Scaledependent bias of galaxies and type distortion of the cosmic microwave background spectrum from singlefield inflation with a modified initial state
Abstract
We investigate the phenomenological consequences of a modification of the initial state of quantum fluctuations of a single inflationary field. While singlefield inflation with the standard BunchDavies initial vacuum state does not generally produce a measurable threepoint function (bispectrum) in the socalled squeezed triangle configuration (where one wavenumber, , is much smaller than the other two, ), allowing for a nonstandard initial state produces an exception. Here, we calculate the signature of an initial state modification in singlefield slowroll inflation as it would appear in both the scaledependent bias of the largescale structure (LSS) and type distortion in the blackbody spectrum of the cosmic microwave background (CMB). We parametrize the initial state modifications and identify certain choices of parameters as natural, though we also note some finetuned choices that can yield a larger bispectrum. In both cases, we observe a distinctive signature in LSS (as opposed to the of the socalled localform bispectrum). As a nonzero bispectrum in the squeezed configuration correlates one longwavelength mode with two shortwavelength modes, it induces a correlation between the CMB temperature anisotropy observed on large scales with the temperatureanisotropysquared on very small scales; this correlation persists as the smallscale anisotropysquared is processed into the type distortion of the blackbody spectrum. While the correlation induced by the localform bispectrum turns out to be too small to detect in near future, a modified initial vacuum state enhances the signal by a large factor owing to an extra factor of compared to the local form. For example, a proposed absolutelycalibrated experiment, PIXIE, is expected to detect this correlation with a signaltonoise ratio greater than 10, for an occupation number of about in the observable modes. Relativelycalibrated experiments such as Planck and LiteBIRD should also be able to measure this effect, provided that the relative calibration between different frequencies meets the required precision. Our study suggests that the CMB anisotropy, the distortion of the CMB blackbody spectrum, and the largescale structure of the universe offer new ways to probe the initial state of quantum fluctuations.
I Introduction
While cosmologists have accumulated extensive evidence for an earlyuniverse inflationary period, the cause and dynamical specifics of that epoch remain unclear. Current and upcoming measurements will provide increasingly precise measurements of the effects of inflation, demanding that theorists persist in relating these observations to inflation’s underlying mechanism. Primordial nonGaussianity is a popular discriminant among the proposed models of inflation (e.g., Bartolo et al. (2004); Komatsu et al. (2009); Chen (2010)).
The scalar curvature perturbation, , which appears in the spacespace part of the metric in a suitable gauge as (where is the RobertsonWalker scale factor), is a convenient quantity relating the observables such as the cosmic microwave background (CMB) and the largescale structure (LSS) of the universe to the primordial perturbations generated during inflation. In particular, this quantity is conserved outside the horizon for singlefield inflation (e.g., Weinberg (2003)). We shall define the twopoint function (power spectrum, denoted as ) and the threepoint function (bispectrum, denoted as ) of in Fourier space as follows:
(1)  
(2) 
The current data constrain the shape of as with Komatsu et al. (2011); Dunkley et al. (2011); *Keisler:2011aw.
The socalled localform bispectrum defined as Gangui et al. (1994); *verde/etal:2000; Komatsu and Spergel (2001)
(3) 
is particularly interesting, both because a detection of the primordial bispectrum at the level of would disfavor singlefield inflation Maldacena (2003); Acquaviva et al. (2003); Creminelli and Zaldarriaga (2004); Creminelli et al. (2011a) and because it is easy to measure the primordial signal since few latetime effects can produce the localform bispectrum. The most important contamination of known to date is due to the lensingIntegrated SachsWolfe (ISW) effect bispectrum Goldberg and Spergel (1999); *Verde:2002mu; *Smith:2006ud; *Serra:2008wc; *Hanson:2009kg; *Junk:2012qt, which can be calculated precisely and removed. The contamination of due to nonlinearity in the photonbaryon fluid has been shown to be at most one Nitta et al. (2009); *Creminelli:2011sq; *Bartolo:2011wb.
The localform bispectrum has the largest signal in the socalled “squeezed triangle configuration,” for which one of the wavenumbers, say, , is much smaller than the other two, . This can be seen from Eq. (3): as for a scaleinvariant spectrum (), the bispectrum is maximized when is taken to be small. In this limit, one finds:
(4) 
for a scaleinvariant spectrum.
