Constraining SUSY with Heavy Scalars – using the CMB
If low-energy SUSY exists, LHC data favors a high mass scale for scalar superpartners (above a TeV), while sfermions and the dark matter can be parametrically lighter – leading to a so-called split-spectrum. When combining this fact with the motivation from fundamental theory for shift-symmetric scalars (moduli) prior to SUSY breaking, this leads to a non-thermal history for the early universe. Such a history implies different expectations for the microscopic properties of dark matter, as well as the possibility of dark radiation and a cosmic axion background. In this paper we examine how correlated and mixed isocurvature perturbations are generated in such models, as well as the connection to dark radiation. WMAP constraints on multiple correlated isocurvature modes allow up to half of the primordial perturbations to be isocurvature, contrary to the case of a single isocurvature mode where perturbations must be dominantly adiabatic. However, such bounds are strongly prior dependent, and have not been investigated with the latest Planck data. In this paper we use the example of a SUSY non-thermal history to establish theoretical priors on cosmological parameters. Of particular interest, we find that priors on dark radiation are degenerate with those on the total amount of isocurvature – they are inversely correlated. Dark radiation is tightly constrained in the early universe and has been used recently to place stringent constraints on string-based approaches to beyond the standard model. Our results suggest such constraints can require more input from theory. Specifically, we find that in many cases constraints on dark radiation are avoidable because the density can be reduced at the expense of predicting an amount of multi-component isocurvature. The latter are poorly constrained by existing probes, and lead to the interesting possibility that such models could have new predictions for the next generation of observations. Our results are not only important for establishing the post-inflationary universe in the presence of SUSY, but also suggest that data from cosmological probes – such as Planck – can help guide model building in models of the MSSM, split-SUSY, and beyond. Our model also demonstrates the utility of UV models in constructing cosmological priors.
The discovery of a GeV Higgs at the Large Hadron Collider (LHC) – and nothing else – has left the relevance of supersymmetry (SUSY) in question, particularly as a mechanism for stabilizing the hierarchy between the Electroweak and Planck scales. Within the minimal SUSY standard model (MSSM), Natural-SUSY still remains possible, but at the cost of increasing the level of complexity of models (see e.g. Randall:2012dm ()). There are a number of alternative ways to reconcile SUSY with the data, including models of Split-SUSY Wells:2003tf (); ArkaniHamed:2004fb (); Arvanitaki:2012ps (), or simply by accepting that some fine-tuning may just be an accident of nature222It is noteworthy that even defining the level of fine-tuning can be an issue, see e.g. Baer:2013gva (). Regardless of one’s viewpoint, it seems that if low-energy SUSY will prevail it will require the existence of a new scale at around TeV. In many models this scale is set by SUSY breaking and scalar superpartners masses will be around this range, whereas fermion superpartners (and dark matter) will be parametrically lighter at around GeV – providing a so-called split spectrum.
Any additional light scalars, or moduli, resulting from beyond the standard model physics would also generically receive masses around the TeV range Acharya:2009zt (). Cosmologically this is very interesting, since if these moduli are only gravitationally coupled to other matter (which is typically the case) this mass range is precisely what is necessary to avoid the cosmological moduli problem (CMP) coughlan1983 (); Watson:2009hw (); Acharya:2008bk (). Moreover, moduli in this mass range will decay early enough to avoid disrupting Big Bang Nucleosynthesis (BBN), and lead to a new non-thermal history for the early universe. This leads to new expectations for early universe cosmology including: altered predictions for confronting inflation models with Cosmic Microwave Background (CMB) data Easther:2013nga (); different expectations for the microscopic properties of dark matter (DM) Acharya:2008bk (); Watson:2009hw (); Cohen:2013ama (); Fan:2013faa (); Allahverdi:2013noa (); enhanced small-scale structure erickcek2011 (); and the possible existence of dark radiation (DR) and a relic cosmic axion background higaki2013 (); Conlon:2013isa ().
It has been said that Planck Ade:2013ktc () cosmological constraints Ade:2013zuv () make for a ‘maximally boring universe’, described with exquisite precision by the six parameter CDM standard cosmological model in which all perturbations are gaussian and adiabatically produced. One phenomenological extension of this model is the possible existence of isocurvature modes in the dark or visible sector bucher2000 (). WMAP Hinshaw:2012aka (); savelainen2013 () and Planck Ade:2013uln () place constraints on the level of isocurvature in various models. The constraints on single-mode isocurvature are strong, and limit the isocurvature fraction at the percent or sub-percent level depending on the model, with correlated models being more tightly constrained. However, when multiple isocurvature modes are allowed, degeneracies between the modes can allow the isocurvature fraction to be almost half bucher2004 (); moodley2004 (); Bean:2006qz (). In the two-mode model with DM and neutrino isocurvature that we will consider in this paper, the fraction is lowered to around 30 to 40%, but is still substantially larger than that allowed for single modes. In such a scenario the universe is certainly not ‘maximally boring’. The key question to be answered is whether such a model is theoretically well motivated and can exist within a UV extension of the SM.
