Cosmological Moduli and the Post-Inflationary Universe:
A Critical Review
We critically review the role of cosmological moduli in determining the post-inflationary history of the universe. Moduli are ubiquitous in string and M-theory constructions of beyond the Standard Model physics, where they parametrize the geometry of the compactification manifold. For those with masses determined by supersymmetry breaking this leads to their eventual decay slightly before Big Bang Nucleosynthesis (without spoiling its predictions). This results in a matter dominated phase shortly after inflation ends, which can influence baryon and dark matter genesis, as well as observations of the Cosmic Microwave Background and the growth of large-scale structure. Given progress within fundamental theory, and guidance from dark matter and collider experiments, non-thermal histories have emerged as a robust and theoretically well-motivated alternative to a strictly thermal one. We review this approach to the early universe and discuss both the theoretical challenges and the observational implications.
- I Introduction
- II String theory, Moduli, and a Non-thermal Universe
III Dark Matter Genesis
- III.1 WIMPs
- III.2 Axions
- IV Baryogenesis and the Cosmic Coincidence
- V Gravitational Constraints: CMB and Large Scale Structure
- VI Challenges and Outlook
The success of Big Bang cosmology in predicting the correct abundances of the light elements and the existence of the Cosmic Microwave Background (CMB) rests upon the assumption that the universe began in a hot, dense state and as the universe expanded it cooled. This thermal history for the early universe can then be used to try to understand the primordial origin of dark matter particles, baryons, and the matter / anti-matter asymmetry. This picture is simple, elegant, and robust, but it can not be complete. Indeed, the primordial density perturbations – indirectly detected through the temperature anisotropies of the CMB and required for the growth of the large-scale structure of the universe – can not be causally generated in a strictly thermal universe. Instead, inflation provides the most compelling model for their origin, and during this epoch the universe was not in thermal equilibrium. Moreover, following inflation the universe must reheat providing the hot big bang. This process can occur far from equilibrium and result in an effectively matter dominated epoch that could last up to the time of Big Bang Nucleosynthesis (BBN).
Given the importance of realizing inflation and reheating, model building has had a rich and long history (for recent reviews see Baumann:2014nda (); Baumann:2009ds ()). ‘Chaotic’ inflation provides one of the simplest examples, where inflation is of the so-called large field type111Small field models of inflation typically require additional fine-tunings of initial conditions to enter the attractor regime (see e.g. Brandenberger:2003py () and references within.). This tuning is in addition to that required for ensuring the flatness of the inflationary potential. with the inflaton evolving over super-Planckian distances in the field space. Some of these models can lead to detectable gravitational wave signals, but they are also sensitive to Planck scale physics and dangerous quantum gravity corrections. As a result, a large effort has been put forth to understand how to construct models in the controlled framework of string theory. One lesson from these studies is that typically additional symmetries (such as Supersymmetry (SUSY)) and additional scalar fields (moduli) are generic and necessary ingredients for constructing realistic models. The former are necessary for protecting the inflaton potential from dangerous UV corrections222In string theory, large field models with approximate shift symmetries providing the flatness of the potential can be constructed, but additional symmetries – such as SUSY – are required to protect this potential from further corrections., whereas moduli are a ubiquitous prediction resulting from both the presence of these new symmetries, as well as the string construction itself. If unstable these moduli can spoil inflation, or lead to undesirable late-time effects such as large entropy production after BBN or fifth force violations of General Relativity. Thus, although string theory provides a framework to construct realistic models of inflation, it appears to introduce a new nuisance – the Cosmological Moduli Problem333The cosmological moduli problem made its first appearance in the form of the Polonyi problem in the earliest versions of spontaneously broken supergravity. This was the observation that hidden sector models of supersymmetry breaking with gravity mediation often contain scalar fields with weak scale masses and gravitational couplings. Just after inflation, these fields would behave like non-relativisitic matter and continue to dominate the energy density of the universe until after BBN. Banks:1995dp (); Banks:1995dt (); Coughlan:1983ci (); deCarlos:1993jw (); Banks:1993en ().
Significant progress has been made on moduli stabilization both during and after inflation (for reviews see Grana:2005jc (); Douglas:2006es ()). Many of the moduli have been shown to be stabilized with string scale masses by accounting for a combination of effects resulting from the presence of branes, fluxes, and/or strong hidden sector dynamics – all of which are expected ingredients of the theory. However, in explicitly constructed examples it is also a common result that many of the moduli will remain parametrically lighter than the string scale. The resulting cosmology differs from that of a strictly thermal post-inflationary history. These fields are displaced from their low-energy minima in the early universe and undergo coherent oscillations, mimicking a matter dominated epoch444This is strictly true only if the mass term in the potential gives the dominant contribution, otherwise the cosmological scaling of the energy density is determined by the dominant term in the potential Turner:1983he (). prior to BBN. The fields typically decay late through gravitational strength couplings, and the universe reheats (again) via the production of relativistic Standard Model and other light (beyond the Standard Model) particles. Depending on the masses and couplings to other fields, such as dark matter and baryons, this can lead to a new expectation for the post-inflationary cosmology prior to BBN.
The rest of the paper is structured as follows. In Section II, we review the cosmological moduli problem and briefly comment on its appearance in effective supergravity descriptions of string / M-theory. In Section III we discuss the effect of cosmological moduli on the thermal history of dark matter. We look at the non-thermal WIMP miracle and its phenomenological status, as well as axions from string theory and their status as dark matter candidates within non-thermal histories. In Section IV, we study the impact of cosmoligical moduli on theories of the baryon asymmetry of the universe, critically reviewing the Affleck-Dine mechanism and the baryon-dark matter cosmic coincidence. In Section V, we study cosmological moduli in the context of the CMB and Large Scale Structure, looking at implications for the matter power spectrum, effective number of neutrino species, and isocurvature constraints. We end with a section on the challenges and future prospects of these scenarios.
Ii String theory, Moduli, and a Non-thermal Universe
ii.1 Moduli Decay and Another Big Bang
To see how a non-thermal history arises from moduli, we consider a single modulus555The case of non-thermal histories with multiple moduli was considered in Acharya:2008bk (). Generically, string / M-theory theories have many moduli with various masses, and the lightest is typically the relevant one. which enjoys a classical shift symmetry . At low energy we expect the symmetry to be broken, resulting in a mass for the field . Moreover, if the presence of this new symmetry is tied to addressing the electroweak hierarchy problem (e.g. in SUSY or extra dimensional approaches) then we expect with GeV the reduced Planck mass. 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. Both thermal and quantum fluctuations can displace the scalar during the early evolution leading to a matter dominated phase Dine:1995kz ().
As an explicit example, consider string moduli in SUSY based approaches where on general grounds we expect that the shift symmetry of the modulus should be broken by both the finite energy density of inflation (which spontaneously breaks SUSY at the high energy) 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 positive666One can argue rather generally for the positivity of based on causality, unitarity, and demanding that the model have a UV completion (although a few known counter-examples exist in gravitational systems). We refer the reader to Adams:2006sv () for details. On the other hand, the choice of positive (leading to a tachyonic mass) is chosen to realize the general expectation that the high energy and low energy minima of the field would not generically be expected to coincide in a gravitational and/or time dependent background Dine:1995uk (). Ideally one would like to calculate this in an explicit, UV complete model. order one constants, 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 physics777This 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 (1) are the dominant terms in the potential during high scale inflation (), this implies the minimum of the field at that time will be
Whereas, later the low-energy SUSY breaking mass will give the dominant contributions to the potential, and this minimum is near . This initial displacement from the low energy minimum eventually leads to coherent oscillations of the field and the formation of a scalar condensate. The amplitude of the oscillations is determined by the initial displacement , where for the potential (1) we have . Hubble friction ceases and oscillations 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. 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 of at most . Most of the field decays888In 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 because the entropy is not conserved. 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 could provide a viable dark matter candidate (more on this in Section III).
