General Analysis of Dark Radiation in Sequestered String Models

We perform a general analysis of axionic dark radiation produced from the decay of the lightest modulus in the sequestered LARGE Volume Scenario. We discuss several cases depending on the form of the Kahler metric for visible sector matter fields and the mechanism responsible for achieving a de Sitter vacuum. The leading decay channels which determine dark radiation predictions are to hidden sector axions, visible sector Higgses and SUSY scalars depending on their mass. We show that in most of the parameter space of split SUSY-like models squarks and sleptons are heavier than the lightest modulus. Hence dark radiation predictions previously obtained for MSSM-like cases hold more generally also for split SUSY-like cases since the decay channel to SUSY scalars is kinematically forbidden. However the inclusion of string loop corrections to the Kahler potential gives rise to a parameter space region where the decay channel to SUSY scalars opens up, leading to a significant reduction of dark radiation production. In this case, the simplest model with a shift-symmetric Higgs sector can suppress the excess of dark radiation $\Delta N_{eff}$ to values as small as 0.14, in perfect agreement with current experimental bounds. Depending on the exact mass of the SUSY scalars all values in the range 0.14 $\lesssim \Delta N_{eff} \lesssim$ 1.6 are allowed. Interestingly dark radiation overproduction can be avoided also in the absence of a Giudice-Masiero coupling.