Recently, Agullo and Parker have shown that a nonstandard initial state of quantum fluctuations generated during singlefield inflation can enhance the bispectrum in the squeezed configuration by a factor of , i.e., Agullo and Parker (2011). This would have profound implications for observations of the bispectrum in the squeezed configuration. For example, the signature in the bispectrum of CMB of this model was investigated in a paper by one of the authors Ganc (2011), who found that the model could produce a measurable local signal in the CMB.
The primordial bispectrum in the squeezed configuration was initially constrained mostly by measurements of the temperature anisotropy of the CMB Komatsu et al. (2002); *Komatsu:2003fd; Komatsu et al. (2011). However, over time, tools for observing the bispectrum have proliferated, providing a variety of ways to compare inflationary models. In this paper, we will explore two such methods:

In the large scalestructure (LSS) of the universe, the localform bispectrum leaves a signature by contributing a scaledependence to the halo bias, Dalal et al. (2008); Slosar et al. (2008); Matarrese and Verde (2008). For the localform bispectrum, the scale dependence goes as ; however, for a modified initial state, this scale dependence can become .

Anisotropy in the socalled type distortions of the blackbody spectrum of the CMB can be correlated with the CMB temperature anisotropy measured on large scales. This correlation can be used to measure the bispectrum in the squeezed configuration but with a larger value of than previously thought possible Pajer and Zaldarriaga (2012).
This paper is organized as follows. In Section II, we review the model under consideration. In Section III, we give the form of the bispectrum and comment on potential uncertainties in the results. In Section IV, we discuss a useful approximation to the bispectrum in the squeezed configuration. In Section V, we calculate the signal of this model in the scaledependent bias of LSS. In Section VI, we calculate the signal of this model in the type distortion of the CMB blackbody spectrum, correlated with the CMB temperature anisotropy on large scales. Finally, we conclude in Section VII.
Throughout this paper, we shall set , and use the cosmological parameters given by the WMAP 5year bestfit parameters (WMAP+BAO+ ML; Komatsu et al. (2009)): , , , , and , unless stated otherwise.
Ii Action and mode function
We consider here singlefield slowroll inflation with a canonical kinetic term, where the action (to lowest order in slowroll) can be written as Maldacena (2003)
(5)  
(6)  
(7) 
We expand the curvature perturbation into creation, , and annihilation, , operators (not to be confused with the RobertsonWalker scale factor, ):
(8) 
Usually, one chooses an initial state so that a comoving observer in the approximately de Sitter spacetime observes no particles (i.e., for this observer ). This implies a positivefrequency mode function given by
(9) 
where is the conformal time; for future reference, we note
(10) 
While this is certainly a reasonable assumption, it is an assumption, and all assumptions must be tested by observations. Thus, a responsible scientist should ask: “If the initial state of was not in this preferred vacuum state (known as the BunchDavies state), what are the implications for observations?” Our goal in this paper is not to construct candidate models of a modified initial state, but to study phenomenological consequences of such a modification, i.e., to let our observations tell us about the initial state of quantum fluctuations.
Once we adopt this approach, the next question is: “How should we parametrize a modified initial state?” We will represent a modified initial state as a Bogoliubov transformation of the above BunchDavies mode function:
(11) 
This is not the most general form one can write down (see, e.g., Kundu (2012)), but it provides us with a reasonable starting point. In line with our previous goal, we will take the Bogoliubov coefficients as given rather than trying to derive them from a fundamental theory. From the commutation relation of creation and annihilation operators, the coefficients and must satisfy . We also find that the occupation number of particles , i.e., the expected number density of particles with momentum , is given by .