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 or
where and is the number of relativistic degrees of freedom at , and this temperature must be larger than around MeV to be in agreement with BBN light element abundances Kawasaki:1999na ().
A comparison between the standard thermal history and that of the non-thermal history appears in Figure 1 for an interesting case where the modulus decays slightly before BBN and inflationary reheating occurs at a high temperature. We now turn to the question of what moduli masses (and so reheat temperatures) are naturally expected from string / M-theory approaches.
ii.2 Fundamental Theory and Moduli Masses
For an interesting departure from a thermal post-inflationary universe the mass of the lightest modulus should be in a window between TeV. For masses much further above the upper bound little departure from a thermal universe is expected, whereas the lower bound is required for consistency with BBN as seen from (5). As a phenomenological approach non-thermal cosmologies have been considered for their implications for dark matter (see Watson:2009hw () for a review). However, lacking theoretical motivation these models are arguably somewhat exotic.
The key question is – should one generally expect a modulus in this mass range? In string / M-theory approaches, inextricably connected with the question of moduli masses is the question of the gravitino mass. This is because moduli stabilization plays a central role in the nature of supersymmetry breaking. The potential of the moduli determines their F-terms dynamically, and along with the Kahler metric of the matter sector, determines the strength of gravity mediation. To extract the mass of the modulus, it is sufficient to write down its scalar potential in the supergravity limit (given in terms of its superpotential and Kahler potential, and )
One requires, at the extremum, that and . Moreover, is required to obtain an almost vanishing cosmological constant. We note that the gravitino mass is defined as where we work in units of for simplicity.
This set of conditions is restrictive enough to give a relation between and , without going into the details of visible sector model-building. Since the modulus mass matrix is positive definite by assumption and both terms in (6) are comparable to due to the condition , one obtains that the smallest eigenvalue of the mass matrix should be . For a single modulus, this trivially implies .
However, the modulus must be redefined to the physical basis with canonical kinetic term. An example Kahler potential is given by , where is the volume form whose expression in terms of the moduli depends on the compactification manifold through the intersection numbers. In a basis with canonical kinetic terms, the lightest modulus will generally have a non-trivial profile along different field directions. However, for many settings that provide realistic phenomenology, the lightest modulus is also the largest, and one can approximate , where . This is the case in single modulus examples like KKLT Kachru:2003aw (), as well as the cases of Large Volume Compactifications in Type IIB Conlon:2013isa () and compactifications of M-theory Acharya:2008zi (). In that case, the canonical normalization yields , where by abuse of notation is now the physical field. One thus obtains
Usually, one finds in the supergravity regime. For example, if the modulus is stabilized supersymmetrically999The relation between moduli and gravitino mass in this case was first noted in LoaizaBrito:2005fa (). so that , then . Then, under quite general circumstances one expects
For TeV, as one would expect in models that at least partially address the electroweak hierarchy problem, we see that falls within the mass range where a non-thermal history results.
We would like to emphasize the interesting result that the lower bound on the mass required to address the cosmological moduli problem provides further motivation for a little hierarchy between the electroweak scale and the scale of new scalar physics. Indeed, the discovery of a GeV Higgs at the Large Hadron Collider (LHC) – and nothing else – appears to have left the relevance of supersymmetry (SUSY) in question, particularly as a mechanism for stabilizing the hierarchy between the Electroweak and Planck scales. Standard Model Light superpartners (‘Natural’) SUSY still remains possible, but with some cost of increasing the level of complexity of models (see e.g. Randall:2012dm ()). Within the Minimal Supersymmetric Standard Model (MSSM), 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 nature101010It 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.
It is then remarkable that in these string / M-theory based approaches with moduli that the scale required to avoid the cosmological moduli problem is of the correct order of magnitude for realistic particle phenomenology111111We note that already in 2006 this mass range for the lightest scalars and gravitino was noted in Acharya:2006ia () long before the Higgs discovery and LHC bounds favoring heavy scalar superpartners. with a GeV Higgs.
Another interesting result of this string-based framework is that the reheat temperature is not a free parameter, but a consequence of the hierarchy between the electroweak and Planck scale (determined by ), which also helps determine the SUSY breaking masses of other sparticles in the theory. In both supergravity and string motivated approaches, the key lesson is that the reheat temperature is intimately connected to other aspects of the theory and not a free and tunable parameter. Given that gravitationally coupled scalars or moduli are generic in high energy UV completions of the Standard Model and in no sense exotic, we see that non-thermal histories are a feasible and robust possibility, and perhaps an inevitable one. Given this motivation from theory, we now turn to the observational consequences of non-thermal histories.
Iii Dark Matter Genesis
The origin of dark matter as a thermal relic is a simple and predictive scenario for its creation, which is also independent of the prior thermal history of the Universe. However, as we have argued so far, one is led on compelling theoretical grounds to a re-evaluation of the history of the Universe prior to BBN. This necessitates looking afresh at the consequences of non-thermal histories on dark matter phenomenology, a topic to which we turn in this section. In our choice of dark matter candidates, we will discuss both Weakly Interacting Massive Particles (WIMPs) and axions.
If the primordial universe is thermalized, massive long-lived particles may freeze-out with a final abundance determined primarily by their mass and annihilation cross-section Bertone:2004pz (). This is the basis of thermal WIMPs, which assumes a weak-scale cross-section, and cms, where is the typical velocity. Alternatively, non-thermal dark matter is produced via the decay of heavier particles into a long-lived final state and does not require thermal equilibrium Moroi:1999zb (); Acharya:2008bk (); Chung:1998rq (); Fornengo:2002db (); Pallis:2004yy (); Gelmini:2006pw (); Gelmini:2006pq () (for reviews, see Watson:2009hw (); Baer:2014eja (); Sinha:2013sxa (), and for recent related work Allahverdi:2013tca (); Hooper:2011aj (); Kelso:2013paa ()). Dark matter models are constrained by both direct detection experiments and searches for astrophysical signals generated by their annihilation products. Non-thermal dark matter can have a higher annihilation cross-section than thermal dark matter so astrophysical signals are potentially stronger for these scenarios, particularly in indirect experiments such as FERMI and AMS-2 Gelmini:2008sh (); Grajek:2008pg (); Grajek:2008jb (); Dutta:2009uf (); Kane:2009if (); Sandick:2011rp (). We now review the thermal and non-thermal WIMP paradigms.
iii.1.1 Thermal WIMPs
The standard picture of the cosmic history assumes that the universe is radiation dominated prior to Big Bang Nucleosynthesis (BBN) and that dark matter is created from the thermal bath created at the end of inflation. The evolution of dark matter is described by the Boltzmann equation
where is the thermally averaged cross-section, is the number density, and is the number density of the species in chemical equilibrium, i.e. , where is a relativistic particle. In the early universe with , the dark matter is relativistic and in thermal equilibrium with the plasma, and its density is determined by the full solution to (9). When , it becomes Boltzmann suppressed, until finally the expansion rate becomes larger than the annihilation rate, leading to thermal ‘freeze-out’. The dark matter density at this time is sensitive only to its mass and annihilation cross-section.