Introduction
According to the cosmological Standard Model (SM), neutrinos were in thermal equilibrium at early times and decoupled at temperatures of order 1 MeV. This decoupling left behind a cosmic neutrino background which has been emitted much earlier than the analogous cosmic microwave background (CMB). Due to the weakness of the weak interactions, this cosmic neutrino background cannot be detected directly, and so goes under the name of 'dark radiation'. Its contribution to the total energy density ρ tot is parameterised in terms of the effective number of neutrino-like species N eff as:  [1] which is in tension at 2.5σ with the Hubble Space Telescope (HST) value H 0 = (73.8 ± 2.4) km s −1 Mpc −1 (68% CL) [2]. Hence the Planck 2013 estimate of N eff with this HST 'H 0 prior' is N eff = 3.62 +0.50 −0.48 (95% CL) [1] which is more than 2σ away from the SM value and gives ∆N eff ≤ 1.07 at 2σ.
However the HST Cepheid data have been reanalysed by [3] who found the different value H 0 = (70.6 ± 3.3) km s −1 Mpc −1 (68% CL) which is within 1σ of the Planck 2015 estimate H 0 = (67.3 ± 1.0) km s −1 Mpc −1 (68% CL) [4]. Hence the Planck 2015 collaboration performed a new estimate of N eff without using any 'H 0 prior' and obtaining N eff = 3.13 ± 0.32 (68% CL) [4] which is perfectly consistent with the SM value and gives ∆N eff ≤ 0.72 at around 2σ. This result might seem to imply that extra dark radiation is ruled out but this naive interpretation can be misleading since larger N eff corresponds to larger H 0 and there is still an unresolved controversy in the direct measurement of H 0 . In fact the Planck 2015 paper [4] analyses also the case with the prior ∆N eff = 0.39 obtaining the result H 0 = (70.6 ± 1.0) km s −1 Mpc −1 (68% CL) which is even in better agreement with the new HST estimate of H 0 performed in [3]. Thus we stress that trustable direct astrophysical measurements of H 0 are crucial in order to obtain reliable bounds on N eff .
N eff is also constrained by measurements of primordial light element abundances. The Planck 2015 estimate of N eff based on the helium primordial abundance and combined with the measurements of [5] is N eff = 3.11 +0.59 −0.57 (95% CL) giving ∆N eff ≤ 0.65 at 2σ [4]. However measurements of light element abundances are difficult and often affected by systematic errors, and so also in this case there is still some controversy in the literature since [6] reported a larger helium abundance that, in turn, leads to N eff = 3.58 ± 0.50 (99% CL) which is 3σ away from the SM value and gives ∆N eff ≤ 1.03 at 3σ. Due to all these experimental considerations, in the rest of this paper we shall consider ∆N eff 1 as a reference upper bound for the presence of extra dark radiation.
Extra neutrino-like species can be produced in any beyond the SM theory which features hidden sectors with new relativistic degrees of freedom (dof ). In particular, extra dark radiation is naturally generated when reheating is driven by the decay of a gauge singlet since in this case there is no a priori good reason to suppress the branching ratio into hidden sector light particles [7][8][9].
This situation is reproduced in string models of the early universe due to the presence of gravitationally coupled moduli which get displaced from their minimum during inflation, start oscillating when the Hubble scale reaches their mass, quickly come to dominate the energy density of the universe since they redshift as matter and finally reheat the universe when they decay [10,11]. In the presence of many moduli, the crucial one is the lightest since its decay dilutes any previous relic produced by the decay of heavier moduli.
Two important cosmological constraints have to be taken into account: (i) the lightest modulus has to decay before BBN in order to preserve the successful BBN predictions for the light element abundances [12]; (ii) the modulus decay to gravitini should be suppressed in order to avoid problems of dark matter overproduction because of gravitini annihilation or modifications of BBN predictions [13]. The first constraint sets a lower bound on the lightest modulus mass of order m mod 30 TeV, while a straightforward way to satisfy the second constraint is m mod < 2m 3/2 .
However in general in string compactifications the moduli develop a mass because of supersymmetry (SUSY) breaking effects which make the gravitino massive via the super Higgs mechanism and generate also soft-terms of order M soft . Because of their common origin, one has therefore m mod ∼ m 3/2 ∼ M soft . The cosmological lower bound m mod 30 TeV then pushes the soft-terms well above the TeV-scale ruining the solution of the hierarchy problem based on low-energy SUSY. An intriguing way-out is given by type IIB string compactifications where the visible sector is constructed via fractional D3-branes at singularities [14][15][16]. In this case the blow-up modulus resolving the singularity is fixed at zero size in a supersymmetric manner, resulting in the absence of local SUSY breaking effects. SUSY is instead broken by bulk moduli far away from the visible sector singularity. Because of this geometric separation, the visible sector is said to be 'sequestered' since the soft-terms can be suppressed with respect to the gravitino mass by ǫ = m 3/2 M P ≪ 1 [17]. A concrete example of sequestered SUSY breaking is given by the type IIB LARGE Volume Scenario (LVS) with D3-branes at singularities, which is characterised by the following hierarchy of masses [17]: This mass spectrum guarantees the absence of moduli decays to gravitini and allows for gaugino masses M 1/2 around the TeV-scale for m mod ∼ 10 7 GeV and m 3/2 ∼ 10 10 GeV.
On the other hand, SUSY scalar masses m 0 are more model dependent since their exact ǫ-dependence is determined by the form of the Kähler metric for visible sector matter fields and the mechanism responsible for achieving a dS vacuum. The general analysis of [18] found two possible ǫ-scalings for scalar masses: (i) m 0 ∼ M 1/2 corresponding to a typical MSSM-like scenario and (ii) m 0 ∼ m mod ≫ M 1/2 resulting in a split SUSY-like case with heavy squarks and sleptons. Following the cosmological evolution of these scenarios, reheating takes place due to the decay of the volume modulus which produces, together with visible sector particles, also hidden sector dof which could behave as extra dark radiation [7,8]. Some hidden sector dof are model dependent whereas others, like bulk closed string axions, are always present, and so give a non-zero contribution to ∆N eff . In fact, as shown in [19], the production of axionic dark radiation is unavoidable in any string model where reheating is driven by the moduli decay and some of the moduli are stabilised by perturbative effects which keep the corresponding axions light. Note that light closed string axions can be removed from the low-energy spectrum via the Stückelberg mechanism only for cycles collapsed to zero size since in the case of cycles in the geometric regime the combination of axions eaten up by an anomalous U (1) is mostly given by open string axions [19].
R-parity odd visible sector particles produced from the lightest modulus decay subsequently decay to the lightest SUSY particle, which is one of the main dark matter candidates. Due to their common origin, axionic dark radiation and neutralino dark matter have an interesting correlation [19]. In fact, by combining present bounds on N eff with lower bounds on the reheating temperature T rh as a function of the dark matter mass m DM from recent Fermi data, one can set interesting constraints on the (N eff , m DM )-plane. [19] found that standard thermal dark matter is allowed only if ∆N eff → 0 while the vast majority of the allowed parameter space requires non-thermal scenarios with Higgsino-like dark matter, in agreement with the results of [20] for the MSSM-like case.
Dark radiation production for the MSSM-like case has been studied in [7,8] which showed that the leading decay channels of the volume modulus are to visible sector Higgses via the Giudice-Masiero (GM) term and to ultra-light bulk closed string axions. The simplest model with two Higgs doublets and a shift-symmetric Higgs sector yields 1.53 ∆N eff 1.60, where the window has been obtained by varying the reheating temperature between 500 MeV and 5 GeV, which are typical values for gravitationally coupled scalars with masses in the range m mod ≃ (1 ÷ 5) · 10 7 GeV. These values of ∆N eff lead to dark radiation overproduction since they are in tension with current observational bounds. 1 Possible way-outs to reduce ∆N eff involve models with either a larger GM coupling or more than two Higgs doublets.
Due to this tension with dark radiation overproduction, different models have been studied in the literature. [21] showed how sequestered LVS models where the Calabi-Yau (CY) volume is controlled by more than one divisor are ruled out since they predict huge values of extra dark radiation of order ∆N eff ∼ 10 4 . On the other hand, [22] focused on non-sequestered LVS models where the visible sector is realised via D7-branes wrapping the large cycle controlling the CY volume. 2 In this way, the decay rate of the lightest modulus to visible sector gauge bosons becomes comparable to the decay to bulk axions, and so the prediction for ∆N eff can become smaller. In fact, the simplest model with a shift-symmetric Higgs sector yields ∆N eff ≃ 0.5 [22]. However this case necessarily requires high-scale SUSY since without sequestering M soft ∼ m 3/2 (up to loop factors), and so from (1.2) we see that the cosmological bound m mod ∼ m 3/2 √ ǫ 30 TeV implies GeV. Moreover in this case the visible sector gauge coupling is set by the CY volume V, α −1 SM ∼ V 2/3 ∼ 25, and so it is hard to achieve large values of V without introducing a severe fine-tuning of some underlying parameters. A possible way-out could be to consider anisotropic compactifications where the CY volume is controlled by a large divisor and a small cycle which supports the visible sector [23,24].
In this paper we take instead a different point of view and keep focusing on sequestered models as in [7,8] since they are particularly promising for phenomenological applications: they are compatible with TeV-scale SUSY and gauge coupling unification without suffering from any cosmological moduli and gravitino problem, they can be embedded in globally consistent CY compactifications [16] and allow for successful inflationary models [25] and neutralino non-thermal dark matter phenomenology [20]. Following the general analysis of SUSY breaking and its mediation to the visible sector performed in [18] for sequestered type IIB LVS models with D3-branes at singularities, we focus on the split-SUSY case where squarks and sleptons acquire a mass of order the lightest modulus mass: m 0 = c m mod with c ∼ O(1). We compute the exact value of the coefficient c for different split-SUSY cases depending on the form of the Kähler metric for visible sector matter fields and the mechanism responsible for achieving a dS vacuum. We find that the condition c ≤ 1/2, which allows the new decay channel to SUSY scalars, can be satisfied only by including string loop corrections to the Kähler potential [27,28]. However this relation holds only at the string scale M s ∼ 10 15 GeV whereas the decay of the lightest modulus takes place at an energy of order its mass m mod ∼ 10 7 GeV. Hence we consider the Renormalisation Group (RG) running of the SUSY scalar masses from M s to m mod and then compare their value to m mod whose running is in practice negligible since moduli have only gravitational couplings. Given that also the RG running of SUSY scalar masses is a negligible effect in split SUSY-like models, we find that radiative corrections do not alter the parameter space region where the lightest modulus decay to SUSY scalars opens up.
We then compute the new predictions for ∆N eff which gets considerably reduced with respect to the MSSM-like case considered in [7,8] since the branching ratio to visible sector particles increases due to the new decay to squarks and sleptons and the new contribution to the decay to Higgses from their mass term. We find that the simplest model with a shift-symmetric Higgs sector can suppress ∆N eff to values as small as 0.14 in perfect agreement with current experimental bounds. Depending on the exact value of m 0 all values in the range 0.14 ∆N eff 1.6 are allowed. Interestingly ∆N eff can be within the allowed experimental window also in the case of vanishing GM coupling Z = 0 since the main suppression of ∆N eff comes from the lightest modulus decay to squarks and sleptons. Given that a correct realisation of radiative Electro-Weak Symmetry Breaking (EWSB) in split SUSY-like models requires in general a large µ-term of order m 0 , the lightest modulus branching ratio into visible sector dof is also slightly increased due to its decay to Higgsinos. However this new decay channel yields just a negligible correction to the final prediction for dark radiation production. This paper is organised as follows. In Sec. 2 we review the main features of sequestered LVS models whereas Sec. 3 contains the main results of this paper since it analyses the predictions for axionic dark radiation. We present our conclusions in Sec. 4.

