CMB Cold Spot from Inflationary Feature Scattering

We propose a"feature-scattering"mechanism to explain the cosmic microwave background cold spot seen from WMAP and Planck maps. If there are hidden features in the potential of multi-field inflation, the inflationary trajectory can be scattered by such features. The scattering is controlled by the amount of isocurvature fluctuations, and thus can be considered as a mechanism to convert isocurvature fluctuations into curvature fluctuations. This mechanism predicts localized cold spots (instead of hot ones) on the CMB. In addition, it may also bridge a connection between the cold spot and a dip on the CMB power spectrum at $\ell \sim 20$.


Introduction
In the recent years, observations of the cosmic microwave background (CMB) radiation fluctuations by the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck satellite have led to a precise measurement of temperature fluctuations on the sky from the largest scales down to arcmin scales [1,2]. The temperature anisotropy is found to be highly Gaussian and "statistically isotropic" in the sense that nearly all statistical proprieties of the temperature anisotropy can be described by the angular power spectrum C TT [3]. However, it was found since WMAP 1-year data that there is a deep cold spot (∆T −120 K) in the southern Galactic hemisphere along the direction (l = 209 o , b = −57 o ) with angular radius θ 10 o [4,5], which is further confirmed by Planck nominal mission data [3]. The cold spot is highly non-Gaussian in the sense that the probability of the cold spot existing in the statistical isotropic Gaussian universe is less than 0.1 percent.
Since then, the cold spot feature in the CMB map has invoked many observational and theoretical investigations. Initially, it was suggested that the unsubtracted foreground contamination might be responsible for the apparent non-Gaussian features [6,7], but later studies [3,8] show that the significance of cold spot is not affected by Galactic residues in the region of the spot. It was also proposed that a spherically symmetric void with radius ∼ 300 Mpc at redshift z = 1 can produce a large and deep CMB cold spot through the late-time integrated Sachs-Wolfe effect (ISW) [9] (also known as the Rees-Sciama effect [10]). Later studies [11,12] with the galaxy survey data [13] do find such a supervoid of size r 195 Mpc with density contrast δ 0 −0.1 at redshift z = 0.16 align with the cold spot direction. However, more detailed following-up studies [14,15] show that the Rees-Sciama effect produced by such a void is several orders of magnitude lower than the linear ISW effect therefore is not able to account for the observed feature.
The interesting non-Gaussian feature of cold-spot also invokes theorists to investigate the plausible explanation from the early universe. By considering various cosmological defects in the early universe, Refs. [16,17] proposed that a cosmic "texture" (i.e. a concentration of stress-energy and a time-varying gravitational potential due to the symmetry-breaking phase transition) can generate hot and cold spots on the last-scattering surface, with the fundamental symmetry-breaking scale found to be φ 0 ∼ 10 15 GeV. However, by applying the Bayesian method to WMAP full-sky data, Ref. [18] does not find strong evidence of the texture model, neither completely rule out the possibility (at 95% confidence level). It was also proposed that cosmic bubble collision, predicted by eternal inflation theories, can induce the density perturbation between our bubble and others, which can give arise to the localized features in the CMB [19]. But more detail data analysis [20] shows that the expected number of bubble collision is too few to account for the features in the CMB. Alternatively, a cold spot may follow from a different trajectory during multi-stream inflation [21,22].
The above physical or astronomical interpretations of cold spot either fail at some level, or require fine tuning or exotic scenarios of the early universe. Economically, some cosmologists would prefer to interpret the cold spot merely as a "3σ" statistical fluke. In this paper, we will provide a natural and physically plausible explanation of the cold spot, through multiple-field inflation. If the inflationary trajectory is scattered by a feature hidden in the isocurvature direction, the inflaton loses some energy and thus inflation tends to be longer. Therefore, it is possible that only a small portion of the sky hits the feature due to stochastic fluctuations, then that local patch of the universe has longer period of inflation and thus produce a cold spot. The mechanism is illustrated in Fig. 1. Figure 1. An illustration of the feature scattering mechanism. The classical inflationary trajectory is featureless. However, because of the quantum fluctuation of the isocurvature direction, hidden features may be encountered. At such an encounter, the inflationary trajectory gets scattered, the inflaton losses kinetic energy and thus isocurvature fluctuation converts to curvature fluctuation. This paper is organized as follows. In Section 2, we provide an explicit example of feature scattering, from massless isocurvature directions. Our predictions are compared with the measurement of the cold-spot in Planck's SMICA map. In Section 3, we consider isocurvature directions with mass m ∼ H. We conclude in Section 4 and discuss possible future directions. Throughout the paper, the unit M pl = 1/ √ 8πG = 1 is used unless otherwise stated.
2 Cold spot from feature scattering