Thus, the thermal paradigm is simple, predictive, and more or less independent of the underlying microscopic physics. Moreover, there is the suggestive fact that the observed relic density is satisfied for cms (or picobarn), which corresponds to weak scale interactions. The weak scale is precisely where new physics is expected to appear, on very different physical grounds. Plausibly, then, dark matter is a Weakly Interacting Massive Particle (WIMP) that is there as a part of new physics at the weak scale, motivated by issues like the gauge hierarchy problem. The thermal WIMP miracle is precisely this combination of simplicity and serendipity. Since the weak scale is also the energy scale that we are currently able to probe experimentally, the thermal WIMP is by far the most actively studied dark matter candidate.
iii.1.2 The Non-thermal WIMP Miracle
Despite its appeal, we saw in Section II that there are compelling reasons to expect that the history of the universe prior to BBN was not as simple as the thermal WIMP paradigm requires. It is worthwhile to revisit the assumptions underlying the arguments above. They are: WIMPs reached chemical equilibrium in the early universe; the universe was radiation dominated at the time of freeze-out; following freeze-out, there was no dilution of the relic density due to entropy production from any sources, and conversely, there was no further production of dark matter from late-time sources.
The above assumptions look difficult to realize from a top-down perspective (and are in increasing tension with data from the bottom-up perspective, as we will discuss in Section III.1.4). Indeed, as we discussed in Section II assumptions such as WIMPs achieving thermal equilibrium, the absence of entropy production following thermal freeze-out, and strictly a radiation dominated universe prior to BBN seem to be in conflict with top-down approaches to model building. In particular, the presence of moduli imply violations of all these conditions.
This by itself does not mean, however, that the thermal WIMP paradigm itself is disfavored, although a purely thermal history of the universe prior to BBN is. Moduli may decay early enough (above the freeze-out temperature of WIMPs) to render a prior non-thermal history irrelevant as far as low-energy observables are concerned. The entropy dilution from such very early decay is likely to erase the prior dark matter density and produce it afresh (by reheating the modulus into the visible sector generally). The dark matter thus produced will then have time to thermalize and give the usual thermal WIMP story.
A non-thermal WIMP is obtained if a modulus decays after the freeze-out temperature but before the onset of BBN. This translates into a window in terms of the decay width, and hence in terms of parameters of the fundamental theory (such as the modulus mass and overall decay constant). Leading up to (8) we have argued that it is natural for moduli to fall within this mass window, and in fact the explicitly studied compactifications in the literature have such moduli.
iii.1.3 Reheating Again and Dark Matter Production
Moduli decay dilutes any previous population of dark matter by a factor , which is typically large enough to render it irrelevant. This scaling can be understood from the fact that the temperature goes121212If there is significant decay of the modulus during the moduli dominated epoch entropy is not conserved and this alters the temperature / scale factor relation Giudice:2000ex (). In this review we will work in the instantaneous decay approximation and refer the reader to Fan:2014zua () for a more rigorous treatment. We note that for our considerations here we do not expect any significant qualitative differences. as , while volumes scale as (where is the scale factor).
The moduli decay is expected to produce dark matter and as a result there are two options:
If the WIMP number density exceeds the critical value , then the WIMPs will quickly annihilate down to this value, which acts as an attractor Acharya:2008bk (). The fixed point value is evaluated at the reheating temperature instead of the freeze-out temperature. This results in a parametric enhancement of the relic density
Since the relic density scales inversely with the annihilation cross section, this implies that WIMP candidates with annihilation cross sections larger than the canonical value by a factor can give the correct relic density in a non-thermal history with the above compensating factor. Examples of such WIMPs are the Winos and Higgsinos of the minimal supersymmetric extension of the Standard Model (MSSM). For future reference, we will call this option the ’annihilation’ scenario.
In more detail, the standard expression for the thermal relic density given by
where in the last line we used the entropy density today with the Hubble parameter in units of km/s/Mpc and we chose some fiducial values with the number of relativistic degrees of freedom at the time of reheating. We note that unlike the thermal result (12) depends on both the properties of the dark matter (mass and annihilation rate) and on the mass of the decaying modulus (). As discussed in Section II the scalar mass (and so reheat temperature) is not a free parameter, but related to the gravitino mass. The mass of the scalar (and so the relic density of dark matter) is controlled by the need for SUSY to generate a hierarchy between the electroweak and Planck scale. We have chosen a fiducial value for the annihilation rate that yields roughly the right amount of dark matter for the hierarchy set by the choice of low-scale SUSY breaking. The cross-section is three orders of magnitude higher than expected with a thermal history with important experimental consequences, as discussed next in Section III.1.4.
The second option we will call the ‘branching scenario’. In this case, the WIMP number density produced from moduli decay is below the fixed point value. This implies that there is no further WIMP annihilations and the final number density is that produced from the decay, i.e., , where is the branching ratio for scalar decay to WIMPs and is the number density of the scalar condensate. With an appropriate choice of (that very much depends on the microphysics), this option can accommodate WIMPs whose annihilation cross section is higher and (crucially) also lower than the canonical value. Note that the latter case (a pure Bino of the MSSM) cannot be accommodated in any thermal setting without substantial fine-tuning of the spectrum.
iii.1.4 Phenomenological Aspects and Future Probes
We have seen that in theories with moduli and TeV scale SUSY breaking that a non-thermal history is attractive. When considering the associated phenomenology, it is non-trivial to avoid this conclusion without introducing fine-tuning. For example, one could raise both the gravitino and modulus mass, re-introducing the gauge hierarchy fine-tuning, or one could decouple the modulus-gravitino correlation in (7) making the modulus much heavier as in racetrack models Kallosh:2004yh (); BlancoPillado:2004ns (); Linde:2007jn (); Allahverdi:2009rm (). This introduces large fine-tuning that is not associated with the cosmological constant.
From a bottom-up perspective, one of the most striking aspects of the thermal WIMP paradigm is how difficult it is to realize even with the large number of parameters within the MSSM. Taking Neutralinos as the WIMP candidates, the Wino and Higgsino – which annihilate very efficiently – must be very heavy ( TeV and TeV, respectively) to give the correct relic density through thermal freeze-out ArkaniHamed:2006mb (). Whereas a thermal Bino typically gives a relic density that is too high, unless its annihilation cross section is boosted through major co-annihilation effects Griest:1990kh () or mixing with the Higgsino ArkaniHamed:2006mb (). Both of these options are fine-tuned and the latter is highly constrained by direct detection Hooper:2013qjx (); Anandakrishnan:2014fia (). However, within non-thermal histories Wino or Higgsino dark matter with masses of GeV can be accommodated within the annihilation scenario discussed above. Moreover, Bino dark matter with a non-thermal history following the ’branching scenario’ can satisfy the relic density constraint, without relying on any other low-energy ingredient. Thus, an experimental confirmation of a pure gauge eigenstate of the MSSM as the dark matter of the universe would seem to suggest that the universe underwent a non-thermal history, since otherwise one would need to accept a large amount of fine-tuning. Direct detection limits are currently unable to probe such pure eigenstates, whose scattering cross-section off of Standard Model nuclei is typically cm, although future facilities like XENON1T and LZ will reach the required sensitivity up to WIMP masses of TeV Cushman:2013zza ().
Indirect detection provides an enormous amount of information by putting upper bounds on the dark matter annihilation cross section. A joint analysis of seven Milky Way dwarf galaxies using a frequentist Neyman construction and Pass 7 data from the Fermi Gamma-ray Space Telescope undertaken in Geringer-Sameth:2014qqa (); GeringerSameth:2011iw () excludes generic thermal WIMP candidates with mass less than GeV annihilating to final states131313The exclusion bounds are similar for other final states.. Current limits using dwarf spheroidals data and the PASS-8 software from Fermi-LAT collaboration rule out the canonical WIMP for masses below GeV. WIMPs in this mass range thus over-satisfy the relic density in a purely thermal history. Since multi-component dark matter is not an option in this case, the only way WIMPs lighter than GeV can constitute the dark matter of the universe is if they followed a non-thermal ’branching scenario’ cosmological history Allahverdi:2012wb (). As this exclusion reach becomes larger, it forces more and more potential candidates squarely into the non-thermal framework.