Sequestered LARGE Volume Scenario
In this section we shall present a brief review of sequestered type IIB LVS models with D3branes at singularities [16][17][18]. After describing the general setup of the N = 1 supergravity effective field theory, we outline the procedure followed to fix all closed string moduli in a dS vacuum which breaks SUSY spontaneously. We then list all the relevant D-and F-terms and the final results for the soft-terms generated by gravity mediation (anomaly mediation contributions can be shown to be negligible [17,18]).

Effective field theory setup
The simplest LVS model which leads to a visible sector sequestered from SUSY breaking is based on a CY manifold whose volume takes the form: is the 'big' divisor which controls the CY size, D s is a 'small' blowup modulus supporting non-perturbative effects and D SM is the 'Standard Model' cycle which collapses to a singularity where the visible sector D3-branes are localised. More precisely, explicit realisations in compact CY manifolds involve two identical shrinkable divisors D 1 and D 2 which are exchanged by the orientifold involution [16]. This gives rise to an orientifold even cycle D + = D 1 + D 2 and an orientifold odd cycle Both τ SM and Re(G) develop a vanishing VEV due to D-term stabilisation while the corresponding axions ψ SM and Im(G) are eaten up by two anomalous U (1)s [15]. The resulting low-energy N = 1 effective field theory is characterised by the following Kähler potential with leading order α ′ correction (without loss of generality we ignore from now on the orientifold odd modulus G): where S is the axion-dilaton,ξ ≡ ξRe(S) 3/2 , K cs (U ) is the tree-level Kähler potential for the complex structure moduli U and K matter is the Kähler potential for visible matter fields C which is taken to be: where the matter metric is assumed to be flavour diagonal and [18]: and the bilinear Higgs mixing term is proportional to the GM coupling Z which we allow to depend on S and U -moduli. Note that additional contributions to (2.2) can come from either an extra sector responsible for achieving a dS vacuum or from higher α ′ and g s corrections [27,28]. The superpotential takes instead the form: where W flux (U, S) is generated by three-form background fluxes, A(U, S) and a s depend on the D-brane configuration which generates non-perturbative effects and the matter superpotential W matter takes the form: where the µ-term and the Yukawa couplings Y αβγ can depend on the T -moduli only at non-perturbative level. Finally, the expression for the gauge kinetic function of the visible sector localised at the singularity τ SM → 0 is: where k a is a singularity-dependent coefficient.

de Sitter moduli stabilisation
The scalar potential receives several contributions from various effects which are suppressed by different inverse powers of the overall volume V. By taking the large volume limit V ≫ 1, the moduli can therefore be stabilised order by order in 1/V. All closed string moduli are stabilisedá la LVS by the following procedure: • At leading 1/V 2 order, the dilaton and the U -moduli are stabilised by background fluxes while τ SM shrinks to zero via D-term stabilisation and ψ SM is eaten up by an anomalous U (1) [15,16].
• At subleading 1/V 3 order, τ s and ψ s are stabilised by non-perturbative corrections to W while τ b is fixed at exponentially large values by the interplay between α ′ and non-perturbative effects. The scalar potential at 1/V 3 order looks like: where we included a model-dependent positive contribution V dS in order to obtain a dS solution. The minimum of this potential is at: with subleading V dS -dependent corrections. Following [18], we shall consider two possible mechanisms to generate V dS : 1. dS 1 case: hidden sector matter fields φ dS living on D7-branes wrapping D b acquire non-zero VEVs because of D-term stabilisation. In turn, their F-term scalar potential gives rise to a positive contribution which scales as V dS ∼ W 2 0 /V 8/3 and can be used to obtain a dS vacuum [16]. 2. dS 2 case: a viable dS vacuum can arise also due to non-perturbative effects at the singularity obtained by shrinking to zero an additional divisor D dS [29]. The corresponding Kähler modulus T dS = τ dS + iψ dS is again fixed by D-terms which set τ dS → 0 (ψ dS is eaten up by an anomalous U (1)). The new contribution to the superpotential W dS = A dS e −a dS (S+k dS T dS ) yields a positive term which scales as V dS ∼ e −2a dS Re(S) /V and can be used to develop a dS vacuum.
• At very suppressed e −V 2/3 /V 4/3 order, also ψ b develops a mass via T b -dependent nonperturbative contributions to W . For V 5·10 3 , the scale of this tiny contribution to V is smaller than the present value of the cosmological constant, and so no additional 'uplifting' term is needed.

