Generating the cosmic microwave background power asymmetry with
Abstract
We consider a higher order term in the expansion for the CMB power asymmetry generated by a superhorizon isocurvature field fluctuation. The term can generate the asymmetry without requiring a large value of . Instead it produces a non-zero value of . A combination of constraints leads to an allowed region in space. To produce the asymmetry with this term without a large value of we find that the isocurvature field needs to contribute less than the inflaton towards the power spectrum of the curvature perturbation.
I Introduction
Inflation is widely accepted as the likely origin for structure in our universe. Its generic predictions of a nearly scale invariant and close to Gaussian primordial curvature perturbation, , have been confirmed with increasing precision by successive Cosmic Microwave Background (CMB) experiments. There are, however, also observational anomalies which are harder to explain within the standard inflationary paradigm. One such anomaly is the hemispherical power asymmetry – the observation that for scales with there is more power in CMB temperature fluctuations in one half of the sky than the other. First identified in the Wilkinson Microwave Anisotropy Probe data Eriksen et al. (2004, 2007); Hansen et al. (2004); Hoftuft et al. (2009), it was later confirmed by the Planck collaboration Ade et al. (2014a) and others Paci et al. (2013); Flender and Hotchkiss (2013); Akrami et al. (2014), although its significance remains disputed Bennett et al. (2011). In this work we treat the asymmetry as a real effect which requires a primordial origin. So far, CMB data has been fitted to a template which models the asymmetry as a spatially linear modulation.
The leading primordial explanation for this asymmetry is the Erickcek-Kamionkowski-Carrol (EKC) mechanism Erickcek et al. (2008a, b), in which a long-wavelength isocurvature perturbation modulates the power on shorter scales. Further work investigating this effect includes Refs. Kobayashi et al. (2015); Liddle and Cortes (2013); Dai et al. (2013); Lyth (2013); Namjoo et al. (2013, 2014); Lyth (2015); Kanno et al. (2013); Abolhasani et al. (2014). The origin of the long wavelength mode may be explicitly realised in the open inflation scenario of Liddle and Cortes (2013) or due to a domain wall, as in, for example, Jazayeri et al. (2014); Kohri et al. (2014).
The formalism provides a convenient expression for the modulation of power by a super-horizon mode, as reviewed below. In principle many terms in this expansion can contribute to the observed asymmetry. Until now, however, most theoretical work has focused on the leading term, which can have the form of a spatially linear modulation.
If the leading term in is responsible for the asymmetry then a further consequence is that the local bispectrum parameter must satisfy the constraint Kobayashi et al. (2015) ^{1}^{1}1Without considering our position within the modulation, and with slightly different numerical values Kanno et al. (2013) earlier found . on the scales that are modulated. A value of can be achieved but only if our observable universe is located at a fine-tuned region within the long-wavelength perturbation Kobayashi et al. (2015), and otherwise can be much larger than one. Combined temperature and polarization data bounds a purely scale-independent local bispectrum as at Ade et al. (2015), while we work with as a rough . The asymmetry appears to be scale dependent Hirata (2009), and hence the non-Gaussianity produced must also be, but there are no direct constraints on such a strongly scale dependent non-Gaussianity. A new parametrisation of the scale-dependence of the non-Gaussianity and its application to the scale-dependence of the asymmetry was given in Byrnes and Tarrant (2015), which includes an accompanying . It is, however, perhaps unlikely that a very large value of could be accommodated by current observations, even if decays with scale.
In this short paper, therefore, we investigate whether the next term in the expression for the asymmetry could instead be responsible. We find it can, without violating any other observational or self-consistency constraints. It contributes a more general modulation of the power, leading to an asymmetry, which does not necessarily only involve a spatially linear modulation ^{2}^{2}2To the best of our knowledge, no current data analysis has been performed using a template involving these more general modulation terms.. Using this higher order term requires a non-zero value of , but allows for a smaller value of than when the linear term alone contributes. If this higher order term is responsible for the asymmetry, then the allowed parameter space indicates the modulating isocurvature field must contribute less than the inflaton towards the total power spectrum of the curvature perturbation on scales which are modulated, and this may be considered a fine-tuning of the model. Related to this, we find that if this higher order term is dominant in our observable patch, then in certain neighbouring patches the linear term will instead be dominant.
