CMB Spectral \mu-Distortion of Multiple Inflation Scenario

CMB Spectral -Distortion of Multiple Inflation Scenario

Gimin Bae School of Undergraduate Studies, College of Transdisciplinary Studies, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea    Sungjae Bae School of Undergraduate Studies, College of Transdisciplinary Studies, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea    Seungho Choe schoe@dgist.ac.kr School of Undergraduate Studies, College of Transdisciplinary Studies, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea    Seo Hyun Lee School of Undergraduate Studies, College of Transdisciplinary Studies, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea    Jungwon Lim School of Undergraduate Studies, College of Transdisciplinary Studies, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea    Heeseung Zoe heezoe@dgist.ac.kr School of Undergraduate Studies, College of Transdisciplinary Studies, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea
July 12, 2019
Abstract

In multiple inflation scenario having two inflations with an intermediate matter-dominated phase, the power spectrum is estimated to be enhanced on scales smaller than the horizon size at the beginning of the second inflation, . We require to make sure that the enhanced power spectrum is consistent with large scale observation of cosmic microwave background (CMB). We consider the CMB spectral distortions generated by the dissipation of acoustic waves to constrain the power spectrum. The -distortion value can be times larger than the expectation of the standard CDM model () for , while the -distortion is hardly affected by the enhancement of the power spectrum.

I Introduction

Inflation provides the seeds of statistical fluctuations for the structure formation of the universe Gliner (1965, 1970); Guth (1981); Linde (1982); Albrecht and Steinhardt (1982) It fits with the large scale observations of the cosmic microwave background (CMB) and large scale structure (LSS) such as the Wilkinson Microwave Anisotropy Probe (WMAP) Spergel et al. (2007), the Planck Ade et al. (2016) and Sloan Digital Sky Survey (SDSS) Tegmark et al. (2006). However, there are many inflation models consistent with those large scale observations, and we should develop proper methods to specify the primordial inflation. One of the possible ways should be probing inflationary power spectrum on small scales by the observations such as ultracompact minihalos Bringmann et al. (2012); Gosenca et al. (2017), primordial black holes Carr (1975); Josan et al. (2009), the lensing dispersion of SNIa Ben-Dayan and Kalaydzhyan (2014); Ben-Dayan (2014); Ben-Dayan and Takahashi (2016), the 21cm hydrogen line at or prior to the epoch of reionization Cooray (2006); Mao et al. (2008) or CMB distortions Chluba et al. (2015a); Chluba and Sunyaev (2012); Chluba et al. (2012a, b).

Multiple inflation scenario, having more than one inflationary periods after the first inflation, could leave characteristic signatures on small scales. Since the double inflation model, or inflation with a break, was introduced to give decoupling the power spectrum on large (CMB) and small (cluster-cluster/galaxy-galaxy) scales Silk and Turner (1987); Mukhanov and Zelnikov (1991); Polarski and Starobinsky (1992), many versions of multiple inflation have been suggested as theoretical possiblities in supersymmetric particle physics models Adams et al. (1997); Kanazawa et al. (2000); Lesgourgues (2000); Yamaguchi (2001); Burgess et al. (2005); Kawasaki and Miyamoto (2011); Lyth and Stewart (1995, 1996); Hong et al. (2015); Cho et al. (2017); Hong et al. (2017); Craig et al. (2017).

