Dark Radiation predictions from general Large Volume Scenarios

Recent observations constrain the amount of Dark Radiation ($\Delta N_{\rm eff}$) and may even hint towards a non-zero value of $\Delta N_{\rm eff}$. It is by now well-known that this puts stringent constraints on the sequestered Large Volume Scenario (LVS), i.e. on LVS realisations with the Standard Model at a singularity. We go beyond this setting by considering LVS models where SM fields are realised on 7-branes in the geometric regime. As we argue, this naturally goes together with high-scale supersymmetry. The abundance of Dark Radiation is determined by the competition between the decay of the lightest modulus to axions, to the SM Higgs and to gauge fields. The latter decay channel avoids the most stringent constraints of the sequestered setting. Nevertheless, a rather robust prediction for a substantial amount of Dark Radiation can be made. This applies both to cases where the SM 4-cycles are stabilised by D-terms and are small"by accident"as well as to fibred models with the small cycles stabilised by loops. Furthermore, we analyse a closely related setting where the SM lives at a singularity but couples to the volume modulus through flavour branes. We conclude that some of the most natural LVS settings with natural values of model parameters lead to Dark Radiation predictions just below the present observational limits. Barring a discovery, rather modest improvements of present Dark Radiation bounds can rule out many of these most simple and generic variants of the LVS.


Introduction
Dark Matter is an essential ingredient of the universe with strong experimental support. While Dark Matter is constituted by hidden non-relativistic particles, there are no reasons a priori why hidden sectors could not exhibit species which remain relativistic at CMB and BBN temperatures. Such a sector of particles is called Dark Radiation and it represents an area of Beyond-the-Standard-Model physics, which is increasingly well tested by experiments. The amount of Dark Radiation is most conventionally defined as a contribution to N ef f -the effective number of neutrino species. The value of N ef f at the CMB time can be measured and any excess ∆N ef f over the Standard Model prediction N ef f,SM = 3.046 is evidence for the existence of Dark Radiation. The most recent result for the effective number of neutrino species is N ef f = 3.30 +0.54 −0.51 (95 % CL; Planck+WMAP polarisation+high L+BAO) [1]. If direct measurements of the Hubble parameter are included the value is modified to N ef f = 3.52 +0. 48 −0.45 (95 % CL; Planck+WMAP polarisation+high L+BAO+H 0 ) [1]. 1 The situation is thus unclear at the moment: while there is a mild preference for some Dark Radiation, the absence of Dark Radiation is currently not excluded by experiment.
Limits on Dark Radiation are powerful tests of Beyond-the-Standard-Model physics and this is particularly true for a popular class of models based on string theory: The scheme of moduli stabilisation known as the Large Volume Scenario (LVS) [4] always leads to a Dark Radiation candidate in form of a light axion-like particle (henceforth axion) [5][6][7][8]. Dark Radiation is a byproduct of reheating which -in cosmologies based on string compactifications -is most naturally caused by the decay of moduli. These dominate the energy density in the post-inflationary period and it is the decay of the lightest modulus which reheats the visible sector but is also responsible for Dark Radiation production. One universal feature of the LVS is the existence of a decay channel for the lightest modulus into light axions, with a decay rate Γ ∼ 1 48π m 3 τ b M 2 P typical for Planck coupled fields. These axions then constitute Dark Radiation which is thus a generic prediction of the LVS.
In particular, the value of ∆N ef f is controlled by the ratio of the decay rates of the lightest modulus τ b into light axions versus Standard Model (SM) particles: 10.75 g * (T d ) where T d is the decay temperature of the modulus τ b and g * (T d ) is the effective number of particle species at that temperature. Thus upper bounds on N ef f can be directly interpreted as lower bounds on the decay rates of τ b into into SM fields. The latter crucially depends on the realisation of the visible sector in the LVS model. Dark Radiation has been studied in detail in the sequestered LVS (cf. [5,6] in particular), where the visible sector arises from a stack of D3-branes at a singularity [9]. 2 The sequestered LVS has the attractive feature that it allows for TeV soft terms while keeping the gravitino and moduli heavy enough to evade the Cosmological Moduli Problem (CMP) [11][12][13]. However, given the absence of superpartner signals at the LHC, a high SUSY breaking scale may also be an option. 3 In the sequestered LVS decays of the lightest modulus into (MS)SM fields are dominated by the decay channel into Higgs scalars: τ b → H u H d . This coupling originates from a Giudice-Masiero term K HH (τ b )(zH u H d + c.c.) in the Kähler potential leading to a decay rate Γ ∼ 2z 2 The upper bounds on N ef f then give rise to bounds on this model [6]. To arrive at a N ef f consistent with experiment one requires a value of z > 1.5. Alternatively, this value can be reduced to z = 1 if one allows for n H > 4 Higgs doublets beyond the MSSM field content.
The question thus remains whether it is possible to evade the bounds on N ef f in more general constructions of the LVS. In particular, it would be important to determine whether there are realisations of the LVS which are consistent with data on Dark Radiation without the need for additional matter beyond the (MS)SM and for natural values of parameters. 2 For instance, in [6] the authors study scenarios with the SM at a singularity and with a certain amount of de-sequestering occurring due to moduli-mixing at 1-loop. This scenario differs from the non-sequestered case (with the SM brane in the geometric regime) which we analyse in this work. Moreover, in [10] LVSlike racetrack models based on poly-instanton effects have been considered. We comment further on those scenarios when comparing them to our models. 3 In addition, the recent measurements of B-modes in the CMB [2] may strengthen this point. Assuming that the LVS scalar potential sets the suggested inflation scale of 10 16 GeV, the compactification volume is expected to be too small to allow for soft masses at O(1) TeV in any LVS model. In addition, if the height of the inflaton potential sets the supersymmetry breaking scale, the high scale of inflation suggested by the BICEP2 is an indication for high scale supersymmetry [14].
The purpose of this paper is to examine more general constructions of the LVS and study their predictions for DR. While we aim to be general, we will be most interested in constructions, which maximise the decay rate of τ b into SM fields compared to the decay rate into light axions and can thus evade stricter bounds on N ef f . As it turns out, settings where reheating proceeds through SM gauge bosons predict, for the most natural parameter values, a DR abundance just below present observational bounds.
Let us now step back and examine the various possibilities to boost the decay rate of the lightest modulus into SM fields: 1. For one, different realisations of the visible sector could lead to a higher decay rate into matter scalars (squarks, sleptons). These fields couple to the lightest Kähler modulus through terms in the Kähler potential of the form K ⊃ τ b CC, leading to a To make this rate comparable to the decay rate into Higgs fields, we would need a mass for matter scalars close to the threshold m sof t = m τ b /2. Unfortunately, we are not aware of a variant of the LVS where m sof t ∼ m τ b is a natural outcome of the construction: In the sequestered case, we have a hierarchy between (see [15,16]). In the non-sequestered case, m sof t is typically much larger than the mass of the lightest modulus. We thus do not study the decay to matter scalars in the following.
2. Further, decays into matter fermions are chirality-suppressed, which is a modelindependent statement [5]. Hence this decay channel is suppressed w.r.t. the decay into Higgs fields regardless of the realisation of the visible sector.
3. Last, decays of the lightest modulus into gauge bosons are suppressed in the sequestered LVS, as the modulus τ b can only decay into visible sector gauge fields at loop level. If there was a tree-level interaction τ b F µν F µν between the modulus and visible sector gauge fields, the decay rate would be increased to Γ ∼ 1 96π , which is comparable to the decay rate into axions or Higgs fields. For this coupling to be present, the visible sector gauge kinetic function has to depend on this modulus. This is the most interesting possibility for increasing the branching ratio of decays of τ b into visible sector fields and, in the following, we will discuss setups which exhibit this property.
This defines the strategy for this paper. To examine LVS setups which can avoid the most stringent constraints of the sequestered setup, we will explore LVS models, where the lightest modulus reheats the SM by dominantly decaying into gauge bosons. There are two main possibilities: 1. On the one hand, the lightest modulus could couple to the visible sector gauge bosons directly.
2. On the other hand, the lightest modulus can decay into gauge bosons which do not belong to the SM gauge group, but under which some SM fields are charged. Such gauge bosons can arise from so-called flavour branes [17], which realise approximate global symmetries of the SM spectrum. The gauge theory on the flavour branes must be broken at some sub-stringy scale where associated gauge bosons become massive. Decays of these gauge bosons then reheat the SM.
In both cases, for a Kähler modulus to couple to gauge bosons at tree-level, we require the cycles supporting the gauge theory on 7-branes to be stabilised in the geometric regime. For case (1.) above, this has immediate consequences for the low energy phenomenology: visible sector cycles in the geometric regime are not sequestered from the source of supersymmetry breaking and, consequently, superpartners typically obtain masses m sof t ∼ m 3/2 . As m 3/2 O(10) TeV to avoid the CMP these setups necessarily require high scale supersymmetry. 4 However, when coupling τ b to the visible sector gauge theory directly, we find the following difficulties. If the cycle supporting the SM gauge group is stabilised supersymmetrically by D-terms, the coupling of τ b F µν F µν automatically implies a coupling a b F µνF µν of the DR candidate a b to the SM. In particular, the axion a b couples to QCD and thus takes over the role of the QCD axion [21,22]. Consequently, unless there is another axion which can play the role of the QCD axion, the DR candidate a b can be identified with the QCD axion for which there are stringent constraints from astrophysics and cosmology (see e.g. [23][24][25][26]). For one, we find that our QCD axion candidate produces too much Dark Matter by the vacuum realignment mechanism unless the inital misalignment angle is tuned to θ i ∼ 10 −2 . Further, if the recent BICEP2 results [2] are explained by primordial gravitational waves, the situation is far more severe: we find that setups with a b as the QCD axion are then ruled out by isocurvature bounds. Similar constraints arise if the cycle supporting the visible sector is stabilised perturbatively by string loop corrections. In this case, there will an additional light axion beyond a b , which will couple to QCD.
The paper is structured as follows. In section 2 we review DR in the sequestered LVS and examine DR predictions if the requirement of TeV SUSY is lifted. In section 3 we study DR predictions for LVS models with visible sectors on D7-branes wrapping cycles in the geometric regime. In particular, we distinguish between setups where ratios of large cycle-volumes are stabilised by D-terms or by loops. Last, we examine DR in LVS models where the SM is reheated via gauge bosons on flavour branes in section 4.