In this paper, as a first step we only focus on one of the higher order terms, but the idea is more general and could be applied to a combination of higher order terms. Satisfying the constraints in that case might be more complicated than the simple use of exclusion plots that we employ here.
Ii Generating the Asymmetry
ii.1 The Formalism
Our calculation is performed within the formalism Sasaki and Stewart (1996); Sasaki and Tanaka (1998); Wands et al. (2000); Lyth et al. (2005); Lyth and Rodriguez (2005) which states that can be associated with the difference in the number of e-folds undergone by neighbouring positions in the universe from an initial flat hypersurface at horizon crossing to a final uniform density one when the dynamics have become adiabatic: . On the flat hypersurface the inflationary fields are not constant, and by writing as a function of the fields, can be written as a Taylor expansion in the horizon crossing field fluctuations.
We consider two scalar fields, though our work easily generalises for more than two fields, and we take both our fields to have canonical kinetic terms. We choose the inflaton field, denoted , to be the direction in field space aligned with the inflationary trajectory at horizon exit, , so that and this implies the derivative of with respect to the inflaton is a constant
(1) |
The isocurvature field orthogonal to is denoted , and the curvature perturbation has contributions from both fields
(2) |
where we have neglected terms with higher order derivatives since they are negligible. The arguments of and its derivatives are usually taken to be the average values of the fields within our observable universe, denoted and , while and contain all fluctuations in and with wavelengths of order the size of our observable universe or less.
The power spectrum of the curvature perturbation is then given by
(3) |
where runs over , the summation convention has been used, and we have neglected higher order and correlators.
Non-Gaussianities in are generated because of the non-linear relationship between and in (2). In particular, one finds for the local bispectrum, , and trispectrum, , parameters that Byrnes et al. (2006); Seery and Lidsey (2007)
(4) | |||
(5) |
In what follows we will only be concerned with the magnitude of and , and , but to avoid clutter we will drop the absolute symbols. We will also use the expression for the tensor-to-scalar ratio
(6) |
and we will find it convenient to define the contribution of to the total power spectrum
(7) |
ii.2 Superhorizon Fluctuation
In addition to the background value of the fields inside our observable universe, and their fluctuations with wavelength inside our observable universe, , the EKC mechanism works by postulating a superhorizon field fluctuation in , denoted , with wavelength, , much larger than the size of our observable universe, this size given by the distance to the last scattering surface, , such that . We assume the leading order behaviour for within our observable universe, where and , and we don’t assume any particular form for the fluctuation outside of our observable patch. Note that in this paper we take to be the maximum variation in across our patch about our observable universe’s average field value as seen in the left panel of Fig 1 ^{3}^{3}3 These properties are in contrast to the of Ref. Kobayashi et al. (2015) which is a long wavelength fluctuation around the background field value, , of the entire universe which is much larger than our observable patch. Our results can be matched to the results of their section 6, with our replacing their , and our replacing their . One might demand which is in fact satisfied by (9) for . .
Superhorizon fluctuations source multipole moments in the CMB, upon which there are constraints from the observed homogeneity of the universe Erickcek et al. (2008b, a). Using the non-linear results of Kobayashi et al. (2015), together with the multipole constraints from Erickcek et al. (2008b), we have the following homogeneity constraints from the quadrupole and octupole respectively ^{4}^{4}4The hexadecapole will also receive contributions from though it appears suppressed by powers of meaning if it satisfies the octupole constraint it will also satisfy the hexadecapole, and similarly for higher derivatives and multipoles.