CMB spectral distortions are a useful technique to probe small scales at Chluba et al. (2015a); Chluba and Sunyaev (2012); Chluba et al. (2012a, b). COBE/FIRAS measurements indicate that the CMB photons are subject to the blackbody spectrum of temperature with the spectral distortions Fixsen et al. (1996); Mather et al. (1994). However, we have many astrophyscial/cosmological sources inducing spectral distortions: decaying or annihilating particles Chluba (2013a); Chluba and Jeong (2014); Hu and Silk (1993); McDonald et al. (2001); Sarkar and Cooper-Sarkar (1984), reionization and structure formation Cen and Ostriker (1999); Hu et al. (1994a); Miniati et al. (2000); Oh et al. (2003); Refregier et al. (2000); Sunyaev and Zeldovich (1972); Zhang et al. (2004); Hill et al. (2015), primordial black holes Carr et al. (2010); Pani and Loeb (2013); Tashiro and Sugiyama (2008), cosmic strings Ostriker et al. (1986); Tashiro et al. (2012a, b, 2013), small-scale magnetic fields Jedamzik et al. (2000); Kunze and Komatsu (2014); Sethi and Subramanian (2005); Chluba et al. (2015b), the adiabatic cooling of matter Chluba and Sunyaev (2012); Ali-Haïmoud et al. (2015), cosmological recombination Dubrovich (1975); Dubrovich and Stolyarov (1997); Rubino-Martin et al. (2006); Chluba and Sunyaev (2006); Sunyaev and Chluba (2009); Chluba and Ali-Haimoud (2016), gravitino decay Dimastrogiovanni et al. (2016), and the dissipation of primordial density perturbations Barrow and Coles (1991); Chluba et al. (2012b); Chluba and Grin (2013); Chluba et al. (2012a); Daly (1991); Ganc and Komatsu (2012); Hu et al. (1994b); Pajer and Zaldarriaga (2013); Sunyaev and Zeldovich (1970); Clesse et al. (2014); Emami et al. (2015); Dimastrogiovanni and Emami (2016); Chluba et al. (2017) which is the main concern of this paper. Types of spectral distortions are characterized by the redshifts at which energy release to CMB photons: At , Compton and double Compton scatterings and Bremsstrahlung can effectively thermalize the energy release maintaining a black body spectrum Sunyaev and Zeldovich (1970); Hu et al. (1994a); Burigana et al. (1991). At , only Compton scattering can efficiently redistribute the energy injected to the CMB. Compton scattering keeps the electrons and photons in kinetic equilibrium forming a chemical potential . We define -distortion as the spectral distortions associated with this chemical potential Sunyaev and Zeldovich (1970). At , Compton scattering becomes inefficient, and the photons diffuse only a little in energy. We define -distortion as the spectral distortion caused by the energy release at this epoch. It is also connected with the Sunyaev-Zeldovich effect on galaxy clusters Zeldovich and Sunyaev (1969).

In this paper, we focus on multiple inflation with an intermediate matter-dominated period (See Figure 1). We calculate the power spectrum, and predict the spectral distortion due to the dissipation of the density perturbations. In Cho et al. (2017); Nakama et al. (2017a), the power spectrum with suppression on small scales is discussed, and its implications on the spectral distortions are already mentioned. However, the power spectrum of multiple inflation with a matter-dominated break turns out to be enhanced on small scales, and its pattern of the spectral distortions should be different from them.

We discuss the evolution of the curvature perturbations generated by the multiple inflation scenario with an intermediate matter domination in Section II. In Section III, we estimate the CMB spectral distortions by Silk damping of acoustic waves, due to the shear viscosity in the baryon-photon fluid. We find that the enhanced power spectrum of multiple inflation scenario could be constrained by the spectral distortion measurements. In Section IV, we summarize our results and discuss possible ways to constrain multiple inflation scenario.

Ii The evolution of cosmic perturbations

Figure 1: Characteristic scales of multiple inflation with an intermediate phase between the primordial and the second inflations. Three characteristic scales , and correspond to the comoving scale of the horizon at each of the era boundaries.

ii.1 Overview

We discuss the evolution of cosmological perturbations through two inflations with an intermediate phase as in Figure 1. The first inflation ends up with an intermediate phase, and then the second inflation begins after the intermediate phase and lasts until the usual radiation domination of Big-bang Nucleosynthesis (BBN) gets started.

The characteristic scales are defined as

(1)

where and are the scale factor and the Hubble parameter at the era boundary . is the comoving scale of the horizon at . The first inflation occurs at generating the perturbations. While modes with remain outside the horizon before the radiation domination and are not affected by the phases after the first inflation, modes with enter and exit the horizon and should evolve differently from those with as the followings:

  • Modes with enter the horizon during the intermediate phase and then exit the horizon during the second inflation.

  • Modes with exit the horizon for the first time during the second inflation.

  • Modes with never exit the horizon and are not our interest.

The power spectrum of the multiple inflation would fit with the large scale observation of CMB and LSS at , but should be clearly different from the expectations of a simple single primordial inflation or CMD model at .