The existence of DR (parameterised by the effective number of neutrino species ) having departures from its canonical value, is also a generic and well-motivated extension of the six parameter CDM model linde1979 (). Constraints on from Planck and other CMB experiments has caused it to receive much attention over the last few years dunkley2010 (); hamann2010a (); fischler2011 (); keisler2011 (); kobayashi2011 (); marsh2011b (); abazajian2012 (); benetti2013 (); calabrese2013 (); Conlon:2013isa (); Weinberg:2013kea (). The interpretation of constraints to can depend on the theoretical model underpinning the departure from the canonical value archidiacono2013 (); angus2013 (), while the constraints themselves can have a dependence on the priors coming from such a model gonzales-morales2011 (); wyman2013 (); Verde:2013cqa ().
Under certain generic assumptions, which we outline below, the moduli of SUSY pick up isocurvature perturbations during inflation. In the subsequent decay of the moduli these perturbations are passed on to the DM, just like in the well-known curvaton scenario enqvist2001 (); lyth2002 (); moroi2001 (). Moduli effectively behave as scalar fields, and in SUSY come partnered with pseudo-scalar axion fields to which they are coupled. The shift symmetry of the axions protects their masses, allowing them to be light compared to the moduli, while also suppressing their couplings and making them long lived. As such, heavy moduli can decay into light, relativistic axions, providing a component of DR that also inherits an isocurvature perturbation correlated to the DM isocurvature. Much of our analysis has overlap with existing studies of curvatons and related toy models (see e.g. Lemoine:2009is (); Ade:2013uln () and references within), but with the added twist of DR, and non-thermal DM.
The cosmological constraints to correlated DM-DR isocurvature are strongly prior dependent bucher2004 (); moodley2004 (), so that such a situation cries out for a UV model able to fix the priors based on other considerations. We will discuss the importance and implications of this in some detail. In the context of our model there is also a prior, and therefore interpretation, for constraints to as an axion background produced by decay of a modulus with certain mass, width, and branching ratios. The goals of this work are to establish the importance of isocurvature constraints for SUSY models, find the cosmological priors implied in a SUSY set up for correlated DM-DR isocurvature, and how the constraints and degeneracies from DR production can complement the isocurvature phenomenology.
The remainder of the paper is organised as follows. In Section II we review the connection between SUSY-based model building after LHC and a non-thermal history for the post inflationary universe – including what this implies for the expected microscopic properties of DM, and how DR can be produced. We discuss how in some cases this can lead to a substantial amount of isocurvature in the primordial temperature fluctuations, which would be in conflict with observations. We then identify which cases are most severely restricted by observations, which we find correspond to cases where the decaying field responsible for the non-thermal history has sub-Hubble mass during inflation, leading to the production of isocurature modes. In Section III we establish the basic formalism for computations in this model, computing cosmological observables and relating them to the CMB spectrum. We present the results of these calculations in Section IV, where we use priors on the modulus parameters in Split and Natural-SUSY to compute the priors on cosmological parameters, discussing how constraints to DR and isocurvature can be complementary. In the last section we conclude and discuss what this implies for the current status of SUSY dark matter of non-thermal origin, and outline future directions, in particular how the results of this work can be used to accurately constraint SUSY using Planck data. The details of our numerical calculation are relegated to Appendix A, while some details of the power spectrum normalisation and spectral indices are given in Appendix B.
Ii SUSY WIMPs, Non-thermal Histories, Inflation and Reheating
The motivation from LHC for higher than anticipated superpartner masses implies that the scale of SUSY breaking should be around GeV, where the gravitino mass TeV sets the mass scale of the scalar superpartners and GeV is the reduced Planck mass. However, superpartner fermions – one of which plays the role of WIMP DM – can be parametrically below this scale due to loop suppression and R-symmetry Moroi:1999zb (); Wells:2003tf (); ArkaniHamed:2004fb (). When this type of framework is required to have a high-energy (UV) completion within supergravity (SUGRA) or string theory (which is necessary for self-consistently), additional scalars with little or no potential (moduli) will naturally appear leading to a non-thermal cosmological history Watson:2009hw (); Acharya:2008bk (); Acharya:2009zt (). We briefly review this in the next section, followed by a discussion of how this can lead to a large generation of isocurvature perturbations in the primordial spectrum, and production of DR.
ii.1 Non-thermal dark matter and cosmological moduli
The non-thermal post-inflationary history of the universe which we outline in this subsection is sketched in Fig. 1.
