (CMB). This ra-diation was emitted at about z ≃ z eq at which the energy den-sities of matter and radiation were equal, see [1, Section6.3] and the references therein for details; for standardvalues, z eq ≃ N eff , and the value inferred from observational datadepends on the data sets chosen. The Planck collabora-tion [1] reports e.g. values of N eff = 3 . +0 . − . at 95%confidence level from combined CMB power spectrumdata sets, N eff = 3 . +0 . − . at 95% confidence level fromcombining these data sets with direct measurements ofthe Hubble constant H and of the power spectrum ofthe three-dimensional distribution of galaxies (so-calledbaryon acoustic oscillation, BAO), and N eff = 3 . ± . N eff >
0. Thus there has been anincreasing interest in models which can explain a poten-tial excess in radiation and thus ∆ N eff , see for instancethe recent surveys [19, 20, 43, 51] and references therein.Most of there models assume additional particles/fields,e.g. a fourth, sterile, neutrino, whereas other considergeometric effects from e.g. modifications of General Rel-ativity. Moreover, in most models ∆ N eff is constant andthus affects BBN and CMB physics alike, while in others,e.g. [29, 35], ∆ N eff is generated only after BBN and thusaffects only CMB physics.In the following we shall propose a new and alterna-tive explanation for Dark Radiation which follows natu-rally from our analysis of the ΛCDM-model in quantumfield theory on curved spacetimes and has the interesting3 FIG. 3: ∆ N eff ( z ) depending on ǫ for z = 10 (BBN, solidline) and z = 3 × (CMB, dashed line). For ǫ < ǫ positive and large enough, the values at the two redshiftscoincide because the maximum value of ∆ N eff ( z ) is reachedalready for z < × in these cases. characteristic that it generates a value of ∆ N eff which increases with z and thus affects BBN physics more thanCMB physics. To our knowledge, this is the first ex-planation for Dark Radiation proposed which has thischaracteristic feature.Following the motivation outlined at the beginning ofthis section, we solve the equation H H = Ω Λ + Ω m a + Ω r a + ǫ J H , (22)which can be rewritten as a second order ordinary dif-ferential equation for H in z , numerically with ΛCDM-initial conditions H ( z = 0) = H , ∂ z H ( z = 0) = H (3Ω m +4Ω r ) /
2. As before, we consider for definitenessΩ m = 0 .
30, Ω Λ = 1 − Ω m − Ω r , because the exact val-ues of these parameters are not essential for our analysis.Looking at the characteristics of the solution to this or-dinary differential equation, it turns out that a non-zero ǫ generates a time-varying ∆ N eff >
0. In more detail, wedefine for the solution H of (22)∆ N eff ( z ) := H H − Ω Λ − Ω m (1 + z ) − Ω r (1 + z ) . z ) , and sample this observable at the redshift z = 10 as-sociated to BBN physics and at the redshift z = 3000associated to CMB physics. We collect our results inFigure 3.As can be inferred from this figure, ∆ N eff ( z ) is mono-tonically increasing in ǫ , where positive and negative val-ues of ǫ result in very different behaviours. For positivevalues of ǫ one finds that ∆ N eff ( z ) vanishes in the limit ofvanishing ǫ , as one would expect. On the other hand, itturns out that for negative values of ǫ , ∆ N eff ( z ) divergesas ǫ approaches zero. While this seems to be puzzling FIG. 4: ∆ N eff ( z ) for ǫ = 2 × − . at first sight, it fits well with previous qualitative anal-yses of the effect of the higher derivative term J . Infact, it is known that the inclusion of this higher deriva-tive term can lead to unstable solutions of the semiclas-sical Einstein equations, where for ǫ < ǫ >
0) theclass of solutions we consider here, effectively fixed bythe ΛCDM initial conditions, turns out to be unstable(stable), see e.g. [2, 28, 34, 46, 58, 72]. Thus, the diver-gence of ∆ N eff ( z ) as ǫ approaches zero from below canbe just interpreted as a sign of this instability.In [14, 79], ǫ = 0 has been chosen on conceptualgrounds in order to discard unstable solutions altogether.However, as we see here a non-zero ǫ can have interestingphenomenological implications. After all, taking quan-tum field theory on curved spacetimes seriously, ǫ is afree parameter of the theory, which we can only fix ina more fundamental theory or by observations. Indeed,we see in Figure 3 that ǫ <
0, corresponding to an un-stable solution of the semiclassical Einstein equation, isalready ruled out by observations because it generallyleads to ∆ N eff ( z ) ≫ N eff ≃ . − . N eff = 1 at both BBN and CMBwe have to choose 0 ≤ ǫ < × − , thus, without per-forming a detailed fit of BBN and CMB data, we can saythat the values for ∆ N eff reported e.g. by the Planck col-laboration in [1] give an upper bound of about 2 × − for ǫ . We plot ∆ N eff ( z ) for redshifts 0 < z < inFigure 4. As already anticipated in Figure 3, one cannicely see how ∆ N eff ( z ) is monotonically growing in z ,with ∆ N eff ( z = 0) = 0 as fixed by our initial conditions.Moreover one can see clearly that if one wants to meetthe bounds on ∆ N eff at the BBN redshift, the excess inthe effective number of neutrinos at the CMB is negligi-ble, which is the characteristic signature of this potentialexplanation for Dark Radiation. We have not consideredthe influence of the initial conditions for (22) on ∆ N eff ,but we expect that for the initial conditions compatiblewith low- z observational data such as supernova type Iadata and baryon acoustic oscillation data, ∆ N eff will not4differ considerably from the form we found as these datawill not allow for large deviations in the initial conditionsfrom the ΛCDM ones we chose.As a further, rather pedagogical remark, we would liketo comment on the fact that, for large absolute values of ǫ , ∆ N eff does not depend on the sign of ǫ , as can be seenfrom Figure. 