# 'Constraint consistency' at all orders in cosmological perturbation theory

## Abstract

We study the equivalence of two—order-by-order Einstein's equation and Reduced action—approaches to cosmological perturbation theory at all orders for different models of inflation. We point out a crucial consistency check which we refer to as 'Constraint consistency' condition that needs to be satisfied in order for the two approaches to lead to identical single variable equation of motion. The method we propose here is quick and efficient to check the consistency for any model including modified gravity models. Our analysis points out an important feature which is crucial for inflationary model building i.e., all 'constraint' inconsistent models have higher order Ostrogradsky's instabilities but the reverse is not true. In other words, one can have models with constraint Lapse function and Shift vector, though it may have Ostrogradsky's instabilities. We also obtain single variable equation for non-canonical scalar field in the limit of power-law inflation for the second-order perturbed variables.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22525507

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Cosmology and Astroparticle Physics

- Additional Journal Information:
- Journal Volume: 2015; Journal Issue: 08; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1475-7516

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; COSMOLOGICAL INFLATION; COSMOLOGY; EQUATIONS OF MOTION; GRAVITATION; INFLATIONARY UNIVERSE; INSTABILITY; LIMITING VALUES; PERTURBATION THEORY; SCALAR FIELDS; VECTORS

### Citation Formats

```
Nandi, Debottam, and Shankaranarayanan, S., E-mail: debottam@iisertvm.ac.in, E-mail: shanki@iisertvm.ac.in.
```*'Constraint consistency' at all orders in cosmological perturbation theory*. United States: N. p., 2015.
Web. doi:10.1088/1475-7516/2015/08/050.

```
Nandi, Debottam, & Shankaranarayanan, S., E-mail: debottam@iisertvm.ac.in, E-mail: shanki@iisertvm.ac.in.
```*'Constraint consistency' at all orders in cosmological perturbation theory*. United States. doi:10.1088/1475-7516/2015/08/050.

```
Nandi, Debottam, and Shankaranarayanan, S., E-mail: debottam@iisertvm.ac.in, E-mail: shanki@iisertvm.ac.in. Sat .
"'Constraint consistency' at all orders in cosmological perturbation theory". United States. doi:10.1088/1475-7516/2015/08/050.
```

```
@article{osti_22525507,
```

title = {'Constraint consistency' at all orders in cosmological perturbation theory},

author = {Nandi, Debottam and Shankaranarayanan, S., E-mail: debottam@iisertvm.ac.in, E-mail: shanki@iisertvm.ac.in},

abstractNote = {We study the equivalence of two—order-by-order Einstein's equation and Reduced action—approaches to cosmological perturbation theory at all orders for different models of inflation. We point out a crucial consistency check which we refer to as 'Constraint consistency' condition that needs to be satisfied in order for the two approaches to lead to identical single variable equation of motion. The method we propose here is quick and efficient to check the consistency for any model including modified gravity models. Our analysis points out an important feature which is crucial for inflationary model building i.e., all 'constraint' inconsistent models have higher order Ostrogradsky's instabilities but the reverse is not true. In other words, one can have models with constraint Lapse function and Shift vector, though it may have Ostrogradsky's instabilities. We also obtain single variable equation for non-canonical scalar field in the limit of power-law inflation for the second-order perturbed variables.},

doi = {10.1088/1475-7516/2015/08/050},

journal = {Journal of Cosmology and Astroparticle Physics},

issn = {1475-7516},

number = 08,

volume = 2015,

place = {United States},

year = {2015},

month = {8}

}