Given a Split-SUSY-like spectrum in theories that contain moduli (such as SUGRA and string theories), a non-thermal cosmological history naturally results Watson:2009hw (); Acharya:2008bk (); Acharya:2009zt (). As an example, consider a modulus or scalar field with a shift symmetry so that naively . If this remains a good symmetry until SUSY breaking, we expect the field to get a mass , where the split spectrum implies
where the constant is typically not more than and in string based examples is frequently related to the hierarchy LoaizaBrito:2005fa (); Watson:2009hw (). However, we note that in models of Split-SUSY (where the electroweak hierarchy is addressed anthropically) the gravitino mass can be significantly higher than the TeV scale, and so in those models the moduli mass will take values that can range all the way up to the Planck scale – we will consider both possible mass ranges in this paper. We will consider ‘Natural-SUSY’ to have masses below TeV, and ‘Split-SUSY’ to have masses up to TeV.
The non-thermal history arises from the observation that there is no a-priori reason why the modulus should initially begin in its low-energy minimum. As an explicit example, we expect on general grounds that the shift symmetry of the modulus should be broken by both the finite energy density of inflation (another source of SUSY breaking) and quantum gravity effects Dine:1995kz (). These considerations imply additional contributions to the effective potential in the form of a Hubble scale mass and a tower of non-renormalizable operators,
where in the absence of special symmetries and are expected to be order one constants (with ), is the Hubble rate during inflation, and is the scale of new physics, e.g. quantum gravity. As this will be important later, we note that the dimension of the leading irrelevant operator that lifts the flat direction is model dependent as well as the scale of new physics333This is a generic expectation in string theories where new thresholds before the Planck scale are common place. Examples include both the compactification (Kaluza-Klein) scale and string scale where is required for consistency of the effective theory Polchinski:1998rq ().. Because the contributions (2) are the dominant terms in the potential during high scale () inflation, this implies the minimum of the field at that time will be
The mass at this high energy minimum, , plays a key role in the generation of isocurvature perturbations. The constant of proportionality is set by the model dependent values of and can be greater or less than unity. Much later, at the time that the low-energy SUSY breaking gives the dominant contributions to the potential, the minimum is near , while the mass is given by Eq. (1). More complicated potentials and contributions are possible, but in this simple case where the mass term dominates at low energy, this displacement from the low energy minimum leads to energy stored in coherent oscillations of forming a scalar condensate. The amplitude of the oscillations is determined by the initial displacement , where for the example potential (2) we have .
Hubble friction ceases and oscillations will set in when the expansion rate satisfies . Because the oscillations scale like matter, they dilute more slowly than the primordial radiation (produced during inflationary reheating). Depending on the initial value , the energy stored in the moduli may quickly come to dominate the energy density of the universe (see e.g. Acharya:2008bk ()). At the time oscillations begin the initial abundance is given by
and once the oscillations become coherent (which typically takes less than a Hubble time) they will scale as pressure-less matter Turner:1983he () with . The universe remains matter dominated until the field decays. Because the field is a modulus we expect it typically to be gravitationally coupled to other particles and so its decay rate is
where we expect and depends on the precise coupling in the fundamental Lagrangian, but typically takes values in the range . Most of the field decays444In most of the literature the moduli decay is treated as instantaneous at . However, this approximation can be misleading. For example, in the case where the radiation energy density significantly drops below the moduli energy density (), the continuous (even though small) amounts of particle decay before reheating can lead to changes in the scale factor - temperature relation. This is because during the decays the entropy is not conserved, but changes as , which follows from the equation of motion for the radiation (42) with , the definition of the entropy density , and the first law of thermodynamics . Using this relation one can show that the scale factor is related to the temperature of the radiation as and so if the modulus dominates the universe so that instead of the entropy preserving values (in a matter dominated universe) and . We refer the reader to Giudice:2000ex () for further discussion. at the time and we expect it to decay democratically to Standard Model particles and their super-partners. Any heavy super-partners produced will typically decay rapidly into the lightest SUSY partner (LSP), which is stable and will provide the dark matter candidate (which we will denote as ).
In addition to dark matter, light standard model particles that are produced will thermalize and ‘reheat’ the universe for a second time (with inflationary reheating occurring early and at high temperature). The corresponding reheat temperature is given by
and this temperature must be larger than around MeV to be in agreement with BBN light element abundances Kawasaki:1999na ().
