Cosmic inflation constrains scalar dark matter1

In a theory containing scalar fields, a generic consequence is a formation of scalar condensates during cosmic inflation. The displacement of scalar fields out from their vacuum values sets specific initial conditions for post-inflationary dynamics and may lead to significant observational ramifications. In this work, we investigate how these initial conditions affect the generation of dark matter in the class of portal scenarios where the standard model fields feel new physics only through Higgs-mediated couplings. As a representative example, we will consider a symmetric scalar singlet coupled to Higgs via . This simple extension has interesting consequences as the singlet constitutes a dark matter candidate originating from non-thermal production of singlet particles out from a singlet condensate, leading to a novel interplay between inflationary dynamics and dark matter properties.


Introduction
New physics beyond the standard model of particle physics (SM) is strongly implied by a number of cosmological observations, such as the existence of dark matter and baryon asymmetry in our universe.
Whenever a theory contains scalar fields, such as the SM Higgs boson, which are light and energetically subdominant during cosmic inflation, the inflationary fluctuations generically displace the fields from their vacuum values generating a primordial scalar condensate (Enqvist, Meriniemi, & Nurmi, 2013;Enqvist, Nurmi, Tenkanen, & Tuominen, 2014;Starobinsky & Yokoyama, 1994). These specific out-of-equilibrium initial conditions may then affect physics also at low-energy scales and lead to significant observational ramifications.

ABOUT THE AUTHOR
Tommi Tenkanen is a member of a group which main research activities concentrate on dynamics of quantum fields during and after cosmic inflation, on (p)reheating and on quantum gravity. The connection between standard model Higgs boson and new physics is of particular interest to the group. The research reported in this paper is based on earlier work conducted by the group and its collaborators.

PUBLIC INTEREST STATEMENT
By studying dynamics of quantum fields in the very early universe, we find a novel connection between the cosmic inflation and dark matter properties. This connection severely constrains some theoretical models which aim to explain the origin of dark matter. The connection also means that the study of gravitational waves may provide an interesting new probe on dark matter properties in the near future.
In this proceeding, based on Nurmi, Tenkanen, and Tuominen (in press) and first presented in From Higgs to Dark Matter 2014 conference, we investigate how the presence of scalar condensates affect the generation of dark matter in the class of portal scenarios where the standard model fields feel new physics only through Higgs-mediated couplings. As a representative example, we will consider a Z 2 symmetric scalar singlet s coupled to Higgs via Φ † Φs 2 .
We show that with these values of the portal coupling, it is possible to slowly produce a sizeable fraction of the observed dark matter abundance via singlet condensate fragmentation already at temperatures above the electroweak (EW) scale. This severely constraints the standard freeze-in scenario and requires earlier model computations to be revisited.