F-and D-terms
The typical feature of models with D3-branes at singularities is the fact that the SM modulus T SM (together with the corresponding orientifold odd modulus G) does not break SUSY since its F-term is proportional to τ SM that shrinks to zero via D-term stabilisation: Therefore there is no local SUSY breaking effect and the visible sector is said to be sequestered from SUSY breaking which takes place in the bulk via the following non-zero F-terms (we write down only the leading order expressions) [18]: where the gravitino mass is m 3/2 = e K/2 |W | ∼ M P /V. Because of subleading S and Udependent corrections to V at 1/V 3 order, also the dilaton and complex structure moduli develop non-zero F-terms of order [18]: Moreover, the fields responsible for achieving a dS vacuum contribute to SUSY breaking via [18]: Further contributions to SUSY breaking come from non-zero D-terms. For non-tachyonic scalar masses, the visible sector D-term potential vanishes whereas for the two dS cases, the hidden sector D-term potential at the minimum scales as [18]: (2.14)

Soft SUSY breaking terms
Gravitational interactions mediate SUSY breaking to the visible sector generating softterms at the string scale. Let us summarise all the main results for the soft-terms.

Gaugino masses
Given the form of the gauge kinetic functions (2.7), the gaugino masses turn out to be universal and read:

Scalar masses
Scalar masses depend on the Kähler matter metric since their supergravity expression reads: It is therefore crucial to determine the exact moduli-dependence ofK α focusing in particular on the dependence on the volume V since F T b is the largest F-term. This dependence can be inferred relatively easily by requiring that the physical Yukawa couplingsŶ αβγ (we neglect here tiny non-perturbative T -dependent contributions to Y αβγ ): The expansion (2.4) ofK guarantees that (2.18) is exact at leading order, as required in the local case. The ultra-local limit is then realised only for c s = 1/3. Let us see how scalar masses get affected by the exact form ofK: • Local limit: in this case scalar masses are universal and do not depend on the sector responsible for getting a dS vacuum. Their leading order expression is generated by F T b and looks like: Note that non-tachyonic scalars require c s ≥ 1/3 and M 1/2 ≪ m 0 leading to a typical split-SUSY spectrum.
• Ultra-local limit: in this case scalar masses depend on the dS sector. Setting c s = 1/3 kills the leading contribution to (2.19) from F T b . Subleading effects depend on F U and F S which are volume-suppressed with respect to F T b since the dilaton and complex structure moduli are fixed supersymmetrically at leading order. Due to this cancellation, D-term contributions to scalar masses turn out to be the dominant effect for the dS 1 case leading to universal and non-tachyonic scalar masses of the form: This is again a split SUSY-like scenario. On the other hand, D-term contributions vanish in the dS 2 case where the leading effects generating scalar masses come from F U and F S that give: where the function c α (U, S) involves derivatives of f α (U, S) with respect to U and S [18]. In this case scalar masses are potentially non-universal and tachyonic depending on the exact functional dependence of f α (U, S). This situation reproduces a standard MSSM-like scenario. Dark radiation production in this case has been studied in [7,8].

µ and Bµ terms
The other soft-terms relevant for the computation of dark radiation production from moduli decays are the canonically normalisedμ and Bμ terms. These two terms receive contributions from both the Kähler potential and the superpotential. The more model-independent contribution from K is induced by a non-zero GM coupling Z in (2.3). It turns out that in each dS caseμ is always proportional to M 1/2 whereas Bμ scales as m 2 0 [18]: where c µ,K (U, S) and c B,K (U, S) are two tunable flux-dependent coefficients. Additional contributions toμ and Bμ can come from model-dependent non-perturbative effects which produce an effective µ-term in the superpotential of the form [18]: if the cycle associated to τ = Re(T ) is in the geometric regime while: if the cycle associated to τ = Re(T ) is in the singular regime, i.e. if τ → 0. These nonperturbative effects generate effectiveμ-and Bμ-terms in the low-energy action which can be parameterised as:μ where c µ,W (U, S) and c B,W (U, S) are flux-dependent tunable coefficients and n is the instanton number. As we shall explain in Sec. 3.4.2, this model-dependent contribution to the µ-term is crucial to reproduce a correct radiative EWSB for most of the parameter space of split SUSY-like models.

Dark radiation in sequestered models
As already argued in the Introduction, the production of dark radiation is a generic feature of string models where some of the moduli are fixed by perturbative effects [19]. In fact, if perturbative corrections fix the real part of the modulus T = τ + iψ, the axion ψ remains exactly massless at this level of approximation due to its shift symmetry, leading to m τ ≫ m ψ . Hence very light relativistic axions can be produced by the decay of τ , giving rise to ∆N eff = 0.

Dark radiation from moduli decays
Following the cosmological evolution of the Universe, during inflation the canonically normalised modulus Φ gets a displacement from its late-time minimum of order M P . After the end of inflation the value of the Hubble parameter H decreases. When H ∼ m Φ , Φ starts oscillating around its minimum and stores energy. During this stage Φ redshifts as matter, so that it quickly comes to dominate the energy density of the Universe. Afterwards reheating is caused by the decay of Φ which takes place when: where Γ Φ is the total decay rate into visible and hidden dof : (3. 2) The corresponding reheating temperature is given by: where ρ vis = (c vis /c tot ) 3H 2 M 2 P with c tot = c vis + c hid . Using (3.1) and (3.2) T rh can be rewritten as: This reheating temperature has to be larger than about 1 MeV in order to preserve the successful BBN predictions.
In the presence of a non-zero branching ratio for Φ decays into hidden sector dof, i.e. for c hid = 0, extra axionic dark radiation gets produced, leading to [7,8]: where T dec ≃ 1 MeV is the temperature of the Universe at neutrino decoupling with g * (T dec ) = 10.75. The factor in brackets is due to the fact that axions are very weakly coupled (they are in practice only gravitationally coupled), and so they never reach thermal equilibrium. Therefore, given that the comoving entropy density g * (T )T 3 a 3 is conserved, the thermal bath gets slightly reheated when some species drop out of thermal equilibrium. Note that the observational reference bound ∆N eff 1 implies: where we have used the fact that g * (T rh ) = 75.75 in the window 0.2 GeV T rh 0.7 GeV while g * (T rh ) = 86.25 for T rh 0.7 GeV.