If the number density of dark matter particles produced in the decay is larger than the critical value approximately given by
then rapid self annihilations will take place until the dark matter abundance reduces to this value, which acts as an attractor value555In cases where the dark matter abundance is subcritical, then no annihilations take place. This case is model-dependent, but usually occurs when the moduli do not come to dominate the energy density and/or if decays to super-partners are highly suppressed. In the former case, this can lead to a large generation of isocurvature as we discuss in the next section.. In the equation above, is the averaged annihilation rate and velocity at the time of decay. Once the fixed point is reached, the resulting abundance of non-thermally produced dark matter is found to be
where is the freeze-out temperature of thermal dark matter (around a few GeV), and is the reheat temperature following the modulus decay and can be as small as a few MeV.
A second possibility is that the yield of dark matter from scalar decay is sub-critical . In this case, the amount of dark matter depends on the initial scalar density. If is the branching ratio for decay to superpartners then the amount of dark matter after decay is , where is the scalar energy density before decay. As an example, if the modulus dominates the energy density before decay, the comoving amount of dark matter will be independent of its cross-section and will depend primarily on the masses . Whether one has the sub-critical or super-critical case, depends in practice on the initial displacement of the modulus as this determines the amplitude of oscillations and the amount of energy stored in the oscillations Moroi:1999zb ().
If the LSP constitutes all of the dark matter we must require Ade:2013zuv (). In a standard cosmology where dark matter has a thermal origin this implies . However, when dark matter has a non-thermal origin this number can be larger.
As an example, if we consider a non-thermal history where the modulus decay reheats the universe to a temperature MeV, which is significantly below the freeze-out temperature of a typical GeV WIMP GeV we find . For a given candidate (like the MSSM neutralino) this leads to new and interesting predictions for experiments probing the microscopic properties of dark matter such as indirect detection, direct detection, and LHC searches.
ii.2 Dark Radiation Production
In addition to moduli decays to standard model particles and their superpartners, there may also be decays to hidden sector fields. Indeed, this is a common expectation in string-based models that give rise to the non-thermal history for dark matter discussed above Higaki:2012ar (); Cicoli:2012aq (); Higaki:2012ba (). If the particles resulting from decay are light (meaning relativistic) at the time of BBN and/or recombination, and non-interacting with MSSM particles, this leads to additional radiation coming from the hidden sector. If these particles contribute substantially to the energy density they will affect the expansion rate changing predictions for both the abundances of primordial elements Steigman:2012ve () and the physics of the CMB Abazajian:2013oma (). Thus, using precision cosmological measurements one can establish constraints on the amount of dark radiation that is permitted within a particular class of models – see Conlon:2013isa () and references within.
During radiation domination after the decay of the lightest modulus the effect of the hidden sector radiation on the Hubble expansion can be understood through the Hubble equation , where the total relativistic contribution to the energy density is
where the sums are over relativistic hidden sector particles with degrees of freedom, the factor of results from Fermi-Dirac statistics of fermions, is the temperature of the hidden sector particles (which importantly need not be equilibrated with standard model radiation), and is the visible sector relativistic degrees of freedom, which in the early universe would be at least in the MSSM, but near the MeV scale only the photons and neutrinos contribute with
where for the photon, for neutrinos, and is the effective number of neutrino species at temperature .
At the time of BBN, the neutrino temperature tracks the photons so that and so with three relativistic neutrinos, , the standard model prediction is . However, because the neutrinos are weakly interacting with GeV Fermi’s constant, they decouple below the temperature of BBN ( MeV) and the entropy in photons increases (so that the total entropy is conserved). At the time of recombination we have
where accounts for a small injection of entropy into neutrinos coming from electron/positron annihilations prior to recombination, as well as energy dependent distortions and and small finite temperature corrections from the plasma Abazajian:2013oma (). The increase in the photon entropy following neutrino decoupling leads to an increase in the temperature of Abazajian:2013oma (). Given the standard model (MSSM) predictions, any new dark radiation would lead to an additional contribution to or . For historic reasons, constraints on new hidden sector radiation are typically expressed through .
Constraints from BBN result from requiring agreement with both the abundances of and Cyburt:2004yc (), which implies at the time of BBN. At the time of recombination, the Planck satellite Ade:2013zuv () provides constraints with the current results implying . If we make the additional assumption that all radiation was initially in equilibrium with standard model photons (so that the temperature of all relativistic species is the same) then this corresponds to the bound , with a final projected sensitivity for of Galli:2010it (). However, as pointed out in Feng:2008mu () these bounds can be significantly relaxed if the hidden sector radiation does not share the photon temperature.
For example, if hidden radiation couples different to decaying moduli (or the inflaton during reheating) than standard model particles this can lead to different temperature for each species and this will be preserved in the absence of interactions between the systems of particles. As we will see in the next section, the situation where and thermal equilibrium with photons is not reached is interesting for the case of isocurvature perturbations. In such a case, the Planck constraint on from recombination ( and using (10) ) implies the bound
where (following convention) we have treated the extra radiation as a neutrino species and subtracted the contribution from standard model neutrinos. Instead we can express this constraint in terms of where
with and if the hidden sector radiation shares a common temperature with photons then . Thus, the upper value from Planck of implies an upper bound on the combination of and the departure in temperature from the standard model thermal bath.