Review: Dark Radiation in the sequestered Large Volume Scenario
In this section we review DR predictions in the sequestered LVS from [5][6][7]. In particular, we comment on how observational results for ∆N ef f constrain this model and interplay with the SUSY breaking scale.
As in [5], the compactification manifold is taken to be a Swiss-Cheese Calabi-Yau with a volume given by (2.1) The LVS procedure then fixes V as well as at least one of the small cycles τ i,s through an interplay of non-perturbative effects as well as α corrections [4], such that V ≈ ατ is exponentially large. This scheme of moduli stabilisation leads to a clear hierarchy of moduli masses. In particular, the real scalar τ b parameterising the bulk volume is the lightest modulus with its axionic partner a b essentially massless: Predictions for DR can then be made by studying the decay rates of τ b into a b compared to SM matter.
In the sequestered LVS, the visible sector is realised by D3-branes at a singularity. This scenario is attractive as gaugino and soft scalar masses are suppressed w.r.t. the gravitino mass: 5 Such a hierarchy allows for TeV soft terms while keeping the gravitino and further moduli heavy enough to evade the CMP. The decay rate of τ b into SM fields depends on the realisation of the visible sector, which thus introduces a model-dependence into DR predictions. Given the Kähler potential and gauge kinetic function of the low-energy effective theory, the decay rates of the lightest modulus into the visible sector as well as into Dark Radiation can then be computed. For the sequestered LVS the relevant terms are where T b = τ b + ia b is the bulk volume modulus superfield, C i are chiral matter superfields and T sa,k are blow-up modes. Given the setting above the rates of decay of τ b into DR and SM fields can be determined. One finds that the two dominant decay modes of the volume modulus are the decay into its axionic partner a b and into Higgs fields.
Decays into DR: Decays into SM: In particular, all other decay channels into visible sector fields are subleading w.r.t. the decay into H u H d as long as z is O(1). This can be understood as follows: 5 The exact scale of soft scalar masses will depend on α corrections to the Kähler matter metric which have not been determined yet. Here, we continue with the hypothesis that soft scalar masses only arise at the scale MP /V 2 . This assumption is strengthened at lowest order in string perturbation theory by a direct calculation [27].
• Gauge bosons and gauginos: Decays into gauge bosons are controlled by the modulidependence of the gauge kinetic function which, in the sequestered setup, is independent of the light modulus T b at tree level. Correspondingly, decays of τ b into gauge bosons can only occur at loop level, leading to a decay rate which is suppressed by a loop factor: • Matter scalars: Decays into matter scalars C i arise from the term K ij C iCj leading to a rate Γ ∼ In the sequestered LVS the soft scale m sof t is parametrically lower than the mass of τ b : where we also allow for a generic number n H of Higgs doublets. Clearly, predictions for ∆N ef f also depend on the exact reheating temperature T d through g * (T d ). This can be determined as T d ∼ √ Γ Φ M P O(0.1)M P /V −9/4 . As the volume V sets all other scales of the setup, including in particular the supersymmetry breaking scale, the prediction for DR will be discussed in terms of this scale.
For TeV scale SUSY one finds a reheating temperature of T d 1 GeV, which corresponds to g * = 247/4. We compare the resulting ∆N ef f to the bound ∆N ef f < 0.79 (95 % CL; Planck+WMAP polarisation+high L+BAO) [1]. If we allow for one pair of Higgs doublets only, the sequestered LVS is consistent with experimental observation if z > 1.5. Alternatively, allowing additional Higgs doublets while fixing z = 1, the sequestered LVS is not ruled out by observation as long as n H > 4. This requires the field content of the visible sector to be extended beyond the MSSM.
The above constraints on the sequestered LVS can be somewhat relaxed if one allows for high scale supersymmetry breaking. For example, for V 10 7 , we obtain m sof t 10 TeV and T d 100 GeV. In this regime we have g * = 106.75 -the maximum number in the SM -and the DR constraints give the following: for ∆N ef f < 0.79 and n H = 2, we now require z > 1.3. If z = 1, experimental bounds can be met as long as n H ≥ 4. While the constraints are less severe, values of z < 1.3 still require the addition of matter beyond the MSSM field content. Further, allowing for two Higgs doublets only, small values for the Giudice-Masiero coupling z < 1 are still excluded.
It is thus apparent that measurements of N ef f impose severe constraints on string models based on the LVS. While there are no a priori reasons why z > 1 should be impossible, it remains an open question whether such values can be obtained and whether such a regime occurs naturally in the string landscape. To give an example for the difficulties, note that the particular value z = 1 can be derived from a shift symmetry in the Higgs sector. In type IIB/F-theory such a symmetry can arise if the Higgs is contained in brane deformation moduli [20,[28][29][30]. However, in this case the Kähler metric is independent of Kähler moduli and the lightest modulus τ b cannot decay to Higgs fields at all.