Light relativistic axions in LVS models
Let us summarise the main reasons why axionic dark radiation production is a typical feature of sequestered LVS models: • Reheating is driven by the last modulus to decay which is τ b since the moduli mass spectrum takes the form (the axion ψ SM is eaten up by an anomalous U (1)): where ǫ = m 3/2 /M P ∼ W 0 /V ≪ 1. Given that gaugino masses scale as M 1/2 ∼ m 3/2 ǫ, TeV-scale SUSY fixes m τ b around 10 7 GeV which in turn, using (3.4), gives T rh around 1 GeV. 3 Note that m τ b ≪ m 3/2 , and so sequestering addresses the gravitino problem since the decay of the volume modulus into gravitinos is kinematically forbidden.
• Given that axions enjoy shift symmetries which are broken only by non-perturbative effects, the axionic partner ψ b of the volume mode τ b is stabilised by non-perturbative contributions to the superpotential of the form W ⊃ A b e −a b T b ∼ e −V 2/3 ≪ 1. These tiny effects give rise to a vanishingly small mass m 2 ψ b ∼ e −V 2/3 ∼ 0. Hence these bulk closed string axions are in practice massless and can be produced from the decay of τ b [7,8].
• Some closed string axions can be removed from the 4D spectrum via the Stückelberg mechanism in the process of anomaly cancellation. However, the combination of bulk axions eaten up by an anomalous U (1) is mostly given by an open string mode, and so ψ b survives in the low-energy theory (the situation is opposite for axions at local singularities) [19].

Volume modulus decay channels
The aim of this section is to compute the ratio c hid /c vis which is needed to predict the effective number of extra neutrino-like species ∆N eff using (3.5).

Decays into hidden sector fields
Some hidden sector dof are model dependent whereas others are generic features of LVS models. As pointed out above, bulk closed string axions are always a source of dark radiation. On top of them, there are local closed string axions which however tend to be eaten up by anomalous U (1)s (this is always the case for each del Pezzo singularity) and local open string axions (one of them could be the QCD axion [16]) whose production from τ b decay is negligibly small [7]. Moreover the decay of τ b into bulk closed string U (1)s is also a subdominant effect [7]. Model dependent decay channels involve light dof living on hidden D7-branes wrapping either D b or D s and hidden D3-branes at singularities which are geometrically separated from the one where the visible sector is localised. However, as explained in [7], the only decay channels which are not volume or loop suppressed are to light gauge bosons on the large cycle and to Higgses living on sequestered D3s different from the visible sector. Given that the presence of these states is non-generic and can be avoided by suitable hidden sector model building, we shall focus here just on τ b decays into bulk closed string axions. The corresponding decay rate takes the form [7,8]: where Φ and a are, respectively, the canonically normalised real and imaginary parts of the big modulus T b . This result can be derived from the tree-level Kähler potential: which gives a kinetic Lagrangian of the form: (3.10) After canonical normalisation of τ b and ψ b : and expanding Φ as Φ = Φ 0 +Φ, the kinetic Lagrangian (3.10) can be rewritten as: which encodes the coupling of the volume modulus to its axionic partner. Integrating by parts and using the equation of motion Φ = −m 2 ΦΦ we obtain the coupling: which yields the decay rate (3.8).

Decays into visible sector fields
The dominant volume modulus decays into visible sector dof are to Higgses via the GM coupling Z. Additional leading order decay channels can be to SUSY scalars and Higgsinos depending respectively on m 0 andμ. On the other hand, as explained in [7,8], τ b decays into visible gauge bosons are loop suppressed, i.e. c Φ→AA ∼ α 2 SM ≪ 1, whereas decays into matter fermions and gauginos are chirality suppressed, i.e. c Φ→f f ∼ (m f /m Φ ) 2 ≪ 1. The main goal of this section is to compute the cubic interaction Lagrangian which gives rise to the decay of the volume modulus into Higgses, Higgsinos, squarks and sleptons.

Decay into scalar fields
Let us first focus on the volume modulus decays into visible scalar fields which are induced by the τ b -dependence of both kinetic and mass terms in the total effective Lagrangian L = L kin − V . L kin is determined by the leading order Kähler potential: where we included only the leading term of the Kähler matter metricK α in (2.3). Writing each complex scalar field as C α = ReC α +iImC α √ 2 , the canonically normalised real scalar fields look like: Keeping only terms which are at most cubic in the fields and neglecting axion-scalarscalar interactions, we can schematically write the kinetic Lagrangian as L kin = L kin,quad + L kin,cubic where: while the cubic part can be further decomposed as L kin,cubic = L Φaa + L Φhh + L ΦCC , with L Φaa given in (3.13) and: and: In addition to the LVS part, the scalar potential contains also the following terms: Since the soft-terms depend on the volume modulus, we can expand them as: where α, β and γ depend on the specific scenario. This expansion leads to new cubic interactions coming from the terms of the scalar potential in (3.16): Including the relevant cubic interactions coming from the kinetic Lagrangian and integrating by parts, we obtain a total cubic Lagrangian of the form: The leading order expressions of the equations of motion are: which have to be supplemented with: Plugging these equations of motion into L cubic , the final result becomes: from which it is easy to find the corresponding decay rates using the fact that: , (3.20)

Decay into Higgsinos
The decay of the volume modulus into Higgsinos is determined by expanding the Higgsino kinetic and mass terms around the VEV of τ b and then working with canonically normalised fields. The relevant terms in the low-energy Lagrangian are: After imposing the equations of motion, we get the following cubic interaction Lagrangian: The corresponding decay rates take the form:

Dark radiation predictions
It is clear from (3.19) and (3.24) that the volume modulus branching ratio into visible sector dof depends on the size of the soft-terms. Hence the final prediction for dark radiation production has to be studied separately for each different visible sector construction.