We close our brief review of hidden sector radiation by considering the example of axions, which represent a well-motivated and simple example of dark radiation (c.f. Conlon:2013isa () and references within). We can revisit the non-thermal history resulting from moduli decay above, but this time allowing for decay to axions as well. If we denote by and the branching fraction to dark matter and radiation, respectively, then the remaining fraction to standard model particles is simply . If we consider for simplicity the case that the moduli dominate before decay (and with no dark matter annihilations), then the density in axions will be , where is the expansion rate at decay given by (5). Comparing this to the energy density in standard model radiation , we can find a constraint on the branching ratios through constraints on .
where and . Identify the axion density with the second term above and inverting the expression we have
Following their production the axion’s entropy remains fixed and so they simply scale with the expansion as
where is the reheat temperature (6). The entropy in photons (MSSM sector) will change following neutrino decoupling, since positrons and electrons will freeze-out so that decreases, while the temperature must increase so that the comoving entropy remains constant666Electron and positron freeze-out occurs on microscopic time scales so that the cosmic expansion during this event is negligible, i.e. we can take in the expression for the entropy. It follows that
Using these expressions in (16), and the expression for the energy density in axions and MSSM radiation at the time of reheating, we have a constraint at the time of recombination
where again we have used at recombination , and assumed that the modulus dominates the energy density.
ii.3 Curvature and Isocurvature Perturbations in Non-thermal Histories
After a brief review of isocurvature perturbations, in this subsection we consider the non-thermal histories discussed above to establish how well existing isocurvature constraints restrict SUSY model building in the presence of moduli and identify the corresponding observationally interesting cases.
One can assign a curvature perturbation to each species, , which is defined such that it is exactly conserved on super-horizon scales in the adiabatic limit, when the expansion is dominated by a single species (i.e. once the universe is radiation dominated after modulus decay):
where is the Newtonian potential and dots denote derivatives with respect to cosmic time (see Appendix A.1 for conventions used). From this we find the total conserved curvature perturbation
Then, a gauge invariant definition of an isocurvature perturbation between two fluids and is given by (e.g. Malik:2008im ())
In connecting with observations it is convenient to instead define the isocurvature contribution of a particular fluid relative to the total curvature, which in the radiation dominated, post modulus decay universe is approximately given by that in radiation , so that
where is the spatial curvature on surfaces of constant standard model (MSSM) radiation density.
During inflation, if the mass of the modulus is lighter than the Hubble scale
then the quasi-deSitter period will result in long-wavelength fluctuations of the field with an average amplitude (e.g. Linde:2005ht ())
This leads to an additional source of cosmological perturbations different from that sourced by the inflaton. Following inflationary reheating – where typically all of the energy and matter of the universe is assumed to be created – the modulus can decay leading to an additional source of radiation and matter. Thus, whereas radiation and matter created during inflationary reheating will inherit the inflaton’s fluctuation , those produced from moduli decay will instead be set by that initially carries no curvature, implying the existence of isocurvature modes.
Although we have seen that the moduli decay is an essential part of a non-thermal history, there are still many ways in which isocurvature perturbations may be observationally irrelevant and lead to no new constraints on model building. Firstly, if the mass of the modulus is above the Hubble scale during inflation , then in a single Hubble time the amplitude of its fluctuations will be exponentially suppressed on large scales by a factor and so the inflaton will be the only relevant source of cosmological fluctuations Linde:2005ht (). Another important observation was made by Weinberg, who demonstrated that even if an isocurvature mode is generated initially, if local thermal equilibrium is reached these modes will become adiabatic Weinberg:2004kf (). And finally, if the modulus comes to dominate the energy density of the universe (determining the cosmic expansion rate) this also has the effect of washing out any existing isocurvature perturbations.
To make some of these ideas more precise and establish the cases of observational interest for the rest of the paper we closely follow the formalism of Langlois:2011zz (). We will be interested in the decay of moduli into MSSM and hidden sector particles. We define the branching ratio of the decay from moduli to species as . Thus, if before the decay the abundance of a species is then the fraction of particles created by the decay is
where is the initial abundance in moduli compared to the total energy density . Thus, if no particles are produced in the decay we have , whereas if all of them are produced then we have .
Assuming the moduli scale as pressureless matter ( with ) we can express the curvature perturbation for a fluid following moduli decay compared to its value before as
where the matrix elements are given by (no trace)
where the sum is over all significant contributions to the energy density prior to decay. Using that for any , we can then express (23) in terms of the matrix elements as
where is the entropy perturbation before the decay.