(8) | ||||
(9) |
where we have assumed no cancellation between terms. We also take the following constraint
(10) |
where Ade et al. (2014b) and is some threshold parameter.
ii.3 Asymmetry
The superhorizon fluctuation modulates the power spectrum on shorter scales, and so it depends on the direction through
(11) |
Since in our patch, and is small, we can Taylor expand in (11) in powers of giving
(12) |
where the round brackets indicate multiplication,
(13) |
and we have used the shorthand that when and its derivatives appear without an argument they are taken to be evaluated at the average field values of the observable universe. Observations indicate a power asymmetry, with the power along the preferred direction being greater than the power on the opposite side of the sky . We note that only the odd terms in (12) can contribute towards an asymmetry of this sort, with the even terms contributing only towards general anisotropy.
Usually only the term is kept, and the data has been fitted to this with the result that Ade et al. (2014a) . The and terms were considered in Byrnes and Tarrant (2015) ^{5}^{5}5Although the authors of Byrnes and Tarrant (2015) claim the limit , which they inferred from Kim and Komatsu (2013), we think Kim and Komatsu (2013) only constrains a term in Fourier rather than real space , and so to the best of our knowledge there is no direct bound on . . Here we consider instead the term, since this can contribute towards asymmetry ^{6}^{6}6One might worry that the second and third order terms in (2) become of comparable size for asymmetry generated by the term and so loop corrections to may be important, changing the expression for in (4). However one can check these loop corrections to are subdominant to the tree level term for . . Ideally a fit to the data with terms should be done to constrain the parameters and . In the absence of this, we will look at the simplest case involving only the term and take ^{7}^{7}7We expect this to be since the area under a cubic curve is less than the area under a linear curve if they share the same endpoints. .
ii.4 Linear Term Asymmetry
It has been noted in e.g Kanno et al. (2013); Lyth (2013); Kobayashi et al. (2015) that a large accompanies the asymmetry when only the term is considered, and we briefly review this now. Differentiating (3) gives
(14) |
We now combine this with constraint (8) giving
(15) |
which is outside of the observational bounds for a local-type non-Gaussianity ^{8}^{8}8Different authors have used slightly different values for the quadrupole and octupole, and the value of , leading to other numbers appearing in (15) ranging from ..
ii.5 Cubic Term Asymmetry
The asymmetry may be due to multiple odd terms in (12). We will now show that postulating the cubic term is dominant over the linear term, and is responsible for the asymmetry, allows the constraint on to be relaxed, but introduces new ones on . Later we will check the self-consistency of ignoring the term compared to the one.
Differentiating (3) three times gives
(16) |
We will be interested in the case where the asymmetry is generated by the term, and we neglect , so that our asymmetry is given by ^{9}^{9}9We consider the term in the conclusion, noting that this term may avoid a large , introducing a non-zero instead.
(17) |
In this case, we now show there is still a lower bound on , but this time it depends on defined in (7). Using (17) together with the octupole constraint (9), we find
(18) |
We see that if is sufficiently small, we can have an acceptably small in this scenario. We will later show that there is a lower bound , and so is the smallest value of allowed from this cubic term alone ^{10}^{10}10Although one can get a value of if both terms contribute equally ., which is an improvement compared to the contribution from the linear term alone.
ii.6 Consistency Checks
For simplicity we assumed that the asymmetry is only due to the term in (12), which then must be larger than the term. We therefore require
(19) |
Even powers of don’t contribute towards the asymmetry but they do still cause more general anisotropy of the power spectrum. Since these anisotropies have not been observed, we also demand the following
(20) | |||
(21) |
where are some threshold parameters.