Dark Radiation beyond the sequestered Large Volume Scenario
In this section we go beyond the sequestered LVS and analyse DR predictions for more general setups. In particular, while the sequestered LVS considers branes on collapsed cycles, here we examine models with visible sectors given by D7-branes wrapping 4-cycles in the geometric regime. There are in principle three ways how such cycles can be stabilised: 1. non-perturbative effects, 2. gauge-flux-induced D-terms and 3. string loop corrections to the Kähler potential.

Large Volume Scenario with D-term stabilisation
Consider a Calabi-Yau orientifold X with several 4-cycles D i whose volumes are given by τ i . For this discussion it will be useful to also introduce the moduli t i , which give the volumes of 2-cycles. In terms of these the overall volume can be written as V = 1 6 k ijk t i t j t k , where k ijk are the triple intersection numbers of X. We also have the relation τ i = ∂V/∂t i = 1 2 k ijk t j t k . In the following, one "smal" 4-cycle as well as the overall volume of X will be stabilised using the LVS procedure. All other 4-cycles will be stabilised in a geometric regime by D-terms. In particular, the visible sector will be realised on D7-branes wrapping one of the 4-cycles stabilised by D-terms.
D-terms are induced due to fluxes on D7-branes wrapping these 4-cycles and give rise to a D-term potential where ξ i are FI-terms and φ j are open string states charged under the anomalous U (1) giving rise to the D-term. The sum over i is over all 7-branes and the sum over j is over all charged open string states. We assume that supersymmetric stabilisation can be achieved, without appealing to VEVs of charged fields, by the simultaneous vanishing of all FI-terms: ξ i = 0 for all i.
The FI-terms are given by an integral over X: whereD i are Poincaré dual 2-forms to the 4-cycles D i , J = t iD i is the Kähler form and F i =f j iD j is the gauge flux. The q ij =f k i k ijk are then the charges of the Kähler moduli T i under the anomalous U (1) [31,32].
From (3.2) it is apparent that FI-terms are linear combinations of 2-cycle volumes t i and hence the requirement ξ i = 0 for all i leads to a linear system of equations for the 2-cycle volumes:

As a result D-terms fix volumes of some 2-cycles in terms of the volumes of other 2-cycles.
To combine D-term stabilisation with the LVS, we proceed as follows. We consider the case where one of the 2-cycles, t 1 , does not appear in the expressions for FI-terms, while all other 2-cycles t j with j = 1 are fixed w.r.t. one another by the system of equations (3.3). As a result, t 1 remains unfixed at this stage and all other 2-cycles can be expressed in terms of one other 2-cycle, say t 2 . Further, we consider geometries where t 1 enters the volume in a diagonal way: k 1jk = 0 for j, k = 1 only, such that t 1 only contributes to the volume as V ⊃ 1 6 k 111 (t 1 ) 3 . Then it follows that where c i is a numerical factor. On the level of 4-cycles this leads to the desired result: D-terms stabilisation leaves two flat directions which we can parameterise by τ 1 and τ 2 . All other 4-cycles are stabilised w.r.t. τ 2 . Thus, after D-term stabilisation the volume depends on τ 1 and τ 2 only. Here, we consider geometries which lead to a volume of Swiss-Cheese form: 6 WLOG we define τ 2 = τ b and τ 1 = τ np . The remaining moduli τ b and τ np are then fixed using the standard LVS procedure. Then τ b is the lightest modulus as before and its axionic partner a b is a nearly massless DR candidate (see (2.2)). An explicit example for a construction of this type is given in [32]. It behoves to describe what this implies for the visible sector 4-cycle with volume τ a . While D-terms fix τ a = c a τ b , the cycle τ a has to be fixed small to produce the correct gauge coupling on the visible sector branes: α −1 SM = τ a ≈ 25 (ignoring flux contributions so far). It follows that the parameter c a has to be tuned such that τ a is small despite τ b 1. This amounts to a potentially severe tuning of fluxes, which cannot be avoided in our construction in this section.
As stabilisation by D-terms is achieved at the supersymmetric locus V D = 0, the effective theory after D-term stabilisation should still be supersymmetric -i.e. it should still be formulated in terms of superfields. Thus the condition τ a = c a τ b on the visible sector cycle should in fact be enhanced to the condition T a = c a T b at the level of superfields. 7 This has the following consequences for the kinetic term of the visible sector gauge theory. Starting with the superfield Lagrangian for the visible sector gauge theory we have: As a result, the lightest modulus τ b now couples to visible sector gauge bosons, which was the objective of the current construction. This opens another channel for τ b to decay into SM particles.
On the other hand, from (3.8) we find that the DR candidate axion a b now also necessarily couples to the topological term of the visible sector gauge theory including QCD. In this case there will be further constraints on our model. While a b remains essentially massless after moduli stabilisation, QCD effects will now generate a potential for a b . This changes the cosmological role of this axion: while a massless axion can only contribute to DR, a massive axion can exhibit both relativistic and non-relativistic populations which can be identified as DR and Dark Matter (DM) respectively. While axions a b produced by modulus decay will form DR, a population of axion DM will be produced through the misalignment mechanism. The amount of axion DM produced crucially depends on the coupling to QCD. The relevant terms in the effective Lagrangian are (see e.g. [33,34]): Canonically normalising all fields, the axion couples to QCD as where g 2 = c a τ b /4π and f a b = K T bTb /2π is the axion decay constant. In this case vacuum realignment production of axion DM exceeds observational bounds if f a b /c a > 10 12 GeV [24]. The bound can be relaxed if the initial misalignment angle is tuned. Alternatively, axion DM could be diluted due to some late time entropy release. There is also a lower bound on f a b /c a : to avert excessive cooling of stars an axion coupling to QCD has to also satisfy f a b /c a > 10 9 GeV. Beyond the gauge kinetic function, the matter Kähler metric will also enter the expressions for decay rates of the bulk volume modulus. For matter living on intersections of D7-branes wrapping cycles τ a , the modulus-dependence of the Kähler metric is expected to be of the form [35] (also see [20]). This Kähler metric is also appropriate for Higgs fields, as long as they arise from chiral superfields.
Before calculating the decay rates of the large cycle modulus into visible sector fields, it is worth checking which decay channels are kinematically allowed. In particular, for the non-sequestered setup considered here, the modulus T a corresponding to the visible sector cycle acquires an F-term and visible sector soft terms are not suppressed w.r.t. the gravitino mass [16,36]. The relevant scales are Correspondingly, decays of the bulk modulus into matter scalars (and the heavy Higgs) are kinematically forbidden. Thus, in the given setup the volume modulus can only reheat the SM by decaying into gauge bosons and the light Higgs. There is another observation which can be made from (3.11). To avoid the Cosmological Moduli Problem, we require all moduli masses including m τ b to satisfy m mod 10 TeV. From (3.11) it then follows that m sof t 10 TeV, and our setup forces us to consider high scale supersymmetry only.