To see the utility of this approach, consider a three fluid system similar to that expected by the non-thermal history discussed above. Again treating the scalar oscillations as pressure-less matter, we will have three fluids , , and , which are energy densities of modulus, DM, and radiation, respectively. The matrix elements (II.3) then become
where the matrix above is written in the basis so that gives the curvatures after the transition and
is the total weighted relic abundance prior to decay.
From this expression we can immediately see the statement earlier that if the modulus comes to dominate the energy density, with , and all of the dark matter and radiation are produced in the decay, , then the isocurvature mode above vanishes, . In the other limit, where radiation and dark matter are not produced in the decay we have
and so the existence of an isocurvature mode depends on whether one was initially imprinted. Thus, if these (prior to decay) sources were produced during inflationary reheating and where thermal equilibrium was established then the isocurvature perturbation vanishes as at the time of inflationary reheating where is the curvature fluctuation of the inflaton. When modulus decay from the mode with is included multiple sources of curvature perturbations are present and can generate a non-zero .
Thus, the cases we will be interested in here correspond to when the modulus does not completely dominate, and/or when the MSSM and any dark sector particles come from multiple sources. As an example of the latter, some dark matter will be produced thermally in the early universe and moduli decay will lead to an additional source of dark matter. If the modulus does not come to dominate, we will see this can generate a substantial isocurvature perturbation. Isocurvature requires the modulus to be subdominant prior to decay and this will require us to consider moduli fields with sub-Planckian displacements – since otherwise complete moduli domination will be inevitable. This corresponds in (2) to the case where the flat direction is lifted by a low dimension operator and/or the scale of new physics is taken significantly below the Planck scale.
Iii Modulus Decay and Correlated Isocurvature
Having reviewed the instances where isocurvature and dark radiation can be generated in non-thermal cosmologies, we now examine these cases in more detail with emphasis on how these two phenomena can provide complementary constraints. We begin by presenting the background equations and then show how we compute cosmological parameters. The background and perturbations define a system of coupled O.D.E.s that we solve numerically. Details of our numerical procedure to treat the evolution of the perturbations, including equations of motion and initial conditions, are relegated to Appendix A.
iii.1 Background Evolution and Parameters
The background is given by unperturbed flat FRW space, in physical time
The evolution of the scale factor, , is fixed by the Friedmann equation
where runs over all species of energy density: all standard model and MSSM (visible sector, VS) radiation, including massless neutrinos 777Using rather than will have only a minor effect on our value of for dark radiation.; dark matter; dark radiation; the lightest modulus field 888Our treatment assumes that the heavier moduli decayed producing no isocurvature. Fixing the amplitude of scalar perturbations using inflationary parameters assumes in addition that they did not alter the curvature spectrum, i.e. that they decayed before or during inflation.. In practice, rather than using time to evolve our equations, we will use the number of e-folds since the initial time, , which absorbs the scale factor normalisation.
We ignore the effects of the baryons. In the background evolution they are sub-dominant, while in the perturbations they are tightly coupled to the photons. Baryon isocurvature modes have the same spectrum as DM isocurvature, and can be accounted for with appropriate scaling Ade:2013zuv ().
We take the universe to be initially dominated by radiation, which can be assumed to have originated either from the decay of the inflaton, or of the next to lightest modulus, and to contain no DM or DR999For numerical stability in our code we begin with tiny amounts of DM and DR.. Following inflationary reheating the modulus field, , has a quadratic effective potential and begins displaced from the minimum at by some initial value and with no initial velocity, .
The energy density and pressure of the modulus field are given by:
At early times the modulus field evolves according to the free Klein-Gordon equation
The field begins frozen at the initial displacement, . Once the mass overcomes the Hubble friction the modulus begins to roll in its potential and oscillates about the minimum. At this point perturbative decay of the modulus begins, which can be taken into account by introducing an additional friction term given by the modulus decay rate, Kofman:1997yn ()101010We do not consider the possibility of parametric resonance during this decay, though it may lead to further interesting phenomenology. Given the criterion for the onset of parametric resonance presented in erickcek2011 (), we are safe with our assumption as the majority of our parameter space will satisfy the required bound for negligible parametric instability. .
We define the time when this term is introduced, , to be given by the first passage of the modulus field through the minimum of the potential.
Once coherent oscillations begin the average pressure in the modulus field goes to zero, causing it to behave like matter Turner:1983he (), while the energy density evolves according to the conservation equation
It is computationally impractical to evolve the two time scales and when is given by Eq. (5). Therefore after we use Eq. (40) rather than Eq. (38) to evolve the modulus energy density. We will use a similar approximation for the perturbations, and show in Appendix A.4 that neither approximation has a substantial effect on our results.