There is a lower limit on coming from , by definition (7). Expanding out to cubic order and neglecting the linear term, we find for .
ii.7 Allowed Parameters
We have six constraints to simultaneously satisfy: (8), (9), (10), (19), (20) and (21). Using (17) to substitute for , and using (4), (5) and (6) the six constraints become, respectively,
(22) | ||||
(23) | ||||
(24) | ||||
(25) | ||||
(26) | ||||
(27) |
In the right panel of Fig 1 we plot the allowed region, left in white, for (22)-(27), with , , and . We find that the cubic term can generate the required asymmetry with a lower value of than from the linear term alone. Moreover it requires a non-zero value of for the smallest allowed values of . Note that if is much bigger than then this pushes the allowed values of and up. The small value of may be considered a fine-tuning required when only the term generates the asymmetry.
ii.8 Outside Our Observable Patch
In the above we neglected the first order term in (12), assuming that this term is small in our observable universe. However, since we are considering a scenario in which and are non-zero, neighbouring regions of the universe with a different background field value may have a larger first order term. This is closely related to a similar effect in inhomogeneous non-Gaussianity Byrnes et al. (2012); Nelson and Shandera (2013); Nurmi et al. (2013); LoVerde et al. (2013); Baytas et al. (2015). If this term is larger in neighbouring patches this would not violate observational bounds, but would imply that our position within neighbouring regions was finely tuned – in the sense that neighbouring regions would instead see a dominant first order term. Although not invalidating the proposed scenario, it would make it less appealing. The biggest change in the average value of is in a neighbouring patch along the direction of the long wavelength mode, where its average value is of order , since . The first order term in these patches is then of order
(28) |
where we have neglected fourth and higher derivatives of . The order term in (28) is related to the zeroth order term by
(29) |
and so these terms are of comparable order for and if (20) and (21) are not hierarchical inequalities. The order term in (28) is related to the zeroth order term using (19)
(30) |
so we see that the first order term in in (12) in these neighbouring patches will actually be of the same order or larger than the cubic one in our own patch which we consider to be repsonsible for the asymmetry. This then implies that in these neighbouring patches the value of is necessarily larger than in our own patch. This agrees with the result of Byrnes et al. (2013) that if in our observable patch, then neighbouring patches will generically have a larger value of than in our own. If the asymmetry in our patch is due to the third order term rather than the linear term, then our patch should be considered fine-tuned compared to its neighbours along the direction of the long wavelength mode.
Iii Conclusion
We have presented a mechanism involving a modulating isocurvature field which can produce the required hemispherical power asymmetry while satisfying the homogeneity constraints, and which produces non-Gaussianity within observational bounds. A novel feature is the non-zero value of required to generate this asymmetry. We note that there are models with a large and small , for example, Elliston et al. (2012) and Enqvist et al. (2014). A requirement on the model is that the isocurvature field contributes a small amount towards the power spectrum of the curvature perturbation, which could be considered a fine tuning. We also note that the large minimal value of required implies our observable patch of the universe has a significantly smaller value of than our neighbours. The observed asymmetry is scale dependent, with a smaller asymmetry on small scales, which this model does not account for.
If the observed asymmetry is due to the higher order term considered in this work, then this will put strong bounds on and . Measurements of and outside of our allowed region would falsify models which use this cubic term to generate the asymmetry.
The cubic term has a different -dependence compared to the first order term. For this paper we assumed , but we would like to see a fit to the data with the terms, in order to properly constrain the parameters and .
This study has shown that a higher order term can generate the required asymmetry, relaxing the constraint on compared to that generated only by the first order. Perhaps the other cubic order term in (16), , may also contribute – although the bound on the non-linear parameter, Suyama and Yokoyama (2011), associated to this term is considerably weaker than that on , and so this term is not as easily falsifiable. Indeed, since the third order term can have a large contribution, other higher order terms (and combinations of them) may also be significant. Our work prompts investigation of the case where can’t be Taylor expanded.
Iv Acknowledgements
We would like to thank Takeshi Kobayashi and Christian T. Byrnes for many useful comments and discussions on a draft of this work. ZK is supported by an STFC studentship. DJM is supported by a Royal Society University Research Fellowship. ST is supported by STFC consolidated grants ST/J000469/1 and ST/L000415/1.
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