Predictions for Dark Radiation
The decay rates can be calculated from the low energy effective Lagrangian. The relevant terms in the Kähler potential and gauge kinetic function are After D-term stabilisation we integrate out T a by replacing T a = c a T b . Rates for decays of the bulk volume modulus can then be determined as Decays into DR: (3.14) Decays into SM: where N g is the number of gauge bosons, β is the angle describing the ratio of Higgs VEVs via tan β and we defined γ ≡ τ a τ a + h Re(S) .
Various values for γ correspond to the following regimes. For the case that that gauge fluxes do not contribute to the gauge kinetic function (h = 0) one finds γ = 1. For |γ| 1 the gauge kinetic function is dominated by the flux-dependent part h Re(S). For γ 1 we require a delicate cancellation between contributions from τ a and h Re(S).
The decay rate into bulk axions is unchanged relative to the sequestered case (2.6) while the decay rate into Higgs fields is slightly modified. Using all the above decay rates we obtain the following expression for the amount of DR: In the following, we set N g = 12, which corresponds to the twelve gauge bosons of the SM and we also assume a minimal matter spectrum with only one light Higgs. As we are mainly interested in setups which minimise ∆N ef f , we choose sin(2β) = 1. This assumptions is further motivated by the fact that high scale supersymmetry suggests sin(2β) = 1 ( [28,[37][38][39]. The prediction for ∆N ef f shows some mild dependence on g * . We summarised possible values for g * together with the corresponding reheating temperatures and moduli masses T d g * 5 · 10 5 50 MeV 10.75 2 · 10 6 300 MeV 61.75 10 7 3 GeV 75.75 ≥ 2 · 10 8 ≥ 200 GeV 106.75 Table 1: Number of degrees of freedom corresponding to different reheating temperatures obtained for different masses of the modulus τ b . The reheating temperature is determined as Most interestingly, the constraints on z due to bounds on ∆N ef f can be relaxed compared to the sequestered scenario if the modulus decays significantly into gauge bosons. Even if decays into the light Higgs are subdominant, z 1, we find that ∆N ef f < 0.79 can be satisfied if γ > 0.8 (for g * = 106.75) or γ > 1.2 (for g * = 10.75). Hence, current bounds on DR can be satisfied as long as the gauge coupling is dominantly set by τ a .
If bounds on ∆N ef f become stricter in the future, the Higgs sector remains unconstrained if we allow for a mild cancellation between τ a and h Re(S) in the gauge kinetic function. For ∆N ef f < 0.1 we require γ > 2.2 for g * = 106.75 (γ > 3.2 for g * = 10.75), corresponding to a fine-tuning between τ a and h Re(S) to 1 part in 2 (3).
On the other hand, if decays into gauge bosons are prohibited, we have γ = 0 and restrictions on the moduli-Higgs couplings are even more severe than in the sequestered case.
There are further constraints as a b becomes massive through its coupling to QCD (3.21) as the axion will contribute to DM through the vacuum realignment mechanism. If the PQ symmetry is broken before inflation, the initial misalignment angle θ i = a b,initial fa b /ca ∈ [−π, π) is homogeneous in our patch. The axion relic density is then (see e.g. [24]): At most the axion density can represents all of cold dark matter, whose density was measured as Ω DM h 2 = 0.1199 [1]. Thus, for generic initial misalignment angles there is an overproduction of axion DM if f a b /c a 10 12 GeV. Using (3.12) we find: and thus we only arrive at an acceptable axion DM relic if θ i is tuned small. This tuning can be justified anthropically [25] and we find that θ i ∼ 10 −2 is sufficient to evade DM bounds.
Bounds on isocurvature perturbations can lead to even stricter constraints for the QCD axion. If the measurement of B-modes in the CMB by the BICEP2 experiment [2] is explained by primordial tensor modes, the QCD axion candidate with f a b /c a ∼ 10 16 GeV will source excessive isocurvature perturbations [23]. In consequence, the scenario described in this section would be ruled out.
A possible way out is the existence of another axionã with a decay constant 10 9 GeV < fã < 10 12 GeV which couples to QCD: In this case the QCD axion is mainly given byã, which can evade all bounds. For one, it leads to an acceptable DM density without the need to tune θ i . Further, the decay constant is lower than the Hubble scale of inflation H I ∼ 10 14 GeV suggested by the BICEP2 results, which implies that the PQ symmetry is intact during inflation. In this case the axion will not source excessive isocurvature perturbations. Beyond the QCD axion there will also be a combination of axions, dominantly given by a b , which will remain light and contribute to DR. This axion is then unaffected by isocurvature and DM bounds.