MSSM-like case
Firstly we consider MSSM-like models arising from the ultra-local dS 2 case where all softterms are suppressed relative to the volume modulus mass: Let us briefly review the results for dark radiation production which for this case have already been studied in [7,8].
Given that all soft-terms are volume-suppressed with respect to m Φ , only the last term in (3.19) gives a non-negligible contribution to the volume modulus branching ratio into visible sector fields. Thus the leading Φ decay channel is to MSSM Higgses via the GM coupling. Using (3.21), we find: (3.26) Plugging this value of c vis together with c hid = 1 (see (3.8)) into the general expression (3.5) for extra dark radiation, we obtain the window: 27) for 0.2 GeV T rh 10 GeV. Clearly this gives values of ∆N eff larger than unity for Z = 1. Using the bound (3.6), we see that we need c vis 3 in order to be consistent with present observational data, implying Z 3/2 ≃ 1.22.

Split SUSY-like case
Let us now analyse dark radiation predictions for split SUSY-like scenarios arising in the dS 1 (both local and ultra-local) and local dS 2 cases. In these scenarios the hierarchy among soft-terms is (considering µ and Bµ-terms generated by K): The main difference with the MSSM-like case is that now Bμ and m 2 0 scale as m 2 Φ . In order to understand if volume modulus decays into SUSY scalars are kinematically allowed, i.e. R ≡ m 2 0 /m 2 Φ ≤ 1/4, we need therefore to compute the exact value of m Φ and compare it with the results derived in Sec. 2.4.2. It turns out that m Φ depends on the dS mechanism as follows: Let us analyse each case separately: • Local and ultra-local dS 1 cases: Even if the F-term contribution to scalar masses (2.19) for the local case can be made small by appropriately tuning the coefficient c s , the D-term contribution to m 2 0 given by (2.20) cannot be tuned to small values once the requirement of a dS vacuum is imposed. Hence for both local and ultra-local cases, m 2 0 cannot be made smaller than (2.20) giving: which is clearly in contradiction with the condition R ≤ 1/4 that has to be satisfied to open up the decay channel of Φ into SUSY scalars. Therefore the decay of Φ into squarks and sleptons is kinematically forbidden.

Decay to Higgses
Similarly to the MSSM-like case, Φ can still decay to Higgs bosons via the GM term in (3.19). Given that m Φ < m 0 , when Φ decays at energies of order m Φ , EWSB has already taken place at the scale m 0 . 4 The gauge eigenstates h i i = 1, ..., 8 given in (3.15) then get rotated into 8 mass eigenstates. 4 Higgs dof which we denote by A 0 , H 0 , H ± remain heavy and acquire a mass of order m 2 H d ≃ m 2 0 , and so the decay of Φ into these fields is kinematically forbidden. The remaining 4 dof are the 3 would-be Goldstone bosons G 0 and G ± which become the longitudinal components of Z 0 and W ± , and the ordinary SM Higgs field h 0 . The Φ decay rate into light Higgs dof can be obtained from the last term in (3.19) by writing the gauge eigenstates in terms of the mass eigenstates as [30]: h 7 = ImG + sin β + ImH + cos β , h 8 = ImG + cos β − ImH + sin β , (3.31) where Since in split SUSY-like models tan β ∼ O(1) in order to reproduce the correct Higgs mass [31,32], the interaction Lagrangian simplifies to: Neglecting interaction terms involving heavy Higgses, (3.32) gives a decay rate of the form: (3.33) • Local dS 2 case: The situation seems more promising in this case since in the local limit the D-term contribution to m 2 0 is volume-suppressed with respect to m 2 Φ since it scales as m 2 0 D ∼ O V −4 [18]. In this case it is therefore possible to tune the coefficient c s to obtain R ≤ 1/4. By comparing the second term in (3.29) with (2.19), this implies that c s has to be tuned so that c s − 1 3 ≤ 9 10 asτs , where a s τ s ∼ 80 in order to get TeV-scale gaugino masses [18]. However the condition m 2 0 > 0 to avoid tachyonic masses translates into c s − 1 3 > 0, giving rise to a very small window: Given that c s should be extremely fine-tuned, it seems very unlikely to open this decay channel. However the total Kähler potential, on top of pure α ′ corrections, can also receive perturbative string loop corrections of the form [27]: where C loop and k loop are two O(1) coefficients which depend on the complex structure moduli. Due to the extended no-scale structure [28], g s effects do not modify the leading order scalar potential, and so the mass of the volume modulus is still given by (3.29). However, in order to reproduce a correct ultra-local limit (2.18), we need to change the parametrisation of the Kähler matter metric from (2.4) to: where we introduced a new coefficient c loop and we neglected k loop -dependent corrections in (3.40) since they are subdominant in the large volume limit τ s ≪ τ b . The new ultra-local limit is now given by c s = c loop = 1/3.
These new c loop -dependent corrections in (3.41) affect the final result for scalar masses and can therefore open up the Φ decay channel to SUSY scalars. In fact, the result (2.19) for scalar masses in the local case gets modified to: The two terms in square brackets are of the same order for g s ≃ 0.1 and V ∼ 10 7 which is needed to get TeV-scale gauginos, and so they can compete to get R ≤ 1 4 . As an illustrative example, if we choose c s = 1/3 and natural values of the other parameters: C loop = a s = 1, ξ = 2 and c loop = 0 (non-tachyonic scalars require c loop < 1/3 for c s = 1/3), the ratio between squared masses becomes: As can be seen from Fig. 1, there is now a wide region of the parameter space where the Φ decay to SUSY scalars is allowed. We finally point out that g s corrections to the Kähler matter metric affect the result for scalar masses only in the local case since in the ultra-local limit m 0 is generated by effects (D-terms for dS 1 and F-terms of S and U -moduli for dS 2 ) which are sensitive only to the leading order expression ofK α .
Let us now analyse the final prediction for dark radiation production for split SUSY-like models where the decay channel of Φ into SUSY scalars is kinematically allowed.