On the other hand, candidates like Winos and Higgsinos with large annihilation cross section are now severely bounded by a combination of continuum and line searches from FERMI data Fan:2013faa (); Cohen:2013ama (). Continuum photons arise from the tree level annihilation process , where denotes the WIMP. Dwarf galaxy data rules out Wino dark matter masses up to around GeV, while galactic center data rules it out up to around GeV for either NFW or Einasto profiles Hryczuk:2014hpa (). These bounds become stronger (weaker) if one considers steeper (core) profiles. Recent discoveries of optical signatures of black holes in dwarf galaxies indicates that low-mass galaxies can host intermediate massive black holes. Accounting for this without spoiling the measured velocity dispersion of stars, the constraints on the dark matter annihilation cross section may become orders of magnitude stronger Gonzalez-Morales:2014eaa (), ruling out thermal Wino dark matter masses up to 1 TeV. Similar exclusion bounds hold for any general scalar dark matter candidate that is a triplet under Queiroz:2014pra ().
CMB measurements are another source of constraint for non-thermal histories. We reserve a full discussion of this for Section V, here we briefly mention how CMB observations can lead to constraints on the self-annihilation rate of non-thermal dark matter. These constraints arise because energy injection from increased dark matter annihilation can alter the recombination history141414This is the period during which the universe cools to a level where CMB photons can free-stream and no longer interact with electrons on average. leading to changes in the temperature and polarization power spectra of the CMB Slatyer:2009yq (). We have seen that non-thermal dark matter can have a larger annihilation rate than thermal WIMPs, and so this would lead to stronger CMB constraints from the additional energy injection. The 2015 results from the PLANCK mission Planck:2015xua () imply a constraint on non-thermal WIMPs with masses GeV of cm s. The constant is a model dependent parameter that depends on the mass of the dark matter and captures the efficiency of the annihilations – it typically takes values – . Although this parameter does introduce model dependence and some uncertainty into the constraints, with the 2015 Planck release it is already possible to place reasonable constraints on the annihilation rates of low-mass thermal WIMPs. Before reaching the cosmic variance limit, this type of constraint will be capable of ruling out non-thermal WIMPs completely when they have electroweak scale masses and comprise all of the cosmological dark matter.
Current collider bounds on Winos and Higgsinos are rather weak, given their low production cross-section at a hadron collider. Approaches to study the dark matter parameter space include effective operators Goodman:2010ku () and simplified models Abdallah:2014hon (), with a mono- observational signature, where may denote monojet Beltran:2010ww (), mono-photon Gershtein:2008bf (), mono- Petriello:2008pu (); Carpenter:2012rg (), mono- Bai:2012xg (), mono-Higgs, mono- Lin:2013sca (), or mono-top Lin:2013sca (). A particularly promising approach to the study of dark matter is the use of vector boson fusion processes Delannoy:2013ata (); Cirelli:2014dsa (); Baer:2014kya (); Gori:2014oua (); Dutta:2012xe (); Dutta:2013gga (); Giudice:2010wb (). The current exclusion prospects for pure Winos and Higgsinos are the following Berlin:2015aba (): taking a background systematic uncertainty of , with fb of data, the 14 TeV LHC is sensitive to Winos of 240 GeV (125 GeV) and Higgsinos of 125 GeV (55 GeV). A future 100 TeV collider with 3000 fb of data would exclude Winos of mass 1.1 TeV (750 GeV) and Higgsinos of mass 530 GeV (180 GeV) taking background uncertainty.
The above limits are in the case when everything except the lightest neutralino dark matter is decoupled from the low-energy spectrum. In specific models of string/M-theory compactifications, other light fields may be present. For example, in cases where other species such as a neutralino NLSP or gluino are accessible, the dark matter exclusion limits would depend on the final states and the decay modes of the NLSP or the colored state to the dark matter candidate. This is clearly more model-dependent. We briefly describe the low-energy spectrum and collider prospects of three of the most well-studied compactification schemes with non-thermal histories: mirage mediation models based on KKLT, Large Volume Scenarios, and the G2-MSSM.
In the case of mirage mediation models based on KKLT compactification, the dark matter candidate may be the Bino, Wino, or Higgsino, depending on the ratio (commonly denoted in the literature by ) of the gravitino mass and the universal gaugino masses at the GUT scale. The mass spacing between the gauginos is determined by the same quantity, which also gives the relative strengths of the moduli and anomaly-mediated supersymmetric contributions. The rest of the spectrum depends on the modular weights of the visible sector. The collider and dark matter phenomenology of these models has been studied most recently in the post-Higgs era in Kaufman:2013oaa () and previously151515These models have also been studied in the context of Yukawa-unified models Anandakrishnan:2013tqa (); Anandakrishnan:2013cwa (). in Dutta:2011kp (); Baer:2007eh (). The canonical case of results in a spectrum with decoupled Bino dark matter, which is largely inaccessible in colliders and may be probed in future direct detection experiments. However, depending on the model parameters (modular weights) a gluino with mass GeV may also be obtained in the spectrum in which case a discovery may be made with 300 fb of data at LHC-14. Changing the uplifting sector leads to other values of which result in either decoupled Higgsinos or Winos with collider prospects described above.
In the case of Large Volume Scenarios, a wide range of phenomenological possibilities are present depending on the local construction of the visible sector (specifically the matter Kahler metric) and the way to achieve a de Sitter vacuum, as found in Aparicio:2014wxa (). In particular, two classes of spectra were obtained by these authors. The first is a split-SUSY spectrum with a decoupled scalar sector, whose collider prospects have been discussed earlier. The second is a CMSSM/mSUGRA scenario, which has been studied in detail in Aparicio:2015sda (). There it was shown that the confluence of collider bounds (LEP, LHC), CMB constraints, direct detection (LUX, XENON100, CDMS, IceCube) and indirect detection (Fermi) bounds result in an allowed parameter space that is characterized by decoupled Higgsino-like dark matter with a mass around 300 GeV.
In the case of the G2-MSSM, the collider prospects have been studied most recently by the authors of Ellis:2014kla (). For a benchmark spectrum with a gravitino mass of TeV, the gluino mass is expected to be about 1.5 TeV, while the Wino (Bino)-like gaugino mass is about 614 (450) GeV. Three and only three production channels should be discoverable at LHC-14: , and where and are respectively Bino and Wino-like. The expected signature of the channel is . The production channel gives the final state , which should be quite a clear channel at the LHC. Given the dark matter mass of 450 GeV, a 1.5 TeV gluino is expected to be discoverable at the 5 level at LHC-14 given 300 fb of data.
In addition to WIMPs, axions provide another well-motivated particle candidate for the cosmological dark matter. The QCD axion provides a possible solution to the Strong CP problem Peccei:1977hh (); Weinberg:1977ma (); Wilczek:1977pj (), whereas string axions arise ubiquitously in string / M-theory constructions upon compactification to four dimensions Svrcek:2006yi (); Arvanitaki:2009fg (); Acharya:2010zx (); Cicoli:2012sz (). In both situations, axions arise as pseudo-Nambu-Goldstone Bosons (PNGB) of a spontaneously broken Peccei-Quinn (PQ) symmetry. The observational consequences of the axion depend primarily on its mass , the axion decay constant , and the initial misalignment angle .
For the QCD axion, the mass is related to the decay constant as the axion enjoys a shift symmetry to all orders in perturbation theory and its mass is assumed to result from non-perturbative QCD instanton corrections , where , and so there is a relation between the mass and decay constant Peccei:1977hh (); Weinberg:1977ma (); Wilczek:1977pj (),
Lab and astrophysical constraints require GeV with stronger model-dependent constraints possible depending on the axion coupling to photons and whether the axion is assumed to comprise all of the cosmological dark matter Agashe:2014kda ().