Potential choice and method of calculation
In this section, we provide an explicit example of feature scattering to illustrate the physical mechanism. The generalization to other inflationary potentials should be straightforward. Consider inflation with two fields, both have standard kinetic term, and the potential is given by The classical initial trajectory is chosen to be on φ direction, i.e. χ = 0. Here V sr (φ) is the slow roll part of the potential. For illustration purpose, we choose a small field potential for V sr (φ): Such a potential can be derived from brane dynamics of string theory [23]. We shall nevertheless not going to import anything other than this potential from string theory, and consider it as a phenomenological example.
To fit the observations, we will take µ = 0.01 and V 0 4.89 × 10 −14 [24]. At the horizon crossing scale of the cold spot, the inflaton field is φ * 0.153 for N * 4 (counting from the start of the observable stage of inflation), and the Hubble parameter is H * 1.28 × 10 −7 . The feature δV is put at φ 0 = φ * .
The δN formalism [29][30][31] shall be used to investigate the cosmological perturbations from the feature scattering. The δN formalism makes use of the observation that different Hubble patches during inflation can be approximated as different local FRW universes. Thus the cosmological observables can be calculated by exploring those local FRW universes. Especially, the curvature perturbation in the uniform energy density slice can be calculated by 1 See the appendix of [32] for clarification of sign conventions.
where δN is the difference of e-folding number between an initial flat slice and a final uniform energy density slice. The curvature perturbation ζ is then converted to the CMB temperature anisotropy following the standard theory of CMB. In this paper we will use the Sachs-Wolfe approximation δT /T −ζ/5 = −δN/5. Intuitively, the relation can be understood as: Inflaton scattered by feature → smaller kinetic energy in φ direction → larger δN → later reheating → less dilution of energy → higher energy density → deeper gravitational potential → lower CMB temperature .
More precise relation between δT /T and ζ can be studied by solving the Boltzmann equations at recombination. Note that δN has two sources: the quantum fluctuation of φ and the quantum fluctuation of χ. Those independent sources can be studied independently, and they needs to be added together to make the total curvature fluctuation. In the numerical calculation, we will focus on investigating δN as a function of χ but we will add back a contribution of Gaussian random field to account for the contribution from φ at the last map-making stage.
The angular size of the cold spot is about 10 degrees. Converting to horizon crossing time of comoving wave number, this corresponds to about = 20. Note that in the observed CMB temperature power spectrum, there is indeed a dip at ∼ 20. It shall be interesting to do a combined analysis on both signals, from feature scattering.
In the following subsections, we shall explore different parameter space of (2.1) with three benchmarks, and calculate the e-folding number as functions of the position of isocurvature field δN = δN (χ).
Before getting to the implications to the late universe, it is intuitive to have a look at what was happening during inflation.
As an example, we consider the χ field value when hitting the feature to be χ * − χ 0 = H * /(2π). The time evolution of ∂ N φ is plotted on the left panel of Fig. 2. One can find from the plot that, at N * 4 (where feature scattering happens) When the inflationary trajectory hits the feature, there is a change in the velocity of the inflaton φ. At first, |dφ/dN | increase because φ falls into a potential well. However, two effects immediately follow -on the one hand, the obtained energy is returned because φ has to climb out of the potential well; and on the other hand, the potential well can be considered as a scattering center, such that a similar amount of energy is transferred to the χ direction (as evidenced in the right panel of Fig. 2). As a result, there is a overall loss of velocity of the inflaton, with the value read from the plot (2.5) The color on the plot denotes e-folding number. One finds that about 0.1% of the φ field kinetic energy transfers to χ. Here we have taken χ * = H * /(2π) right before hitting the feature.
In words, the inflaton field has lost 0.1% of its kinetic energy because of hitting the feature in the potential. Note that the inflaton always lose its kinetic energy because of scattering off a feature. Thus a cold spot instead of a hot spot is predicted 2 .
Note that the loss of kinetic energy takes of order one e-fold to recover, because this is the time scale to reach the inflationary attractor solution.
To be more precise, as shown in the left panel of Fig. 2, it takes about roughly 0.2 e-fold for the inflaton to recover its kinetic energy. As a result 3 , From the Sachs-Wolfe approximation, the temperature fluctuation δT /T −ζ/5 = −δN/5. Thus we get δT −100µK.
Carrying on the above analysis for general values of χ * −χ 0 , numerically the temperature fluctuation as a function of χ * − χ 0 is plotted in Fig. 3. We then simulate 1000 maps and rotate those maps such that they center at the cold spot. In the left panel of Fig. 4, we bin the pixels of the map as a function of θ, and then plot the 0.5, 0.16 and 0.84 quantiles (corresponding to the central value and 1σ bars if the probability distribution were Gaussian) 2 When the potential is positive, the inflaton actually gains kinetic energy at the very beginning, when it falls into the potential well. However, the gained energy is returned after a time interval that is much smaller than Hubble time. Thus the effect of gaining the kinetic energy is negligible when we integrate to get the e-folding number. On the other hand, the loss of kinetic energy through feature scattering needs of order 1 e-fold to recover, and thus is the dominate effect. 3 The δN here is from the quantum fluctuation of χ. The part from φ shall be added at the map-making stage.
of the binned pixel temperature. In the right panel of Fig. 4, we plot the top five best-fitting cold spot profiles. Some best-fitting examples are given in Fig. 5.