3. This phenomenon can be understood asfollows. Naturally, for large absolute values of ǫ , the otherterms in (22) become negligible and one effectively solvesfor J = 0. The solution of this ordinary differentialequation with initial conditions H ( z = 0) = c , ∂ z H ( z =0) = d is H = c / d / (cid:0) cd − z ) (cid:1) / / and thus, inserting the ΛCDM initial conditions H ( z =0) = H , ∂ z H ( z = 0) = H (3Ω m + 4Ω r ) / z N eff ( z ) ≃ as in Figure 3.While our analysis is to our knowledge the first at-tempt to bound ǫ with cosmological observations, it isnot the first attempt to determine it with observationsat all. In fact, the effect of higher-derivative correctionsto General Relativity has already been analysed in thepast, and since the tensors I µν and J µν in (4) can be ob-tained as variational derivatives with respect to the met-ric of the Lagrangeans √− gR and √− gR µν R µν , theyhave been considered in these analyses as well. To wit,the Lagrangean L = √− g (cid:18) R πG + c R + c R µν R µν (cid:19) leads to the Newtonian potential of a point mass m [73] φ = − mGr (cid:18) e − m r − e − m r (cid:19) (23) m = 1 p πG ( − c − c ) m = 1 √ πGc . Using recent data [41] from torsion-balance experimentsto test the gravitational inverse-square law at ∼ − mand assuming that the two Yukawa corrections don’t can-cel each other at this length scale, one obtains − c , c < [13]. To compare this with our results, we recall thatin our treatment these higher curvature terms appear onthe right hand side of the semiclassical Einstein equationand that we have computed in units of H , thus we have ǫ = ( − c − c )8 πGH ≃ ( − c − c )10 − which would imply ǫ < − and thus a stronger boundthen the one we inferred from cosmological observations.Of course such a low value of ǫ leads to ∆ N eff ≪ ǫ is completely independent from the one inferred from lab-oratory experiments and can thus be considered as an ad-ditional confirmation of those results. Moreover, it is stillpossible that the Yukawa corrections in (23) cancel eachother on the length scales relevant for the experimentsdescribed in [41], such that ǫ could be as large as ourupper bound, which in this case would give a real boundon one and hence both Yukawa corrections. Finally, thebounds inferred from [41] and from our analysis stemfrom phenomena on completely different length scales.As a rough estimate we note that the diameter of our ob-servable universe, which today is about 6 /H ≃ m,was at e.g. z = 10 still 10 m and thus much larger thanthe submillimeter scales relevant for the torsion-balanceexperiments. Thus it could be that effects we have notconsidered so far, e.g. state-dependent effects which aredue to the small-scale structure of the quantum states wehave fixed only on cosmological scales so far, affect thecomparison between the two different sources of input forthe determination of ǫ . VII. CONCLUSIONS
We have demonstrated that it is possible to under-stand the cosmological evolution for redshifts z < as described in the ΛCDM-model entirely in terms ofquantum field theory in curved spacetime, by comput-ing the energy density in generalised thermal quantumstates and showing that the state and field parameterscan be chosen such as to match the energy density in theΛCDM-model up to small corrections.One of these corrections, quantified by a parameter ǫ , occurred due to higher-derivative terms appearing asrenormalisation freedom of the energy density of anyquantum state. We have demonstrated that this correc-tion can constitute a natural explanation for Dark Radia-tion with the characteristic signature of leading to a time-varying effective number of neutrino families N eff whichdecays in time and have obtained the bound ǫ < × − by comparison with experimental data, which is compat-ible with the ΛCDM-value ǫ = 0. A conservative inter-pretation of laboratory experiments leads to a strongerbound ǫ < × − which cancels any Dark Radiation ef-fects, but we have argued that there are possibilities toevade this stronger bound. Thus we believe that it isworth to include this new parameter in further analysesof the ΛCDM-model parameters.An additional correction to the ΛCDM-model appearsdue to the so-called trace anomaly. This contribution tothe energy density is negligible for redshifts z < butcan have considerable impact on the cosmological evolu-tion at larger redshifts as discussed already in the frame-work of the so-called Starobinski-inflation [72]. Notethat, in contrast to the renormalisation freedom of theenergy density quantified by the in principle free model5parameter ǫ , the trace anomaly is fixed by the field con-tent and thus predicted by quantum field theory in curvedspacetime, if one accepts the validity of the semiclassi-cal Einstein equations up to the regimes where the traceanomaly becomes important.Finally, potential further corrections to the ΛCDM-model can come from specifics of the quantum state wehave neglected in our analysis. We have chosen the quan-tum states in our discussion such that their characteris-tic energy density was entirely of thermal nature, but wehave seen that also pure, non-thermal states can havecontributions to the energy density which scale like a − ,cf. Figure 2. It could be that there exist states which arecompatible with observations and have sizable energy-density contributions of this kind; these states would thenprovide a further alternative explanation for Dark Radi- ation which does not call for the introduction of newparticles respectively fields. Acknowledgments
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