Generic features of the power spectrum of multiple inflation with breaking intermediate periods are extensively studied in Namjoo et al. (2012). It is shown that the power spectrum is being enhanced with oscillations on small scales when the intermediate phase is matter dominated. Hence, we expect that the enhanced power spectrum could be explored by the CMB spctral distortions, and this is the motivation of this paper.

In our paper, however, we adopt the scheme of multicomponent perturbation calculation in Hong et al. (2015) to refine the calculation on the curvature perturbation by making the background smoothly changing from the matter domination to the second inflation. It turns out that our result is consistent with Namjoo et al. (2012). From now on, we consider only matter domination as the intermediate phase, and discuss the power spectrum and its spectral distortions.

ii.2 During the first inflation

The scalar parts of the perturbed metric Kodama and Sasaki (1984) is

(2)

We define

(3)

where is the intrinsic curvature perturbation on comoving hypersurfaces of Eq. (2) and is an inflaton field. Its mode functions are calculated by

(4)

where with the slow-roll parameters and (e.g. (Stewart and Lyth, 1993)), and describe the evolution of the perturbation during the first inflation. From Eq. (4), we express the curvature perturbation at as

(5)

where we assume throughout the first inflation and is a constant depending on the inflation model, and . For large scale, i.e. ,

(6)

and the power spectrum is

(7)

where and will be fixed by Planck normalization Ade et al. (2016).

ii.3 Matter-dominated intermediate phase

Now we calculate the evolution of the curvature perturbation on constant (matter) energy hypersurface during the matter-dominated intermediate phase by using the scheme of multicomponent perturbations introduced in Hong et al. (2015)111In Hong et al. (2015), the authors discuss the evolution of curvature perturbation on constant energy hypersurface during the (moduli) matter domination. However, their result is very different from ours because they should calculate it in terms of radiation hypersurface which is relevant to describe the thermal inflation period.. We assume that the second inflation vacuum energy becomes dominating as the matter energy density is gradually redshifted. The energy density is

(8)

where is the matter energy density and is the vacuum energy density driving the second inflation. The characteristic scale is naturally identified by the comoving scale at which the expansion rate of universe is changed

(9)

and hence

(10)

where is the Hubble parameter during the second inflation and assumed to be constant. Note that Eqs. (8) and (9) give

(11)

The curvature perturbation on constant matter energy hypersurface is defined by

(12)

whose mode functions should be matched with at by requiring

(13)
(14)

From Hong et al. (2015), the governing equation is given by

(15)

whose solution is reduced to

(16)

where

(17)

with and

(18)
(19)

ii.4 The second inflation

As the vacuum energy density of the second inflation gets dominating, the curvature perturbation

(20)

with the inflaton for the second inflation would be matched with at by

(21)
(22)

We treat the evolution of in the same manner of the first inflation in Eq. (4), but have to consider both and for the second inflation as in Namjoo et al. (2012). We express the full solution by using Bessel functions, instead of Hankel functions, to have concise forms.

(23)

where with the slow-roll parameters and and

(24)
(25)

Therefore, the curvature perturbation should be calculated at by

(26)

ii.5 Transfer Function and Power Spectrum

The evolution of the curvature perturbation after the first inflation is summarized by a transfer function

(27)
Figure 2: Transfer Function for , and .

One example of transfer functions is plotted in Figure 2. For , the transfer function goes to and the power spectrum on those scales is determined mainly by the first inflation. The dips and peaks of the transfer function are related with zeros of Bessel functions. For example, the first dip is found at the first zero of in Eq. (25), i.e. .

The resultant power spectrum is expressed by

(28)

We follow Planck normalization and at in (Ade et al., 2016). We can assume in Eq. (26) to include generic models for the second inflation, but have to fix in Eq. (7) to recover the correct value of .

Figure 3: Power Spectrum for , and . The red-dotted line represents the power spectrum of CDM model, i.e. .