Large Volume Scenario with D-term stabilisation and string loop corrections
In the previous case we employed D-terms to stabilise the SM branes in the geometric regime. However, to arrive at the correct gauge coupling, this came at the expense of a potentially severe fine-tuning of fluxes. In the following, we will examine scenarios where some cycles are stabilised by string loop effects. Explicit examples of LVS models involving string loop stabilisation have been studied for fibred Calabi-Yau manifolds, and it is these spaces which we will analyse in this section. Thus, to be specific, we consider Calabi-Yau three-folds with volume of the form If some cycles have been stabilised by D-terms, we assume that this is the volume after the moduli stabilised by D-terms have been integrated out [32,40]. The overall volume V and τ np are stabilised using the standard LVS procedure, such that V ∼ = √ τ 1 τ 2 is large. There remains one flat direction corresponding of simultaneous changes in τ 1 and τ 2 such that V is unchanged. It is this mode (denoted by χ in what follows) which is fixed by string loop corrections as in [41].
To study reheating in this setup we need to specify the visible sector. Here, we model the visible sector by D7-branes on the fiber τ 1 . To arrive at an acceptable value for the physical gauge coupling, the volume of the fiber has to be small compared to τ 2 given that V ∼ = √ τ 1 τ 2 is large. Correspondingly, we need to study the above setup in the "anisotropic limit" τ 2 τ 1 τ np also discussed in [32,40]. Alternatively, the visible sector could be realised by D7-branes on a further cycle τ a , whose volume is coupled to the size of τ 1 by D-terms (see [32] for such a realisation). In this case the constraints on the volume of τ 1 can be relaxed, as long as the D-term conditions lead to the correct size of τ a . This corresponds to a tuning of fluxes possibly much weaker than in the previous section. In any case, as the visible sector gauge coupling depends on τ 1 in both cases, our results of this section will be the same for both situations.
Reheating proceeds via decays of the lightest modulus. For the case of the fibred Calabi-Yau considered here there are two moduli which are lighter than all the other closed string moduli. These are the bulk volume and the mode χ orthogonal to V with masses respectively [36]. If both moduli have comparable masses, we would need to examine the decay of both moduli simultaneously, which complicates the analysis considerably. To simplify the situation, we will thus only consider the case where χ is lighter than the volume mode, such that the latter can be integrated out. For this to be the case, τ 1 cannot be fixed too small. There is a second reason why τ 1 should not be chosen too small. While τ 1 is a fiber modulus and thus should not give rise to a non-perturbative superpotential of the form e −aT 1 due to its zero mode structure, this is not necessarily true in the presence of fluxes. These could lift some of the zero modes such that a non-perturbative superpotential is generated. Thus τ 1 should be large enough that contributions of the form e −aτ 1 in the scalar potential can be safely ignored. There is one more point worth mentioning before we study predictions for ∆N ef f in this setup. For every cycle τ i that is stabilised by string loop corrections its associated axion a i will remain light, as perturbative effects cannot generate a potential for axions due to the shift symmetry. Thus, in setups where some cycles are stabilised using stringloop effects, there will be additional light axions which can only contribute positively to ∆N ef f . Specifically, for the setup considered here, there will be two light axions a 1 = Im(T 1 ) and a 2 = Im(T 2 ). As we saw in the previous section, if one of the axions couples to visible sector gauge fields and, in particular, to QCD, there will be further constraints on this setup. As we realise the visible sector on D7-branes wrapping τ 1 , the axion a 1 will couple to visible sector gauge fields at tree-level since After canonically normalising all fields we have: where f a 1 = K T 1T1 /2π. On the other hand, the axion-like particle a 2 will remain light and only contribute to DR.

Predictions for Dark Radiation
We begin by analysing the decay rates of the lightest modulus χ into axions, which can be derived from the Kähler potential K = −2 ln V. As there are two light axions in the given case, we present this in some more detail. In particular, we find the following kinetic terms for the moduli τ 1 , τ 2 and their corresponding axions a 1 and a 2 : To arrive at a Lagrangian involving the mode χ orthogonal to the bulk volume, we integrate out the stabilised volume by setting τ 2 = α −1 Vτ −1/2 1 . Up to volume-suppressed terms the fields are then canonically normalised by (3.28) In the following, we will drop all primes on the canonically normalised axions. From this Lagrangian the decay rate of χ into the two light axions can be easily derived.
The decay rate into Higgs fields is obtained from the following part of the Kähler potential: where we realised the visible sector by D7-branes wrapping the cycle τ 1 . 8 Decays into any other Standard Models fields are suppressed as before.
Overall, we find the following decay rates for the modulus χ: Decays into DR:

34)
8 For a visible sector given by D7-branes wrapping a different cycle τa the gauge kinetic function has to be modified as fvis = Ta + hS. However, if the cycle τa is stabilised w.r.t. τ1 via D-terms, such that Ta = cT1, there will still be a direct coupling between χ and visible sector gauge fields.  .35)) for an LVS model with cycles stabilised by string loop corrections and D-terms, for (a) g * = 10.75 and (b) g * = 106.75. As before, the plots have been produced choosing sin(2β) = 1, but they can be reinterpreted for general values of sin(2β). To this end we define an effective parameter z 2 = sin 2 (2β)z 2 and relabel z →z on the horizontal axis.
where γ is defined as before: Thus, for γ = 1, the gauge coupling is dominantly set by τ 1 alone. For |γ| < 1 the gauge kinetic function is dominated by the flux-dependent part h Re(S). For γ > 1 we require a cancellation between contributions from τ a and h Re(S). Using the above we arrive at the following expression for ∆N ef f : where we have set N g = 12, which corresponds to the twelve gauge bosons of the SM. As before, we also assumed a minimal matter spectrum with only one light Higgs and set sin(2β) = 1 to maximise the decay rate into the light Higgs. The effective number of species g * depends on the reheating temperature and thus on the modulus mass, and it can take values between g * = 10.75 and g * = 106.75. We present contour plots of ∆N ef f for the two extreme values for g * as a function of z and γ in figure 2.
As a result we find that constraints on the Higgs sector from DR can be significantly relaxed if the SM can be reheated via decays of the lightest modulus into gauge bosons.
To obtain ∆N ef f < 0.79, we require γ > 0.9 (for g * = 106.75) or γ > 1.3 (for g * = 10.75) if all constraints on z are lifted. Consequently, this scenario satisfies current DR bounds when the visible sector gauge coupling is dominantly set by τ 1 .
If bounds on ∆N ef f become stricter in the future, the Higgs sector remains unconstrained if we allow for a mild cancellation between τ 1 and h Re(S) in the gauge kinetic function. For ∆N ef f < 0.1 we require γ > 2.5 for g * = 106.75 (γ > 3.6 for g * = 10.75), corresponding to a fine-tuning between τ a and h Re(S) to 1 part in 3 (4).
However, as the DR candidate axion a 1 couples to QCD, there will also be further constraints. The discussion is analogous to the one in section 3.1, so we do not give details here. In particular, for a generic initial misalignment angle, the vacuum realignment mechanism produces too much axion DM for decay constants f a 1 10 12 GeV. Here we have: As the visible sector is realised on τ 1 , we require τ 1 ∼ 25 for a realistic gauge coupling and the decay constant is f a 1 ∼ 10 16 GeV. Thus our setup requires a tuning of the initial misalignment angle to avert DM overproduction, which can be justified anthropically.
If the BICEP2 results [2] are explained by primordial tensor modes, the situation is far more drastic. In this case the setup described in this section is ruled out, as a QCD axion with f a 1 ∼ 10 16 GeV leads to excessive isocurvature perturbations [23]. This can be avoided if the model gives rise to an additional axion with a lower decay constant which takes over the rôle of the QCD axion as described before in section 3.1.