Prior to modulus decay, the other components evolve according to the free conservation equations, while during decay they are sourced by the modulus:
where is the equation of state for the species and gives the branching ratio of the modulus to species . By conservation of energy . The term accounts for particle production of species by modulus decay. In addition to decays, dark matter annihilations can also play an important role particularly if the modulus dominates the energy density prior to decay and the branching ratio to dark matter is large. The effect of annihilations can be captured in an “effective decay rate”, which is how we will deal with them here. As the annihilations happen in less than a Hubble time, their primary effect is to simply reduce the amount of dark matter to that given by (7) and increase the amount of radiation.
The term proportional to represents energy density in species produced by modulus decay, while represents energy density present already (e.g. from inflationary reheating). When the modulus is totally dominant in the energy density, the component is universal across species, and dominates their evolution, while as the modulus goes between dominance and sub-dominance the species dependent term dominates. These scalings will effect the sensitivity of isocurvature observables, which depend on modulus energy density fraction, to the modulus parameters.
The branching fractions to standard model radiation, DM and DR give the reheat temperature and abundances, which we define some number of e-foldings after the initial time, when the modulus has decayed completely. After this time the conservation of energy conserves the abundances as in a standard thermal cosmology. is defined by
when the modulus has decayed such that it is sub-dominant to the DM. Since the modulus is initially dominant over the DM, which to give a standard cosmology with BBN and equality at the correct temperatures is itself substantially sub-dominant to the radiation, this condition guarantees that modulus decay has been completed. The DM abundance, reheat temperature, and amount of dark radiation parameterised by are all evaluated at .
The effective value of sets the DM abundance by giving the value of . While this should not vary too much around its central Planck value set by , it is strictly a free parameter and we should expect its central value to change in any non-standard cosmology. We choose small to give a reasonable DM abundance, and find that changing its value in any sensible range does not affect the evolution of the perturbations. Firstly, since DM is always sub-dominant prior to the actual DM abundance does not affect the expansion rate. Secondly, by assumption we consider only cases where decay of the modulus sources practically all of the DM, either by sub-critical or super-critical production depending on the cross-section. Therefore in Eq. (32) is always close to unity and does not affect the isocurvature observables (we do not assume the same for radiation: modulus dominance or sub-dominance affects ). and the DM abundance will play no further role in our analysis, except to stress again that when the modulus dominates the energy density and the DM is by necessity non-thermal in origin.
The ‘reheat temperature’111111Since we allow for decay when the modulus is sub-dominant this is a slightly liberal use of the term., , is found from the energy density of radiation at :
For any finite range of we assume to be a constant and the inversion to find is trivial. Our reheat temperatures are often low, GeV, and so we take to be given by the Particle Data Group, Ref. Beringer:1900zz (). Taking this model for only affects and, as we discuss below, , and does not affect the isocurvature observables defined below.
We choose to reject all models where MeV. This is a hard cut if the modulus is dominating the energy density just prior to decay and is the origin of the Cosmological Moduli Problem (CMP) (see Watson:2009hw (); Acharya:2008bk () and references therein): a non-thermal universe with decaying moduli can ruin the successful predictions of BBN. However, if the modulus is sub-dominant at the time of decay there is a continuous region in parameter space connecting our model to effects in the late universe where decay can occur much later or not at all, such as is the case for (early) dark energy and axion dark matter121212For example, the modulus can decay in a radiation (or DM) dominated universe at very low temperature long after BBN has completed. Such effects should be thought of as decaying Dark Energy, and since cosmological modes will then be entering the horizon these effects should be computed using a Boltzmann code. (e.g. marsh2011 (); marsh2012 ()). In such a case, the choice taking in our model is just a matter of definition separating models of ‘initial conditions’ from models affecting late universe physics.
As discussed in Section II.2, the dark radiation abundance is parameterized by evaluated at neutrino decoupling. This can be obtained from the DR energy density at the time of modulus decay at , given , using the fact that (e.g. Choi:1996vz ())
Since we allow for the possibility that the modulus does not dominate the energy density at the time of decay we cannot set the ratio of dark to standard model radiation evaluated at equal simply to the ratio of branching ratios, as in Eq. 19. We must therefore evaluate numerically for each choice of parameters. However, when the modulus is dominant at the time of decay, the amount of DR produced can be computed analytically and approaches the asymptotic value:
The parameters of the background evolution are specified by , from which we compute and . Once the initial conditions for the perturbations are fixed by inflationary parameters giving normalisation and spectral indices of the power spectra, the background evolution determines the evolution of the perturbations. Therefore the final amplitudes and correlations between the isocurvature modes are fixed by these same basic parameters of the background evolution.