Whereas, for string theory axions the relation (13) need not apply as the mass can be generated by a number of different non-perturbative effects. Stated another way, this implies challenging modeling building constraints for realizing a solution to the Strong CP problem in string theory – notably QCD instantons need to provide the dominant contribution to the low-energy effective mass of the QCD axion. Yet, investigating the UV completion of the PQ symmetry seems essential given the expectation that global (PQ-like) symmetries are expected to be broken by quantum gravity effects Holman:1992us (); Kallosh:1995hi (); Kamionkowski:1992mf (). Regardless of whether string theory can accommodate the QCD axion these theories do predict PNGBs that can act as all or part of the cosmological dark matter and may have other interesting observational implications.
iii.2.1 Axions from String Theory
Within string / M-theory constructions, PNGB’s naturally arise from the compactification of higher dimensional gauge fields to four dimensions Witten:1984dg (); Svrcek:2006yi (). The higher dimensional gauge invariance of these fields leads to an approximate shift symmetry in the low-energy effective theory. This shift symmetry can be lifted by both quantum and non-perturbative effects that depend on the details of the compactification Svrcek:2006yi (). As an example, consider axions arising from compactifications of Type IIB string theory. To see how axions arise in the theory, we consider the dimensional reduction of the theory to four dimensions by starting from the D action in the string frame given by Polchinski:1998rr ()
where is the ten dimensional string frame metric and and are the NS-NS and RR three-form fluxes, respectively, with and the corresponding gauge potentials and is the string tension. The model independent axion and dilaton are combined as the axio-dilaton where is the string coupling and we will take to be fixed and instead concentrate on the model dependent axions arising from the compactification of the form fields. The additional terms represented by dots include higher form fields such as (we will concentrate on axions).
The zero modes of and are independent of the co-ordinates of the compact dimensions and can be integrated over chosen two-cycles of the internal geometry giving rise to axions in the four dimensional theory. To make this explicit, consider compactifying on a Calabi-Yau 3-fold () and for the form field we make the ansatz Svrcek:2006yi ()
where the are only functions of the four non-compact space-time dimensions and labels the two-cycle. We have introduced the basis forms to describe the internal geometry and they obey the normalization condition with the two cycles giving a basis of the dual homology . The normalization factors of are chosen for later convenience. Making a similar ansatz for and using this in (14) we have
where Greek indices run over the four non-compact dimensions and lower-case latin indices denote the compact dimensions. At the classical level, the gauge invariance of the higher dimensional gauge potential implies that the axions can only be derivatively coupled and so we have a shift-symmetric pseudo-scalar in the low energy theory. This symmetry can be broken in a number of ways, which we will discuss shortly. Denoting both types of axions as , upon dimensional reduction we find
where is the Einstein frame metric and the reduced Planck mass is
with the string frame161616 Another common convention is to instead work with the Einstein frame volume. The D string frame is related to the Einstein frame by the Weyl rescaling and working in units of the string length the two volumes are related by . volume of .
For the RR axion the provide the axion decay constants and depend on the internal geometry as Svrcek:2006yi ()
whereas for the NS axion one gets the same result multiplied by an extra factor of .
Once the specifics of the internal geometry are known one can calculate the to find the corresponding axion decay constants. This is a non-trivial task, which requires a full specification of the compactification geometry in a Calabi-Yau manifold. In Svrcek:2006yi () the were calculated in a variety of string models assuming compactifications sufficiently symmetric to be amenable to estimates.
Generating a potential. Although the higher dimensional gauge invariance guarantees a low energy shift symmetry to all orders in perturbation theory, there are a number of non-perturbative effects that can generate a potential – e.g. branes wrapped on the corresponding cycle of the axion McAllister:2008hb (); Douglas:2006es () and/or various instanton and condensation effects Kallosh:1995hi (); Becker:1995kb (); Dine:1986zy (); Dine:1987bq (); Acharya:2010zx (). These effects can completely lift the axion from the low energy spectrum or result in a cosine-like potential just as in the case of the QCD axion, albeit at vastly different scales. In general, the range of axion masses expected from these effects can range over an entire spectrum from very heavy, to string instantons generating masses far smaller than the QCD axion. The large range of model building possibilities has interesting observational implications for many different types of experiments and the resulting theoretical picture has been termed the "axi-verse" Arvanitaki:2009fg ().
Moduli Stabilization, the Saxion and a Non-thermal History. As familiar from inflationary model building in string theory, it is difficult to introduce a single degree of freedom without examining other ingredients of the fundamental theory. String / M-theory compactifications come with a number of additional ingredients, which is the cost for working with a UV complete approach. In particular, one must take into consideration the additional moduli, fluxes, and branes within a chosen framework. We will not review these difficulties in detail here (see e.g. Acharya:2010zx (); Cicoli:2012sz () and references within). However, we note that moduli stabilization with light axions has a generic challenge as its SUSY partner – the saxion – can lead to a number of difficulties. For example, if the saxion is stabilized supersymmetrically (as Kahler moduli are in KKLT setups before the uplift to deSitter) then the saxion and axion are both expected to have comparable masses. Thus, if the axion is to remain light and the saxion receives a comparable mass then this can lead to an example of the cosmological moduli problem – discussed in Section II. Whether this is the case or if there can be a large mass splitting is somewhat model dependent. However, this must be addressed within a given setup to guarantee a successful axion dark matter candidate. As an example, for a saxion receiving a mass of -TeV this once again alleviates the cosmological moduli problem and motivates a non-thermal history since the saxion would decay slightly before BBN. Depending on the initial conditions for the axion this may or may not lead to a successful model for axion dark matter and agreement with the relic dark matter abundance.
iii.2.2 Axion Dark Matter Density in a Non-thermal History
Given motivation for axions from both string / M-theory and the Strong CP problem in QCD we now consider the role of the axion as dark matter in a non-thermal universe. We will not review the standard case of axions in a thermal universe, instead we refer the reader to existing literature Raffelt:2002zz (); Kolb:1990vq (); Kim:1986ax (). However, the standard calculation for the relic density of axion dark matter will hold, even in the non-thermal history with moduli decay, if the axion mass is such that axion oscillations begin following moduli decay – i.e. if with the moduli decay rate given in (4). This typically holds true for axion masses eV, and then axion oscillations will begin following moduli decay and the critical density in axions today is Acharya:2010zx ()
where is the density of axions today normalized to the critical density and this expression
is accurate up to corrections due to effects such as anharmonicity Fox:2004kb (). We note that for the special case of a QCD axion (20) can be further simplified by making use of (13). As mentioned above, the result (20) depends on the axion mass and decay constant, as well
as the misalignment angle of the axion from its minimum . In most of the literature the initial value is taken as random variable with an average squared value . However, the actual value depends on a number of assumptions, including the status of the PQ symmetry during inflation Dine:2004cq (). In this review we will leave this value unspecified to be as general as possible, invoking the standard average when numeric answers are required. We see from (20) that for a decay constant in the vicinity of the GUT or string scale ( GeV) we must either have extremely light axions , or fine-tune the initial misalignment angle , in order not to over-close the universe, i.e. to be consistent with cosmological observations which imply the abundance of axions today must be .
Axion oscillations before moduli decay. Next we consider when the axions begin oscillating prior to moduli decay requiring eV. In this case the moduli decay and the entropy produced will dilute the relic abundance as first discussed in Lazarides:1990xp (). For completeness, we now briefly review this calculation of the relic abundance as it appears in Fox:2004kb ().