Large Volume Scenario and stabilisation by non-perturbative effects
We will now describe DR predictions for the case where the visible sector cycle is stabilised by non-perturbative effects. We will find that DR bounds are more restrictive on this scenario than in the sequestered case. Correspondingly, our analysis here will be less detailed than our examinations in the previous sections.
Typically there is a conflict between the presence of both a chiral visible sector and a non-perturbative effect on the same 4-cycle [42]. Chiral intersections between the visible sector brane and the non-perturbative effect induce superpotential terms for visible sector fields, which generate VEVs for these fields and break the visible sector gauge theory. However, in [43] it was pointed out that D-brane instantons carrying flux can relieve this tension: fluxes can render the instanton superpotential e −T gauge-invariant without the presence of any visible sector fields. Thus it is in principle possible to realise a chiral visible sector on a cycle stabilised by non-perturbative effects.
Here, we will analyse a simple toy model which nevertheless exhibits all the necessary features. To be specific, we consider a compactification with a volume of Swiss-Cheese type: Generalisations to setups with more than one cycle of type τ s are straightforward. The visible sector will be realised by D7-branes wrapping τ s . At the same time τ s will be wrapped by an E3 instanton or D7-branes exhibiting gaugino condensation, thus giving a non-perturbative contribution W np to the superpotential: Here a s is a model-dependent parameter which depends on the non-perturbative effect wrapping τ s : For the case of an E3 instanton we have a s = 2π while for for gaugino condensation on a stack of N D7-branes we have a s = 2π N . The prefactor A s depends on the dilaton and the complex structure moduli and is a constant at this stage.
The moduli τ b and τ s will be stabilised by the standard LVS procedure. As before, the lightest modulus is τ b and its axion partner is essentially massless (2.2).

Dark Radiation predictions
Here we will analyse the rates for decays of τ b into axions a b vs. SM fields.
For one, realising the visible sector on τ s leads to superpartners which are heavier than τ b : m 1/2 ∼ m sof t ∼ M P /V (see e.g. [15]). Hence decays into matter scalars, the heavy Higgs and gauginos are kinematically forbidden.
As before, decays into the light Higgs will arise through the Giudice-Masiero term with a decay rate of the form The exact expression for decay rate into the light Higgs will depend on the moduli dependence of the Kähler metric for Higgs fields. We do not give more detail as we do not expect any improvement compared to previous sections. We will be most interested in the decay rate into gauge bosons, which can be calculated given the gauge kinetic function f vis = T s + hS . To study decays of τ b we need the effective theory for τ b which is obtained by integrating out τ s . By minimising the F-term potential w.r.t. τ s one obtains (see e.g. [4,44]): 9 Most importantly, this introduces a dependence of the tree-level visible sector gauge coupling on ln V ∼ ln τ b . The rates for the decay channels of interest are then: Decays into DR: Decays into SM: where γ = τs τs+h Re(S) is defined as in the previous sections, and N g is the number of generators of the SM gauge group. 9 Here, the F-term potential is given by (see e.g. [44]) The decay rate into axions is unchanged compared to the sequestered scenario. Further, we find that the decay rate into Higgs fields is not parametrically different from the expressions found in previous sections. In contrast, the decay rate into SM gauge bosons is suppressed by (a s τ s ) −2 compared to the cases where the visible sector cycle was stabilised by D-terms or string loops.
It is then easy to see that this construction is more constrained by DR bounds than the setups considered in 3.1 and 3.2. As the visible sector is realised on τ s , we require τ s ∼ 25 for an acceptable gauge coupling. In addition a s = 2π or 2π N depending on the non-perturbative effect sourced by τ s . Also, unless there is tuning between τ s and h Re(S) we have γ ∼ 1. Then it follows that the decay rate into gauge bosons is typically suppressed w.r.t. the decay rate into axions. This conclusion would only fail if we have N 100, corresponding to a gauge group SU (N 100) on the stack of D7-branes exhibiting gaugino condensation. Alternatively, one could tune γ to adjust the amount of DR. As the amount of DR is not expected to have an effect on the development of life, such a tuning cannot be justified anthropically.

Comparison with results from previous work
Further extensions of LVS models have been suggested in [45] and [10], where a nonsequestered LVS is combined with poly-instanton corrections to the superpotential. The authors consider a scenario with a swiss-cheese CY with volume V = (η 1 τ 1 ) 3/2 − (η 2 τ 2 ) 3/2 − (η 3 τ 3 ) 3/2 . Two separate stacks of D7-branes wrapping the 4-cycle τ 2 yield a superpotential with terms e −f i (race-track model), where the gauge-kinetic function is of the form f i = T 2 + C i e −2πT 3 + S due to gaugino condensations and Euclidean D3-instantons on the nonrigid cycle τ 3 . Additionally, the VEV of the flux-superpotential is assumed to be zero.
This race-track model allows to construct (large volume) minima by integrating out the heaviest modulus T 2 near the supersymmetric locus, ∂ T 2 W np = 0. Hence, one is left with an effective superpotential W ef f which is small due to its exponential suppression by the vev of τ 2 . The stabilisation of T 1 and T 3 then proceeds as in the usual LVS.
In our paper we do not consider this scenario any further since no substantial improvement of DR bounds relative to more conventional LVS constructions is expected. The main hope for such an improvement is associated with modulus decays to gauginos which, according to [10], are very light. However, the corresponding rate Γ 1/2 ∼ where M 1/2 = g/f is the physical gaugino mass, around the VEV of V: V = V 0 + δV: We use the fact that both f and g do not depend on V more strongly than through some power, f ∼ V α and g ∼ V β . This implies that, at most f (V) ∼ f (V)/V, and similarly for g. Furthermore, one has to recall that the modulus V and the corresponding canonically normalised field φ are related by V ∼ exp(φ/M p ). With the expansion φ = φ 0 + δφ one then finds the following parametric form of the Lagrangian relevant for the three-particle vertex and hence for the decay: Now, we normalise canonically, λ → λ/ √ f , and use equations of motion in first term, i / ∂λ = M 1/2 λ. This gives a contribution of the order of from both the first and second term above. From here, one can read off the decay rate (up to some numerical factors). Due to its suppression via the mass hierarchy, one can safely neglect this channel.