Fig. 2 shows the background evolution in an example model where the modulus decays while it dominates the energy density, with TeV, , (exaggeratedly small for illustration), 131313The value of can be computed in explicit models. For example in angus2013 () it lies in the range 0.3 to 0.5. Our value is chosen semi-arbitrarily. It is and as we will see later gives variation of over a range interesting for isocurvature.. The DR is sub-dominant to the radiation right up until modulus decay has completed, and the value of freezes in. Being sourced entirely by modulus decay, both the DM and the DR scale in the same way with e-folding , as in terms of scale factor. The SM radiation joins them once the modulus becomes dominant as we see looking more closely in Fig. 3 (Left Panel). The two scalings of the radiation during modulus decay, Eq. (43), when the modulus is dominant, and when it is in transition from sub-dominant to dominant, are clearly visible.
Fig. 3 (Right Panel) compares the previous model to another the same except with (exaggeratedly large for illustration), so that the modulus decays while it is sub-dominant. Sub-dominant modulus decay does not affect the scaling of the dominant SM radiation, although the branching ratios in the two models are the same. With sub-dominant decay the amount of DR produced is considerably smaller, . By analogy with the curvaton and from the results of Section II.3, the sub-dominant decay produces large amounts of isocurvature, while the dominant decay does not. The detailed understanding of this in our model is the focus of the rest of this paper. The plots shown in Fig. 3 thus serve as cartoons to aid in understanding the entire model.
iii.2 CMB Observables
Here we build upon the results reviewed in Section II.3, defining precisely our CMB observables and how we use the curvature perturbation to compute them. Details of the perturbed equations of motion, numerical method, and initial conditions are given in Appendix A, while the power spectra are discussed in Appendix B.
In the initially radiation dominated universe the curvature is primarily due to radiation and the adiabatic condition implies for the perturbations ‘inf’ laid down by the inflaton that , with and . With isocurvature initial conditions ‘mod’ seeded by the modulus we have that , while sources non-zero .
We follow the evolution of all species, keeping track of their ’s, up until the modulus has decayed and we have entered radiation domination at , when all freeze-in and set the initial conditions for computation of the CMB power spectrum. Unless otherwise stated, correlators are evaluated at .
The two point correlations between the total are easy to compute, since by assumption the ‘inf’ and ‘mod’ initial condition modes (see Appendix A.2) are uncorrelated with one another, but totally correlated with themselves. This implies that the correlation matrix is given by
Using this correlation matrix we can compute any other correllators of total or . Defining the power spectrum by
then, for example, the total curvature perturbation power spectrum, . We construct the observables for the isocurvature fraction, , its correlation with the curvature perturbation, , and the cross correlation between any two isocurvature modes, , all evaluated at the pivot scale, :
where there is no sum implied over repeated indices, unless stated. The total power from isocurvature is given by considering the total isocurvature, , and total scalar power, :
Clearly is not independent of the ’s and ’s, yet it is useful to compute since it gives an overall measure of the isocurvature power. In the limit of pure isocurvature we have that and , with the further contribution to from the correlation of DM and DR proportional to , as we will see below. That is, in this limit, and are numerically equal to one another.
Our variables are used to construct the total CMB power spectrum, , as follows
The overall normalisation is given by
The unit CMB spectra, , are computed with unit normalisation at the pivot scale, , from the adiabatic (ad), CDI and DRI initial conditions bucher2000 (), where ‘CDI’ and ‘DRI’ refer to the CDM and DR density isocurvature modes. The DRI mode is related to the more familiar neutrino density isocurvature mode, NDI, by
The factor of takes into account that in our model the DRI mode is not sourced by the standard model neutrinos 141414Use of NDI and DRI also avoids confusion about the production mechanism, which should be contrasted to that of e.g. savelainen2013 (). The physical difference between NDI and DRI is that with standard model neutrinos (rather than sterile neutrinos or axions) NDI can only be produced if the modulus decays after neutrino decoupling, while the DR by assumption decoupled at very high temperatures and so DRI is produced by modulus decay at any temperature. The NDI and DRI do, however, produce the same CMB spectra, as can be seen from the equations of motion, which are not sensitive to the fermionic or bosonic character of radiation bertschinger1995 ().. In contrast to the effect of varying the axion contribution to DM in the axion CDI mode 5yearWMAP (), here constraints to mean this factor can be determined, just like the ultra-light axion contribution can be determined to break a similar degeneracy as discussed further in Ref. marsh2013 () (see also the next subsection).
The contribution to CMB power from correlations, , can be calculated given the values of , and . It is given by
Again we must be careful with DR versus neutrinos and note that