Dark radiation results
We start by parameterising the scalar mass m 0 in terms of the volume mode mass m Φ as m 0 = c m Φ and theμ-term as µ =c m Φ so that the corresponding kinematic constraints for Φ decays into SUSY scalars and Higgsinos become c ≤ 1 2 andc ≤ 1 2 . Parameterising also the Bμ-term as in (2.22) and using the fact that for split SUSY-like models we have in (3.17) β = γ = 9/2 and α = 4, 6 the leading order cubic Lagrangian is given by the sum of (3.19) and (3.23): Contrary to the MSSM-like case, now the decay of the volume modulus into squarks and sleptons through mass terms is kinematically allowed and also the decay rate into Higgses is enhanced due to mass terms and Bμ couplings. Using (3.20), the total decay rate into squarks and sleptons reads: (3.45) where N = 90 is the number of real scalar dof of the MSSM, 7 except for the Higgses.
On the other hand, the decay rate into Higgs bosons receives contributions from both mass and GM terms. Using (3.20) we obtain: The decay rate into Higgsinos is given again by (3.37) and thus the total Φ decay rate into visible sector fields becomes: The final prediction for dark radiation production is then given by (3.5) with c hid = 1, g * (T dec ) = 10.75 and g * (T rh ) = 86.25 for T rh 0.7 GeV. 8 The results are plotted in Fig. 2 where we have set c B,K = 1, Z = 1 and we are considering a conservative case in which the decay into Higgsinos is negligible, i.e.c = 0. For c > 0.2, the vast majority of the parameter space yields ∆N eff 1, in perfect agreement with present experimental bounds with a minimum value ∆N eff | min ≃ 0.14 at c ≃ 1/ √ 5. It is interesting to notice that, contrary to the MSSM-like case, dark radiation overprodution can now be avoided if the GM term is absent or it is very suppressed. In fact even for Z = 0, ∆N eff 1 if c 0.23, as a consequence of the fact that in this region of the parameter space almost the whole suppression of ∆N eff is due to the decay into scalar fields. The predictions for ∆N eff for different values of the GM coupling Z = 0 (blue line), Z = 1 (red line) and Z = 2 (green line) are shown in Fig. 3. Forc = 0 ∆N eff is even further suppressed than what is shown in Fig. 2 and 3 but the correction is at the percent level in the interesting region where the decay into scalars dominates Γ vis . For example including the effect of decays into Higgsinos and settingc ≃ 1/ √ 10 to maximise the decay rate into Higgsinos, the correction δ∆N eff | min to ∆N eff | min turns out to be:  of squarks and sleptons are given by: where c α,a is the weak hypercharge squared for each SUSY scalar and the RG running contributions K a are proportional to gaugino masses [30]. Given that in split SUSY-like models gaugino masses are hierarchically lighter than scalar masses, the RG running of first and second generation squarks and sleptons is a negligible effect. Thus we can consider their mass at the scale m Φ as still given by m 0 to a high level of accuracy. The situation for the third generation is slightly trickier since there are additional contributions from large Yukawa couplings. Using mSUGRA boundary conditions, the relevant RG equations become (ignoring contributions proportional to M 1/2 ) [30]: which are coupled to those involving Higgs masses: The quantities X i look like: 55) where y i are the Yukawa couplings and a i are the only sizable entries of the A-term couplings. Given that for sequestered scenarios the A-terms scale as M 1/2 [18], the contribution 2|a i | 2 can be neglected with respect to the first term in each X i . Moreover X t , X b and X τ are all positive, and so the RG equations (3.52) and (3.53) drive the scalar masses to smaller values at lower energies. This has a two-fold implication: • When m 0 > m Φ , RG running effects could lower m 0 to values smaller than m Φ /2 so that the decay channel to SUSY scalars opens up at the scale m Φ . However this never happens since RG effects are negligible.
• When m 0 ≤ m Φ /2, no one of the scalars becomes heavier than the volume modulus if R < 1 4 at the boundary energy scale. On the other hand, RG running effects could still lower the scalar masses too much, suppressing the Φ decay rate to SUSY scalars. However this does not happen since RG effects are negligible.
In split SUSY-like models a correct radiative realisation of EWSB requires a low value of tan β [32], which implies y b , y τ ≪ y t . In turn, X b and X τ give rise to a tiny effect, and so the running of m 2 , m 2 L 3 and m 2 e 3 turns out to be negligible. In the end, the only relevant RG equations become: We performed a numerical computation of the RG running, using as boundary conditions m 0 = Bμ 1/2 =μ = 10 7 GeV and M a = 10 3 GeV at the GUT scale M GUT = 2 × 10 16 GeV and tan β ≃ 1.4. We also used that the stop left-right mixing is given by χ t = A t −μ cot β ≃ −m 0 /tan β, being A t ≃ M 1/2 ≪ m 0 . We used SusyHD [33] to run the Yukawa couplings from the top mass scale up to m 0 combined with SARAH [34] to run them from m 0 up to the GUT scale. 9 These runnings have been computed at order one loop. Using the values of the Yukawa couplings obtained at the GUT scale, we have been able to compute the running of scalars,μ, Bμ and the GM coupling Z down to the scale of the decay m Φ . Fig. 4 shows the running of scalar masses while Fig. 5 showns the running of m 2 Hu and m 2 H d . The running ofμ and Bμ is almost negligible. We clarify that our purpose here is not to study EWSB in full detail but to understand which kind of behaviour we should expect for the running of soft-terms from the GUT scale to m Φ using boundary conditions which are consistent with EWSB.
Due to RG running effects each of the scalars has a different mass at the scale m Φ , and so the exact prediction for ∆N eff becomes: where: The decay rate into squarks and sleptons is given by: 61) 9 We are grateful to J. P. Vega for useful discussions about this point.  where the index α runs over all squarks and sleptons m α = mQ L , mũ R , md R , mL, mẽ R while κ α is the number of dof for each scalar. 10 On the other hand the decay rate into Higgs dof is given by (we focus on the case whereμ ≪ m 0 since, as we have seen in the previous section, a largeμ-term gives rise just to a negligible correction to the final dark radiation prediction): All quantities in (3.61) and (3.62) have to be computed at the decay scale m Φ . As already explained in the previous section, Bμ does not explicitly contribute to (3.62) since it gets reabsorbed into the Higgs masses due to EWSB. The decay of Φ into heavy Higgses A 0 , H 0 , H ± through the mass term can instead contribute to Γ Higgs , provided that m A 0 ,H 0 ,H ± /m Φ ≤ 1/2. The mass of the heavy Higgses in the limit m Z , m W ± ≪ m A 0 can be written as [30]: The decay rate into Higgsinos instead reads: where we added the two contributions in (3.24) and we used α = 4. In (3.64)μ/m Φ =c, sinceμ is computed at the decay scale m Φ . We computed ∆N eff for different values of m Φ , keeping the boundary conditions fixed at m 0 = Bμ 1/2 =μ = 10 7 GeV and M a = 10 3 GeV. The qualitative behaviour of ∆N eff is the same as in the previous section where RG flow effects have been ignored. The results are shown in Tab. 1 which shows that the dominant contribution to ∆N eff is given by the decay into squarks and sleptons while the suppression coming from decay into Higgsinos is always subdominant. If m Φ ≃ 2.2 × 10 7 GeV, corresponding to m 0 /m Φ ≃ 1/ √ 5, ∆N eff ≃ 0.15 which is only slightly larger then ∆N eff | min = 0.14 computed in the previous section without taking into account RG flow effects. This is due to the fact that the running of the SUSY scalars is negligible as can be clearly seen from Fig. 4 Table 1. Values of ∆N eff corresponding to different masses m Φ of Φ for fixed boundary condition m 0 = 10 7 GeV at the GUT scale. We also indicate the relative importance of the various decay channels. In the case denoted by a ( * ) the only non-vanishing contribution to Γ Higgs is due to the decay into light Higgs dof through the GM coupling, since the decay into heavy Higgs dof turns out to be kinematically forbidden as a consequence of the RG flow: 2m A 0 ,H 0 ,H ± > m Φ at the decay scale m Φ . The decay into Higgsinos is always a subleading effect.
GeV which leads to a reheating temperature of order T rh ∼ 1 GeV. The gravitino mass is much larger than m Φ (m 3/2 ∼ 10 10 GeV), so avoiding any gravitino problem [13]. Lowenergy SUSY can still be achieved due to sequestering effects that keep the supersymmetric partners light. Gauginos are around the TeV-scale whereas squarks and sleptons can either be as light as the gauginos or as heavy as the lightest modulus Φ depending on the moduli dependence of the matter Kähler metric and the mechanism responsible for achieving a dS vacuum [18]. The final prediction for dark radiation production due to the decay of the volume modulus into ultra-light bulk closed string axions depends on the details of the visible sector construction:

MSSM-like case:
MSSM-like models arise from the ultra-local dS 2 case where the leading visible sector decay channel of Φ is to Higgses via the GM coupling Z. The simplest model with two Higgs doublets and Z = 1 gives 1.53 ∆N eff 1.60 for 500 GeV T rh 5 GeV [7,8]. Values of ∆N eff smaller than unity require Z 1.22 or more than two Higgs doublets.
2. Split SUSY-like case with m 0 > m Φ /2: Local and ultra-local dS 1 cases give rise to split SUSY-like scenarios where scalar masses m 0 receive a contribution from D-terms which cannot be made smaller than m Φ /2. Thus the decay of Φ into squarks and sleptons is kinematically forbidden. The leading visible sector decay channel of Φ is again to Higgses via the GM coupling Z. However, given that EWSB takes place at the scale m 0 which in these cases is larger than the decay scale m Φ , the volume mode Φ can decay only to the 4 light Higgs dof. For a shift-symmetric Higgs sector with Z = 1, the final prediction for dark radiation production is 3.07 ∆N eff 3.20. Consistency with present experimental data, i.e. ∆N eff 1, requires Z 1.68. In most of the parameter space of split SUSY-like models a correct radiative EWSB can be achieved only if theμ-term is of order the scalar masses. Hence, depending on the exact value ofμ, the decay of Φ into Higgsinos could not be mass suppressed. However it gives rise just to a negligible contribution to ∆N eff .
3. Split SUSY-like case with m 0 ≤ m Φ /2: Given that in the local dS 2 case the D-term contribution to scalar masses is negligible, the decay of Φ into SUSY scalars can become kinematically allowed. In fact, thanks to the inclusion of string loop corrections to the Kähler potential [27,28], a large region of the underlying parameter space features m 0 ≤ m Φ /2. Hence the final prediction for ∆N eff gets considerably reduced with respect to the previous two cases since, in addition to decays into Higgses via the GM term, leading order contributions to the branching ratio to visible sector particles involve decays into squarks and sleptons, decays into heavy Higgses induced by mass terms and possible decays into Higgsinos depending on the exact value of theμ-term. Depending on the exact value of m 0 , the simplest model with Z = 1 gives 0.14 ∆N eff 1.6. Hence these models feature values of ∆N eff in perfect agreement with present observational bounds. Note that dark radiation overproduction can be avoided even for Z = 0 due to the new decay channels to squarks and sleptons.
We finally studied corrections to these results due to RG flow effects from the string scale M s ∼ 10 15 GeV to the volume mode mass m Φ ∼ 10 7 GeV where the actual decay takes place. However these corrections do not modify our predictions since the RG running of SUSY scalar masses is a negligible effect in split SUSY-like models and radiative corrections to m Φ are tiny since moduli are only gravitational coupled.