Large Volume Scenario with flavour branes
In the previous sections we examined how the lightest modulus of LVS constructions can be coupled most effectively to the visible sector. Here we will examine the situation where the lightest modulus reheats the visible sector fields via intermediate states. We will argue that gauge bosons arising from the worldvolume theory of so-called flavour branes are ideal candidates for such intermediaries. Flavour branes are (stacks of) 7-branes in the geometric regime going through the singularity at which the SM is geometrically engineered. They are known since the very early days of 'model building at a singularity' [46] and can also be viewed as a tool for generating (approximate) global flavour symmetries of the SM. 10 Flavour branes wrap bulk cycles such that for a large bulk volume the gauge theory on their worldvolume is extremely weakly coupled. There will be visible sector states charged under the flavour brane gauge group. This gauge theory has to be spontaneously broken such that, at low scales, a global symmetry of visible sector states emerges. For state-of-the-art string model building employing flavour branes see [17].
The setup which we are considering in this case is as follows. The Calabi-Yau exhibits a large bulk cycle and a small blow-up cycle giving rise to a non-perturbative effect. These cycles are stabilised by the standard LVS procedure. The visible sector is realised by D3branes at a singularity as in the sequestered case. However, in addition there are flavour branes, which wrap the bulk cycle but also intersect the singularity. A globally consistent realisation of such a setup in Calabi-Yau orientifolds is described in [17]. As we model the visible sector by D3-branes at a singularity, supersymmetry breaking is sequestered and gravity-mediated soft terms are suppressed w.r.t. the gravitino mass: m sof t ∼ M P /V 2 ∼ m 3/2 /V. However, flavor branes may affect these soft terms.

Reheating in LVS models with flavour branes
We now review the important steps in the cosmological history of the universe, which lead to the reheating of the SM in our setup.
1. As in the previous scenarios, the energy density of the universe after inflation is dominated by the lightest modulus, which is the volume modulus τ b in LVS models 11 . In the following, we wish to reheat the visible sector fields via gauge bosons A µ on flavour branes. If this scenario is to reheat the visible sector more efficiently (and thus lead to a lower ∆N ef f ) than in the sequestered setup without flavour branes, the decay of τ b into pairs of A µ should be the dominant decay channel of τ b . In the following we will proceed under this assumption. The decay rate Γ τ b →AµAµ can be determined from the tree-level interaction between τ b and A µ captured by the supersymmetric Lagrangian term f f l W α W α , where the gauge kinetic function for flavour branes wrapping a bulk cycle is given by f f l = T b . 12 The resulting decay rate is given by where N f is the number of generators of the flavour brane gauge theory. After τ b has decayed the energy density of the universe is then dominated by the gauge bosons A µ and the axions.
2. The subsequent evolution of the universe then crucially depends on the mass m A of the flavour brane gauge bosons. Hence we will now examine the bounds on m A .
(a) The upper bound on the flavour brane gauge boson mass is given by m A = m τ b /2, as A µ are then produced at threshold. To determine the subsequent development of the universe, we determine the decay rate of A µ into SM particles.
In particular, there are SM fermions which are charged under the flavour brane gauge group and the decay rate of A µ into these fermions is given by where α f ∼ 1/τ b ∼ V −2/3 . One can now easily check that for a wide range of masses m A below threshold the flavour brane gauge bosons decay into SM fields as soon as they are produced by decays of τ b : The flavour brane gauge bosons then decay into SM fields instantaneously.
(b) If m A V −7/3 m τ b , the flavour brane gauge bosons will not decay instantaneously, but form a population of highly relativistic particles carrying a significant fraction of the energy density of the universe. In the end these particles still have to reheat the SM. An interesting question for the following evolution of the universe is whether the flavour brane gauge bosons become non-relativistic before they decay into SM degrees of freedom. If they become non-relativistic, their energy density will scale as matter with time, while any DR produced by the decay of τ b earlier will scale as radiation. As a consequence the fraction of the energy density in A µ over the energy density in DR would grow. As the flavour brane gauge bosons would eventually decay into SM fields, the relic abundance of SM fields would be enhanced with respect to the relic abundance of the axionic Dark Radiation. As a result ∆N ef f could be further suppressed.
However, one can show that the population of A µ will always remain relativistic until they decay. Initially the population of flavour brane gauge bosons is relativistic with energy density ρ A = π 2 30 g * ,A (T )T 4 . If the temperature falls to T ∼ m A the gauge bosons become non-relativistic. One can now check that T d at which the gauge bosons decay into SM fields is always higher than m A . To determine T d we note that A µ will decay when Γ Aµ→ff = H. As the gauge bosons are highly relativistic initially, we need to correct the decay rate into SM fermions by multiplying by a time-dilation factor for relativistic particles. This can be justified a posteriori, as we will show that A µ stay relativistic until they decay. The decay rate (4.2) is modified as The decay temperature T d can then be determined using the following equations: leading to We recall that for the gauge bosons not to decay instantly when produced their mass had to be small: It then follows that T d > m A and flavour brane gauge bosons always remain relativistic.
(c) While the upper bound on the gauge boson mass m A is set by the kinematics of the decay of τ b we want to examine whether there is a cosmological lower bound on m A . In particular, we will require that when reheating the SM through the decay of flavour brane gauge bosons, the reheating temperature of the SM is T SM O(1) MeV to allow for standard BBN. To determine the decay temperature of the SM we recall that A µ will decay into SM fields when Γ Aµ→ff = H.
We have To relate T d,A to T d,SM we recall that the comoving entropy density s = g * a 3 T 3 is conserved when A µ decays: (4.10) Putting (4.8) and (4.9) together one finds Standard BBN requires T d O(1) MeV and thus we find the following lower bound on m A : Given that we require V 10 14 to evade the Cosmological Moduli Problem, it is clear that the cosmological lower bound on m A is very low. As a result, our setup can successfully reheat the SM -for a wide range of masses from threshold to the lower limit shown above.
While constraints from reheating allow very light weakly coupled vector bosons there will be further constraints on the parameter space of such particles from collider experiments and precision measurements.
Beyond cosmological constraints, there are also consistency conditions on the string construction. The setup of visible sector and flavor branes has to satisfy local tadpole cancellation conditions. In addition, the local D-brane charges of the flavour branes at the intersection locus with the visible sector have to originate from restrictions of charges of globally well-defined D7-branes. While these consistency conditions do not determine a unique setup of allowed flavour branes, they constrain the number of flavour branes allowed given a particular visible sector [17].