These Bogoliubov coefficients, and , encode information about physics on scales where we have limited information; thus, they can vary widely without inconsistency. However, we can place some constraints on the coefficients by demanding that the theory reproduce the observed power spectrum (including the spectral tilt, ) and that the energy in the fluctuations not backreact on the background inflaton dynamics Anderson et al. (2005); Boyanovsky et al. (2006); Kundu (2012). These requirements can be satisfied in a fairly natural way if we suppose that the coefficients are such that , where the cutoff momentum must be specified. The values allowed for depend on the value of Boyanovsky et al. (2006); for , i.e., the scale of inflation, can be of order unity. Additionally, if we suppose that the smallest primordial scales observable today come from momenta sufficiently smaller than , then , i.e., roughly constant in . Remembering that and that only the relative phase between and is significant, we parametrize
(12) 
There is still uncertainty with respect to . As explained further in Ganc (2011), we identify two scenarios as plausible behaviors: 1) , where for relevant , and 2) . In the latter scenario, one can tune the value of to give larger effects; we will generally show results that assume the value of that gives the largest signal. In this sense (and for another reason discussed in Ganc (2011)), we consider the former scenario to be more conservative.
Iii Power spectrum and bispectrum
The power spectrum of on superhorizon scales, , which seeds the observed fluctuations, is given simply by Ganc (2011), i.e.,
(13) 
which becomes (using Eq. (12))
(14) 
The calculation of the bispectrum requires more thought. Formally, it is given by Maldacena (2003)
(15)  
where the interaction Hamiltonian, , is given by with given by Eq. (5). We would then specify the initial state at the initial time, , or equivalently at the initial conformal time, . For the action given by Eq. (5), one finds
(16)  
In this paper, dots will denote derivatives with respect to and primes will denote derivatives with respect to . For the standard calculation, we take the BunchDavies initial vacuum state, given by and , for all modes into the infinite past, (i.e., ). For this case, there is an accepted prescription for calculations: we take , giving an imaginary component when its absolute value is large Maldacena (2003). The exponential terms in the integrand like (see (10) for their origin) would ordinarily oscillate rapidly at very early times but are suppressed by the imaginary part of . Note that this suppression depends on .
However, when we allow for a more general initial state, we can have resulting in terms like , , etc. Furthermore, one may object (e.g., for reasons of renormalizability) to setting initial conditions in the infinite past, especially if some of the modes are excited (i.e., ); instead, one might prefer that initial conditions be set at some finite time. If we ignore this objection for a moment, one can still suppose that . By triangle inequalities (e.g., , etc.), the exponentials are still suppressed except at the precise folded limit (note that this would result in Eq. (17) but without the exponentials).
In this paper, however, we will generally take the objection seriously and suppose that initial conditions were not set infinitely far in the past. Unfortunately, this draws us into an area of active research which does not offer a definite formalism for calculations. Here, as in Holman and Tolley (2008) (though see Meerburg et al. (2009, 2010); *Ashoorioon:2010xg), we will adopt the “Boundary Effective Field Theory” approach to nonBunch Davies initial conditions Schalm et al. (2004); *GreeneBEFT, which like the other available approaches is not without problems (e.g., Easther et al. (2005)). In this approach, one cuts off the integral given in Eq. (15) at a finite , where the initial conditions are set.
We shall assume that for excited modes (i.e., where ), so that was deep inside the horizon at the initial time. This can be explained as expressing the requirement that the process for mode excitation was causal and thus, could only excite subhorizon modes.
Performing the integral given in Eq. (16), one obtains the bispectrum Ganc (2011)
(17)  
(18)  
(19)  
(20)  
(21) 
where
(22)  
(23) 
for , ; gives information about the initial conditions at . Note that we ignore a field redefinition term (derived in Maldacena (2003)) that is negligibly small for the purposes of this paper.
First, note that we recover the standard BunchDavies result Maldacena (2003) if we set , , and . In the squeezed limit, and we get , where is the slowroll parameter, which is equivalent to (see Eq. (4)). If we restore the fieldredefinition piece we ignored, we obtain the full standard squeezedlimit bispectrum: Maldacena (2003).
Since we have assumed , the exponentials in the bispectrum (17) oscillate rapidly and can, to a decent approximation, be ignored. Then, one sees that the bispectrum peaks in the socalled “folded triangle configuration,” where one of the wavenumbers is approximately equal to the sum of the other two, i.e., , , or ; this was noted earlier by Chen et al. (2007); Holman and Tolley (2008); Meerburg et al. (2009). Since the local bispectrum has no corresponding peak, this regime provides a way to distinguish the shape of this bispectrum from a purely local form. We shall come back to this point in Section VII.