Benchmark 2: Negative long feature
In this subsection we consider a negative long feature. Inside a "long" feature, there is enough time for the χ field to oscillate. We set A = −1.4 × 10 −5 V 0 µ 4 /φ 4 * , σ φ = 50H * /(2π), σ χ = 0.7H * /(2π) and χ 0 = 5.2H * /(2π). Note that the feature size in the φ direction is 5 times longer than that of Benchmark 1. Finer structures develop in the case of long feature. If we have chosen an initial value χ * − χ 0 = H * /(2π), we find similar behavior to Fig. 2. However, for some other initial values, for example, χ * − χ 0 = 1.4H * /(2π), the χ field oscillates inside the feature. The situation is illustrated in Fig. 6. As a result, some amount of χ kinetic energy is returned to φ. For some special values, almost all the kinetic energy are returned. Thus oscillatory patterns develop, as shown in Fig. 7. The oscillations in δT (χ) leads to ring objects in δT (x). The cold spot in this parameter space is not completely cold, but instead has nested rings of cold and hot patterns. There is no evidence of such patterns in the actual CMB cold spot. There has been a debate if there are other ring patterns in the sky [33]. The Benchmark 2 parameters provides an explanation for such rings (though the shape is not perfectly spherical). However, it is shown that the appearance of such rings are due to an inappropriately chosen power spectrum, and thus not actually in the sky [34]. Thus we shall not tune the parameters to fit such features here.
The statistics, best-fitting statistics, and sample spots are plotted in Figures 8 and 9, respectively.
The δT (χ) dependence is not as sharp as the previous two benchmarks. Thus the cold spot does not have as clear a boundary as those previous cases. Because of the not-so-sharp δT (χ) dependence, to get a cold enough center of the spot, we have to increase A. As a result, the cold spots generated by Benchmark 3 typically (though not always) have very cold tinny cores.
The statistics, best-fitting statistics, and sample spots are plotted in Figures 11 and 12, respectively.

Massive isocurvature directions
The feature scattering not only applies to massless isocurvature directions, but also to marginally massive ones. As long as the mass of the isocurvature direction is not much greater than H, the energy scale for inflationary fluctuations, the isocurvature direction can still be excited to explore its field space. During inflation, fields with m χ ∼ H arises naturally [25][26][27][28]. Thus it is worthy to investigate feature scattering of such marginally massive fields, with an additional mass term in the potential Note that the process of feature scattering happens on a time scale much shorter than Hubble. Thus as long as m χ is not much great than H * , m χ does not enter the dynamics of feature scattering. However, the spectral index of a massive field is significantly different from a scale invariant spectrum, with spectral index Thus the mass of the field controls how many e-folds can the isocurvature direction randomly travel. For example, when m = 0.9H * , the χ field has a spectrum n χ − 1 = 0.3. In the numerical example, we set m = 0.9H * , A = −1.1×10 −5 V 0 µ 4 /φ 4 * , σ φ = 10H * /(2π), σ χ = 0.7H * /(2π) and χ 0 = 4.4H * /(2π). In other words, we have only tuned χ 0 to be slightly greater, and consider a massive χ field. Otherwise the parameters coincide with the Benchmark 1 of massless case.
The statistics, best-fitting statistics, and sample spots are plotted in Figures 13 and 14, respectively.
As one can observe from (14), there are more small scale structures compared to the case of massless Benchmark 1.