The equation of motion for the axion in the background of the oscillating moduli is given by
where in the effectively matter dominated epoch and is the axion mass which can now depend on the temperature Fox:2004kb (). The energy density of the axion is and once coherent oscillations begin (typically in much less than a Hubble time, ) then the average energy density evolves as and the pressure then vanishes. We then have
where the time dependence of the mass follows from the temperature dependence and so the energy density evolves as
where is the scale factor (not to be confused with the axion). The temperature dependence of the mass implies that the energy density does not scale like matter but that the number density () does. Thus, if the axions start to oscillate during the moduli phase their number density compared to that of the moduli (which we recall also scale as ) is constant
where and denote the Hubble parameter and temperature at the onset of axion oscillations and is the axion displacement including various finite temperature and other corrections which can be found as a function of the zero temperature displacement Fox:2004kb (). The amount of axions after moduli decay and at the time of reheating is
where is the entropy density and we assume that the moduli do not decay into axions (we will discuss other cases soon). Comoving energy is conserved during the decay so for instant reheating we can relate the moduli density to the radiation after reheating and , so that
where we used and . Using that (26) is conserved until today, the critical density is
where is the ratio of the finite temperature mass to the zero temperature mass and is typically an order one number Fox:2004kb (). This result is interesting in that as long as eV the resulting abundance is independent of the mass of the axion. We see from (28), for low reheat temperatures and axion decay constants near the GUT scale GeV, the initial displacement angle would require some level of fine-tuning () in order to be consistent with the observed dark matter abundance. This presents a challenge for many string and M-theory based setups containing axions. For example, in Acharya:2010zx () it was found that for M-theory approaches with gauge coupling unification that the masses of the axions would be distributed logarithmically between eV and about an eV, but that typically should be near the GUT scale. However, it is again important to stress that a knowledge of the nature of the PQ symmetry during and after inflation is important for establishing the likely values of the displacement Dine:2004cq ().
iii.2.3 Observational Constraints on Axions
In addition to bounds on axions from the dark matter relic density, their behavior both during inflation and at the time of moduli decay can provide important constraints. For axions with masses comparable or lighter than the Hubble scale () during inflation axions can generate an isocurvature component in the primordial perturbations. Whereas, axions which are relativistic near the epoch of BBN can act as an additional source of dark radiation which is also constrained by both CMB observations and BBN. Both of these constraints can be quite stringent and depend on assumptions of the cosmic history. We will discuss both of these constraints and their connection with the moduli phase in Section V.2.
Many of the constraints on axions are rather independent of whether the post-inflationary history is thermal or non-thermal. We will mention these briefly and refer the interested reader to the literature for more details – see e.g. Arvanitaki:2009fg () and references within. The relevant constraints depend primarily on the mass range of the axions and on the strength of the coupling to photons . In Arvanitaki:2009fg (), it was shown axions with masses eV (and with appreciable electromagnetic coupling) lead to rotation of the CMB and the level of constraint will depend on the strength of the coupling – of course for axion dark matter candidates this coupling must be quite weak. For axions in the mass range eV eV step-like features in the matter power spectrum can become important for structure formation on small scales. For larger masses in the range eV eV axions can form bound states with black holes leading to a spin-down of the black hole and a mass gap in the spectrum. Finally, for axions with masses larger than around eV and with a significant coupling to photons, axion-photon conversion can occur in strong magnetic fields resulting in signals both in galaxies and in more compact objects such as pulsars.
Iv Baryogenesis and the Cosmic Coincidence
String cosmology has been most widely studied in the contexts of inflationary, axionic, and dark matter physics. On the other hand, the baryon asymmetry typically relies on model-building in the visible sector. While realistic vacua that approximate the Standard Model have been constructed in different corners of M-theory, it is safe to say that extensions required for baryon asymmetry have not been a model-building priority. This is understandable, since processes such as electroweak baryogenesis Cohen:1993nk () or leptogenesis Fukugita:1986hr () are relatively independent of a particular UV completion, according to the common lore.
There is every reason to expect that the viability of canonical processes like electroweak baryogenesis and leptogenesis does, in fact, have an unavoidable dependence on UV physics. The decay of moduli will dilute any previous population of baryons. Since electroweak baryogenesis depends on processes at scales GeV, and moduli decay at scales GeV, this dilution factor can and should be large.
There are then two options. The first is to rely on a high scale baryogenesis process such as Affleck-Dine (AD) baryogenesis which produces a large baryon asymmetry that can withstand subsequent dilution. The second is to implement baryogenesis at a post-sphaleron stage. We discuss these options in turn.
iv.1 AD Baryogenesis: UV Constraints and Challenges
Affleck-Dine (AD) baryogenesis Dine:1995kz (), Dine:1995uk (), Affleck:1984fy () relies on very general properties of supersymmetric theories: the existence of flat directions. An inflationary sector produces coherent oscillations along a supersymmetric flat direction , and the interaction between the inflationary sector and the flat direction leads to dynamics in field space that ultimately breaks CP and baryon number. This interaction occurs through Planck-suppressed operators. The setting is effective supergravity, and string phenomenology plays a definite role.
where and correspond to a Hubble-induced mass and -term, whereas and are the soft-SUSY breaking mass and -term, respectively. The last term in the potential corresponds to a higher dimensional operator in the superpotential, ( is the cutoff scale where new physics appears).
The curvature along is dominated by the Hubble-induced mass term for . If , the field sits at the origin. However, if , the minimum is non-zero and the field tracks this minimum until . The field settles into one of the discrete vacua given by the Hubble induced -term. When , the new minimum develops at the origin and the field begins to oscillate around it. The soft -term becomes dominant over the Hubble-induced -term, and the field acquires an angular motion to settle into a new phase. The baryon number violation becomes maximal at this time and imparts an asymmetry to the condensate.
The different components of (29) are non-trivial to justify from a top-down perspective. We describe these challenges in turn.
Negative Hubble-induced mass. The foremost requirement is a tachyonic Hubble-induced mass term for the field . This requires a condition on the geometry of the moduli space. The Kahler potential and superpotential can be written as
where denotes a generic visible sector field. The soft mass is given by
with the holomorphic bisectional curvature being given by
where denotes the Riemann curvature tensor of the moduli space.
Since during inflation we can take , the requirement of a tachyonic Hubble-induced mass reduces to
This condition has been explored in a wide variety of contexts within type IIB string theory Dutta:2010sg (); Dutta:2012mw (). For a non-local modulus , it was found that the Hubble-induced masses are generally positive, with
making AD baryogenesis difficult to satisfy. If the modulus corresponds to deformations of the local cycle on which the visible sector is constructed, on the other hand, then the induced masses are also typically positive, although for suitable choices of a tachyonic mass can be generated. If the visible sector construction is at a singularity, then the
induced mass depends on appropriate choice of fluxes on the mirror type IIA manifold.
Aspects of this problem have recently been discussed in Garcia:2013bha (); Marsh:2011ud (); Kamada:2012bk (); Higaki:2014eda ().
Hubble-induced A-terms. To obtain successful AD bayogenesis, the presence of a Hubble-induced -term is necessary. The -term is given by
where is the connection on moduli space, and the -th modulus acquires the dominant -term during inflation (we will simply call it henceforth).
We need the presence of a Yukawa coupling in the superpotential in (IV.1). To obtain Hubble-induced terms during inflation, the various contributions in (35) should acquire a non-zero value. Generally, the dependence of the Yukawa coupling on the modulus sector depends on the details of the local model building. Consider a visible sector localized near a small four-cycle . Due to holomorphy and the shift symmetry of the Kahler moduli, they cannot appear at any level in perturbation theory in , and hence, importantly, in of (IV.1). The entire dependence on Kahler moduli appears in the normalized Yukawa couplings, given by
This should only depend on local geometric data and complex structure moduli , but not the overall volume or other Kahler moduli.