Iii CMB Spectral Distortions

Now we are ready to calculate the CMB spectral distortions generated from Eq. (28) by the full thermalization Green’s function Chluba (2013b, 2015)222We use the numerical code of Green function method which has been developed by Jens Chluba.. The dissipation of acoustic waves with adiabatic initial conditions heats up the CMB photons generating the spectral distortions. One can calculate this heating rate over the redshift , and then the spectral distortion at a given frequency is estimated from the heating rate by

(29)

where includes the relevant thermalisation physics, which is independent of the energy release scenario. The spectral distortions is expressed in terms of the temperature shift , and contributions

(30)

where with for , and .

We clarify the characteristic scales , and to estimate the spectral distortions.

  • First, we need to make sure that the enhancement of the power spectrum around is not inconsistent with large scale observation such as CMB Ade et al. (2016). -distortion is sensitive to the power spectrum at Chluba et al. (2012b), but the power spectrum is enhanced at . Then, -distortion is affected by the dissipation of the modes within only small portion of -space, i.e. . We expect that -distortion hardly depends on the value of .

  • Second, we assume to avoid the difficulties in discussing the evolution of the modes with . These modes can be severly affected by the details of the reheating process depending on the inflation model. Since -distortion is affected by the modes with Chluba et al. (2012b), the condition of guarantees that the major contributions to spectral -distortion comes from the enhancement on the scales .

- and -distortions are shown in Figure 4 and Figure 5, where we take , and ( is estimated by Eq. (11)). For -distortion, we include a small correction due to the energy extraction from photons to baryons as CMB photons heat up the non-relativistic plasma of baryons by Compton scattering Chluba and Sunyaev (2012). Both - and -distortions are safely below than the limit of COBE/FIRAS for the whole range of .

-distortion clearly depends on the enhancement of the power spectrum in Figure 4, while -value hardly changes over the range in Figure 5. For , -value is slightly less than the expectation of the standard CDM model , because the power spectrum of our scenario is suppressed with respect to that of CDM around the first dip, i.e. (See Figure 3). For , the enhancement of the power spectrum at contributes -distortion, and the value can be 10 times larger than . Therefore, the observation of -value is a key to test the enhancement of power spectrum on small scales in our multiple inflation scenario, just as it is in other scenarios including Salati (1985); Dimastrogiovanni et al. (2016); Cho et al. (2017); Nakama et al. (2017a); Diacoumis and Wong (2017).

Figure 4: -distortion for and . The red-dotted line represents the -value of CDM model.
Figure 5: -distortion for and .

Iv Discussion

In this paper, we consider multiple inflation scenario with having an intermediate matter domination between inflations in Figure 1, and find that the power spectrum is enhanced at where is defined as the comoving horizon scale at the beginning of the second inflation by Eq. (9). If we require , the enhancement of power spectrum at has no trouble with large scale observations such as CMB. For , the spectral distortions of the multiple inflation scenario is expected to be similar to that of the standard CDM model. For , however, -distortion value would be allowed within the COBE/FIRAS, but could be 10 times larger than the expectation of CDM model which future distortion experiments may be able to test.

We do not cover various topics on the astrophysical and cosmological applications of the multiple inflation scenario in this paper. First, there could be constraints on the energy scale of the second inflation from particle physics nad cosmology, especially in the context of supersymmetric model German et al. (2001); Ross and German (2010); Kaneta et al. (2017). Two inflations at different energy scales may produce a special profile of gravitational waves background Zelnikov and Mukhanov (1991); Polarski and Starobinsky (1995); Choi and Kyae (2014). Second, the enhanced power spectrum may impact on the halo abundance and galaxy substructure as in Hong et al. (2017). This is entangled with the issues such as primordial black holes, minihalos and 21cm observations, which may also give feedbacks to -distortions Tashiro and Sugiyama (2008); Pani and Loeb (2013); Kohri et al. (2014); Gong and Kitajima (2017); Nakama et al. (2017b). If we consider those issues altogether, then multiple inflation scenario would be more constrained. We leave these aspects as future work.

Acknowledgements.
The authors thank to Ewan Stewart, Kihyun Cho, Sungwook Hong and Jens Chluba for helpful discussions. We use Jens Chluba’s Green function code to estimate CMB spectrl distortions. This work is supported by the DGIST Undergraduate Group Research Project (UGRP) grant.

References

Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
109689
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description