Conclusion and discussion
To conclude, we propose a scattering mechanism of inflaton due to the features in the inflation potential, during which process isocurvature fluctuations are converted into curvature fluctuations. The curvature fluctuations direction loses kinetic energy due to the scatters at isocurvature direction, therefore the number of e-folds becomes bigger in some region of the universe. We find that the cold spot can be well explained by such a mechanism, and the spot profile reasonably fits the CMB cold spot without fine tuning of the inflationary parameters. Before ending up the paper, we would like to mention a few future directions: • Beyond the Sachs-Wolfe approximation. Currently we have not used the full Boltzmann code to calculate the CMB transfer function. The Sachs-Wolfe approximation works well for exploring the coarse-grained cold-spot profile but is not enough for probing the fine structures inside the spot. We hope that we can make use of the full radiation transfer function to carry on a more detailed analysis in the future.
• Verifying the approximations of the δN formalism. The δN formalism assumes that the horizon-crossing amplitude of the φ and χ quantum fluctuations are Gaussian and uncorrelated. In the case of feature scattering, this is not rigorously true because of the turning of trajectory. However, note that the transfer of the inflaton kinetic energy is tinny (0.1% in the studied example), the correction from sub-horizon physics should be suppressed by this small fraction.
Having that said, it remains interesting to see if the calculation from δN formalism can be verified by first principle calculation. However, such a calculation is challenging. The first principle calculation of cosmological perturbations is known as the in-in formalism. But in the in-in formalism, the primary calculatables are the correlation functions. In other words, one starts from the Gaussian two point correlation function and study small departure from that. But the cold spot is highly non-Gaussian and localized object, and thus is not easily captured by the correlation functions from the in-in formalism.
• Extra species/symmetry point (ESP) [35,36] as scattering centers. In our current examples, the inflaton kinetic energy is lost into the collective motion of the isocurvature direction. It may also be possible that the kinetic energy of the inflaton is lost into hidden ESPs in the isocurvature direction. It is interesting to explore such possibilities.
• Connection between the feature scattering regime and the multi-stream regime: By carefully arranging the position of the feature, it is possible that different part of the universe follows different classical trajectories, separated by temporary domain walls. This is known as the multi-stream inflation [21,22], which is also a possible explanation of the cold spot. Feature scattering and multi-stream corresponds to different regimes of the multi-field parameter space. If the features on the inflationary potential is random, we expect the feature scattering to be more typical (whereas highly random features in multi-stream inflation cause disasters and provide a constraint on multi-field inflation [37]).
• Connection between the cold spot and the dip of CMB temperature power spectrum at ∼ 20. We argue they may come from the same origin -feature scattering during inflation. It would be interesting to carry out a combined analysis with CMB map and temperature power spectrum.
• String theory model building. It is believed that string theory has a landscape of complicated vacuum structures [38,39]. It would be interesting to build string landscape models for feature scattering.
is about 2.5σ away from σ χ . Those two possibilities are illustrated in the orange/purple and red/green regions of Fig. 15, respectively. Practically, the former tends to produce a number of small cold spots and the latter tends to produce one or a few large cold spot (which is the case of our current interest). Figure 15. Probability distribution of χ * as a Gaussian random field, and 0.76% unlikely regions.
To be more explicit, the relation between σ χ and χ 0 is where c ≡ χ 2 * . Here we have used "∼" because there are uncertainties from both different realization of random variables, and detailed profile of the inflationary potential (considering the feature does not sharply start at −σ χ and sharply end at σ χ ). Equation (A.1) do not have analytic solution. The numerical solution is plotted in Fig. 16, and can be fitted to good precision by

B Appendix: Selection of spots
In the numerical simulation of the CMB maps, the cold spot is selected with the following pipeline: • The simulated sky is a 200×200 square degrees. This approximately resembles the whole CMB sphere. Note that we shall eventually focus on the spot, which is about 10×10 square degrees, thus the flat sky approximation can be applied here for technical simplicity.
• For each map, we generate a mask, where the unmasked pixels have temperature δT < −100µK.
• The mask is smoothed with a Gaussian filter with a width of 2 degrees. After Gaussian smoothing, very close-by spots gets connected.
• The largest connected (after smoothing) spot on the mask is selected.
• We crop the real image according to the spot position of the mask, with a radius of 30 degree. If the spot is too close to the boundary of the whole image, we drop this image.
Some examples of spot shapes are illustrated in Fig. 17.