Field dynamics during and after inflation
The scalar sector of the model is specified by the potential where Φ is the usual standard model Higgs doublet and s is a Z 2 -symmetric real-singlet scalar.
Usually, the exact value of the self-interaction s is considered to be irrelevant for the dark matter production but we shall see that it plays an important role in determining the total dark matter yield.
If the scalar fields are light 2 during cosmic inflation and their energy density is subdominant, the mean fields are subject to acquire large fluctuations around the minima of their potential. Using the so-called stochastic approach (Starobinsky & Yokoyama, 1994), we find that the typical scalar field values at the onset of the post-inflationary era are Enqvist et al. (2014) provided that sh ≲ √ s h . Here H * is the Hubble parameter value at the end of inflation.
We take the results (2) as inflationary predictions for the initial values of scalar condensates.
Assuming an instant reheating, the Higgs acquires a large thermal mass m 2 h ≃ 0.1T 2 and quickly decays into other SM particles (Enqvist et al., 2014).
As the scalars relax toward their vacuum values, they open up additional channels for the production of singlet particles. As we shall see, these channels may easily compete with the low-energy particle production. In the following, we will concentrate on the regime where the portal coupling take a value sh ≲ 10 −7 and where the singlet never thermalizes above the EW scale.
As the singlet s becomes massive, 3 s s 2 ≃ H, it starts to oscillate about the minimum of its potential. Ignoring the decay processes, the equation of motion in a flat FRW space reads where V � ≡ dV∕ds and s 0 denotes the envelope field value of the homogeneous condensate. (1) When the singlet oscillates in a s s 4 potential, the solution to Equation 3 is where r is the tensor-to-scalar ratio measuring the energy scale of inflation and T is the SM bath temperature.
In a m 2 s s 2 potential, the solution for the envelope reads As the singlet oscillates about the minimum of its potential, a transition from the quartic to quadratic regime may take place. This happens approximatively as 3 ss 2 0 ∼ m 2 s , corresponding to If the singlet is light, m s ≲ 0.5 GeV, the potential is essentially given by s s 4 down to the EW scale.
The dominant decay channel for the singlet condensate is the perturbative production of singlet particles directly from the condensate s 0 (t). In the quartic regime, 3 s s 2 0 ≫ m 2 s , the corresponding singlet particle production rate is given by (see e.g. Ichikawa, Suyama, Takahashi, and Yamaguchi, 2008); In the quadratic regime, the singlet particle production directly from the condensate is kinematically blocked.

Dark matter production
The total number of produced particles can be calculated by writing the effective Boltzmann equation for the number density of singlet particles, n s , Here s 0 is the singlet condensate background field value given by Equation 4 in the quartic regime and by Equation 5 in the quadratic regime, Γ s 0 is the condensate decay rate given by Equation 7, f s and f h are the singlet and Higgs phase space densities, respectively, and f s,bg is the singlet condensate phase space density chosen such that s 0 = m s,eff n s,bg . The effective mass m 2 s,eff is equal to 3 s s 2 0 (T) in the quartic regime and to m 2 s in the quadratic part of the potential.
The corresponding present abundance is where T = T EW is a natural cut-off for high-temperature processes. Note that the potential is given by s s 4 down to the EW scale only if m s ≲ 0.5 GeV.
In the quadratic regime, the solution to Equation 8 can be written as: where (in units of GeV) and where we fix the coefficient C 1 so that the two solutions, Equations 10 and 13, match at the moment of transition (6). The corresponding present abundance then is In order to correctly determine the present DM abundance, one should also take the energy density in the singlet condensate into account. We find that the condensate's contribution to the observed value of DM abundance is given by The final dark matter yield above the EW scale is depicted in Figure 1 for different values of m s and the parameter r. The result severely constrains the viable region where a frozen-in scalar can act as a DM particle.

Conclusions
In this work, we have studied the implications of formation of scalar condensates during cosmic inflation. The formation of such condensates is a generic feature of a theory where the scalar fields We have found severe constraints on the freeze-in scenario where the total dark matter abundance is given by presence of a singlet scalar condensate and singlet particles, produced mainly by out-of-equilibrium decay of a singlet condensate. Usually, the energy density of the present remnants of the condensate is the dominant contribution to the observed DM relic density. For the maximum observationally allowed inflationary scale, r ≃ 0.1 (BICEP2 Collaboration & Planck Collaboration, 2015), we find that the possibility of a frozen-in scalar as a DM candidate is strictly constrained. We also find that in many cases, the portal coupling sh between the singlet scalar and SM Higgs is required to be super-feeble, sh ≲ 10 −12 , in order not to produce too much dark matter particles already at high temperatures.
Contrary to the standard freeze-in scenario, the singlet self-coupling s is found to play a crucial role in determination of the correct DM abundance. The result constrains also those models in which the frozen-in scalar acts only as a mediator and decays further to the actual DM particle. For these reasons, all the standard freeze-in scenarios need to be revisited.
The study of formation of scalar condensates and its implications on post-inflationary dynamics has also revealed a novel connection between inflationary dynamics and observed dark matter abundance, meaning that the study of primordial tensor perturbations may provide an interesting new probe on dark matter properties in the near future. 3