On Loops in Inflation (0912.2734v1)

Abstract: We study loop corrections to correlation functions of inflationary perturbations. Previous calculations have found that the two-point function can have a logarithmic running of the form log(k/mu), where k is the wavenumber of the perturbation, and mu is the renormalization scale. We highlight that this result would have profound consequences for both eternal inflation and the predictivity of standard inflation. We find a different result. We consider two sets of theories: one where the inflaton has a large cubic self-interaction and one where the inflaton interacts gravitationally with N massless spectator scalar fields. We find that there is a logarithmic running but of the form log(H/mu), where H is the Hubble constant during inflation. We find this result in three independent ways: by performing the calculation with a sharp cutoff in frequency-momentum space, in dimensional regularization and by the simple procedure of making the loop integral dimensionless. For the simplest of our theories we explicitly renormalize the correlation function proving that the divergencies can be reabsorbed and that the correlation function for super-horizon modes does not depend on time (once the tadpole terms have been properly taken into account). We prove the time-independence of the super-horizon correlation function in several additional ways: by doing the calculation of the correlation function at finite time using both the regularizations and by developing a formalism which expresses loop corrections directly in terms of renormalized quantities at each time. We find this last formalism particularly helpful to develop intuition which we then use to generalize our results to higher loops and different interactions.

• The paper revises the running of the two-point function from log(k/μ) to log(H/μ), clarifying energy scale dependencies in inflation.
• It employs three independent techniques—sharp cutoff, dimensional regularization, and dimensionless loop integrals—to achieve consistent results.
• The analysis confirms that super-horizon modes remain stable, supporting the reliability of cosmic microwave background predictions.

Overview of "On Loops in Inflation"

The paper "On Loops in Inflation" by Leonardo Senatore and Matias Zaldarriaga presents an analysis of loop corrections to correlation functions of inflationary perturbations. The authors aim to investigate the implications of such corrections on the predictability of inflationary models, especially in the contexts of eternal inflation and standard inflation.

Key Findings and Numerical Results

The paper focuses on two theoretical models: one where the inflaton exhibits significant cubic self-interactions, and another where the inflaton interacts gravitationally with $N$ massless spectator scalar fields. The notable outcome of the research is a revision of the running of the two-point function. Unlike previous calculations that proposed a logarithmic running of $\log(k/\mu)$, the authors derive a running of $\log(H/\mu)$, with $H$ representing the Hubble constant during inflation.

This finding is consistent with three independent approaches: employing a sharp cutoff in frequency-momentum space, using dimensional regularization, and reformulating the loop integral to be dimensionless. The paper demonstrates that these different methodologies converge to the same interpretation of the running.

In quantitative terms, for the scenario with a strong $\dot{\pi}^3$ self-interaction, the one-loop corrections to the two-point function of the inflaton field are calculated to be suppressed by factors indicative of the weak coupling of the system, aligning with expectations given the perturbative nature of the interactions.

Theoretical Implications

The implications of these revised loop corrections are significant for the understanding of inflationary dynamics. The robustness of the $\log(H/\mu)$ running contributes to the theoretical consistency of models concerning the predictability of inflation. This form of logarithm illustrates that the energy scales relevant to inflationary processes are inherently tied to the Hubble parameter, clarifying ambiguities presented by the $\log(k/\mu)$ form.

The analysis further establishes that the correlation functions of super-horizon modes do not exhibit long-term time dependence, contingent upon appropriate renormalization of divergences. This invariance is critical, as it ensures that inflationary predictions remain stable against the small-scale fluctuations captured by loop corrections.

Practical and Future Implications

From a practical standpoint, these findings impact experiments probing the Cosmic Microwave Background (CMB), as they provide theoretical backing for the expectation that inflationary perturbations remain constant across significant spans of cosmic evolution. This constancy underpins the reliability of connecting early-universe models to observable cosmic structures.

Going forward, these insights prompt further exploration into higher-loop corrections and interactions in more complex inflaton frameworks, such as those involving multiple fields or non-canonical kinetic terms. The paper highlights the importance of accurately contextualizing energy scales in the examination of primordial quantum fluctuations, guiding future theoretical and observational pursuits in cosmology.

Conclusion

"On Loops in Inflation" delivers a thorough examination of loop corrections within inflationary models, reframing our understanding of their impact on cosmological predictions. By addressing the limitations of prior assumptions and providing a coherent framework for considering loop effects, the authors reinforce the foundational underpinnings of inflationary theory. Their work directs attention toward further investigations into the nuanced interplay between quantum corrections and large-scale cosmic observations.