Predictions for Dark Radiation
Here we determine the decay rates of the lightest modulus into Dark Radiation and Standard Model fields. The lightest modulus is the bulk volume modulus as in the sequestered case or in section 3.1. The rate of decays of τ b into its associated axion can be determined from K = −3 ln T b +T b and gives the familiar result obtained before (2.6).
As argued before, decays of τ b into gauge bosons on flavour branes lead to a direct reheating of the Standard Model. As flavour branes wrap bulk cycles, there is a treelevel coupling between τ b and the gauge bosons on the flavour brane through the kinetic Last, there can also be direct decays of the volume modulus into visible sector matter fields. As described in section 2 the dominating decay channel is given by the interaction of τ b with Higgs fields, which arises from the Giudice-Masiero term (zH u H d + h.c.) in the Kähler potential. In contrast to the non-sequestered setups studied before, here both Higgs scalars are light enough to be produced by decays of τ b leading to a decay rate (2.7).
Overall, we find the following decay rates for the volume modulus: Decays into DR: Decays into SM: (4.14) 15) where N f is the number of generators of the flavour brane gauge group. Thus we find the following expression for the effective number of neutrino species: ∆N ef f = 43 7 10.75 g * (T d )  We plot ∆N ef f as a function of z and N f , with g * = 75.75, in figure 3. One finds that the bound ∆N ef f < 0.79 can be achieved without any restrictions on the Higgs sector as long as N f ≥ 9. For z = 1 the DR bound ∆N ef f < 0.79 requires at least N f ≥ 5 gauge bosons. Thus for a number of flavour branes as small as n < 10 current bounds on DR can be easily met.
If bounds on ∆N ef f become more restrictive, we have the following options. If we do not wish to impose constraints on the Higgs sector of the model, we require a larger number of generators on flavour branes. A table of the minimum numbers of gauge bosons needed for fixed z given an upper bound on ∆N ef f is shown in 2. However, conditions for globally consistent models restrict the maximum number of flavour branes for a given visible sector [17] and thus place an upper limit on the allowed number of gauge bosons. The exact constraints on the numbers of flavour branes will depend on the details of the individual model. While we cannot be more specific it follows that lower bounds on ∆N ef f cannot necessarily be evaded by simply introducing more flavour branes.

Conclusion
In this paper we examined predictions for the amount of Dark Radiation (DR) for string models employing the scheme of moduli stabilisation known as the Large Volume Scenario (LVS). By analysing the ratio of decays of the lightest modulus into axions vs. Standard Model particles in these setups we study contributions to the effective number of neutrino species ∆N ef f produced during reheating.
We find that DR bounds on LVS models are considerably relaxed if the lightest modulus can reheat the Standard Model by decaying into gauge bosons. We consider setups where the modulus couples directly to visible sector gauge bosons and we also examine models where the modulus decays into gauge bosons on flavour branes which subsequently reheat the Standard Model.
In the first case we find that models can evade current DR bounds for natural values of parameters. In particular, we find ∆N ef f < 0.79 as long as the visible sector gauge coupling 4π/g 2 = τ vis +h Re(S) is dominantly set by τ vis . However, if bounds on DR become stricter in the future, the parameter space is increasingly constrained. In particular, the amount of DR can be reduced if one allows for a cancellation between τ vis and h Re(S). For a mild tuning of 1 part in 2 between τ vis and h Re(S) the amount of DR can be reduced to ∆N ef f ∼ 0.1 − 0.2.
We also observe that when coupling the lightest modulus Φ to visible sector gauge bosons, the model is necessarily non-sequestered and m sof t m Φ . As we require m Φ > O(10) TeV to solve the Cosmological Moduli Problem we have m sof t O(10) TeV, so that we are led to a regime of high scale supersymmetry.
In addition, by coupling the lightest modulus -a saxion -to visible sector gauge bosons, the DR candidate axion is coupled to the topological term of the visible sector gauge theory and takes the rôle of the QCD axion. In particular, the axion a couples to QCD as leading to an overproduction of axion Dark Matter through the misalignment mechanism. This can be averted if the initial misalignment angle is tuned to θ i ∼ 10 −2 . Further, if the recent BICEP2 results [1] are explained by primordial gravitational waves setups with a as the QCD axion are ruled out by isocurvature bounds. A possible way out is the existence of an additional axion with a decay constant below the scale of inflation which can take over the rôle of the QCD axion.
For models with flavour branes, we find that current DR bounds (∆N ef f < 0.79) can be satisfied if flavour branes give rise to N g = 5 − 9 gauge bosons. This scenario requires that the gauge theory on the flavour branes is broken in such a way, that the gauge bosons stay light enough to be produced by decays of the lightest modulus. The DR axion does not couple to QCD in this case and there is no immediate problem with overproduction of axion DM or isocurvature bounds. However, a realistic model would require an additional axion which would take the rôle of the QCD axion.
One possibility to further enhance the decay rate of the lightest modulus to the SM and thus to evade possible stronger DR bounds is the following: Recall that the special role of the Higgs in the decays to the SM arises because the supersymmetric Higgs sector allows for a Giudice-Masiero type term in the Kähler potential. Such a term cannot arise for other SM fields due to chirality. Instead of duplicating the Higgs sector, one might consider singlets, which come naturally in many string constructions and can also play the rôle of right-handed neutrinos. Allowing for many such fields with appropriate Giudice-Masiero terms and couplings to Higgs and lepton-doublets has the potential to naturally open up further decay chanels to the SM.
Overall, Dark Radiation is a powerful tool to constrain string models of particle physics based on the LVS. Moreover, as we have seen in great detail, some of the most natural settings with natural values of model parameters lead to Dark Radiation predictions just below the present observational limits.
This paper was submitted simultaneously to the related work [48].