Illustrate mean and standard deviation in ggplot2

I'm trying to construct a plot where I plot normally distributed variables showing their mean on the x-axis and the standard deviation (SD) on the y-axis. Kinda like a density plot, but instead of having the density on the y-axis I want to have the SD (value).

I'm working with the data below,

set.seed(1)
mu1 <- rnorm(10^5, mean = 1, sd = 1)
mu3 <- rnorm(10^5, mean = 3, sd = 2)

two normally distributed variables. Here their mean and sd,

# install.packages("tidyverse", dependencies = TRUE)
require(tidyverse)
tibble(mu1, mu3) %>% summarise_all(funs(mean, sd))
#> # A tibble: 1 x 4
#>    mu1_mean mu3_mean    mu1_sd   mu3_sd
#>       <dbl>    <dbl>     <dbl>    <dbl>
#> 1 0.9993454 3.000825 0.9982848 1.998234

I've played around with ggplot2, and other tidyverse packages, to get closer to what I want. I've also tried copying this function from a box-plot doing something similar, having succeeded yet.

Here is my start,

tibble(mu1, mu3) %>% gather() %>% ggplot() + 
  geom_density(aes(x = value, colour = key)) + 
  labs(x = 'mean', y = 'currently density, but I would like sd')

The mean and standard deviation are measured on the x-scale, so you'd need to plot them along the x-axis. The y-axis is the density of points within a given x-interval, and is analogous to the height of the bars in a histogram.

Maybe this will give you something like what you were looking for: The code below adds a horizontal line that spans the standard deviation of each density plot, along with droplines to mark their location on the x-axis. The sd line is located at y-value where the width of the distribution is equal to the standard deviation. If you wish, you could in addition (or instead) fill the region spanned by the standard deviation.

library(dplyr)

# Densities
n = 2^10
df = data.frame(x = c(density(foo,n=n)$x, density(bar,n=n)$x),
                y = c(density(foo,n=n)$y, density(bar,n=n)$y),
                group=rep(c("foo","bar"), each=n))

## Mean and SD
msd =  melt(data.frame(foo=foo, bar=bar)) %>% 
         group_by(group=variable) %>% summarise(mean=mean(value), sd=sd(value))

# Find y value (of density) where sd has same width as density
msd$y = unlist(lapply(unique(df$group), function(g) {
  d = df[df$group==g,]
  d$y[which.min(abs(d$x - (msd$mean[msd$group==g] - msd$sd[msd$group==g])))]
}))

ggplot(df, aes(x=x, y=y, colour=group)) + 
  geom_line() + labs(x = NULL) +
  geom_segment(data=msd, aes(y=y,yend=y, x=mean - sd, xend=mean + sd), lty="21") +
  geom_point(data=msd, aes(y=y, x=mean)) +
  geom_segment(data=msd, aes(x=mean-sd, xend=mean-sd, y=0, yend=y), alpha=0.5, lty="21") +
  geom_segment(data=msd, aes(x=mean+sd, xend=mean+sd, y=0, yend=y), alpha=0.5, lty="21")