We can also investigate the squeezed configuration ; this configuration is in fact a special case of the folded limit when we additionally suppose that is much smaller than or . In this limit, the third and fourth terms are larger than the first and second by a factor of ; the bispectrum becomes ). Note that this is enhanced relative to the local form in the squeezed configuration Agullo and Parker (2011). (with a proportionality factor
We should highight that the exponential terms cannot be completely ignored Meerburg et al. (2009) because they prevent the bispectrum from blowing up in the folded limit. In particular, the factor , which seems to blow up in the folded limit if one ignores the exponential, actually goes as . Accounting for this behavior plays a role in the usefullness of the approximation we demonstrate in the next section.
Iv Approximation to the bispectrum in the squeezed configuration
While the full form of the bispectrum given by Eq. (17) is complicated, the observables that we shall discuss in this paper (the scaledependent halo bias in LSS and the anisotropy in the type distortion of the CMB blackbody spectrum) depend primarily on the squeezed configuration, . Therefore, it is useful to find an accurate approximation to the bispectrum in the squeezed configuration.
In Ganc (2011), the author expanded to the lowest order in ( here is equal to in Ganc (2011)) after averaging over the exponential. Specifically, he approximated
(24) 
This result is also consistent with a prescription of ignoring oscillating terms by taking for large , as discussed in the previous section.
When we do not ignore oscillating terms, the approximation demonstrates the correct scaling on large scales but it is off by a factor. This arises because the approximation does not properly account for the oscillatory behavior of Eq. (24) at small .
Fortunately, we can come up with a better approximation. Observe that, when calculating observables, the bispectrum is usually multiplied by a function and then integrated over some of the wavenumbers (see, e.g., Eq. (48) below). Let us focus on the integral over . Note that the limits of integration for are ; in the squeezed limit, the function multiplying the bispectrum will vary little over this small range, while the oscillatory terms like the left hand side of Eq. (24) will vary very rapidly. Thus, we can perform the integral only over the rapidly oscillating term, e.g.,
(25)  
(26) 
For this integral, we find
where is Euler’s constant, is the cosine integral, and is the sin integral (which is for ); in the last line, we have dropped the second term since it becomes increasingly unimportant for large .
If we perform this new approximation, we find that, for , the chief contributor to the squeezed bispectrum looks like
(28)  
(29)  
(30) 
For , we find
(32)  
(33)  
(34)  
(35) 
These two equations provide useful approximations to the bispectrum from a modified initial state in the squeezed configuration.
Note that we can also view this approximation as finding a sort of average for the left hand side of (24), i.e., that
Thus, ignoring the oscillating terms in the bispectrum (as in Ganc (2011)) is equivalent to neglecting a factor , so that the regime investigated in Ganc (2011) differs from the one here by a factor of order unity.
V Scaledependent bias
How can we measure observationally? Obvious observables are the bispectrum of the CMB temperature and polarization anisotropy, and that of the matter density distribution in LSS. These observables are (in linear theory) related to in a straightforward way Komatsu and Spergel (2001); Sefusatti and Komatsu (2007).
A much less obvious observable is the power spectrum of dark matter halos (in which galaxies and clusters of galaxies would be formed). Dark matter halos are formed only at the locations of peaks of the underlying matter distribution. While the power spectrum of the underlying matter distribution is insensitive to the bispectrum, the power spectrum of peaks is sensitive to the bispectrum as well as to higherorder correlation functions Grinstein and Wise (1986); *Matarrese:1986et. This leads to a remarkable prediction: one can use the observed power spectrum of the distribution of galaxies (and of clusters of galaxies) to measure the bispectrum of primordial fluctuations Dalal et al. (2008); Slosar et al. (2008); Matarrese and Verde (2008).
In general, as the power spectrum of peaks (hence halos) is different from that of the underlying matter distribution, we say that halos are biased tracers of the underlying matter distribution Kaiser (1984). The degree of bias is often parametrized by the socalled “bias factor,” , defined as
(37) 
Alternatively, one may define as the ratio of the matterhalo cross power spectrum to the matter power spectrum.