The first term in (35) requires that the Yukawa couplings depend on the modulus () that acquires the dominant term during inflation. This term vanishes in general since the Kahler modulus corresponding to deformations of the cycle on which the visible sector is constructed is typically not the dominant source of supersymmetry breaking during inflation. Obtaining a non-zero value for the other two terms in (35) depends both on the geometry as well as the specific details of inflation. A detailed study of these issues is merited.
The AD baryogenesis scenario also has a number of correlated predictions coming from the presence of a light scalar field during the inflationary era. One important issue is the back-reaction on the inflaton. The AD field () makes a transition from the origin to its non-zero vacuum configuration in field space when the Hubble-induced tachyonic mass term becomes dominant. Depending on when this transition takes place, there will be back-reaction on the inflationary potential, leading to features on the power spectrum, which will be constrained by data Marsh:2011ud (). The presence of a scalar field coupling to the inflationary sector should also lead to constraints from non-Gaussianity. Both of these issues deserve further study.
iv.2 Moduli and the Cosmic Coincidence Problem
Cosmological observations not only determine precisely the relic abundance of dark matter and baryons, but also imply an interesting connection between their relative amounts leading to what some have called the ‘Cosmic Coincidence Problem’. In Kane:2011ih () it was argued that moduli decay in non-thermal histories might address this ‘coincidence’ – the primary feature being that both dark matter and baryons can be produced from the decay of the lightest modulus. Keeping in mind that moduli decay will dilute any initial abundance of particles, it is advantageous in non-thermal histories that AD baryogenesis typically over-produces baryons. Indeed in Campbell:1998yi () the authors pointed out that the late decay of a Polonyi field could act to dilute the baryon asymmetry to an acceptable level. More recently, in Kane:2011ih () it was noted that because dark matter can also be produced in the decay, this creates a subtle connection between the two relic densities that could account for the baryon-dark matter coincidence problem. To understand the connection it is useful to consider the ratio of the baryon to moduli co-moving densities
where is the amplitude at the start of the moduli oscillation, and the subscript and superscript denote the values before and after modulus decay, respectively.
To obtain the baryon number density after decay, the dilution factor must be used. We obtain
which demonstrates that the baryon abundance in this approach is intimately related to the ratio of the initial amplitudes of the AD field and the modulus. The AD fields displacement depends nontrivially on the dimension of the non-renormalizable operator that lifts the flat direction. Since larger leads to larger displacement, and therefore larger contribution to the baryon asymmetry, one can focus on the flattest directions in MSSM that require the largest to get lifted171717We assume all non-normalizable operators that are allowed by gauge invariance and R-parity are generated.. As showed in Ref. Gherghetta:1995dv (), the flattest direction (one of the combinations) corresponds to . Assuming that the non-renormalizable operator is generated at the reduced Planck scale and taking GeV we find GeV. For the next flattest direction (one of the combinations) – which is not lifted until – we have GeV. So we can see that these directions in the MSSM naturally have amplitudes two or three order of magnitudes smaller than , and it follows that the correct baryon asymmetry can be obtained for MeV, TeV, and .
The modulus will also decay and provide dark matter, and the relic density will depend on whether the number density produced is above (’annihilation scenario’) or below (’branching scenario’) the critical number density. In either case, the final baryon-dark matter relic density ratio will depend on the factors in (38) and the annihilation cross section of the dark matter or the branching ratio of into dark matter. By a reasonable choice of parameters, one would obtain .
We note that similar approaches to the baryon-dark matter coincidence problem that depend on moduli physics have been undertaken recently Kawasaki:2015cla (); Doddato:2012ja (); Cheung:2011if (); Bell:2011tn (); Shoemaker:2009kg (); Roszkowski:2006kw (). A more direct solution to the coincidence problem would result if the both the baryons and the dark matter resulted from the decay of the modulus. This idea was first proposed in Kitano:2008tk (), and it would be very interesting to see if this approach could be UV completed in String or M-theory.
iv.3 Baryogenesis at Late Times
Another approach to baryogenesis in non-thermal histories can be implemented by a leptogenesis-like mechanism resulting from modulus decay. This approach invokes additional model building, but it implies an intriguing connection between the need for to be a small number and that baryogenesis requires small number densities . This small value can be driven by the smallness of , with the remainder factor of coming from branching ratios and loop factors Allahverdi:2010rh (); Allahverdi:2010im ().
As an explicit model, we can consider the following extension of the MSSM, containing two flavors of singlets and a single flavor of colored triplets (with hypercharges respectively). The superpotential is given by
The interference between the tree-level and one-loop diagrams in the decay generates a baryon asymmetry, in a manner reminiscent of leptogenesis. The asymmetry per decay is loop suppressed and naturally for phases and couplings. Taking , the correct baryon asymmetry can be obtained for a branching of the modulus into the species given by . From counting of the degrees of freedom this branching ratio appears to be somewhat natural.
The modulus will generally produce dark matter in addition to the the field required for baryogenesis. Considering that the number density of the dark matter produced is below the critical density required for further annihilation, one has for the baryon and dark matter density ratio
The branching scenario requires . One then obtains the correct coincidence for GeV, but other values of can also be accommodated.
V Gravitational Constraints: CMB and Large Scale Structure
In this section we consider possible implications of a non-thermal history on the CMB, BBN and the Matter Power Spectrum. Thus far, many of the observational consequences of a non-thermal history have relied on a connection with interactions or decays to Standard Model particles. In the case of dark matter, we saw that non-thermal WIMPs can have unexpected properties (e.g. a stronger self annihilation cross section than the thermal case) and as a result we saw that (in)direct detection of dark matter experiments could help constrain the reheat temperature. However, these probes rely on the assumption that dark matter is composed of WIMPs and have a connection to electroweak-scale interactions.
What if dark matter is not directly associated with weak-scale physics? We discussed one example – axions – and we saw that if these particles are light they could prove difficult to detect. If part of the cosmological dark matter has negligible interactions with (or decays to) the Standard Model, this would present a substantial challenge to identifying its particle nature. Moreover, as LHC prepares to probe energies in excess of TeV, there is, as yet, no evidence for physics beyond the Standard Model. If nature has decides to play a cruel trick on us and the scale of new physics lies significantly above the electroweak scale, our only near-term possibility for probing new physics may rely on cosmological observations. In this section we present implications of the non-thermal history resulting from string / M-theory approaches on gravitational and cosmological based experiments. In particular, we consider what the early matter dominated phase could imply for the formation of primordial structures (such as mini-halos and primordial black holes), we also discuss implications for the matter power spectrum, and different ways in which the matter dominated phase can influence observations of the CMB.
v.1 Implications for the Matter Power Spectrum
v.1.1 Enhanced Structure Growth on Small Scales
In conventional models for structure formation growth of structure only becomes significant following matter-radiation equality. This is because dark matter perturbations that enter the horizon only grow logarithmically with the scale factor in a radiation dominated universe , whereas they grow linearly in a matter dominated universe . However, in non-thermal cosmologies we have seen that the universe is matter dominated prior to BBN. This suggests an early period of growth and a new scale at smaller wavelengths in the matter power spectrum. This possibility was first studied in the absence of annihilations in Erickcek:2011us () and later more generally in Fan:2014zua (). The results are summarized in Figure 2. It was found that the moduli phase would introduce a new period of growth for dark matter perturbations and that the size of the horizon at reheating could provide a new scale for determining the size of the smallest possible dark matter substructures. Indeed, for reheat temperatures near the limit set by BBN it was shown that a detectable level of Earth-sized, ultra-compact mini-halos may result. Of course, whether such structures can survive depends on a number of details, including whether free-streaming and kinetic decoupling effects would destroy the initial formation process. In Fan:2014zua (), it was found that for typical WIMP candidates in string-based non-thermal cosmologies that free-streaming and kinetic decoupling would instead determine the smallest structures. However, as discussed in Erickcek:2011us (); Fan:2014zua (), there are exceptions pointing to new directions for model building and a possible signature of the non-thermal period. One example is if the WIMPs are produced non-relativistically in the moduli decays (so that free-streaming is negligible) and if these particles become kinetically decoupled before reheating. Whether such models can be realized from fundamental theory remains an open question, although if such structures do survive they could lead to new expectations for indirect detection experiments.