On large scales, where the matter density fluctuations are still in the linear regime, approaches a constant for Gaussian matter density fluctuations, . However, the presence of the primordial bispectrum leads to a nontrivial dependence in , and this is called a “scaledependent bias.”
Building on the previous work on the peak statistics Grinstein and Wise (1986); *Matarrese:1986et; Matarrese and Verde (2008), Desjacques, Jeong, and Schmidt arrived at the following formula for Desjacques et al. (2011a, b):
where , , is related to the mass of halos under consideration as , and is the presentday critical density of the universe. The various functions are defined by
(39)  
(40)  
(41)  
(42) 
where is the linear transfer function normalized such that for , and is the growth factor of linear density fluctuations normalized such that during the matter era. (For example, for , , and .)
Before we show the numerical calculations, we will first try to analytically explore Eq. (LABEL:eq:bias) for the case of a modified initial state, allowing us to estimate the bispectrum shape.
An important observation in what follows is that oscillates rapidly for , so that the integral of is dominated by . Therefore, when we are interested in , the integral of is dominated by the squeezed configuration, and is approximated as
(44) 
We can insert the squeezed configuration local bispectrum and calculate
(45) 
The second term in the parenthesis in Eq. (LABEL:eq:bias) vanishes for this case and we find Dalal et al. (2008); Slosar et al. (2008); Matarrese and Verde (2008).
For the bispectrum for a modified initial state, which goes as , we instead find
(46) 
where
(47) 
One may interpret as a characteristic wavenumber for the shortwavelength mode in the squeezed configuration. Thus, we expect the modifiedstate bispectrum to produce a scaledependent bias which grows faster (by a factor of ) for small values of than that for the localform bispectrum.
What about the second term in the brackets in Eq. (LABEL:eq:bias)? If we note that the extra factor in the integrand of Eq. (47) (as compared with the integrand for ) is evaluated at roughly , we get and .
On the other hand, is dominated by the power spectrum of matter density fluctuations at . Approximating the power spectrum of matter density fluctuations as a powerlaw near , i.e., , one obtains . For example, , , and for , 5, and 10 Mpc (or , , and ), respectively.
Therefore, while this second term changes the amplitude of by a factor of , it does not change the dependence of . We thus expect the dependence of the scaledependent bias for a modified initial state to be given by . This scaling was also predicted by Chialva (2011).
In principle, the second term can change the amplitude of by a large factor for lowmass halos whose bias is closer to unity Desjacques et al. (2011a); *Desjacques:2011jb. Nevertheless, as we are focused on the shape of rather than on the amplitude, we will ignore this factor. Then, Eq. (LABEL:eq:bias) simplifies to
(48)  
(49)  
(50)  
(51)  
(52) 
which agrees with the formula first derived by Matarrese and Verde (2008). Here, , which is independent of .
To evaluate Eq. (48) we will use Mpc (corresponding to ). Also, in order to determine the factor which appears in the bispectrum, we use the WMAP 5year normalization, for Komatsu et al. (2009), in Eq. (14):
(53) 
where is the slowroll parameter; for , the term in parenthesis simply becomes . This relation gives for a given , , and .
We now insert the full bispectrum (Eq. (17)) into Eq. (48) and numerically integrate for the halo bias. Fig. 1 shows the results of the numeric integration for , , and for both scenarios ( and , with chosen to maximize ). We do find the expected scaling, which can also be seen by comparison with the local form in Fig. 2.
Fig. 1 also shows the halo bias as calculated from the approximation given in Eq. (28), as well as from the earlier approximation from Ganc (2011) (which is Eq. (28) without the last line). One sees that the full calculation and the new approximation are very similar except on very small scales (near the smoothing scale); by contrast, the old approximation is off in absolute scale. (There is also a slight change in shape, due to the terms that depend on ).