In addition to enhanced growth of dark matter perturbations, it was also found in Erickcek:2011us (); Fan:2014zua () that there can be interesting effects on radiation perturbations. These perturbations also grow during the moduli dominated phase, and then following reheating those that entered the horizon will oscillated with an amplitude suppressed relative to the situation in a strictly thermal history. As seen in Figure 3, the suppression in the radiation perturbations arises at the peak of scalar decay when most of the radiation is created by both dark matter annihilations and decays of the moduli. The suppression of the radiation perturbations can lead to a new critical scale for dark matter structures if dark matter is produced thermally following reheating Erickcek:2011us (); Fan:2014zua (), since the suppression would prevent acoustic oscillations on that scale from wiping out structure Loeb:2005pm (). However, again it was argued in Fan:2014zua () that this would require non-standard WIMPs, since for the interesting case of low reheat temperatures ( MeV) this temperature would typically be far below the thermal freeze-out temperature ( GeV).
In summary, it was found in Erickcek:2011us (); Fan:2014zua () that non-thermal histories do not face any challenges in regards to disrupting structure formation. Moreover, if we consider less typical SUSY WIMP dark matter candidates it way be possible in those models to distinguish between non-thermal and thermal scenarios through future observations.
v.1.2 Primordial Blackhole Constraints
Above we primarily focused on the growth of dark matter and radiation perturbations during the moduli phase. However, both of these components are typically sub-dominant in string-based approaches and instead the moduli significantly dominate the energy density. We argued above that the transfer of these perturbations into dark matter and radiation can have interesting effects in some cases. But one can ask if the growth of moduli perturbations during the early matter domination can lead to interesting constraints? In particular, it was realized many years ago that an early matter dominated (dust) phase could lead to a significant production of primordial black holes (PBHs) Polnarev:1986bi (). The predicted number of PBHs relies sensitively on the scalar tilt of the primordial power spectrum as this sets the initial amplitude of the perturbations and PBHs are expected to form when the relative density contrast approaches the non-linear regime . A matter or moduli dominated phase differs in some important ways compared to the typical estimate of PBH formation in a thermal history. In particular, during a matter dominated phase there is no pressure to prevent the collapse of sub-horizon perturbations to evolving into black holes. This means that unlike the thermal case, perturbations can collapse to form PBHs long after crossing the Hubble radius – the Jean’s radius vanishes and so there is no pressure to prevent collapse. However, in Polnarev:1986bi () (see also doroshkevich1970spatial ()) it was stressed that in a matter (dust) phase the resulting number of PBHs will depend on the probability for perfectly spherical collapse (so that angular momentum doesn’t disrupt the process) and the duration of the period. In gsw () this analysis was adapted to the case of moduli domination (similar analysis with different motivation appeared in Carr:1994ar (); Green:1997pr ()). One important consideration is that scalar fields on small scales can cease to be homogeneous and this can introduce a gradient pressure that prevents collapse hu1998structure (); Hu:2000ke (). However, it was shown in gsw () that for the range of moduli masses of interest that this effect is negligible. It was also shown that the most relevant PBH constraint is requiring that PBH production does not exceed the critical density. Given, a priori, the scalar tilt of the primordial power spectrum the corresponding constraints on the reheat temperature (and so moduli mass) were found in gsw (). There it was also shown that for a fixed duration of moduli domination and a specified reheat temperature (both which follow from the moduli mass), constraints on the primordial scalar tilt can be established. For high scale inflationary reheating and the constraint was found. However, given that at the time CMB modes excited the horizon ( e-foldings before the end of inflation) the scalar tilt is constrained to be strictly red () one can question the importance of this constraint. This implies that only in models where significant changes in result would one be able to place strong constraints on the reheat temperature181818This result may change given an extremely long period of matter domination, but it was found in gsw () that even for high scale inflationary reheating that requiring radiation domination by the time of BBN forces the epoch to be too short in duration to place constraints on a red spectrum.. Examples of such the blue spectra on small scales have appeared in the literature, for example in models of hybrid inflation (see Young:2014oea () for a recent discussion of other possibilities and references). In summary, PBH constraints do not lead to strong constraints on non-thermal histories unless the scalar tilt becomes blue () on sub-CMB scales.
v.2 CMB Constraints
CMB observations are not only sensitive to the physics present during inflation, but also that during the post-inflationary history. Significant work has gone into establishing predictions from inflation for both signatures in the temperature power spectrum and those associated with non-gaussianity (see e.g. Baumann:2014nda ()). In fact, it has been suggested that non-gaussianity arising during inflation may be useful in probing models with a split-SUSY spectrum Craig:2014rta (). In this section we instead focus on the implications of the post-inflationary epoch on CMB observations.
v.2.1 Consequences for Inflationary Constraints from the CMB
CMB data has reached an impressive level of accuracy and can be used to place stringent constraints on purposed inflation models. However, to infer which models of inflation are most likely given the data, we match observations today to the predictions of the inflation model and this matching requires a knowledge of the post-inflationary expansion. The equation of state (e.g. thermal or non-thermal ) during the post-inflationary epoch determines the expansion rate, and thus the rate at which primordial perturbations (re)enter the Hubble horizon. For a spectral index that is not strictly scale-invariant () this difference in expansion rate will lead to different predictions for the power spectrum Liddle:2003as (); Kinney:2005in (); Peiris:2008be (); Adshead:2010mc (); Mortonson:2010er (); Easther:2011yq (); Norena:2012rs (); Martin:2010kz (); Martin:2006rs (); Cook:2015vqa (). The matching and the differences that can arise due to two different post-inflationary histories are shown in Figure 4. As shown in the figure, given a mode of comoving wavenumber , we must trace back to the instant this mode exited the Hubble horizon Liddle:2003as (); Kinney:2005in (); Peiris:2008be (); Adshead:2010mc (); Mortonson:2010er (); Easther:2011yq (); Norena:2012rs (); Martin:2010kz (); Martin:2006rs (). This occurs when where and denote the Hubble parameter and scale factor respectively, and a subscript labels values at horizon crossing. Where with the number of e-folds before the end of inflation and the scale factor at the end of inflation. Assuming single-field slow-roll inflation for the purposes of illustration and characterizing the post-inflationary expansion by an effective equation of state . One finds the matching equation Liddle:2003as ()
where is the value of the energy density at the end of inflation, is the inflaton potential as the th mode leaves the horizon, and is the energy density at which the universe is assumed to become thermalized. The first two terms in (41) are model independent. For GUT scale inflation the third term is roughly . The fourth term is typically order unity given that the value of the inflaton potential necessarily evolves slowly as inflation proceeds. Finally, if the universe thermalizes promptly the last term is negligible, and we recover the familiar result that for modes contributing to the CMB.
In Easther:2013nga () the difference between a SUSY non-thermal history and thermal history was explored and using (41) it was shown that the change in equation of state from the thermal case () to the moduli phase leads to a shift
with for a given potential. If (in fact for the moduli phase we typically have ), is negative, since with the energy density at the end of the moduli phase. For simplicity we can assume inflationary (p)reheating was instantaneous and following Easther:2013nga () we make the substitution , using . At the onset of oscillations and (42) becomes