Vi type distortion of the blackbody spectrum of CMB
vi.1 Motivation and background
Diffusion damping of acoustic waves heats CMB photons and creates spectral distortions in the blackbody spectrum of the CMB Sunyaev and Zeldovich (1970); *1970ApSS...9..368S; *1991ApJ...371...14D. However, this distortion is erased, maintaing a blackbody spectrum for the CMB, as long as photon nonconserving processes are effective. According to Hu and Silk (1993), doubleCompton scattering () is an effective thermalization process for . After this epoch, however, this process shuts off and the spectral distortions from diffusion damping cannot be smoothed from the CMB spectrum. Since elastic Compton scattering () continues to be effective until , the photons can still achieve equilibrium but with a conserved photon number. The result is a BoseEinstein distribution with a nonzero chemical potential, (rescaled by to be dimensionless), an effect known as the “type distortion” of the blackbody spectrum of the CMB, and it affects the distribution by
(54) 
a positive reduces the number of photons at low frequencies. Finally, after , even elastic Compton scattering is inefficient and photons fall out of kinetic equilibrium with electrons, leaving only the socalled “type distortion” Zeldovich and Sunyaev (1969); *Sunyaev:1972eq. As it would be difficult to distinguish among the distortions created by the heating of CMB photons due to diffusion damping, by the cosmic reionization (), and by the thermal SunyaevZel’dovich effect Zeldovich and Sunyaev (1969); *Sunyaev:1972eq from groups and clusters of galaxies () Refregier et al. (2000), we shall focus on the type distortion in this paper.
Diffusion damping occurs near the damping scale given as follows. Over the redshifts of interest, , the expansion rate of the universe is dominated by radiation, , and the effect of baryon density on the photonbaryon fluid is negligible. Therefore, the damping scale, , is given by Silk (1968); *1983MNRAS.202.1169K; *weinberg:COS
(55) 
which gives
(56) 
Meanwhile, the heat generated by diffusion damping, , is given by
(57) 
where is the photon energy density and is the photon energy density contrast. The coefficient merits further explanation. Naively, it would be , where is the sound speed of the photon fluid. However, a recent computation using secondorder perturbation theory Chluba et al. (2012a) reveals that we need an additional factor of , yielding the number above. This heat is then converted into as
(58) 
The diffusion damping scales at and are given by and , respectively. Therefore, the type distortion is created by the (squared) photon density perturbation on very small scales. This property allows us to probe the power spectrum on such small scales Sunyaev and Zeldovich (1970); *1970ApSS...9..368S; *1991ApJ...371...14D; Hu et al. (1994); *Chluba:2011hw; *Khatri:2011aj; *Chluba:2012we.
Pajer and Zaldarriaga recently pointed out that a nonzero bispectrum in the squeezed configuration makes the distribution of on the sky anisotropic, and that this anisotropy of is correlated with the temperature anisotropy of the CMB, which measurements are on scales much larger than the damping scale Pajer and Zaldarriaga (2012). This allows us to measure the bispectrum in the squeezed configuration with a larger value of than previously thought possible. The smallest possible wavenumber one can measure from the CMB anisotropy in the sky corresponds to the quadrupole, i.e., , where Mpc is the comoving distance to the last scattering surface at . This gives , which is far greater than that accessible from the temperature anisotropy of the CMB in , i.e., , or that accessible from the scaledependent bias of LSS: for (the lowest wavenumber that can be plausibly measured from the LSS data in near future) and .
vi.2 Crosspower spectrum of CMB temperature anisotropy and type distortion
First, we decompose the CMB temperature anisotropy on the sky into spherical harmonics: . The spherical harmonics coefficients are related to the primordial curvature perturbation as
(59) 
where is the radiation transfer function. Our sign and normalization are such that in the SachsWolfe limit. In other words, in the SachsWolfe limit. However, we will not use the SachsWolfe limit (except for comparison), and instead use as computed from a linear Boltzmann code ^{1}^{1}1A code for calculating the radiation transfer function is available at http://www.mpagarching.mpg.de/~komatsu/CRL/. This code is based on CMBFAST Seljak and Zaldarriaga (1996).
Next, we similarly decompose the distribution of measured on the sky, , into spherical harmonics: . Following Pajer and Zaldarriaga (2012), we write
(60)  
(61) 
with . (Note that the coefficient of our expression is instead of because of the factor of , mentioned earlier, from Chluba et al. (2012a)). Here, is a filter function; is the scale over which the damped acoustic waves are averaged to give heat (and which we will take to be equal to to obtain a lower bound on the distortion);